Publishing type：Research paper (scientific journal)  

This paper concerns the universality of the two-layer neural network with the k-rectified linear unit activation function with k=1,2,. with a suitable norm without any restriction on the shape of the domain in the real line. This type of result is called global universality, which extends the previous result for k=1 by the present authors. This paper covers k-sigmoidal functions as an application of the fundamental result on k-rectified linear unit functions.

DOI： 10.1155/2024/3262798

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A singular limit problem for the Cauchy problem of the Keller–Segel equation is considered in a critical function space based on Morrey spaces. A solution to the Keller–Segel system in a scaling critical function space converges to a solution to the drift-diffusion system of parabolic-elliptic type in the critical space strongly as the relaxation time tends to infinity. The key ingredient is a Banach space-valued Lorentz space in time. By the careful use of the real interpolation, a sharp estimate for the heat semigroup is obtained.

DOI： 10.1016/j.jmaa.2023.127207

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Abstract: Bourgain–Morrey spaces, generalizing what was introduced by J. Bourgain, play an important role in the study related to the Strichartz estimate and the nonlinear Schrödinger equation. In this article, via adding an extra exponent, the authors creatively introduce a new class of function spaces, called Besov–Bourgain–Morrey spaces, which is a bridge connecting Bourgain–Morrey spaces with amalgam-type spaces. By making full use of the Fatou property of block spaces in the weak local topology of, the authors give both predual and dual spaces of. Applying these properties and the Calderón product, the authors also establish the complex interpolation of. Via fully using fine geometrical properties of dyadic cubes, the authors then give an equivalent norm of having an integral expression, which further induces a boundedness criterion of operators on. Applying this criterion, the authors obtain the boundedness on of classical operators including the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.

DOI： 10.1134/S0081543823050152

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In this paper, we investigate Kantorovich operators on Morrey spaces. We first recall the main tool in the proof of the uniform boundedness of Kantorovich operators in Morrey spaces, namely a pointwise estimate of Kantorovich operators by the Hardy–Littlewood maximal operator. We also investigate the convergence of Kantorovich operators in Morrey spaces under weaker assumptions that is also true for the endpoint case. In addition, we obtain the rate of convergence of Kantorovich operators in Morrey spaces under the condition of Hölder continuity. Our result can be applicable to the function spaces in a recent result by Zeren, Ismailov and Karacam.

DOI： 10.1007/s11117-023-01004-5

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In this paper we prove that wMqp(Rn)\Mqp(Rn) is spaceable in the weak Morrey space wMqp(Rn) , where 1 ≤ q≤ p< ∞ .

DOI： 10.1007/s13226-023-00440-z

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In this paper, we show that under the condition 1 < p-, q-, p+, q+< ∞, the space ℓq(·)(Lp(·)) is reflexive as long as ℓq(·)(Lp(·)) is a Banach space. In this way we give an answer to the open problem posed by Hästö in 2017 about the reflexivity of the variable mixed Lebesgue-sequence spaces ℓq(·)(Lp(·)). What is important here is that the dual space of ℓq(·)(Lp(·)) is specified. As its direct corollary, we show that the corresponding Besov space Bp(·)q(·)s(·) is reflexive.

DOI： 10.1007/s13163-022-00447-w

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In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third author’s paper (Shi et al in J Math Pures Appl 101:394–413, 2014) significantly. In particular, the new proof for Shi et al (J Math Pures Appl 101:394–413, 2014, Theorem 1.1) is quite short and, unlike the original proof, does not rely on the properties of the “Marcinkiewicz function”. This allows us to get a precise linear dependence on Dini constants with a subsequent application to Littlewood–Paley operators by well-known techniques. In other words, we relax the log-Dini condition in the pointwise bound to the classical Dini condition. This solves an open problem (see e.g. Cao and Yabuta in J Fourier Anal Appl 25(3):1203–1247, 2019, pp. 37–38). Our method can be applied to the multilinear case.

DOI： 10.1007/s00041-023-10030-6

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This paper considers the atomic decomposition in martingale Hardy spaces generated by quasi-Banach function lattices. The results in this paper unify and extend the existing ones. Based on the Doob inequality, we make the paper self-contained except for weighted inequalities of martingale transforms. We obtain the vector-valued Doob maximal inequality for a large class of Banach function lattices over probability spaces as a by-product.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-V C H VERLAG GMBH  

This paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood-Paley characterization, and wavelet decomposition.

DOI： 10.1002/mana.202200248

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The uniform closure of the neural networks with Heaviside activation is specified.

DOI： 10.1515/jaa-2022-2012

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Publishing type：Part of collection (book)  

The purpose of the present note is to slightly shorten the proof of the atomic de-composition based on the paper by Dekel et al. The atomic decomposition in the present paper is applicable to Hardy spaces based on the convexification of ball Banach spaces. The decomposition is rather canonical although it does not depend linearly on functions. Also, this decomposition is applicable under a rather weak condition as we will see.

DOI： 10.1515/9783110792720-007

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

A class of function spaces related to Bourgain–Morrey spaces was recently introduced. Here, this class is investigated from the viewpoints of harmonic analysis and functional analysis. Specifically, this paper is oriented to certain inclusion, approximation, and interpolation properties as well as the boundedness of operators such as the Hardy–Littlewood maximal operator, fractional integral operators, fractional maximal operators, and singular integral operators. In particular, the embedding result by Bégout and Vargas is refined. Another important feature of this class of function spaces is that it can describe the convolution property. In addition, we give a description of the dual of Bourgain–Morrey spaces and we combine this with existing results to conclude the reflexivity of these spaces.

DOI： 10.1016/j.jfa.2022.109720

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The aim of this paper is to discuss the maximal regularity of the heat equation in Morrey spaces based on the 2010 paper by Ogawa and Shimizu. What differs from their work is that the Fourier multipliers are used instead of interpolation. Some recent studies on the real interpolation show that Morrey spaces do not interpolate well. The estimate needed in this paper depends on the local means. The norm estimate of local means is transformed into the form which is suitable in the context of the maximal regularity. As an application, the Cauchy problems for the Keller–Segel system are studied. The function spaces used in this paper correspond to the scaling critical case for the Keller–Segel system. An observation shows why Besov–Morrey spaces are suitable for the study of maximal regularity and that some other related spaces are not.

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The goal of this paper is to develop the Littlewood–Paley theory of discrete Morrey spaces. As an application, we establish the boundedness of martingale transforms. We carefully justify the definition of martingale transforms, since discrete Morrey spaces do not contain discrete Lebesgue spaces as dense subspaces. We also obtain the boundedness of Riesz potentials.

DOI： 10.4418/2023.78.2.4

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.jmaa.2022.126508

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In the present paper, we investigate the interpolation of weighted grand Lebesgue spaces by using the complex methods. We establish some inclusion results of complex interpolation between weighted grand Lebesgue spaces. In addition, we show that the first complex interpolation between modified weighted grand Lebesgue spaces coincides with the closure of essentially bounded functions with respect to the norm of the intermediate space. We also show that modified weighted grand Lebesgue spaces are closed under the second complex interpolation method. We also establish the complex interpolation results for (unweighted) grand Lebesgue spaces for distinct parameters τ, τ1, and τ. Our results extend the complex interpolation of grand Lebesgue spaces in Hakim et al. (Monatsh Math 184:245–272, 2017).

DOI： 10.1007/s00009-022-02168-2

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DOI： 10.1007/s00041-022-09956-0

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Our goal of this paper is to show that Ho's vector-valued Morrey spaces can be realized as a special case of pointwise multiplier spaces. This extends Lemarié- Rieusset's theorem. One cannot extend his theorem directly, because we are handling Banach lattices instead of Lebesgue spaces. It turns out that mixed Morrey spaces, Lorentz-Morrey spaces, and Orlicz-Morrey spaces fall under the scope of the framework in this paper. This work can be located as a continuation of the huge study of multiplier spaces collected in the book of Maz'ya.

DOI： 10.1215/21562261-2022-0008

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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：CMV Verlag  

DOI： 10.52547/ijmsi.17.1.219

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Authorship：Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Cellule MathDoc/CEDRAM  

DOI： 10.5802/aif.3416

CiNii Research

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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

DOI： 10.1007/s00020-021-02672-2

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Language：English  

DOI： 10.1007/s10958-022-05889-7

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Publishing type：Research paper (scientific journal)  

In this paper, we establish necessary and sufficient conditions for boundedness of m-linear p-adic integral operators with general homogeneous kernel on p-adic Lebesgue spaces and p-adic Morrey spaces, respectively. In each case, we obtain the corresponding operator norms. Also, we deal with some particular examples and compare them with the previously known from the literature.

DOI： 10.2298/FIL2203801B

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Authorship：Lead author   Publishing type：Research paper (scientific journal)   Publisher：Element d.o.o.  

DOI： 10.7153/jmi-2022-16-24

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Authorship：Corresponding author   Publishing type：Research paper (scientific journal)  

The goal of this paper is to define local weighted variable Sobolev spaces of fractional and negative order and their characterization by wavelets. We first consider local weighted variable Sobolev spaces by means of weak derivatives and obtain a wavelet characterization for these spaces. Using the Bessel potentials, we next define local weighted variable Sobolev spaces of fractional order. We show that Sobolev spaces obtained by weak derivatives and those by the Bessel potentials coincide. Finally, using duality, we define local weighted variable Sobolev spaces with negative order. We also show that local weighted variable Sobolev spaces are closed under complex interpolation. Some examples are given including the applications to weighted uniformly local Lebesgue spaces with variable exponents and periodic function spaces as a by-product, although the exponent is constant.

DOI： 10.1007/s00365-022-09573-6

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

Abstract

We study a characterization of the precompactness of sets in variable exponent Morrey spaces on bounded metric measure spaces. Totally bounded sets are characterized from several points of view for the case of variable exponent Morrey spaces over metric measure spaces. This characterization is new in the case of constant exponents.

DOI： 10.1007/s00009-021-01828-z

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Other Link： https://link.springer.com/article/10.1007/s00009-021-01828-z/fulltext.html

Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

DOI： 10.1007/s00025-021-01490-7

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Publishing type：Research paper (scientific journal)   Publisher：Wiley  

DOI： 10.1002/mana.202000513

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Other Link： https://onlinelibrary.wiley.com/doi/full-xml/10.1002/mana.202000513

Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)  

DOI： 10.3390/math9212754

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s12220-020-00513-z

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Other Link： https://link.springer.com/article/10.1007/s12220-020-00513-z/fulltext.html

Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER SINGAPORE PTE LTD  

In this paper, we give some equivalent conditions to have an inclusion between two generalized Orlicz spaces. One of these conditions does not depend on generalized Young functions. The obtained results improve the famous ones on Lebesgue spaces.

DOI： 10.1007/s41980-020-00437-y

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER  

In this paper, applying the new concept of aperiodic functions, we characterize disjoint hypercyclic and disjoint topologically mixing weighted translation operators on general Banach lattices. The obtained results in addition to previous modes of Lebesgue and Orlicz spaces, also cover new classes such as some closed subspaces of Morrey spaces.

DOI： 10.1016/j.topol.2021.107709

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We prove that block spaces defined on ℝn with an arbitrary Radon measure, which are known to be the preduals of Morrey spaces, are closed under the first and the second complex interpolation method. The proof of our main theorem uses the duality theorem in the complex interpolation method, the complex interpolation of certain closed subspaces of Morrey spaces, a characterization of the preduals of block spaces, and some formulas related to the Calderón product.

