We introduce mixed Morrey spaces and show some basic properties. These properties extend the classical ones. We investigate the boundedness in these spaces of the iterated maximal operator, the fractional integtral operator and singular integral operator. Furthermore, as a corollary, we obtain the boundedness of the iterated maximal operator in classical Morrey spaces. We also establish a version of the Fefferman--Stein vector-valued maximal inequality and some weighted inequalities for the iterated maximal operator in mixed Lebesgue spaces. We point out some errors in the proof of the existing literature.
In this paper, we obtain predual spaces of local mixed Morrey-type spaces, characterize mixed Hardy local Morrey-type spaces. Further also, investigate nonsmooth decomposition of local mixed Morrey-type spaces. As an application, we consider the Hardy operators on local mixed Morrey-type spaces.
In this paper, the authors define the mixed $λ$-central Morrey spaces and the mixed $λ$-central $BMO$ spaces. The boundedness of the fractional integral operators $T_α$ and its commutators $[b, T_α]$ are established on the mixed $λ$-central Morrey spaces, respectively. Furthermore, we also extend these results to the generalized mixed central Morrey spaces.
In this paper, we introduce homogeneous mixed Herz-Morrey spaces $M\dot{K}_{p,\vec{q}}^{α,λ}(\mathbb{R}^n)$ and show it's some properties. Firstly, the boundedness of sublinear operators, fractional type operators in homogeneous mixed Herz-Morrey spaces is investigated. In particular, the above results are still valid for Calder$\acute{o}$n-Zygmund operators and fractional maximal operators. Lastly, the boundedness of their commutators in homogeneous mixed Herz-Morrey spaces is obtained.
In this paper, we introduced the local and global mixed Morrey-type spaces, and some properties of these spaces are also studied. After that, the necessary conditions of the boundedness of fractional integral operators $I_α$ are studied respectively in mixed-norm Lebesgue spaces and the local mixed Morrey-type spaces. Last but not least, the boundedness of $I_α$ is obtained by Hardy operators' boundedness in weighted Lebesgue spaces. As corollaries, we acquire the boundedness of $I_α$ in mixed Morrey spaces(\hyperlink{Corollary 5.6}{Corollary 5.6}) and mixed Lebesgue spaces(\hyperlink{Corollar…
In this paper, we study the separable and weak convergence of mixed-norm Lebesgue spaces. Furthermore, we prove that the block space $\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$ is the Köthe dual of the mixed Morrey space $\mathcal{M}_{\vec{p}}^{p_0}(\mathbb{R}^n)$ by the Fatou property of these block spaces. The boundedness of the Hardy--Littlewood maximal function is further obtained on the block space $\mathcal{B}_{\vec{p}\,'}^{p'_0}(\mathbb{R}^n)$. As applications, the characterizations of $BMO(\mathbb{R}^n)$ via the commutators of the fractional integral operator $I_α$ on mixed Morrey …
In this paper, the author studies the boundedness for a large class of sublinear operator $T_α, α\in[0,n)$ generated by Calder{ó}n-Zygmund operators ($α=0$) and generated by fractional integral operator ($α>0$) on generalized mixed Morrey spaces $M^φ_{\vec{q}}(\Bbb R^n)$. Moreover, the boundeness for the commutators of $T_α, α\in[0,n)$ on generalized mixed Morrey spaces $M^φ_{\vec{q}}(\Bbb R^n)$ is also studied. As applications, we obtain the boundedness for Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operators, fractional integral operator, fractional maximal operato…
In this paper we introduce Besov-Morrey spaces with all indices variable and study some fundamental properties. This includes a description in terms of Peetre maximal functions and atomic and molecular decompositions. This new scale of non-standard function spaces requires the introduction of variable exponent mixed Morrey-sequence spaces, which in turn are defined within the framework of semimodular spaces. In particular, we obtain a convolution inequality involving special radial kernels, which proves to be a key tool in this work.
In this paper we calculate some geometric constants for Morrey spaces and small Morrey spaces, namely generalized Von Neumann-Jordan constant, modified Von Neumann-Jordan constants, and Zbáganu constant. All these constants measure the uniformly nonsquareness of the spaces. We obtain that their values are the same as the value of Von Neumann-Jordan constant for Morrey spaces and small Morrey spaces.
In this paper, we study the boundedness for a large class of multi-sublinear operators $T_m$ generated by multilinear Calder{ó}n-Zygmund operators and their commutators $T^{b}_{m,i}~(i=1,\cdots,m)$ on the product generalized mixed Morrey spaces $M^{φ_1}_{\vec{q_1}}({\Bbb R}^n)\times\cdots\times M^{φ_m}_{\vec{q_m}}({\Bbb R}^n)$. We find the sufficient conditions on $(φ_1,\cdots,φ_m,φ)$ which ensure the boundedness of the operator $T_m$ from $M^{φ_1}_{\vec{q_1}}({\Bbb R}^n)\times\cdots\times M^{φ_m}_{\vec{q_m}}({\Bbb R}^n)$ to $M^φ_{\vec{q}}({\Bbb R}^n)$. Moreover, the sufficient conditions for …
In this paper, we prove that the inclusions between Morrey spaces, between weak Morrey spaces, and between a Morrey space and a weak Morrey space are all proper. The proper inclusion between a Morrey space and a weak Morrey space is established via the unboundedness of the Hardy-Littlewood maximal operator on Morrey spaces of exponent 1. In addition, we also give a necessary condition for each inclusion. Our results refine previous inclusion properties studied in [Gunawan et al, \emph{Math. Nachr.} {\bf 290} (2017), 332--340].
