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 $\mathbb{R}^n$ with finite side length. The latter constructed function is inspired by the striking function constructed by Dafni et al. [J. Funct. Anal. 275 (2018), 577--603]. All the proofs of these results strongly depend on some exquisite geometrical analysis on cubes of $\mathbb{R}^n$. As an application, the relationship between Riesz--Morrey spaces and Lebesgue spaces is completely clarified on all indices.
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.
Boundedness of the maximal function and the Caldeón-Zygmund singular integrals in central Morrey-Orlicz spaces were proved in papers by the second and third authors. The weak-type estimates have also been proven. Here we show boundedness of the Riesz potential in central Morrey-Orlicz spaces and the corresponding weak-type version.
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…
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.
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 paper we introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized 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 as well as the boundedness of Calderón-Zygmund potential type operators. Boundedness of Riesz type potential operators are also obtained in the …
The aim of this paper is to obtain boundedness conditions for the maximal function Mf and to prove the necessary and sufficient conditions for the fractional maximal oparator Ma in the Lorentz Morrey spaces which are a new class of functions. We get our main results by using the obtained sharp rearrangement estimates. The obtained results are applied to the boundedness of particular operators such as the Bochner Riesz operator and the Schrodinger type operators in the Lorentz Morrey spaces, where the nonnegative potential V belongs to the reverse Holder class.
We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\Rn)$ including their weak versions $WM_{Φ,φ}(\Rn)$. In these spaces we prove the boundedness of the Riesz potential from $M_{Φ,φ_1}(\Rn)$ to $M_{Ψ,φ_2}(\Rn)$ and from $M_{Φ,φ_1}(\Rn)$ to $WM_{Ψ,φ_2}(\Rn)$. As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(φ_{1},φ_{2})$, which do not assume any assumption on monotonicity of $φ_{…
Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…
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.
In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator $\mathcal{L}=-Δ+ \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential $\V$ considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the…
In this paper we find the condition on function $ω$ and weight $v$ which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ${\mathcal M}_{p,ω}({\mathbb R}^n,v)$ and generalized weighted central Morrey spaces $\dot{\mathcal M}_{p,ω}({\mathbb R}^n,v)$, when $v$ belongs to Muckenhoupt $A_{\infty}$-class.
In a previous work, "compact versions" of Rubio de Francia's weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of an linear operator from just one space to the full range of weighted Lebesgue spaces, where this operator is bounded. In this paper, we extend these results to the setting of weighted Morrey spaces. As applications, we easily obtain new results on the weighted compactness of commutators of Calderón--Zygmund singular integrals, rough singular integrals and Bochner--Riesz multipliers.
In this paper, we give the definition of local variable Morrey Lorentz spaces which are a new class of functions. Also, we prove the boundedness of the Hardy Littlewood maximal operator M and Calderon Zygmund operators T on these spaces. Finally, we apply these results to the Bochner Riesz operator, identity approximation and the Marcinkiewicz operator on these spaces.
We study the boundedness of some sublinear operators on weighted Morrey spaces under certain size conditions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operator, Bochner-Riesz means at the critical index, oscillatory singular operators, singular integral operators with oscillating kernels and so on. As applications, the regularity in weighted Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients are established.
We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calderón-Zygmund operators suitably defined on the functions of the space. In the case of the fractional maximal operator and the fractional integral we obtain a characterization valid for exponents satisfying the Sobolev relation. For power weights we get sharp results in a larger range of exponents for the usual versions of weighted Morrey spaces.
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.
This manuscript addresses Muckenhoupt $A_{p}$ weight theory in connection to Morrey and BMO spaces. It is proved that $ω$ belongs to Muckenhoupt $A_{p}$ class, if and only if Hardy-Littlewood maximal function $M$ is bounded from weighted Lebesgue spaces $L^{p}(ω)$ to weighted Morrey spaces $M^{p}_{q}(ω)$ for $1<q< p<\infty$. As a corollary, if $M$ is (weak) bounded on $M^{p}_{q}(ω)$, then $ω\in A_{p}$. The $A_{p}$ condition also characterizes the boundedness of the Riesz transform $R_{j}$ and convolution operators $T_ε$ on weighted Morrey spaces. Finally, we show that $ω\in A_{p}$ if and only …
In this article, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on $\mathbb{R}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón--Zygmund operators, Littlewood--Paley $g$-functions, Lusin area functions, Littlewood--Paley $g^\ast_λ$-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) bou…
Let $p,q\in [1,\infty]$, $α\in{\mathbb{R}}$, and $s$ be a non-negative integer. In this article, the authors introduce a new function space $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ of John-Nirenberg-Campanato type, where $\mathcal{X}$ denotes $\mathbb{R}^n$ or any cube $Q_{0}$ of $\mathbb{R}^n$ with finite edge length. The authors give an equivalent characterization of $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ via both the John-Nirenberg-Campanato space and the Riesz-Morrey space. Moreover, for the particular case $s=0$, this new space can be equivalently characterized by both maximal functio…
In this paper we develop a new set of results based on a nonlocal gradient jointly inspired by the Riesz s-fractional gradient and Peridynamics, in the sense that its integration domain depends on a ball of radius delta > 0 (horizon of interaction among particles, in the terminology of Peridynamics), while keeping at the same time the singularity of the Riesz potential in its integration kernel. Accordingly, we define a functional space suitable for nonlocal models in Calculus of Variations and partial differential equations. Our motivation is to develop the proper functional analysis framewor…
We introduce and study multivariate generalizations of the classical BV spaces of Jordan, F. Riesz and Wiener. The family of the introduced spaces contains or is intimately related to a considerable class of function spaces of modern analysis including BMO, BV, Morrey spaces and those of Sobolev of arbitrary smoothness, Besov and Triebel-Lizorkin spaces. We prove under mild restrictions that the BV spaces of this family are dual and present constructive characterizations of their preduals via atomic decompositions. Moreover, we show that under additional restrictions such a predual space is is…
