In this article, we show constructively that Morrey spaces are not uniformly non-$\ell^1_n$ for any $n\ge 2$. This result is sharper than those previously obtained in \cite{GKSS, MG}, which show that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the $n$-th James constant $C_{\rm J}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{\rm NJ}^{(n)}(X)$ for a Banach space $X$, and obtain that both constants for any Morrey space $\mathcal{M}^p_q(\mathbb{R}^d)$ with $1\le p<q<\infty$ are equal to $n$.
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.
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 note we prove that the $n$-th Von Neumann-Jordan constant and the $n$-th James constant for discrete Morrey spaces $\ell^p_q$ where $1\le p<q<\infty$ are both equal to $n$. This result tells us that the discrete Morrey spaces are not uniformly non-$\ell^1$, and hence they are not uniformly $n$-convex.
In this paper, we calculate four geometric constants for discrete Morrey spaces. The constants are generalized von Neumann-Jordan constant, modified von Neumann-Jordan constant, von Neumann-Jordan type constant, and Zbäganu constant. The four constants measure uniformly nonsquareness of the above spaces. We obtain that the value of each of the four constants for the above spaces is two, which means that the spaces are NOT uniformly nonsquare.
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.
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…
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.
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 $p$.
There are two versions of Orlicz-Morrey spaces (on $\mathbb{R}^n$), defined by Nakai in 2004 and by Sawano, Sugano, and Tanaka in 2012. In this paper we discuss the inclusion properties of these two spaces and compare the results. Computing the norms of the characteristic functions of balls in $\mathbb{R}^n$ is one of the keys to our results. Similar results for weak Orlicz-Morrey spaces of both versions are also obtained.
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 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.
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 existin…
We define a scale of weighted Morrey spaces which contains different weighted versions appearing in the literature. This allows us to obtain weighted estimates for operators in a unified way. In general, we obtain results for weights of the form $|x|^αw(x)$ with $w\in A_p$ and nonnegative $α$. We study particularly some properties of power-weighted spaces and in the case of the Hardy-Littlewood maximal operator our results for such spaces are sharp. By using extrapolation techniques the results are given in abstract form in such a way that they are automatically obtained for many operators.
We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $C_0^\infty(\mathbb{R}^n)$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.
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.
The Hardy-Morrey spaces related to Laplace-Bessel differential equations are introduced in terms of maximal functions. The atomic decomposition theory which has the same cancellation properties of the $H^{p}_{Δ_ν}(\mathbb{R}^{n}_{+})$ Hardy spaces is established.
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 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.
In this paper, some boundedness for commutators of fractional integrals are obtained on Herz-Morrey spaces with variable exponent applying some properties of varible exponent and $\BMO$ function.
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.
The Rademacher functions are investigated in the Morrey spaces M(p,w) on [0,1] for 1 \le p <\infty and weight w being a quasi-concave function. They span l_2 space in M(p,w) if and only if the weight w is smaller than the function log_2^{-1/2}(2/t) on (0,1). Moreover, if 1 < p < \infty the Rademacher sunspace R_p is complemented in M(p,w) if and only if it is isomorphic to l_2. However, the Rademacher subspace is not complemented in M(1,w) for any quasi-concave weight w. In the last part of the paper geometric structure of Rademacher subspaces in Morrey spaces M(p,w) is described. It turns out…
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 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 the class of bounded functions and the closure of classical Morrey space in grand variable exponent Morrey space 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 Lebesghue space defined on quasi-metric measure spaces with sigma-finite measure are obtained, which is new even when un…
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.
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…
Endpoint and Interpolation Estimates for Higher-Order Commutators of Rough Fractional Maximal Operators with Variable Kernels on Variable Exponent Morrey Spaces
In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,μ)$. The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if $\mathscr{X}$ supports a weak Poincaré inequality and the measure $μ$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Hajłasz gradient, the authors also introduce the Hajłasz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Hajłasz-Morr…
In the recent years so-called Morrey smoothness spaces attracted a lot of interest. They can (also) be understood as generalisations of the classical spaces $A^s_{p,q} (\mathbb{R}^n)$, $A\in \{B,F\}$, in $\mathbb{R}^n$, where the parameters satisfy $s\in \mathbb{R}$ (smoothness), $0<p \le \infty$ (integrability) and $0<q \le \infty$ (summability). In the case of Morrey smoothness spaces additional parameters are involved. In our opinion, among the various approaches at least two scales enjoy special attention, also in view of applications: the scales $\mathcal{A}^s_{u,p,q} (\mathbb{R}^n)$, wit…
In this paper we consider smoothness and decay properties of radial functions belonging to smoothness spaces related to Morrey spaces (Sobolev-Morrey spaces, Besov-type spaces and Besov-Morrey spaces). Within this framework we prove generalizations of the Radial Lemma of Strauss.