DOI： 10.1515/gmj-2019-2070

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The purpose of the present paper is to discuss the integrability of the Fourier transform of L∞-functions subject to a very weak decay condition. This will include the negative power of the logarithm. For a start, the case when d= 1 is investigated by the use of the second mean value theorem. Based on the discussion of this case, a passage to the weighted Hankel transform is done, which covers the integrability of the d-dimensional Fourier transform of the radial functions.

DOI： 10.1007/s00041-021-09831-4

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Authorship：Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Wiley  

The generalized fractional integral operators are shown to be bounded from an Orlicz–Hardy space (Formula presented.) to another Orlicz–Hardy space (Formula presented.), where Φ and Ψ are generalized Young functions. The result extends the boundedness of the usual fractional integral operator (Formula presented.) from (Formula presented.) to (Formula presented.) for (Formula presented.) and (Formula presented.), which was proved by Stein and Weiss in 1960.

DOI： 10.1002/mana.201900052

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

Recently, many researchers investigated the description of the complex interpolation of Morrey spaces. Among others, the second complex interpolation of Morrey spaces turned out to be the Calderón product of Morrey spaces. In this paper as an application of this fact, we propose an improvement of the Adams theorem asserting that the fractional integral operator maps Morrey spaces to other Morrey spaces.

DOI： 10.1007/s11118-020-09827-7

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In this paper, we consider the compactness and the spectrum of the generalized difference operator Δab on the Banach sequence spaces ℓ∞, of bounded sequences, and bv, of bounded variation sequences, which allows us to generalize and extend some existing results for the operator Δab . Furthermore, the results for the operator Δab on ℓ∞ are new even in the case where Δab is reduced to the difference operator Δ .

DOI： 10.7153/oam-2021-15-62

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Publishing type：Research paper (scientific journal)   Publisher：Informa {UK} Limited  

An investigation is made on the spectrum and fine spectrum of the generalized difference operator (Formula presented.) on the space (Formula presented.) of convergent series. Some related results, concerning the spectrum of the adjoint operator (Formula presented.) on the space (Formula presented.) of bounded variation sequences, are also derived. Further, some special cases of the main results are investigated deeper.

DOI： 10.1080/03081087.2019.1617231

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The goal of this note is to propose an expansion of real numbers, which generalizes the binary and ternary expansions. As an application of this expansion, the ternary Cantor function is generalized to the one adapted to the expansion. This generalized Cantor function can be adjusted to have the set of discontinuity of any Hausdorff dimension.

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In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As this specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.

DOI： 10.1186/s13660-021-02599-7

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Publishing type：Research paper (scientific journal)  

This work extends the theory of Rychkov, who developed the theory of Aploc weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class Ap(⋅)loc is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

DOI： 10.1186/s13660-021-02601-2

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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (international conference proceedings)   Publisher：Springer International Publishing  

The goal of this note is to consider the application of the complex interpolation space of Morrey spaces. Actually, the boundedness of Riesz potentials acting on Morrey spaces, which is obtained by Adams, is refined by means of the complex interpolation.

DOI： 10.1007/978-3-030-77493-6_29

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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Scuola Normale Superiore - Edizioni della Normale  

DOI： 10.2422/2036-2145.201807_001

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Authorship：Corresponding author   Publishing type：Research paper (scientific journal)  

Our aim is to study the modular inequalities for some operators, for example, the Bergman projection in Lebesgue spaces with a variable exponent. Under proper assumptions on the variable exponent, we prove that the modular inequalities hold, if and only if the exponent almost everywhere is equal to a constant. In order to get the main results, we establish a lower pointwise bound for these operators of a characteristic function.

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：Wiley  

Recently, a non-smooth atomic decomposition in Morrey spaces was obtained by the author, Iida and Tanaka. This note refines the above existing result using the notion of sparse families. As an application, we prove the equivalence between fractional integral operators and fractional maximal operators under the Morrey norm.

DOI： 10.1002/mma.6112

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Other Link： https://onlinelibrary.wiley.com/doi/full-xml/10.1002/mma.6112

Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER GMBH  

The predual spaces of generalized grand Morrey spaces over non-doubling measure spaces are investigated. The case of the grand Lebesgue spaces is covered, which is also new. An example shows that the modification of Morrey spaces is essential.

DOI： 10.1515/gmj-2018-0068

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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Elsevier {BV}  

Our aim in this paper is to characterize local Muckenhoupt weighted Lebesgue spaces with variable exponent by compactly supported smooth wavelets. We also investigate necessary and sufficient conditions for the corresponding modular inequalities to hold. One big achievement is that the weights with exponential growth can be handled in the framework of variable exponents.

DOI： 10.1016/j.na.2020.111930

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arXiv

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Other Link： http://arxiv.org/pdf/1912.03400v1

Authorship：Corresponding author   Publishing type：Research paper (scientific journal)  

Recently Grafakos et al., have proved that the bilinear rough singular integral operators are bounded on Lebesgue spaces. Here we aim to show that these operators extend to bounded linear operators on Morrey spaces as well. Although the proof hinges on the boundedness due to Grafakos et al., the proof seems to give a general technique to prove or extend the boundedness of the operators.

DOI： 10.1007/s41478-019-00206-z

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In this article, we discuss the complex interpolation of various closed subspaces of Morrey spaces. We have been considering some closed subspaces of Morrey spaces in our earlier works. The main property that we need is the lattice property but in connection with the diamond spaces defined by Yuan et al. (2015), it seems to be natural to consider the convolution property as well. Our result will extend the results by Hakim and Sawano (2017) and Hakim et al. (2017).

DOI： 10.1007/s11425-017-9318-0

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER WIEN  

This paper concerns the spectrum and the fine spectrum of the discrete generalized Cesàro operator Ct, where 0 ≤ t< 1 , on Banach sequence spaces close to ℓ1 and ℓ∞. We derive some compactness results for the operator Ct to describe the spectrum. Our technique involves standard operator theory and summability theory.

DOI： 10.1007/s00605-020-01376-w

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS  

This note is based on a series of lectures delivered at Kyoto University in 2015. This note surveys the homogeneous Besov space Bspq on ℝn with 1 < p, q < ∞ and s ∈ ℝ in a rather self-contained manner. Among other results, we show that S′∞ and S′/P are isomorphic, and we also discuss the realizations in Bspq. The fact that S′∞ and S′/P are isomorphic can be found in textbooks. The realization of Bspq can be found in works by Bahouri, Chemin, and Danchin and by Bourdaud for example. Here, we prove these facts using fundamental results in functional analysis such as the Hahn-Banach extension theorem.

DOI： 10.1215/21562261-2019-0038

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER GMBH  

This paper proves the boundedness of the generalized Riesz potentials I ρ, μ, τ ⁢ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L (1, φ; κ) ⁢ (G) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X) {X)}, as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

DOI： 10.1515/forum-2019-0140

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：DE GRUYTER POLAND SP Z O O  

The aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy-Littlewood maximal operator for general Radon measure on Rn. Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.

DOI： 10.1515/agms-2020-0113

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Other Link： https://www.degruyter.com/document/doi/10.1515/agms-2020-0113/xml

Publishing type：Research paper (scientific journal)  

We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than 7=4. No renormalization has to be done for this model.

DOI： 10.4064/dm806-7-2020

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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Hindawi Limited  

In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in function spaces. There are many works that handle the convergence over compact sets. In the present paper, we consider a global convergence by introducing a norm suitably, so that our results will be uniform over any compact set.

DOI： 10.1155/2021/6637220

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Other Link： https://dblp.uni-trier.de/db/journals/corr/corr2011.html#abs-2011-10225

Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV NIS, FAC SCI MATH  

The Szász–Mirakjan–Kantorovich operators and the Baskakov–Kantorovich operators are shown to be controlled by the Hardy–Littlewood maximal operator. The Szász–Mirakjan–Kantorovich operators and the Baskakov–Kantorovich operators turn out to be uniformly bounded in Lebesgue spaces and Morrey spaces with variable exponents when the integral exponent is global log-Hölder continuous.

DOI： 10.2298/FIL2007109S

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Our aim in this paper is to give a Gagliardo-Nirenberg inequality for generalized Riesz potentials Iρ,µ,κf of functions in Musielak-Orlicz spaces over quasimetric measure spaces, as an extension of work of Mizuta et al. (2012). What is new in this paper is that ρ depends on (Formula Presented) and that the underlying measure µ is not doubling.

DOI： 10.4064/cm7535-4-2019

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We discuss the boundedness of linear and sublinear operators in two types of weighted local Morrey spaces. One is defined by Natasha Samko in 2008. The other is defined by Yasuo Komori-Furuya and Satoru Shirai in 2009. We characterize the class of weights for which the Hardy-Littlewood maximal operator is bounded. Under a certain integral condition it turns out that the singular integral operators are bounded if and only if the Hardy-Littlewood maximal operator is bounded. As an application of the characterization, the power weight function [pipe] · [pipe] is considered. The condition on α for which the Hardy-Littlewood maximal operator is bounded can be described completely.

DOI： 10.5186/aasfm.2020.4504

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER WIEN  

In the original article, in Theorem 7.4 (in all statements from (1) to (9)): the symbol C0 should be replaced by bv.

DOI： 10.1007/s00605-020-01401-y

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Publishing type：Research paper (scientific journal)  

In this note, the Morrey spaces and the Sobolev–Morrey spaces are considered. In particular, the K-functional with respect to these spaces is estimated from above and below. As an application, we characterize the Nikol'skii–Besov–Morrey spaces via real interpolation.

DOI： 10.1016/j.jat.2019.105295

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Publishing type：Research paper (scientific journal)  

The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the form ‖∑jλj|eitΔfj|2‖LptLxq≲‖λ‖ℓα for orthonormal systems (fj)j of initial data in L2, firstly in work of Frank–Lewin–Lieb–Seiringer and later by Frank–Sabin. The primary objective is identifying the largest possible α as a function of p and q, and in contrast to the classical case, for such estimates the critical case turns out to be [Formula presented]. We consider the case of orthonormal systems (fj)j in the homogeneous Sobolev spaces H˙s for [Formula presented] and we establish the sharp value of α as a function of p, q and s, except possibly an endpoint in certain cases. Furthermore, at the critical case [Formula presented] for general s, we show the veracity of the desired estimates when α=p if we consider frequency localised estimates, and the failure of the (non-localised) estimates when α=p; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting.

DOI： 10.1016/j.aim.2019.106736

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Publishing type：Research paper (scientific journal)  

The Morrey boundedness is proved for the Riesz transform and the inverse operator of the nondegenerate elliptic differential operator of divergence form generated by a vector-function in (L ∞ )n 2 , and for the inverse operator of the Schrödinger operators whose nonnegative potentials satisfy a certain integrability condition. In this note, our result is not obtained directly from the estimates of integral formula, which reflects the fact that the solution of the Kato conjecture did not use any integral expression of the operators. One of the important tools in the proof is the decomposition of the functions in Morrey spaces based on the elliptic differential operators in question. In some special cases where the integral kernel comes into play, the boundedness property of the Littlewood-Paley operator was already obtained by Gong. So, the main novelties of this paper are the decomposition results associated with elliptic differential operators and the result in the case where the explicit formula of the integral kernel of the heat semigroup is unavailable.

DOI： 10.15352/aot.1812-1443

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We investigate the direct and inverse theorems for trigonometric polynomials in the Morrey space Mp(·),λ(·) with variable exponents. For this space, we obtain estimates of the K-functional in terms of the modulus of smoothness and the Bernstein type inequality for trigonometric polynomials.

DOI： 10.1007/s13324-018-0231-y

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

In the present paper, we will characterize the boundedness of the generalized fractional integral operators Iρ and the generalized fractional maximal operators Mρ on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators Mρ and Iρ on generalized Orlicz–Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators Mρ and Iρ on generalized Orlicz–Morrey spaces.

