Sobolev-Morrey Spaces Research Articles

On some differential properties of functions in generalized grand Sobolev-Morrey spaces

In this paper we introduce a generalized grand Sobolev-Morrey spaces. Some differential and differential-difference properties of functions from this spaces are proved by means of the integral representation.

The mixed boundary value problems for uniformly elliptic equations in modified local generalized Sobolev-Morrey spaces

The mixed boundary value problems for uniformly elliptic equations in modified local generalized Sobolev-Morrey spaces

Variational inequalities for hypersingular integrals with variable kernels

We systematically study variational inequalities for hypersingular integral operators. More precisely, we show the variational inequalities for the families 𝒯α:={Tα,𝜀}𝜀>0 of truncated hypersingular integrals with variable kernels, which are defined by Tα,𝜀f(x)=∫|x−y|>𝜀Ω(x,x−y)|x−y|n+αf(y)dy, where α≥0 and the kernel Ω belongs to L∞(ℝn)×L1(𝕊n−1). We first prove that the variation of the hypersingular integral with variable kernel is bounded from the Sobolev space L˙α2 to the Lebesgue space L2 when Ω∈L∞(ℝn)×Lq(𝕊n−1) for q>max{1,2(n−1)∕(n+2α)} and satisfies some cancellation condition in its second variable. The result is sharp in the sense that the (L˙α2,L2) boundedness of 𝒯α fails if q≤2(n−1)∕(n+2α). After strengthening the smoothness of Ω(x,z′) in its second variable, we give the weighted boundedness of the variation of the hypersingular integrals with smooth variable kernels from L˙αp(w) to Lp(w) for 1<p<∞ and w∈Ap. Finally, we extend the result to the Sobolev–Morrey spaces.

The Calderón–Zygmund estimates for a class of nonlinear elliptic equations with measure data

AbstractWe study a class of nonlinear elliptic equations involving measure data where μ is a Radon measure. Under the main assumption of that there exists a constant such that where for some integrable index , we obtain the Calderón–Zygmund estimates in the Sobolev–Morrey spaces for refined fractional‐order derivatives of distributional solutions depending on α.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains

Abstract In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.

Some differential properties of grand generalized Sobolev-Morrey spaces

Some differential properties of grand generalized Sobolev-Morrey spaces

On embeddings of grand grand Sobolev-Morrey spaces with dominant mixed derivatives

In this paper it is constructed a new grand grand Sobolev-Morrey $S_{p),\varkappa ),a,\alpha }^{l}W(G)$ spaces with dominant mixed derivatives. With help integral representation of generalized mixed derivatives of functions, defined on $n$-dimensional domains satisfying flexible horn condition, an embedding theorem is proved. In other works, the embedding theorem is proved in these spaces and belonging of the generalized mixed derivatives of functions from these spaces to the Holder class, was studied.

On the equivalence of the [formula omitted]-functional and the modulus of continuity on the Morrey spaces

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.

On the equivalence of the K-functional and the modulus of continuity on the periodic variable exponent Morrey spaces

In this work, we show that the modulus of smoothness in the periodic variable exponents Morrey spaces and the K-functional with respect to the periodic variable exponents Morrey spaces—the periodic variable exponents Sobolev–Morrey spaces are equivalent. As an application we use this characterization for obtaining an analogue of Whitney’s theorem in the spaces $${{\mathcal {M}}}_{p(\cdot ),\lambda (\cdot )}(I_0)$$.

On Stein's extension operator preserving Sobolev–Morrey spaces

AbstractWe 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.

REMARKS ON SOBOLEV-MORREY-CAMPANATO SPACES DEFINED ON C_{0;\gamma} DOMAINS

We discuss a few old results concerning embedding theorems for Campanato and Sobolev-Morrey spaces adapting the formulations to the case of domains of class $C^{0,\gamma}$, and we present more recent results concerning the extension of functions from Sobolev-Morrey spaces defined on those domains. As a corollary of the extension theorem we obtain an embedding theorem for Sobolev-Morrey spaces on arbitrary $C^{0,\gamma}$ domains.

Some properties of grand Sobolev-Morrey spaces with dominant mixed derivatives

Some properties of grand Sobolev-Morrey spaces with dominant mixed derivatives

Some differential properties of anisotropic grand Sobolev–Morrey spaces

In this paper an anisotropic grand Sobolev–Morrey spaces are introduced. With the help of integral representation we study differential and differential-difference properties of functions from these spaces.

Dirichlet boundary value problems for uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces

In this paper, we study the boundedness of the sublinear operators, generated by Calderon–Zygmund operators in local generalized Morrey spaces. By using these results we prove the solvability of the Dirichlet boundary value problem for a polyharmonic equation in modified local generalized Sobolev–Morrey spaces. We obtain a priori estimates for the solutions of the Dirichlet boundary value problems for the uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces defined on bounded smooth domains.

A criterion for embedding of anisotropic Sobolev–Morrey spaces into the space of uniformly continuous functions

We prove the embedding theorems of the Sobolev–Morrey spaces into the space of uniformly continuous functions so extending the classical Sobolev Theorems.

On Burenkov's extension operator preserving Sobolev–Morrey spaces on Lipschitz domains

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.

Weighted Morrey and weighted fractional Sobolev–Morrey spaces estimates for a large class of pseudo-differential operators with smooth symbols

In this paper, we study the boundedness results for a large class of pseudo-differential operators with smooth symbols on weighted Morrey and Weighted fractional Sobolev–Morrey spaces, respectively.

The radial lemma of Strauss in the context of Morrey spaces

In this article, the authors consider smoothness and decay properties of radial functions belonging to smoothness spaces related to Morrey spaces (Sobolev–Morrey spaces, Besovtype spaces and Besov–Morrey spaces). Within this framework, some generalizations of the radial lemma of Strauss are obtained.

Compact embeddings of radial and subradial subspaces of some Besov-type spaces related to Morrey spaces

In this paper, the authors give a sufficient condition, on the parameters s0, p0, p1, u0 and u1, which ensures that the embeddings of radial and subradial subspaces of Besov-type spaces Bp0,∞s0,1p0−1u0(Rn) and Bp0,1s0,1p0−1u0(Rn) into Morrey spaces Mp1u1(Rn) are compact. As corollaries, some sufficient conditions for the compactness of the embeddings from the radial and the subradial subspaces of Sobolev–Morrey spaces into Morrey spaces are also obtained.