We investigate some smoothness properties for a transport-diffusion equation involving a class of non-degerate L{é}vy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some H{ö}lder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey-Campanato space for which the regularizing properties of the L{é}vy operator suffice to obtain global H{ö}lder continuity.
To each weighted Dirichlet space $\mathcal{D}_p$, $0<p<1$, we associate a family of Morrey-type spaces ${\mathcal{D}}_p^λ$, $0< λ< 1$, constructed by imposing growth conditions on the norm of hyperbolic translates of functions. We indicate some of the properties of these spaces, mention the characterization in terms of boundary values, and study integration and multiplication operators on them.
Let $[b,\mathcal T_α]~(0\leqα<n)$ be the commutators generated by $BMO(\mathbb R^n)$ functions and a class of sublinear operators satisfying certain size conditions. The aim of this paper is to study the endpoint estimates of these commutators in the weighted Morrey spaces and in the generalized Morrey spaces, under the assumptions that $[b,\mathcal T_α]~(0\leqα<n)$ satisfy (weighted or unweighted) endpoint inequalities on $\mathbb R^n$ and on bounded domains. Furthermore, as applications of our main results, we will obtain, in the endpoint case, the boundedness properties of many important op…
Within the global setting of singular drifts in Morrey-Campanato spaces presented in [6], we study now the H{ö}lder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a Besov stability property. We will see how this Besov information is relevant and how it allows to improve previous results. Moreover, in some particular cases we show that as the nonlinear drift becomes more regular, in the sense of Morrey-Campanato spaces, the additional Besov stability property will be less useful.
In this article, the authors completely answer an open question, presented in [Banach J. Math. Anal. 15 (2021), no. 1, 20], via showing that the Riesz--Morrey space is truly a new space larger than a particular Lebesgue space with critical index. Indeed, this Lebesgue space is just the real interpolation space of the Riesz--Morrey space for suitable indices. Moreover, the authors further show the aforementioned inclusion is also proper, namely, this embedding is sharp in some sense, via constructing two nontrivial spare functions, respectively, on $\mathbb{R}^n$ and any given cube $Q_0$ of $\m…
Let $M_{Ω,\a}$ and $I_{Ω,\a}$ be the fractional maximal and integral operators with rough kernels, where $0 < \a < n$. In this paper, we shall study the continuity properties of $M_{Ω,\a}$ and $I_{Ω,\a}$ on the generalized local Morrey spaces $LM_{p,φ}^{x_0}$. The boundedness of their commutators with local Campanato functions is also obtained.
In this paper, applying some properties of variable exponent analysis, we first dwell on Adams and Spanne type estimates for a class of fractional type integral operators of variable orders, respectively and then, obtain variable exponent generalized Campanato estimates for the corresponding commutators on the vanishing generalized Morrey spaces $VL_{Π}^{p\left( \cdot \right) ,w\left( \cdot \right) }\left( E\right) $ with variable exponent $p(\cdot )$ and bounded set $E$
We refine the definition of generalized Stummel classes and study inclusion properties of these classes. We also study the inclusion relation between Stummel classes and other function spaces such as generalized Morrey spaces, weak Morrey spaces, and Lorentz spaces. In addition, we show that these inclusions are proper. Our results extend some previous results in \cite{CRR, RZ}.
We prove well-posedness results for the Dirichlet problem in $\mathbb{R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)$ and in generalized Morrey-Campanato spaces $\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ under certain assumptions on the growth function $ω$. We also identify a class of growth functions $ω$ for which $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)=\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and for which the aforementioned well-posed…
Let ${\mathcal X}$ be a space of homogeneous type in the sense of Coifman and Weiss and ${\mathcal D}$ a collection of balls in $\cx$. The authors introduce the localized atomic Hardy space $H^{p, q}_{\mathcal D}({\mathcal X})$ with $p\in (0,1]$ and $q\in[1,\infty]\cap(p,\infty]$, the localized Morrey-Campanato space ${\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ and the localized Morrey-Campanato-BLO space $\widetilde{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$ with $\az\in{\mathbb R}$ and $p\in(0, \infty)$ and establish their basic properties including $H^{p, q}_{\mathcal D}({\mathcal X…
Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ with $q\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. Assume that $\{e^{-s\mathcal L}\}_{s>0}$ is the heat semigroup generated by $\mathcal L$. The Littlewood-Paley function $\mathfrak{g}_{\mathcal L}$ and the Lusin area integral $\mathcal{S}_{\mathcal L}$ associated with the Schrödinger operator $\mathcal L$ are defined, respectively…