DOI： 10.1007/s11117-018-0635-9

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Language：English   Publishing type：Research paper (scientific journal)  

We consider the modular inequalities for some linear operators on Lebesgue spaces with variable exponent on the complex plane. The main results show that the variable exponent must be constant if modular inequalities hold.

DOI： 10.1134/S0001434619070265

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Publishing type：Research paper (scientific journal)  

Necessary and sufficient conditions for weight norm inequalities on Lebesgue spaces to hold are given in the scale of Orlicz spaces for the fractional Orlicz maximal operators which generalizes the fractional maximal operators. A similar argument for the Orlicz maximal operators is due to Pérez, who generalizes for the Fefferman-Stein inequality. The main result is the Fefferman-Stein inequality for the fractional maximal operators of the Sawyer type and the Hardy-Littlewood-Sobolev type. In this paper, we establish that the Lp -boundedness and the Fefferman-Stein type inequality of Orlicz maximal operator are essentially equivalent to the Saywer type inequality for the fractional Orlicz maximal operators. These inequalities are stronger than the Hardy-Littlewood-Sobolev type inequalities. More generally, we consider several mixed strong type inequalities for the ordinary and generalized fractional Orlicz maximal operators. As an application, we investigate the weight norm inequalities of the commutator [b, Iα], where b ∈ BMO(ℝn), and Iα the fractional integral operator.

DOI： 10.7153/jmi-2019-13-26

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

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Publishing type：Research paper (scientific journal)   Publisher：Polish Mathematical Society  

DOI： 10.14708/cm.v58i1-2.6381

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Our aim in this paper is to deal with the boundedness of generalized Riesz potentials I ρ ,μ ,τ f of functions in Morrey spaces L (1, ϕ;κ ) (G) over non-doubling measure spaces, as an extension of [4, 6, 9, 12, 19]. The local integrability is assumed to be minimal, so that the results can not be obtained by the Hardy-Littlewood maximal operator. What is new in this paper is that ϕ depends on x ∈ X and that the underlying measure μ is not doubling.

DOI： 10.7153/mia-2019-22-41

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Publishing type：Research paper (scientific journal)  

In this paper we calculate three geometric constants, namely the von Neumann-Jordan constant, the James constant, and the Dunkl-Williams constant, for Morrey spaces and discrete Morrey spaces. These constants measure uniformly nonsquareness of the associated spaces. We obtain that the three constants are the same as those for L1 and L∞ spaces.

DOI： 10.4134/BKMS.b190010

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In this paper, we are concerned with the boundedness of the Hardy-Littlewood maximal operator on the Orlicz space Lp(·) (log L)q(·)(X) of two variable exponents over unbounded quasi-metric measure spaces, as an extension of [Math Scand. 116 (2015), pp. 5–22]. The result is new even for the variable exponent Lebesgue space Lp(·) (X) in that the underlying spaces need not be bounded and that the underlying measure need not be doubling.

DOI： 10.1090/proc/14225

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

Our goal is to obtain the John–Nirenberg inequality for ball Banach function spaces X, provided that the Hardy–Littlewood maximal operator M is bounded on the associate space X′ by using the extrapolation. As an application we characterize BMO, the bounded mean oscillation, via the norm of X.

DOI： 10.1186/s13660-019-2220-6

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Authorship：Lead author,　Corresponding author   Publishing type：Research paper (scientific journal)   Publisher：Mathematical Sciences Publishers  

We study interpolation spaces between global Morrey spaces and between local Morrey spaces. We prove that for a wide class of couples of these spaces the upper complex Calderón spaces are not described by the K-method of interpolation. A by-product of our results is that couples of Morrey spaces belonging to this class are not Calderón-Mityagin couples. A Banach couple (X0,X1) is said to have the universal K-property if all relative interpolation spaces from any Banach couple to (X0,X1) are relatively K-monotone. A couple of local Morrey spaces is proved to have the universal K-property once it is a Calderón-Mityagin couple.

DOI： 10.2140/apde.2019.12.1465

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CiNii Research

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：INDONESIAN MATHEMATICAL SOC  

Morrey spaces can complement the boundedness properties of operators that
Lebesgue spaces can not handle. Morrey spaces which we have been handling are
called classical Morrey spaces. However, classical Morrey spaces are not
totally enough to describe the boundedness properties. To this end, we need to
generalize parameters $p$ and $q$, among others $p$.

DOI： 10.22342/jims.1.1.819.41-112

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CiNii Research

arXiv

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Other Link： http://arxiv.org/pdf/1812.08394v1

Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

In this paper, the Kantorovich operators Kn, n∈ N are shown to be uniformly bounded in Morrey spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference Kn(f) - f for functions f of regularity of order 1 measured in Morrey spaces. One of the key tools is the pointwise inequality for the Kantorovich operators and the Hardy–Littlewood maximal operator, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

DOI： 10.1007/s11117-018-0561-x

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

A predual of Bσ-spaces is investigated. A predual of a predual of Bσ-spaces is also investigated, which can be used to investigate the boundedness property of the commutators. The relation between Herz spaces and local Morrey spaces is discussed. As an application of the duality results, one obtains the boundedness of the singular integral operators, the Hardy-Littlewood maximal operators and the fractional integral operators, as well as the commutators generated by the bounded mean oscillation (BMO) and the singular integral operators. What is new in this paper is that we do not have to depend on the specific structure of the operators. The results on the boundedness of operators are formulated in terms of Ḃσ-spaces and Bσ-spaces together with the detailed comparison of the ones in Herz spaces and local Morrey spaces. Another application is the nonsmooth atomic decomposition adapted to Bσ-spaces.

DOI： 10.1007/s11425-017-9187-3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Nature  

We define Morrey-type spaces generated by a basis of measurable functions. The main result in this paper gives the description of the Köthe dual of these spaces. As an application, we recover many known cases including Mock Morrey spaces defined by David R. Adams.

DOI： 10.1007/s40840-016-0382-7

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Cambridge University Press  

We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.

DOI： 10.1017/S0004972717001216

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arXiv

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Other Link： http://arxiv.org/pdf/1802.06531v1

Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

In this note some structural properties of grand variable exponent Lebesgue/Morrey spaces over spaces of homogeneous type are obtained. In particular, it is proved that the closure of L∞ and the closure of Lp(·),λ(·) in grand variable exponent Morrey space Lp(·),λ(·),θ coincide if the measure of the underlying space is finite. Moreover, we get two different characterizations of this class. Further, duality and preduality of grand variable exponent Lebesgue spaces defined on quasi-metric measure spaces with σ-finite measure are obtained, which is new even when μ is finite.

DOI： 10.1007/s00009-018-1145-5

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arXiv

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Other Link： http://arxiv.org/pdf/1710.02383v1

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Walter de Gruyter GmbH  

In this paper, we aim to deal with the boundedness and the weak-Type boundedness for the generalized fractional integral operators on generalized Orlicz-Morrey spaces of the second kind over non-doubling metric measure spaces, as an extension of [Y. Sawano and T. Shimomura, Boundedness of the generalized fractional integral operators on generalized Morrey spaces over metric measure spaces, Z. Anal. Anwend. 36 2017, 2, 159-190], [Y. Sawano and T. Shimomura, Generalized fractional integral operators over non-doubling metric measure spaces, Integral Transforms Spec. Funct. 28 2017, 7, 534-546] and [I. Sihwaningrum, H. Gunawan and E. Nakai, Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces, Math. Nachr., to appear].

DOI： 10.1515/gmj-2018-0018

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer New York LLC  

A necessary condition and a sufficient condition for one weight norm inequalities on Morrey spaces to hold are given for the fractional maximal operator and the fractional integral operator. The difference between the behavior of the fractional maximal operator and one of the fractional integral operator which originates from the structure of Morrey spaces will be clarified. The necessary and sufficient conditions are also verified for the power weights.

DOI： 10.1007/s12220-017-9876-2

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arXiv

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Other Link： http://arxiv.org/pdf/1612.00663v1

Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Taylor and Francis Ltd.  

The upper and lower complex interpolation of BuW(Lp)-spaces and BuW(Mpq)-spaces are considered. When u0, u1 and u are finite, we prove that [Bu0w0(Lp0), Bu1w1(Lp1)]θ = [Bu0w0(Lp0), Bu1W1(Lp1)]θ = BuW(Lp). Otherwise, the space [Bu0w0(Lp0), Bu1w1(Lp1)]θ is a closed subspace of BuW(Lp). We also establish interpolation results of certain closed subspaces of BuW(Lp)-spaces and BuW(Mpq)-spaces, which are the closure of linear subspaces satisfying the lattice condition.

DOI： 10.1080/17476933.2017.1327954

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier Ltd  

In this paper, the size of the embedding constant from weak Morrey spaces into Morrey spaces is specified. As a by-product, the difference between weak Morrey spaces and Morrey spaces is clarified.

DOI： 10.1016/j.na.2017.11.005

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Duke University Press  

We introduce new function spaces to handle the Fourier transform on Morrey spaces and investigate fundamental properties of the spaces. As an application, we generalize the Stein-Tomas Strichartz estimate to our spaces. The geometric property of Morrey spaces and related function spaces will improve some well-known estimates.

DOI： 10.1215/17358787-2017-0014

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Wiley-VCH Verlag  

The main object of this investigation is to study weak Morrey spaces. Block spaces, which are preduals of weak Morrey spaces, are characterized. Besides, the Fatou property of block spaces is proved. Finally, as an application, we study the boundedness of singular integral operators in weak Morrey spaces.

DOI： 10.1002/mana.201700001

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Other Link： https://dblp.uni-trier.de/db/journals/corr/corr1805.html#abs-1805-07517

Authorship：Lead author,　Corresponding author   Publishing type：Research paper (international conference proceedings)  

This article overviews recent topic on Morrey spaces as well as the fundamental properties of Morrey spaces. The topics to be covered are: (1)The origin of Morrey spaces–An observation made by C. Morrey(2)Examples of the functions in Morrey spaces(3)Morrey spaes from the viewpoint of functional analysis(4)Interpolation of Morrey spaces(5)Weighted Morrey spaces(6)Various related spaces.

DOI： 10.1007/978-3-030-00874-1_5

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

Morrey norms, which are originally endowed with two parameters, are considered on general metric measure spaces. Volberg, Nazarov and Treil showed that the modified Hardy-Littlewood maximal operator is bounded on Legesgue spaces. The modified Hardy-Littlewood maximal operator is known to be bounded on Morrey spaces on Euclidean spaces, if we introduce the third parameter instead of adopting a natural extension of Morrey spaces. When it comes to geometrically doubling, as long as an auxiliary parameter is introduced suitably, the Morrey norm does not depend on the third parameter and this norm extends naturally the original Morrey norm. If the underlying space has a rich geometric structure, there is still no need to introduce auxiliary parameters. However, an example shows that this is not the case in general metric measure spaces. In this paper, we present an example showing that Morrey spaces depend on the auxiliary parameters.

DOI： 10.14492/hokmj/1520928055

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Publishing type：Part of collection (book)  

As an application of what we have been gathering, we investigate partial differential equations. In Sect. 5.1, we develop a theory of function spaces on domains so as to consider partial differential equations on domains. To consider some solution operators we take up the pseudo-differential operators in Sect. 5.2. We apply what we have obtained to various equations such as heat equations, Schrödinger equations and wave equations in Sect. 5.3.

DOI： 10.1007/978-981-13-0836-9_5

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Publishing type：Part of collection (book)  

In recent decades, the subject of function spaces has undergone a rapid diversification and expansion, though the decomposition of functions and operators into simpler parts, mentioned in Chap. 4, remains a central tool and theme. One of the benefits in the theory of function spaces is that we can adjust many problems of function spaces in analysis when we handle some problems after familiarizing ourselves with the fundamental theory. In this book, we have been studying the huge development of the rich theory of function spaces.