We will prove that under certain conditions on the parameters the operators $T^{+}f = \max(f,0)$ and $ Tf = |f| $ are bounded mappings on the Triebel-Lizorkin-Morrey and Besov-Morrey spaces. Moreover we will show that some of the conditions we mentioned before are also necessary. Furthermore we prove that for $p < u $ in many cases the Triebel-Lizorkin-Morrey spaces do not have the Fubini property.
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces. To do this we prove the weak-weak type modular inequality of the Hardy-Littlewood maximal operator with respect to the Young function. Orlicz-Morrey spaces contain $L^p$ spaces ($1\le p\le\infty$), Orlicz spaces and generalized Morrey spaces as special cases. Hence we get necessary and sufficient conditions on these function spaces as corollaries.
In this paper, we consider the atomic decomposition for Morrey-Lorentz spaces and applications. Morrey-Lorentz spaces, which have structures of Morrey spaces, Lorentz spaces and their weak-type spaces, are introduced by M. A. Ragusa in 2012. Our study gave some extension of the atomic decomposition to Morrey-Lorentz spaces. As an application, the Olsen inequality can be obtained more sharpness.
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 $p$.
The goal of this paper is to extend Nakai's generalized Morrey spaces to a wider function class, the one-sided Muckenhoupt weighted case. Morrey matching Muckenhoupt enables us to study the weak and strong type boundedness of one-sided sublinear operators satisfying certain size conditions on the one-sided weighted Morrey spaces. We also establish one-sided Fefferman-Stein inequalities on one-sided weighted Morrey spaces in this paper. Meanwhile, the boundedness and compactness of Riemann-Liouville integral operators on locally one-sided weighted Morrey space are considered. As applications, w…
A new subspace of Morrey spaces whose elements can be approximated by infinitely differentiable compactly supported functions is introduced. Consequently, we give an explicit description of the closure of the set of such functions in Morrey spaces. A generalization of known embeddings of Morrey spaces into weighted Lesbesgue spaces is also obtained.
Orlicz--Morrey spaces are generalizations of Orlicz spaces and Morrey spaces which were first introduced by Nakai. There are three versions of Orlicz--Morrey spaces, i.e: Nakai's (2004), Sawano--Sugano--Tanaka's (2012), and Deringoz--Guliyev--Samko's (2014) versions. On this article we will discuss the third version of weak Orlicz--Morrey space which is seen as an enlargement of third version of (strong) Orlicz--Morrey space. Similar to its first version and second version, the third version of weak Orlicz-Morrey space is considered as a generalization of weak Orlicz spaces, weak Morrey spaces…
We prove that Stein's Extension Operator preserves Sobolev-Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n-dimensional Euclidean space.
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 …
We discuss the Hardy-Littlewood maximal operator on discrete Morrey spaces of arbitrary dimension. In particular, we obtain its boundedness on the discrete Morrey spaces using a discrete version of the Fefferman-Stein inequality. As a corollary, we also obtain the boundedness of some Riesz potentials on discrete Morrey spaces.
The bidual of the closure of smooth functions with respect to the Morrey norm coincides with the Morrey space. This assertion is generalized to some Muckenhoupt weighted Morrey spaces. We combine this fact with basic extrapolation techniques due to Rubio de Francia adapted to weighted Morrey spaces. This leads to new results on the boundedness of operators even for the unweighted case.
In this article, we compute Von Neumann-Jordan constant, James constant, and Dunkl-Williams constant for small Morrey spaces. Our approach can also be seen as an alternative way in computing the three constants for the (classical) Morrey spaces. In addition, we prove constructively that Morrey spaces are not uniformly non-octahedral.
We give a necessary condition for inclusion relations between discrete Morrey spaces which can be seen as a complement of the results in \cite{GKS,HS2}. We also prove another inclusion property of discrete Morrey spaces which can be viewed as a generalization of the inclusion property of the spaces of $p$-summable sequences. Analogous results for weak type discrete Morrey spaces is also presented. In addition, we show that each of these inclusion relations is proper. Some connections between inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.
We introduce grand Morrey spaces and establish the boundedness of Hardy--Littlewood maximal, Calderón--Zygmund and potential operators in these spaces. In our case the operators and grand Morrey spaces are defined on quasi-metric measure spaces with doubling measure. The results are new even for Euclidean spaces.
In this note we reprove generalized Hölder's inequality in weak Morrey spaces. In particular, we get sharper bounds than those in \cite{gunawan2}. The bounds are obtained through the relation of weak Morrey spaces and weak Lebesgue spaces.
