Embedding Theorems between Variable-Exponent Morrey Spaces

Abstract and Applied Analysis, 2010

We deepen the study of some Morrey type spaces, denoted by M p,λ Ω , defined on an unbounded open subset Ω of R n . In particular, we construct decompositions for functions belonging to two different subspaces of M p,λ Ω , which allow us to prove a compactness result for an operator in Sobolev spaces. We also introduce a weighted Morrey type space, settled between the abovementioned subspaces. endowed with the norm defined in 2.2 .

arXiv (Cornell University), 2022

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 ⊂ 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 first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of ℓr type, 1 ≤ r ≤ ∞.

arXiv (Cornell University), 2016

In this paper embeddings between weighted complementary local Morrey-type spaces c LM pθ,ω (R n , v) and weighted local Morrey-type spaces LM pθ,ω (R n , v) are characterized. In particular, two-sided estimates of the optimal constant c in the inequality are obtained, where p 1 , p 2 , q 1 , q 2 ∈ (0, ∞), p 2 ≤ q 2 and u 1 , u 2 and v 1 , v 2 are weights on (0, ∞) and R n , respectively. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted local Morrey-type and complementary local Morrey-type spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of the iterated Hardy-type inequalities.

Journal of Mathematical Analysis and Applications, 2019

We introduce variable exponent versions of Morreyfied Triebel-Lizorkin spaces. To that end, we prove an important convolution inequality which is a replacement for the Hardy-Littlewood maximal inequality in the fully variable setting. Using it we obtain characterizations by means of Peetre maximal functions and use them to show the independence of the introduced spaces from the admissible system used.

We prove the boundedness of the weighted Hardy-Littlewood maximal operator and the singular integral operator on variable Morrey spaces L p(•),λ(•),|•| γ (Ω) over a bounded open set Ω ⊂ R n and a Hardy-Littlewood-Stein-Weiss type L p(•),λ(•),|x−x 0 | γ (Ω) to L q(•),λ(•),|x−x 0 | µ (Ω)-theorem for the potential operators I α(•) , x 0 ∈ Ω , also of variable order. In the case of constant α , the limiting case is also studied when the potential operator I α acts into BM O |•| γ .

Journal of Mathematical Inequalities, 2015

In this paper, the embeddings between weighted local Morrey-type spaces and weighted Lebesgue spaces are investigated. 2000 Mathematics Subject Classification. 42B35, 47B38. Key words and phrases. local Morrey-type spaces; weighted Lebesgue spaces; Hardy-type inequalities. We would like to thank Professor V.I. Burenkov for making available the reference [2] in preprint form. When w ≡ 1 on A, we write simply L p (A) and • p,A instead of L p (A, w) and • p,A,w , respectively. We adopt the following usual conventions. Convention 1.1. (i) Throughout the paper we put 0/0 = 0, 0 • (±∞) = 0 and 1/(±∞) = 0. (ii) We put

Journal of Approximation Theory, 2015

In this paper, the authors prove some Franke-Jawerth embedding for the Besovtype spaces B s,τ p,q (R n) and the Triebel-Lizorkin-type spaces F s,τ p,q (R n). By using some limiting embedding properties of these spaces and the Besov-Morrey spaces N s u,p,q (R n), the continuity envelopes in B s,τ p,q (R n), F s,τ p,q (R n) and N s u,p,q (R n) are also worked out. As applications, the authors present some Hardy type inequalities in the scales of B s,τ p,q (R n), F s,τ p,q (R n) and N s u,p,q (R n), and also give the estimates for approximation numbers of the embeddings from B s,τ p,q (Ω), F s,τ p,q (Ω) and N s u,p,q (Ω) into C(Ω), where Ω denotes the unit ball in R n .

Bulletin of Mathematical Sciences

The paper is the first part of a program devoted to the study of the behavior of operatorvalued multipliers in Morrey spaces. Embedding theorems and uniform separability properties involving E-valued Morrey spaces are proved. As a consequence, maximal regularity for solutions of infinite systems of anisitropic elliptic partial differential equations are established.

Constructive Approximation, 2019

Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type have enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces m u, p = m u, p (Z d), 0 < p ≤ u < ∞, which have yet been considered almost nowhere. They are defined as natural generalizations of the classical p spaces. We consider some basic features, embedding properties, a predual, a corresponding version of Pitt's compactness theorem, and further characterize the compactness of embeddings of related finite-dimensional spaces.

Mediterranean Journal of Mathematics

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.