<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll">mml:msubmml:milmml:miq-structure of variable exponent spaces (original) (raw)

Modular Geometric Properties in Variable Exponent Spaces

Mathematics

Much has been written on variable exponent spaces in recent years. Most of the literature deals with the normed space structure of such spaces. However, because of the variability of the exponent, the underlying modular structure of these spaces is radically different from that induced by the norm. In this article, we focus our attention on the progress made toward the study of the modular structure of the sequence Lebesgue spaces of variable exponents. In particular, we present a survey of the state of the art regarding modular geometric properties in variable exponent spaces.

An extension of a Theorem of V. Sverak to variable exponent spaces

In 1993, V.Šverák proved that if a sequence of uniformly bounded domains Ωn ⊂ R 2 such that Ωn → Ω in the sense of the Hausdorff complementary topology, verify that the number of connected components of its complements are bounded, then the solutions of the Dirichlet problem for the Laplacian with source f ∈ L 2 (R 2) converges to the solution of the limit domain with same source. In this paper, we extendŠverák result to variable exponent spaces, in particular to solutions of the p(x)−laplacian.

An extension of a Theorem of V. v{S}ver 'ak to variable exponent spaces

2013

In 1993, V.Šverák proved that if a sequence of uniformly bounded domains Ωn ⊂ R 2 such that Ωn → Ω in the sense of the Hausdorff complementary topology, verify that the number of connected components of its complements are bounded, then the solutions of the Dirichlet problem for the Laplacian with source f ∈ L 2 (R 2 ) converges to the solution of the limit domain with same source. In this paper, we extendŠverák result to variable exponent spaces.

On Weak Compactness of Variable Exponent Spaces

2023

This work shows some refined necessary and sufficient conditions placed on the subsets of variable exponent Lebesgue spaces to satisfy the axiom of weak compactness. We also present some results in connection with conditions for all separable variable exponent spaces to be weakly Banach-saks. That is, some results on the Banach-Saks property in variable exponent spaces are given.

LOCAL-TO-GLOBAL RESULTS IN VARIABLE EXPONENT SPACES

2008

In this article a new method for moving from local to global results in variable exponent function spaces is presented. Several applications of the method are also given: Sobolev and trace embeddings; variable Riesz potential estimates; and maximal function inequalities in Morrey spaces are derived for unbounded domains.

A Note on Variable Exponent Hörmander Spaces

Mediterranean Journal of Mathematics, 2013

In this paper we introduce the variable exponent Hörmander spaces and we study some of their properties. In particular, it is shown that B c p(•) (Ω) is isomorphic to B loc p (•) (Ω) (Ω open set in R n , p − > 1 and the Hardy-Littlewood maximal operator M is bounded in L p(•)) extending a Hörmander's result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hörmander spaces are given.

ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES

Double system of exponents with complex-valued coefficients is considered. Under some conditions on the coefficients, we prove that if this system forms a basis for the Morrey-Lebesgue type space on [−π, π], then it is isomorphic to the classical system of exponents in this space.