DOI： 10.1007/978-981-13-0836-9_6

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Publishing type：Part of collection (book)  

Although the definition of function spaces is complicated, we are still fascinated with Besov spaces and Triebel–Lizorkin spaces. One of the reasons, as we have alluded to in the preface, is that these scales realize many important function spaces.

DOI： 10.1007/978-981-13-0836-9_3

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Publishing type：Part of collection (book)  

The field of harmonic analysis dates back to the 19th century; the branch of function spaces dates back to the first half of 20th century, and has its roots in the study of the decomposition of functions using Fourier series and the Fourier transform. This aspect will be stressed for Besov spaces and Triebel–Lizorkin spaces. One of the elementary tools in harmonic analysis is to decompose functions or distributions into linear sums of elementary units.

DOI： 10.1007/978-981-13-0836-9_4

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Publishing type：Part of collection (book)  

Now we elaborate a fundamental theory of harmonic analysis. We aim here to collect fundamental facts on analysis and the tools which we use throughout this book.

DOI： 10.1007/978-981-13-0836-9_1

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Publishing type：Part of collection (book)  

Having set down elementary facts in the previous chapter, we take a detailed look at Besov spaces and Triebel–Lizorkin spaces, which are the main theme of this book. Chapter 2 is devoted to the introduction of elementary definitions together with some fundamental properties. First we define the Besov space Bspq(ℝn) with 1 ≤ p, q ≤∞ and s∈ ℝ in the spirit of Peetre, although Besov introduced Besov spaces in [171, 172]. After the Besov space Bspq(ℝn) for such a restricted case we define Aspq(ℝn), which unifies the Besov space Bspq(ℝn) with 0 < p, q ≤∞ and s∈ ℝ and the Triebel–Lizorkin space Fspq(ℝn) with 0 < p < ∞, 0 < q ≤∞ and s∈ ℝ.

DOI： 10.1007/978-981-13-0836-9_2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：NATURE PUBLISHING GROUP  

During olfactory appetitive learning, animals associate an odor, or conditioned stimulus (CS), with an unconditioned stimulus (US), often a sugar reward. This association induces feeding behavior, a conditioned response (CR), upon subsequent exposure to the CS. In this study, we developed a model of this behavior in isolated Drosophila brains. Artificial activation of neurons expressing the Gr5a sugar-responsive gustatory receptor (Gr5a GRNs) induces feeding behavior in starved flies. Consistent with this, we find that in dissected brains, activation of Gr5a GRNs induces Ca2+ transients in motor neurons, MN11 + 12, required for ingestion. Significantly, activation of Gr5a GRNs can substitute for presentation of sugar rewards during olfactory appetitive learning. Similarly, in dissected brains, coincident stimulation of Gr5a GRNs and the antennal lobe (AL), which processes olfactory information, results in increased Ca2+ influx into MN11 + 12 cells upon subsequent AL stimulation. Importantly, olfactory appetitive associations are not formed in satiated flies. Likewise, AL-evoked Ca2+ transients in MN11 + 12 are not produced in ex vivo brains from satiated flies. Our results suggest that a starved/satiated state is maintained in dissected brains, and that this ex vivo system will be useful for identification of neural networks involved in olfactory appetitive learning.

DOI： 10.1038/s41598-017-17955-1

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PubMed

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

Recently more and more attention has been paid to subspaces of Morrey spaces. The description of interpolation results for many of these spaces is found. However, the ones for smoothness Morrey subspaces are missing. The aim of this paper is to describe the output by the first and the second complex interpolations of these smoothness Morrey subspaces.

DOI： 10.1007/s00365-017-9392-4

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Language：English   Publishing type：Research paper (scientific journal)  

The Morrey space Mq̃p is disproved dense in Mqp if 1<q<q̃<p.

DOI： 10.1016/j.trmi.2017.05.001

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MAIK NAUKA/INTERPERIODICA/SPRINGER  

Compactness of the commutator generated by fractional integral operators and Lipschitz functions is characterized, while its boundedness has already been characterized by Shirai.

DOI： 10.1134/S0001434617110086

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER BIRKHAUSER  

In this paper, we are going to describe the first and second complex interpolations of closed subspaces of Morrey spaces, based on our previous results in [11]. Our results will be general enough because we are going to deal with abstract linear subspaces satisfying the lattice condition only. We also consider the closure in Morrey spaces on bounded domains of the set of smooth functions with compact support. Here, we do not require the smoothness condition on domains.

DOI： 10.1007/s00041-016-9503-9

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER WIEN  

The aim of this paper is to obtain the description of the first and second complex interpolations of grand Lebesgue spaces. We also investigate the complex interpolation of closed subspaces satisfying the lattice property.

DOI： 10.1007/s00605-017-1022-5

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Nature  

We develop and apply a decomposition theory for generic local Morrey-type spaces. Our result is nonsmooth decomposition, which follows from the fact that local Morrey-type spaces are isomorphic to Hardy local Morrey-type spaces in the generic case. As an application of our results, we consider the Hardy operator.

DOI： 10.1007/s11117-016-0463-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer International Publishing  

We make a further step in the study of regularizing properties of the strong solutions to the equation [Equation not available: see fulltext.]where Ω is a bounded domain in Rn with smooth boundary. We are interested in estimating the second order derivatives of the solutions and in showing that solutions belong to the class C1,α. The relevant assumptions on the leading coefficients are uniform ellipticity and their membership to the class L∞∩ VMO. Moreover, the assumptions on the lower order term are optimal in the sense that they are assumed to belong to some generalized Morrey classes.

DOI： 10.1007/s40879-017-0168-y

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS LTD  

We shall deal with the boundedness and weak-type boundedness for the generalized fractional integral operators on generalized Orlicz-Morrey spaces LΦ, ϕ;2(x: μ) of the second kind over non-doubling metric measure spaces in this paper. We also discuss a necessary condition for the boundedness of the generalized fractional integral operators.

DOI： 10.1080/10652469.2017.1318281

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier {BV}  

The Muckenhoupt–Wheeden conjecture is disproved for fractional integral operator. Both the strong (p,p) and the weak (p,p) type conjectures are proved to be false. The arguments rely upon a property of the characteristic function of an approximating sequence of the Cantor set.

DOI： 10.1016/j.jmaa.2016.11.074

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER-VERLAG ITALIA SRL  

In this paper, we give necessary conditions and sufficient conditions respectively for the boundedness of the singular integral operator on the weighted Morrey spaces. We observe the phenomenon unique to the case of Morrey spaces; the Aq-theory by Muckenhoupt and Wheeden does not suffice. We also discuss the boundedness of the commutators. The difference between the maximal operator and the singular integral operators will be clarified. Further, the property of the sharp maximal operator is investigated.

DOI： 10.1007/s13348-017-0193-7

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELEMENT  

On the product of m weighted Morrey spaces, some m-linear operators are shown to be bounded. The operator norm is calculated explicitly. It may be interesting to compare the results for the Hardy operator and the ones for the Hardy-Littlewood maximal operator. In the end of this article, some concrete examples are presented.

DOI： 10.7153/mia-20-20

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Authorship：Lead author,　Corresponding author   Publishing type：Part of collection (book)   Publisher：Springer International Publishing  

DOI： 10.1007/978-3-319-48812-7_51

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN  

This article unifies the theory for Hardy spaces built on Banach lattices on ℝn satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy–Lorentz spaces, Hardy–Herz spaces, Hardy–Orlicz spaces, Hardy–Morrey spaces, Musielak–Orlicz–Hardy spaces, variable Hardy spaces and variable Hardy–Morrey spaces. Among them, Hardy–Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy–Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy–Littlewood maximal operator satisfies certain Fefferman–Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak–Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.

DOI： 10.4064/dm750-9-2016

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：American Mathematical Society  

We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.

DOI： 10.3934/era.2017.24.006

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Part of collection (book)   Publisher：Springer International Publishing  

The spaces S′/P equipped with the quotient topology and S′∞ equipped with the weak-* topology are known to be homeomorphic, where P denotes the set of all polynomials. The proof is a combination of the fact in the textbook by Treves and the well-known bipolar theorem. In this paper by extending slightly the idea employed in [5], we give an alternative proof of this fact and then we extend this proposition so that we can include some related function spaces.

DOI： 10.1007/978-3-319-51911-1_11

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：EUROPEAN MATHEMATICAL SOC  

Our aim in this paper is to deal with the boundedness of the generalized fractional integral operators on generalized Morrey spaces Lp;Ø;2(X; μ) over metric measure spaces. We also discuss a necessary condition for the boundedness of the generalized fractional integral operators. As applications, we establish new results for the predual spaces.

DOI： 10.4171/ZAA/1584

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：EUROPEAN MATHEMATICAL SOC  

This paper is an off-spring of the contribution [Z. Anal. Anwend. 36 (2017)(1), 17-35]. We propose a way to consider the decomposition method of generalized Hardy-Morrey spaces. Generalized Hardy-Morrey spaces emerged from generalized Morrey spaces. By means of the grand maximal operator and the norm of generalized Morrey spaces, we can define generalized Hardy-Morrey spaces. With what we have culminated for the Hardy-Little wood maximal operator, we can easily refine the existing results. As an application, we consider bilinear estimates, which is the so-called Olsen inequality. In particular, our results complement the one in the 2014 paper by Iida, the fourth author and Tanaka [Z. Anal. Anwend. 33 (2014)(2), 149-170]; there were two mistakes. One lies in the decomposition result and another lies in the proof of the Olsen inequality.

DOI： 10.4171/ZAA/1582

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Authorship：Lead author,　Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：EUROPEAN MATHEMATICAL SOC  

The generalized Morrey space Mp;φ(Rn) was defined by Mizuhara 1991 and Nakai in 1994. It is equipped with a parameter 0 < p < 1 and a function φ : Rn × (0) → (0). Our experience shows that Mp,φ(Rn) is easy to handle when 1 < p < 1. However, when 0 < p ≤ 1, the function space Mp,φ(Rn) is difficult to handle as many examples show. We propose a way to deal with Mp,φ(Rn) for 0 < p ≤ 1, in particular, to obtain some estimates of the Hardy-Littlewood maximal operator on these spaces. Especially, the vector-valued estimates obtained in the earlier papers are refined. The key tool is the weighted dual Hardy operator. Much is known on the weighted dual Hardy operator.

DOI： 10.4171/ZAA/1577

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Authorship：Lead author,　Corresponding author   Language：Japanese   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER BASEL AG  

We present a multidimensional integral inequality related to the Hilbert-type inequality and the Carlson-type inequality. As an application, we obtain a sharper form of the Hilbert-type inequality. Carlson-type inequality is also considered.

DOI： 10.1007/s00009-016-0717-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TOHOKU UNIVERSITY  

We obtain an estimate of the operator norm of the weighted Kakeya (Nikodým) maximal operator without dilation on L2(ω). Here we assume that aradial weight w satisfies the doubling and supremum condition. Recall that, in the definition of the Kakeya maximal operator, the rectangle in the supremum ranges over all rectangles in the plane pointed in all possible directions and having side lengths a and aN with N fixed. We are interested in its eccentricity N with a fixed. We give an example of a non-constant weight showing that √log N cannot be removed.