We presents the study the separability properties for differential-operator equations in Morrey spaces. We prove that the corresponding differential operator is a generator of analytic semigroup in vector-valued Morrey spaces. Moreover, maximal regularity properties of corresponding parabolic equation ıs obtained. In applications, the maximal regularity properties of Wentzell-Robin type problem for elliptic equations and mixed value problem for degenerate parabolic equations in Morrey spaces are derived.
We prove that Burenkov's Extension Operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n-dimensional Euclidean space.
We improve our results on boundedness of the Riesz potential in the central Morrey-Orlicz spaces and the corresponding weak-type version. We also present two new properties of the central Morrey-Orlicz spaces: nontriviality and inclusion property.
In this paper, we define for the first time the local Morrey-type space associated with ball quasi-Banach function spaces and show the related series of properties. In addition, Hardy-Littlewood maximal operator's boundedness is proved. We investigate nonsmooth decomposition of the local Morrey-type space associated with ball quasi-Banach function spaces via the Hardy local Morrey-type spaces associated with ball quasi-Banach function spaces. And we consider Hardy operator's boundedness.
We consider the commutators $[b,T]$ and $[b,I_ρ]$ on Orlicz-Morrey spaces, where $T$ is a Calderón-Zygmund operator, $I_ρ$ is a generalized fractional integral operator and $b$ is a function in generalized Campanato spaces. We give a necessary and sufficient condition for the boundedness of the commutators on Orlicz-Morrey spaces. To do this we prove the boundedness of generalized fractional maximal operators on Orlicz-Morrey spaces. Moreover, we introduce Orlicz-Campanato spaces and establish their relations to Orlicz-Morrey spaces.
We study embeddings between generalised Besov-Morrey spaces. Both sufficient and necessary conditions for the embeddings are proved. Embeddings of the Besov-Morrey spaces into the Lebesgue spaces are also considered. Our approach requires a wavelet characterisation of the spaces which we establish for the system of Daubechies wavelets.
In this paper, we prove the Spanne-type boundedness of the generalized Riesz potential operator from the one generalized weighted local Morrey spaces to the another one, and from the generalized weighted local Morrey spaces to the weak generalized weighted local Morrey spaces. We also prove the Adams-type boundedness of the operator from the weighted spaces to the another one, and from the weighted spaces to the weak weighted spaces.
We discuss discrete Morrey spaces and their generalizations, and we prove necessary and sufficient conditions for the inclusion property among these spaces through an estimate for the characteristic sequences.
We discuss here a weak and strong type estimate for fractional integral operators on Morrey spaces, where the underlying measure $μ$ does not always satisfy the doubling condition.
We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain $Ω\subset {\mathbb R}^d$. This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation numbers. We now prove the fi…
Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces $m_{u,p}=m_{u,p}(\mathbb{Z}^d)$, $0<p\leq u<\infty$, which have yet been considered almost nowhere. They are defined as natural generalizations of the classical $\ell_p$ spaces. We consider some basic features, embedding properties, the pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterize the compactness of embeddings of related finite-dimensional spaces.
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 $L^1$ and $L^\infty$ spaces.
In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results.
In this paper, we establish multilinear BMO estimates for commutators of multilinear fractional maximal and integral operators both on product generalized Morrey spaces and product generalized vanishing Morrey spaces, respectively. Similar results are still valid for commutators of multilinear maximal and singular integral operators.
We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidea…
This paper includes new bounds concepting the vanishing generalized weighted Morrey space. In this sense, it is outlined improved bounds about the a class of fractional type rough higher order commutators on vanishing generalized weighted Morrey spaces.
The aim of this paper is to prove the boundedness of the oscillation and variation operators for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces.
Let $Ω\subset \mathbb{R}^n$ be an unbounded open set. We consider the generalized weighted Morrey spaces $\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ and the vanishing generalized weighted Morrey spaces $V\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ with variable exponent $p(x)$ and a general function $φ(x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces $\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ and $V\mathcal{M}^{p(\cdot),φ}_ω(Ω)$. This result generalizes several existing results for Riesz potential and its commutators on Morrey type spaces.…
In this note we introduce grand grand Morrey spaces, in the spirit of the grand Lebesgue spaces. We prove a kind of \textit{reduction lemma} which is applicable to a variety of operators to reduce their boundedness in grand grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund operators in the framework of grand grand Morrey spaces.
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
Recently it turned out that the paraproduct plays the key role in some highly singular partial differential equations. In this note the counterparts for Besov--Morrey spaces are obtained. This note is organized in a self-contained manner.
Boundedness of composition operator in Orlicz-Morrey spaces
Graduate School of Information Science and Technology, The University of Osaka（大阪大学大学院信息科学技术研究科）；Center for Advanced Intelligence Project RIKEN（理化学研究所先进智能项目中心）；Center for Science Adventure and Collaborative Research Advancement (SACRA), Graduate School of Science, Kyoto University（京都大学理学研究院科学冒险与合作研究推进中心）；Center for Mathematics and Data Science, Gunma University（群马大学数学与数据科学中心）
It is proved that both oscillatory integral operators and fractional oscillatory integral operators are bounded on weighted Morrey spaces. The corresponding commutators generated by $BMO$ functions are also considered.