DOI： 10.2748/tmj/1486177220

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

We deal with the generalized Orlicz–Morrey space Mϕ,Φ(Rn) of the third kind and consider the decomposition method. Also we characterize its predual space. Some maximal estimates for generalized Orlicz–Morrey spaces of the third kind are also obtained by using the weighted Hardy operators. As an application, we consider the Olsen inequality, which is a bilinear estimate on the fractional integral operator. As an appendix, we consider a general form of the vector-valued boundedness of the Hardy–Littlewood maximal operator, where ϕ in the definition of Mϕ,Φ(Rn) depends on x as well. This paper contains a remedy for the mistake in the proof of the Olsen inequality of the 2014 paper by the second author (Iida et al. in Z. Anal. Anwend. 33(2):149–170, 2014).

DOI： 10.1007/s10440-016-0052-7

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Language：English   Publishing type：Research paper (scientific journal)  

The present paper supplements our earlier works. Our goal in the present paper is to establish the boundedness of generalized fractional integral operators from the Morrey space L1,φ(X; µ) to the Campanato space L1,ψ (X; µ) over non-doubling measure spaces (X, d, µ). What is new in the present paper is that µ satisfies a minimal condition; 0 = µ({x}) < µ(B(x, r)) < x for all x ∈ X and r > 0. We first review some elementary facts on the fractional integral operators, generalized Morrey spaces, and analysis on metric measure spaces.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER-VERLAG ITALIA SRL  

In this paper, we shall establish a theory of interpolation of generalized Morrey spaces. We use the complex interpolation methods. Our results extend the interpolation results for Morrey spaces which are discussed by Lu et al. (Can Math Bull 57:598–608, 2014), and also Lemarié-Rieusset (2014). We establish the interpolation of generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also consider the closure of the functions which are essentially bounded and have compact support. The second interpolation of such spaces will yield a class of closed spaces; we describe the second interpolation of the closure of the functions which are essentially bounded and have compact support. This result will carry over to generalized Morrey spaces, generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also give several examples that explain the subtlety of proving the interpolation of Morrey spaces.

DOI： 10.1007/s13163-016-0192-3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SCIENCE PRESS  

The theory of generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which Mizuhara and Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of the atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.

DOI： 10.1007/s11425-015-5096-z

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Other Link： http://arxiv.org/pdf/1605.08500v1

Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER-VERLAG ITALIA SRL  

In the present paper, we shall give a necessary and sufficient condition for the weak/strong boundedness of generalized fractional maximal operators on generalized Orlicz–Morrey spaces. We also give necessary and sufficient conditions for the vector-valued inequalities of the Hardy–Littlewood maximal operator, generalized fractional maximal operators and singular integral operators on these function spaces.

DOI： 10.1007/s13163-015-0178-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS  

In this article, the authors introduce Hardy spaces with variable exponents, H*,p(·)(X), on RD-spaces with infinite measures via the grand maximal function. Then the authors characterize these spaces by means of the non-tangential maximal function or the dyadic maximal function. Characterizations in terms of atoms or Littlewood-Paley functions are also established. As applications, the authors prove an Olsen inequality for fractional integral operators and the boundedness of singular integral operators and quasi-Banach valued sublinear operators on these spaces. Finally, a duality theory of these spaces is developed.

DOI： 10.4064/dm744-9-2015

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SUOMALAINEN TIEDEAKATEMIA  

Our aim in this paper is to establish various norm inequalities in Musielak-Orlicz spaces. We give a generalization of a result due to Cruz-Uribe, Fiorenza, Martell and Pérez and apply it to obtain norm inequalities for classical operators as well as an Olsen inequality in Musielak- Orlicz spaces.

DOI： 10.5186/aasfm.2016.4148

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN  

In this article, the authors introduce Hardy spaces with variable exponents, H*(,p(.)) (chi), on RD-spaces with infinite measures via the grand maximal function. Then the authors characterize these spaces by means of the non-tangential maximal function or the dyadic maximal function. Characterizations in terms of atoms or Littlewood-Paley functions are also established. As applications, the authors prove an Olsen inequality for fractional integral operators and the boundedness of singular integral operators and quasi-Banach valued sublinear operators on these spaces. Finally, a duality theory of these spaces is developed.

DOI： 10.4064/dm744-9-2015

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Language：English   Publishing type：Research paper (international conference proceedings)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATHEMATICAL SOC REP CHINA  

Let b ∈ BMO(Rn) and MΩ be the Marcinkiewicz integral operator with kernel (Formula Presented), where Ω is homogeneous of degree zero, integrable and has mean value zero on the unit sphere Sn−1. In this paper, by means of Fourier transform estimates and approximation to the operator MΩ with integral operators having smooth kernels we show that if b ∈ CMO(Rn) and Ω satisfies a certain weak size condition, then the commutator MΩ, b = [b, MΩ] generated by b and MΩ is a compact operator on Lp(Rn) for some 1 < p < ∞.

DOI： 10.11650/tjm.19.2015.5656

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

There have been known two generalized Orlicz-Morrey spaces. One is defined earlier by Nakai and the other is by Sugano, the second and third authors. In this paper we investigate differences between these two spaces in some typical cases. The arguments rely upon property of the characteristic function of the Cantor set.

DOI： 10.1016/j.jat.2015.04.001

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-V C H VERLAG GMBH  

In the present paper, we consider the non-smooth atomic decomposition of generalized Orlicz-Morrey spaces. The result will be sharper than the existing results. As an application, we consider the boundedness of the bilinear operator, which is called the Olsen inequality nowadays. To obtain a sharp norm estimate, we first investigate their predual space, which is even new, and we make full advantage of the vector-valued inequality for the Hardy-Littlewood maximal operator.

DOI： 10.1002/mana.201400138

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELEMENT  

In this paper we shall provide a unified treatment of Hilbert-type inequalities involving the Hardy operator with a general homogeneous kernel. Furthermore, we establish some new Hilbert-type inequalities involving the integral operator. We shall show that the constants in our results are best possible.

DOI： 10.7153/mia-18-60

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER-VERLAG ITALIA SRL  

Zorko proved that Morrey spaces are known to have preduals. In the present paper such predual spaces are shown to be characterized by means of the Littlewood–Paley operators. Recently, many function spaces are shown to admit the Littlewood–Paley characterization. As for the predual spaces of Morrey spaces, it had not been clear that a pointwise increasing norm-bounded sequence has a least bound. However, the author proved in the earlier paper that it actually has a least bound. In the present paper, by taking advantage of this property, a characterization by means of Littlewood–Paley operators is proposed.

DOI： 10.1007/s13163-014-0158-2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SUOMALAINEN TIEDEAKATEMIA  

In this paper, we characterize weighted Lebesgue spaces with variable exponent in terms of wavelet. Also, we disprove some weighted modular inequalities when the exponent is not a constant one without using the A(infinity)-condition on weights. As a byproduct, we shall obtain the vector-valued maximal inequalities in the weighted setting.

DOI： 10.5186/aasfm.2015.4032

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Language：English   Publishing type：Research paper (scientific journal)  

Around thirty years ago, block spaces, which are the predual of Morrey spaces, had been considered. However, it seems that there is no proof that block spaces satisfy the Fatou property. In this paper the Fatou property for block spaces is verified and the predual of block spaces is characterized.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TUSI MATHEMATICAL RESEARCH GROUP  

We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces MΦω,α (Rn). In all the cases, the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights ' without assuming any monotonicity property of α(x,·) with x fixed.

DOI： 10.15352/bjma/09-2-5

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Other Link： http://arxiv.org/pdf/1406.4957v1

Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS LTD  

In this paper we are concerned with Sobolev embeddings for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over non-doubling measure spaces. We deal with the boundedness of the generalized maximal operators, Sobolev's inequality, Trudinger's inequality and continuity for Riesz potentials of variable order. We prove the results in full generality: our function spaces will cover Morrey spaces, variable Lebesgue spaces as well as Lebesgue spaces with a Radon measure μ. Also, we can readily handle function spaces on metric measure spaces. Our results will cover the ones in our earlier paper [Sawano Y, Shimomura T. Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. Collect. Math. 2013;64:313–350]. We shall show that the Trudinger exponential integrability is available in our generalized setting, whose counterpart on Lebesgue spaces was not considered in Mizuta et al. [Maximal functions, Riesz potentials and Sobolev embeddings on Musielak–Orlicz–Morrey spaces of variable exponent in Rn. Rev. Mat. Complut. 2012;25:413–434.]

DOI： 10.1080/10652469.2014.955099

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AKADEMIAI KIADO RT  

The paper concerns characterization of BMO in terms of Banach function spaces. In particular, we are interested in characterizing BMO by using the variable Lebesgue norm. © 2014 Akadémiai Kiadó, Budapest, Hungary.

DOI： 10.1007/s10476-014-0304-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

In this paper, we obtain a sufficient condition for the convergence of positive linear operators in Banach function spaces on ℝn and derive a Korovkin type theorem for these spaces. Also, we generalized this result via statistical sense. © 2013 Springer Basel.

DOI： 10.1007/s11117-013-0264-2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SCIENCE PRESS  

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions. The class will be wider than the class of all the N-functions. In particular, we consider the non-smooth atomic decomposition. The relation between Orlicz-Hardy spaces and their duals is also studied. As an application, duality of Hardy spaces with variable exponents is revisited. This work is different from earlier works about Orlicz-Hardy spaces H Φ(ℝn) in that the class of admissible functions φ is largely widened. We can deal with, for example, with p 1, p 2 ∈ (0,∞) and q 1, q 2 ∈ (-∞,∞), where we shall establish the boundedness of the Riesz transforms on H Φ(ℝn). In particular, φ{symbol} is neither convex nor concave when 0 < p 1 < 1 < p 2 < ∞, 0 < p 2 < 1 < p 1 < ∞ or p 1 = p 2 = 1 and q 1, q 2 > 0. If Φ(r) ≡ r(log(e+r))q, then H Φ(ℝn) = H(log H)q(ℝn). We shall also establish the boundedness of the fractional integral operators I α of order α ∈ (0,∞). For example, I α is shown to be bounded from H(log H)1 - α/n (ℝn) to L n/(n - α)(log L)(ℝn) for 0 < α < n. © 2014 Science China Press and Springer-Verlag Berlin Heidelberg.

DOI： 10.1007/s11425-014-4798-y

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

In this paper, the authors introduce Morrey-type spaces on the locally doubling metric measure spaces, which means that the underlying measure enjoys the doubling and the reverse doubling properties only on a class of admissible balls, and then obtain the boundedness of the local Hardy-Littlewood maximal operator and the local fractional integral operator on such Morrey-type spaces. These Morrey-type spaces on the Gauss measure space are further proved to be naturally adapted to singular integrals associated with the Ornstein-Uhlenbeck operator. To be precise, by means of the locally doubling property and the geometric properties of the Gauss measure, the authors establish the equivalence between Morrey-type spaces and Campanato-type spaces on the Gauss measure space, and the boundedness for a class of singular integrals associated with the Ornstein-Uhlenbeck operator (including Riesz transforms of any order) on Morrey-type spaces over the Gauss measure space. © 2012 Mathematica Josephina, Inc.

DOI： 10.1007/s12220-012-9362-9

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELEMENT  

We present some characterizations for the boundedness of the generalized fractional integral operators on Morrey spaces. The characterizations follow from two key estimates, one for the norm of some functions in Morrey spaces, and another for the values of the corresponding fractional integrals. We prove three theorems about necessary and sufficient conditions. We show that these theorems are independent by giving some examples. We also obtain counterparts for the weak generalized Morrey spaces.

DOI： 10.7153/mia-17-56

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS LTD  

Consider the quasi-geostrophic equations with the initial data. Let and be two weak solutions with the same initial value. If where is the Orlicz-Morrey space (for a definition of this space, see Definition), then we have. In view of the embedding with and, we see that our result improves the previous result of Dong and Chen. This is an extension of earlier regularity results in the Serrin's type space. © 2013 © 2013 Taylor & Francis.

DOI： 10.1080/00036811.2013.772582

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-V C H VERLAG GMBH  

The Brézis-Gallouët-Wainger inequality describes a subtle embedding property into L∞. The relation between the Brézis-Gallouët-Wainger inequality and the real interpolation functor together with the sharpness of the results is discussed in the present paper. As our first main results shows, it turns out that there are two intermediate terms between L∞ and the logarithmic boundedness, which is supposed to be the right-hand side of the Brézis-Gallouët-Wainger inequality. As the second result, the first result is extended to inequalities which reflect the meaning of the second index of Besov spaces and the interpolation theorem. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

DOI： 10.1002/mana.201100324

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer International Publishing  

In this paper, we discuss the boundedness of the fractional type Marcinkiewicz integral operators associated to surfaces, and we extend a result given by Chen et al. (J. Math. Anal. Appl. 276:691-708, 2002). They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spaces (Formula presented.) to (Formula presented.). Recently the second author, together with Xue and Yan, greatly weakened their assumptions. In this paper, we extend their results to the case where the operators are associated to the surfaces of the form (Formula presented.). To prove our result, we discuss a characterization of the homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences.

DOI： 10.1186/1029-242X-2014-232

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Language：English   Publishing type：Research paper (scientific journal)  

We develop and apply a decomposition theory for generic local Morrey spaces. Our results are smooth and nonsmooth decompositions, which follows from the fact that local Morrey spaces are isomorphic to local Hardy-Morrey spaces and local Triebel-Lizorkin-Morrey spaces in the generic case. As an application of our results, we consider a bilinear estimate for the fractional integral operators.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：EUROPEAN MATHEMATICAL SOC  

The Hardy space Hp(ℝn) substitutes for the Lebesgue space Lp(ℝn). When p > 1, then the Hardy space Hp(ℝn) coincides with the Lebesgue spaces Lp(ℝn). This is shown by using the reexivity of the function spaces. The atomic decomposition is readily available for H p(ℝn) with 0 < p < ∞. This idea can be applied to many function spaces. As example of such an attempt, we now propose here a non-smooth decomposition of Morrey spaces. As applications, we consider the Olsen inequality. In the end of this article, we compare our results with existing ones and propose some possibility of extensions, which are left as future works. © European Mathematical Society.

DOI： 10.4171/ZAA/1504

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

DOI： 10.1155/2014/867192

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER-VERLAG ITALIA SRL  

The target is potential theory in connection with Morrey spaces on general metric measure spaces. The present paper is oritented to investigating Sobolev's inequality, Trudinger exponential integrability and continuity for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. A counterexample shows that our results are reasonable. In addition to the example above, what is new about this paper is that everything can be developed once the underlying measure does not charge any point. © 2013 The Author(s).

DOI： 10.1007/s13348-013-0082-7

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Language：English   Publishing type：Research paper (scientific journal)  

As applications of atomic decomposition results of Hardy spaces with variable exponents, we shall prove the boundedness of commutators and the fractional integral operators as well as the Hardy operators. There are many ways to prove such boundedness. For example, the boundedness of commutators can be proved by the sharp maximal inequalities. But here, we propose a different method based upon our atomic decomposition. © 2013 Springer Basel.

DOI： 10.1007/s00020-013-2073-1

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In the present paper, we consider sampling theorems on Banach function spaces. Here we obtain a necessary and sufficient condition. For the latter half of the paper, we consider sampling theorems in terms of wavelets.

DOI： 10.3836/tjm/1374497524

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS LTD  

In this article, we show that if a Leray solution (u, b) to the 3D magneto-hydrodynamic equation belongs to L2/1-r((0,T);M3/rL2logPL(ℝ3)), with r ∈ (0, 1) and P > 1, then (u, b) is the unique Leray solution on (0, T). © 2013 Copyright Taylor and Francis Group, LLC.

DOI： 10.1080/00036811.2011.636357

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Our main purpose in this article is to establish a Gagliardo-Nirenberg type inequality in the critical Sobolev-Morrey space HMn/pp,q(ℝn) with n∈ℕ and 1<q≤p<∞, which coincides with the usual critical Sobolev space HMn/pp,q(ℝn in the case of q=p. Indeed, we shall show the following interpolation inequality. If q<p, there exists a positive constant Cp,q depending only on p and q such that for all u∈HMn/pp,q(ℝn) and for all p≤r<∞. In the case of q=p, that is, the case of the critical Sobolev space, the corresponding inequality was obtained in Ogawa (Nonlinear Anal. 14:765-769, 1990), Ogawa-Ozawa (J. Math. Anal. Appl. 155:531-540, 1991) and Ozawa (J. Func. Anal. 127:259-269, 1995) with the growth order r1-1/p as r→∞. The inequality (GN) implies that the growth order as r→∞ is linear, which might look worse compared to the case of the critical Sobolev space. However, we investigate the optimality of the growth order and prove that this linear order is best-possible. Furthermore, as several applications of the inequality (GN), we shall obtain a Trudinger-Moser type inequality and a Brézis-Gallouët-Wainger type inequality in the critical Sobolev-Morrey space. © 2012 Springer Science+Business Media, LLC.

DOI： 10.1007/s00041-012-9223-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

We show the boundedness of the Hardy-Littlewood maximal operator, singular and fractional integral operators, and more general sublinear operators on B σ -Morrey-Campanato spaces. These function spaces have been introduced recently to unify central Morrey spaces, λ-central mean oscillation spaces and usual Morrey-Campanato spaces. Using the B σ -Morrey-Campanato spaces, we can study both local and global regularities of functions simultaneously, and unify a series of results on the boundedness of operators on several classical function spaces. © 2011 Revista Matemática Complutense.

DOI： 10.1007/s13163-011-0091-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER INTERNATIONAL PUBLISHING AG  

Our aim in this paper is to give Sobolev's inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces. The main result is oriented to the outrange of the well-known Adams theorem. © 2013 Sawano and Shimomura; licensee Springer.

DOI： 10.1186/1029-242X-2013-12

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Language：English   Publishing type：Research paper (scientific journal)  

Our aim in this paper is to give Sobolev's inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces. The main result is oriented to the outrange of the well-known Adams theorem. © 2013 Sawano and Shimomura
licensee Springer.

DOI： 10.1186/1029-242X-2013-12

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

Our aim in this paper is to prove the Gagliardo-Nirenberg inequality for Riesz potentials of functions in variable exponent Lebesgue spaces, which are called Musielak-Orlicz spaces with respect to Φ(x, t) = t p(x)(log(c0 + t))q(x) for t > 0 and x ε ℝn, via the Littlewood-Paley decomposition. © 2013 The Mathematical Society of Japan.

DOI： 10.2969/jmsj/06520633

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

We introduce generalized Morrey-Campanato spaces of martingales, which generalize both martingale Lipschitz spaces introduced by Weisz (1990) and martingale Morrey-Campanato spaces introduced in 2012. We also introduce generalized Morrey-Hardy and Campanato-Hardy spaces of martingales and study Burkholder-type equivalence. We give some results on the boundedness of fractional integrals of martingales on these spaces. © 2013 Eiichi Nakai et al.

DOI： 10.1155/2013/690258

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In this paper, we will provide an alternative proof to characterize the pointwise multipliers which maps a Sobolev space Ḣr (ℝd) to its dual Ḣr (ℝd) in the case 0 < r < d2 by a simple application of the definition of fractional Sobolev space. The proof relies on a method introduced by Maz'ya-Verbitsky [9] to prove the same result. © 2013 The Korean Mathematical Society.

DOI： 10.4134/BKMS.2013.50.3.915

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN  

In this paper, the authors propose a new framework under which a theory of generalized Besovtype and Triebel-Lizorkin-type function spaces is developed. Many function spaces appearing in harmonic analysis fall under the scope of this new framework. The boundedness of the Hardy-Littlewood maximal operator or the related vector-valued maximal function on any of these function spaces is not required to construct these generalized scales of smoothness spaces. Instead, a key idea used is an application of the Peetre maximal function. This idea originates from recen fndings in the abstract coorbit space theory obtained by Holger Rauhut and Tino Ullrich. In this new setting, the authors establish the boundedness of pseudo-differential operators based on atomic and molecular characterizations and also the boundedness of Fourier multipliers. Characterizations of these function spaces by means of differences and oscillations are also established. As further applications of this new framework, the authors reexamine and polish some existing results for many different scales of function spaces.

DOI： 10.4064/dm489-0-1

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Dyadic fractional integral operators are shown to be bounded on Morrey spaces and their preduals. It seems that the proof of the boundedness by means of dyadic fractional integral operators is effective particularly on the preduals. In the present paper the commutators are proved to be bounded as well. © SAS International Publications.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KOSSUTH LAJOS TUDOMANYEGYETEM  

By using a geometric structure of the Euclidean space, the theory of generalized Morrey spaces is shown to be available in the non-doubling setting. Some classical operators are established to be bounded in the generalized spaces defined in the present paper.

DOI： 10.5486/PMD.2013.5508

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

The boundedness of the various operators on (B) over dot(sigma)-Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization of (B) over dot(sigma)-Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization of (B) over dot(sigma)-Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators on (B) over dot(sigma)-Lipschitz spaces is discussed.

DOI： 10.1155/2013/859402

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.

DOI： 10.1155/2013/809704

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Language：English   Publisher：Kyoto University  

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Language：English   Publisher：Kyoto University  

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Language：English   Publishing type：Research paper (scientific journal)  

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spaces L (1, φ) (G) over nondoubling measure spaces. © 2013 Yoshihiro Sawano and Tetsu Shimomura.

DOI： 10.1155/2013/984259

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Hindawi Publishing Corporation  

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spaces L-(1,L-phi)(G) over nondoubling measure spaces.

DOI： 10.1155/2013/984259

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Language：English   Publishing type：Research paper (scientific journal)  

The boundedness of the various operators on B ̇ σ -Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization of B ̇ σ -Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization of B ̇ σ -Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators on B ̇ σ -Lipschitz spaces is discussed. © 2013 Yasuo Komori-Furuya et al.

DOI： 10.1155/2013/859402

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER BIRKHAUSER  

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces B S,τp,q(ℝ n) and Triebel-Lizorkin-type spaces F S,τp,q(ℝ n) with p∈(0,∞], s∈ℝ, τ∈[0,∞), and q∈(0,∞], as well as their preduals, the Besov-Hausdorff spaces BH S,τp,q(ℝ n) and Triebel-Lizorkin-Hausdorff spaces FH S,τp,q(ℝ n), in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of Rauhut (Stud. Math. 180:237-253, 2007) and discretizations via biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as F S∞,q(ℝ n) for s∈ℝ, q∈(0,∞] (including BMO(ℝ n) when s=0 and q=2), the Q space Q α(ℝ n), the Hardy-Hausdorff space HH -α(ℝ n) for α∈(0,min{n/2,1}), the Morrey space M up(ℝ n) for 1 < p ≤ u < ∞, and the Triebel-Lizorkin-Morrey space E Supq(ℝ n) for 0 < p ≤ u < ∞, s ∈ ℝ and q∈(0,∞], some of these results are new. © 2012 Springer Science+Business Media, LLC.

DOI： 10.1007/s00041-012-9234-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

In this paper, we are going to characterize the space BMO(ℝ n) through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space BMO(ℝ n) by using various function spaces. For example, Ho obtained a characterization of BMO(ℝ n) with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known John-Nirenberg inequality which can be seen as an extension to the variable Lebesgue spaces. © 2012 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.

DOI： 10.1007/s10587-012-0042-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-BLACKWELL  

In this work, we are interested in the study of regularity for the three-dimensional magneto-micropolar fluid equations in Orlicz-Morrey spaces. If the velocity field satisfies u∈L21-r0,T;ML2logPL3rR3 with r∈(0,1) and P>1 or the gradient field of velocity satisfies ∇u∈L22-r0, T;ML2logPL3rR3 with r∈(0,2) and P>1, then we show that the solution remains smooth on [0,T]. In view of the embedding L3r⊂Mp3r⊂ML2logPL3r with 2 < p < 3 / r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 John Wiley & Sons, Ltd.

DOI： 10.1002/mma.2535

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

In the present paper we obtain and extend the boundedness property of the Adams type for multilinear fractional integral operators. Also, we deal with the Olsen type inequality. © 2012 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.

DOI： 10.1007/s10114-012-0617-y

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

A multi-Morrey norm which is strictly smaller than m-fold product of the Morrey norms is introduced and the boundedness property of the Adams type for multilinear fractional integral operators is obtained. The optimality of the results is also discussed. © 2011 Springer Basel AG.

DOI： 10.1007/s11117-011-0129-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In the present paper we define Hardy spaces with variable exponents on Rn by the grand maximal function, and then investigate their several properties. The present paper will connect harmonic analysis with function spaces with variable exponents. We obtain the atomic decomposition and the molecular decomposition. With these decomposition proved, we investigate the Littlewood-Paley characterization. Also, we specify the dual spaces of Hardy spaces with variable exponents. They will turn out to be Campanato spaces with variable growth conditions. The present paper covers local Hardy spaces with variable exponents. © 2012 Elsevier Inc.

DOI： 10.1016/j.jfa.2012.01.004

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

By taking an interest in a natural extension to the small parameters of the trace inequality for Morrey spaces, Orlicz-Morrey spaces are introduced and some inequalities for generalized fractional integral operators on Orlicz-Morrey spaces are established. The local boundedness property of the Orlicz maximal operators is investigated and some Morrey-norm equivalences are also verified. The result obtained here sharpens the one in our earlier papers. © 2011 Springer Science+Business Media B.V.

DOI： 10.1007/s11118-011-9239-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

This paper concerns the complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents. © 2011 Elsevier Inc.

DOI： 10.1016/j.jmaa.2011.09.023

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BIRKHAUSER VERLAG AG  

The aim of this paper is to give a Gagliardo-Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces over spaces of homogeneous type. © 2012 Springer Basel AG.

DOI： 10.1007/s00013-012-0362-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BIRKHAUSER VERLAG AG  

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems. © 2010 Springer Basel AG.

DOI： 10.1007/s11785-010-0083-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-BLACKWELL  

Rychkov defined weighted Besov spaces and weighted Triebel-Lizorkin spaces coming with a weight in the class Aploc, which is even wider than the class Ap due to Muckenhoupt. In the present paper we are concerned with the atomic decomposition of these spaces. As an application, we obtain some key theorems in the theory of function spaces. Finally we conclude this paper with some helpful examples showing the difference between weighted spaces and unweighted spaces. © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

DOI： 10.1002/mana.200910201

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PUBLICATIONS L INSTITUT MATHEMATIQUE MATEMATICKI  

We revisit the Stummel class and its relation with Morrey spaces. We reformulate a result of Ragusa and Zamboni [11] and then discuss its generalization, as proposed by Eridani and Gunawan [4]. An improvement of the results previously obtained by Eridani and Gunawan is obtained and some extensions are presented.

DOI： 10.2298/PIM1206127G

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Publishing type：Part of collection (book)  

First of all, we will be concentrated in some particular but very important inequalities. Namely, for a real-valued absolutely continuous function on [0,1], satisfying f(0)=0 and $\int:{0}^{1}f'(x)^{2}\,dx<1$, we have, by using the theory of reproducing kernels $$\int:0^1\left(\frac{f(x)}{1-f(x)}\right)^{\prime\,2}(1-x)^2\,dx \le\frac{\int_0^1f^{\prime\,2}(x)\,dx}{1-\int_0^1f^{\prime\,2}(x)\,dx}. $$ A. Yamada gave a direct proof for this inequality with a generalization and, as an application, he unified the famous Opial inequality and its generalizations. Meanwhile, we gave some explicit representations of the solutions of nonlinear simultaneous equations and of the explicit functions in the implicit function theory by using singular integrals. In addition, we derived estimate inequalities for the consequent regularizations of singular integrals. Our main purpose in this paper is to introduce our method of constructing approximate and numerical solutions of bounded linear operator equations on reproducing kernel Hilbert spaces by using the Tikhonov regularization. In view of this, for the error estimates of the solutions, we will need the inequalities for the approximate solutions. As a typical example, we shall present our new numerical and real inversion formulas of the Laplace transform whose problems are famous as typical ill-posed and difficult ones. In fact, for this matter, a software realizing the corresponding formulas in computers is now included in a present request for international patent. Here, we will be able to see a great computer power of H. Fujiwara with infinite precision algorithms in connection with the error estimates.

DOI： 10.1007/978-3-0348-0249-9_6

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Language：English   Publishing type：Research paper (scientific journal)  

As an application of our previous paper, we give a sufficient condition that Brézis- Gallouët-Wainger type inequalities in higher order critical Sobolev spaces fails.

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Language：English   Publishing type：Research paper (scientific journal)  

We propose new constructions of the approximate solutions of discrete linear differential equations and their inverse source problems. This is mainly based on a reproducing kernel Hilbert spaces approach where different types of spaces are naturally considered as a consequence of the method. Here, the major influence will be given by the Paley-Wiener and Sobolev spaces. © 2012, by Oldenbourg Wissenschaftsverlag, München, Germany. All rights reserved.

DOI： 10.1524/anly.2012.1104

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：GLOBAL SCIENCE PRESS  

The theory of reproducing kernel Hilbert spaces is applied to a minimization problem with prescribed nodes. We re-prove and generalize some results previously obtained by Gunawan et al. [2,3], and also discuss the Hölder continuity of the solution to the problem. © 2011 Global-Science Press.

DOI： 10.4208/eajam.050511.310711a

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Language：English   Publishing type：Research paper (scientific journal)  

In the present paper, two new operation on reproducing kernel Hilbert spaces are proposed. One is the pasting and another is the wedge product. As applications of these operations, we reconsider the Cauchy-Binet formula. © 2011 Universidad de Jaén.

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Language：English   Publishing type：Research paper (scientific journal)  

The action of the generalized fractional integral operators and the generalized fractional maximal operators is investigated in the framework of Morrey spaces. A typical property of the functions which belongs to Morrey spaces under point wise multiplication by the generalized fractional integral operators and the generalized fractional maximal operators is established. The bounded ness property of the fractional integral operators on the predual of Morrey spaces is shown as well. A counterexample concerning the Fefferman-Phong inequality is given by the use of the characteristic function of the Cantor set. © 2011 American Mathematical Society.

DOI： 10.1090/S0002-9947-2011-05294-3

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS  

A weighted theory describing Morrey boundedness of fractional integral operators and fractional maximal operators is developed. A new class of weights adapted to Morrey spaces is proposed and a passage to the multilinear cases is covered. © Instytut Matematyczny PAN, 2011.

DOI： 10.4064/sm205-2-2

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS  

The aim of this paper is to develop a theory of decomposition in the weighted modulation spaces Ms,Wp,q with 0 < p, q ≤ ∞, s ε R and W ε A∞, where W belongs to the class of A∞ defined by Muckenhoupt. For this purpose we shall define molecules for the modulation spaces. As an application we give a simple proof of the boundedness of the pseudo-differential operators with symbols in M∞0,min(1, p,q). We shall deal with dual spaces as well.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-BLACKWELL  

The purpose of this paper is to develop a theory of the Besov-Morrey spaces and the Triebel-Lizorkin-Morrey spaces on domains in Rn. We consider the pointwise multiplier operator, the trace operator, the extension operator and the diffeomorphism operator. Not only to domains in Rn we extend our definition of function spaces to compact oriented Riemannian manifolds. Among the properties above, the result for the trace operator is in particular interesting, which reflects the property of the parameters p, q in the Morrey space Mpq. © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

DOI： 10.1002/mana.200710151

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let p∈(1,∞), q∈[1,∞), s∈R and τ∈[0,1- 1/max{p,q}]. In this paper, the authors establish the φ-transform characterizations of Besov-Hausdorff spaces BḢp,qs,τ(Rn) and Triebel-Lizorkin-Hausdorff spaces FḢp,qs,τ(Rn) (q>1); as applications, the authors then establish their embedding properties (which on BH p,qs,τ(Rn) is also sharp), smooth atomic and molecular decomposition characterizations for suitable τ. Moreover, using their atomic and molecular decomposition characterizations, the authors investigate the trace properties and the boundedness of pseudo-differential operators with homogeneous symbols in BḢp,qs,τ(Rn) and FḢp,qs,τ(Rn) (q>1), which generalize the corresponding classical results on homogeneous Besov and Triebel-Lizorkin spaces when p∈(1,∞) and q∈[1,∞) by taking τ=0. © 2010 Elsevier Inc.

DOI： 10.1016/j.jmaa.2010.04.021

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MICHIGAN MATHEMATICAL JOURNAL  

DOI： 10.1307/mmj/1272376028

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Element D.O.O.  

We shall introduce natural regularizations for singular integrals from the viewpoint of numerical treatments and establish very good error estimates (fundamental inequalities for potentials) for the regularizations. © ELEMENT Zagreb.

DOI： 10.7153/mia-13-23

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In this paper, the authors prove that Besov-Morrey spaces are proper subspaces of Besov-type spaces over(B, ̇)p, qs, τ (Rn) and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces over(F, ̇)p, qs, τ (Rn). The authors also establish an equivalent characterization of over(B, ̇)p, qs, τ (Rn) when τ ∈ [0, 1 / p). These Besov-type spaces over(B, ̇)p, qs, τ (Rn) and Triebel-Lizorkin-type spaces over(F, ̇)p, qs, τ (Rn) were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces over(B, ̇)p, qs, τ (Rn) and over(F, ̇)p, qs, τ (Rn), the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos-Torres by taking τ = 0. © 2009 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2009.08.002

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In this paper we consider the Haar wavelet on weighted Herz spaces. Our weight class, whose name is Ap-dyadic local, is the one defined by the first author (2007). We shall investigate the class of Ap-dyadic weights in connection with the maximal inequalities. After obtaining the properties of weights in the first half of the present paper, we consider the Haar wavelet on weighted Herz spaces in the latter half. We shall show that the Haar wavelet basis is an unconditional basis. We also show that the Haar wavelet is not greedy except for the trivial case, that is, the Haar wavelet is greedy if and only if the Herz space under consideration is a weighted Lp space. © 2009 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jmaa.2009.08.005

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

The present paper concerns the Sobolev embedding in the endpoint case. It is known that the embedding W1,n (ℝn) → L ∞(ℝn) fails for n≥2. Brézis- Gallouët-Wainger and some other authors quantified why this embedding fails by means of the Hlder-Zygmund norm. In the present paper we will give a complete quantification of their results and clarify the sharp constants for the coefficients of the logarithmic terms in Besov and Triebel-Lizorkin spaces. Copyright © 2010 Kei Morii et al.

DOI： 10.1155/2010/584521

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Adam Mickiewicz University Press  

In the present note we characterize the function space Xr, which is the set of pointwise multipliers which map L2 into H-r. To this end, we use wavelets and capacity.

DOI： 10.7169/facm/1291903392

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS  

The aim of the present paper is to obtain an inequality of Brézis-Gallouët-Wainger type for Besov-Morrey spaces. We investigate these spaces in a self-contained manner. Also, we verify that our result is sharp. © Instytut Matematyczny PAN, 2010.

DOI： 10.4064/sm196-1-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PERGAMON-ELSEVIER SCIENCE LTD  

In the present paper we consider weighted modulation spaces. Weights of exponential growth will be admissible. With the molecular decomposition we shall obtain an unconditional basis. As applications, we obtain the greedy basis of coorbit spaces. © 2009 Elsevier Ltd. All rights reserved.

DOI： 10.1016/j.na.2009.03.036

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We obtain wavelet characterizations of Besov spaces and the Triebel-Lizorkin spaces associated with A∞loc-weights. These characterizations are used to show that our wavelet bases are also greedy. © 2009 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jat.2008.12.003

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY-V C H VERLAG GMBH  

We define Morrey type Besov-Triebel spaces with the underlying measure non-doubling. After defining the function spaces, we investigate boundedness property of some class of the singular integral operators. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

DOI： 10.1002/mana.200610818

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AUSTRALIAN MATHEMATICS PUBL ASSOC INC  

We discuss here the boundedness of the fractional integral operator I a and its generalized version on generalized nonhomogeneous Morrey spaces. To prove the boundedness of Ia, we employ the boundedness of the so-called maximal fractional integral operator Ia,k*. In addition, we prove an Olsen-type inequality, which is analogous to that in the case of homogeneous type. © Australian Mathematical Publishing Association Inc. 2009.

DOI： 10.1017/S0004972709000343

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

In the present paper, we obtain three independent results on the Besov-Morrey spaces and the Triebel-Lizorkin-Morrey spaces. That is, we are going to obtain the characterization of local means, the boundedness of pseudo-differential operators and the characterization of the Hardy-Morrey spaces. By using the maximal estimate and the molecular decomposition, we shall integrate and extend the known results on these spaces. © 2009 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH.

DOI： 10.1007/s10114-009-8247-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：HINDAWI PUBLISHING CORPORATION  

We show some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces. Copyright © 2009 Yoshihiro Sawano et al.

DOI： 10.1155/2009/835865

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In the present paper, we investigate the predual of the morrey spaces with non-doubling measures. we also study the modified maximal function, singular integrals and commutators on the predual spaces. © 2009 International Academic Printing Co. Ltd. All rights reserved.

DOI： 10.3836/tjm/1264170244

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BIRKHAUSER VERLAG AG  

In this paper, we define the generalized Morrey spaces on ℝ with the measure μ non-doubling. After defining the space, we shall investigate the properties of maximal operators, fractional integral operators and the singular integral operators. And we shall allude to the vector-valued extension of these operators. © 2008 Birkhäuser Verlag Basel/Switzerland.

DOI： 10.1007/s00030-008-6032-5

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In this paper we apply quarkonial decomposition to the ordinary differential equation of delay type f′(x) = f(x -l), x ≥1. We shall derive an explicit formula in terms of the quarkonial decomposition. © 2008 WILEY-VCH Verlag, GmbH & Co. KGaA.

DOI： 10.1002/mana.200510716

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Language：English   Publishing type：Research paper (scientific journal)  

In this paper, we define the generalized Morrey spaces on ℝ with the measure μ non-doubling. After defining the space, we shall investigate the properties of maximal operators, fractional integral operators and the singular integral operators. And we shall allude to the vector-valued extension of these operators. © 2008 Birkhäuser Verlag Basel/Switzerland.

DOI： 10.1007/s00030-008-6032-5

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In the present paper, we present a natural extension of the modulation spaces Mp, qs with 0 < p < ∞, 0 < q ≤ ∞ and s ∈ R. The weight we take up here belongs to the class A∞loc, which is even wider and contains weights of exponential growth. After defining the function spaces, we investigate their atomic and molecular decomposition as well as some elementary properties. © 2008.

DOI： 10.1016/j.jmaa.2008.04.034

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BIRKHAUSER VERLAG AG  

We investigate compactness of linear operators associated with the real inversion formulas of the Laplace transform, coming with weighted Sobolev reproducing kernel Hilbert spaces on the half line R +. We present concrete reproducing kernels along with several typical examples. © 2008 Birkhaeuser.

DOI： 10.1007/s11785-007-0041-y

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Adam Mickiewicz University Press  

In the present paper we characterize the Besov-Morrey spaces and the Triebel- -Lizorkin-Morrey spaces in terms of wavelet. As an application we investigate some embedding properties and diversity of the function spaces.

DOI： 10.7169/facm/1229624654

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER GMBH  

We study multi-commutators on the Morrey spaces generated by BMO functions and singular integral operators or by BMO functions and fractional integral operators. We place ourselves in the setting of Rd coming with a Radon measure µ which satisfies a certain growth condition. The Morrey-boundedness of commutators is established by M. Yan and D. Yang. However, the corresponding assertion of Morrey-compactness is still missing. The aim of this paper is to prove that the multi-commutators are compact if one of the BMO functions can be approximated with compactly supported smooth functions. © 2008, Heldermann Verlag. All rights reserved.

DOI： 10.1515/GMJ.2008.353

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

The aim of this paper is to define the Besov-Morrey spaces and the Triebel- Lizorkin-Morrey spaces and to present a decomposition of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato. © 2007 Springer-Verlag.

DOI： 10.1007/s00209-007-0150-3

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CiNii Research

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Other Link： https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-05J11230/

Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATHEMATICAL SOC REP CHINA  

In this paper, we establish a sharp maximal inequality for Morrey spaces with a Radon measure μ satisfying certain growth condition, which is not necessarily doubling. As an application, we obtain the boundedness of commutators generated by singular integral or fractional integrals with RBMO functions in Morrey spaces.

DOI： 10.11650/twjm/1500404805

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CiNii Research

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：POLISH ACAD SCIENCES INST MATHEMATICS  

X. Tolsa defined a space of BMO type for positive Radon measures satisfying some growth condition on ℝd. This new BMO space is very suitable for the Calderón-Zygmund theory with non-doubling measures. Especially, the John-Nirenberg type inequality can be recovered. In the present paper we introduce a localized and weighted version of this inequality and, as applications, we obtain some vector-valued inequalities and weighted inequalities for Morrey spaces. © Instytut Matematyczny PAN, 2007.

DOI： 10.4064/sm181-2-3

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Language：English   Publishing type：Research paper (scientific journal)  

CiNii Research

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGEROPEN  

We show the boundedness of fractional integral operators by means of extrapolation. We also show that our result is sharp. Copyright © 2006 Yoshihiro Sawano et al.

DOI： 10.1155/JIA/2006/92470

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CiNii Research

J-GLOBAL

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Language：English   Publishing type：Research paper (scientific journal)  

We consider the vector-valued extension of the Fefferman-Stein- Strömberg sharp maximal inequality under growth condition. As an application we obtain a vector-valued extension of the boundedness of the commutator. Furthermore, we prove the boundedness of the commutator. © 2006, Heldermann Verlag. All rights reserved.

DOI： 10.1515/GMJ.2006.153

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

The authors give a natural definition of Morrey spaces for Radon measures which may be non-doubling but satisfy certain growth condition, and investigate the boundedness in these spaces of some classical operators in harmonic analysis and their vector-valued extension. © Springer-Verlag 2005.

DOI： 10.1007/s10114-005-0660-z

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Language：English   Publishing type：Research paper (scientific journal)  

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Language：English   Publishing type：Research paper (scientific journal)  

In this paper we consider the modified maximal operator on the separable metric space. Define [Formula] and [Formula] respectively. We investigate in what parameter k the weak (1, 1)-inequality holds for Mk and Mk, uc in general metric space and Euclidean space. The proofs are sharper than the method of Vitali’s covering lemma. This attempt is partially done by Yutaka Terasawa [9] before. When we investigate Rd, we prove a new covering lemma of Rd. We also show that our condition on parameter k is sharp. In connection with this we consider the dual inequality of Stein type and its applications. © 2005 by the University of Notre Dame. All rights reserved.

DOI： 10.14492/hokmj/1285766231

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Total pages：xxiii, 945 p.   Language：English  

ASIN

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Total pages：ii, 239p  

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Total pages：xviii, 452 p.   Language：English  

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Total pages：203  

ASIN

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Total pages：viii, 203p   Language：Japanese  

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Total pages：195  

ASIN

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Total pages：ix, 195p   Language：Japanese  

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Total pages：xii, 426p   Language：Japanese  

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Total pages：1冊   Language：English  

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Composition operators have been extensively studied in complex analysis, and<br />
recently, they have been utilized in engineering and machine learning. Here, we<br />
focus on composition operators associated with maps in Euclidean spaces that<br />
are on reproducing kernel Hilbert spaces with respect to analytic positive<br />
definite functions, and prove the maps are affine if the composition operators<br />
are bounded. Our result covers composition operators on Paley-Wiener spaces and<br />
reproducing kernel spaces with respect to the Gaussian kernel on ${\mathbb<br />
R}^d$, widely used in the context of engineering.

DOI： 10.1016/j.jmaa.2022.126048

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arXiv

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Authorship：Corresponding author  

We obtain the boundedness from a product of Lebesgue or Hardy spaces into Hardy spaces under suitable cancellation conditions for a large class of multilinear operators that includes the Coifman–Meyer class, sums of products of linear Calderón–Zygmund operators and combinations of these two types.

DOI： 10.1002/mana.201800252

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arXiv

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Other Link： http://arxiv.org/pdf/1702.02229v1

Authorship：Lead author,　Corresponding author  

The paraproduct plays a key role in some highly singular partial differential equations. For example, that for Hölder–Zygmund spaces is used to solve stochastic differential equations, as demonstrated by Gubinelli, Imkeller, Perkowski, and Hairer. In this note, which is organized in a self-contained manner, the counterparts for Besov Morrey spaces are obtained.

DOI： 10.1007/978-3-030-26748-3_8

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arXiv

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Other Link： http://arxiv.org/pdf/1812.08379v1

Language：Japanese   Publisher：京都大学数理解析研究所  

本研究は関数空間の補間についてである. 特にモレー空間を調べる. 第二補間空間の見える化をすることが目的である. この研究はDenny Ivanal Hakim氏, Miestyslaw Mastylo氏, 中村昌平氏, 曽布川拓也氏との共同研究である. とくに, 現時点では投稿中のMicstyslaw Mastylo氏との研究成果について説明する.

CiNii Books

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Other Link： http://hdl.handle.net/2433/251699

Language：English   Publishing type：Internal/External technical report, pre-print, etc.  

We study the stochastic dissipative quasi-geostrophic equation with
space-time white noise on the two-dimensional torus. This equation is highly
singular and basically ill-posed in its original form. The main objective of
the present paper is to formulate and solve this equation locally in time in
the framework of paracontrolled calculus when the differential order of the
main term, the fractional Laplacian, is larger than $7/4$. No renormalization
has to be done for this model.

arXiv

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Other Link： http://arxiv.org/pdf/1806.00564v3

Publishing type：Internal/External technical report, pre-print, etc.  

We prove that the global minimum of the backpropagation (BP) training problem<br />
of neural networks with an arbitrary nonlinear activation is given by the<br />
ridgelet transform. A series of computational experiments show that there<br />
exists an interesting similarity between the scatter plot of hidden parameters<br />
in a shallow neural network after the BP training and the spectrum of the<br />
ridgelet transform. By introducing a continuous model of neural networks, we<br />
reduce the training problem to a convex optimization in an infinite dimensional<br />
Hilbert space, and obtain the explicit expression of the global optimizer via<br />
the ridgelet transform.

arXiv

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Language：Japanese   Publisher：日本数学会  

CiNii Books

CiNii Research

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Language：English   Publisher：京都大学数理解析研究所