On a weighted variable spaces $L_{p(x), \omega}$ for $0< p(x)< 1$ and weighted Hardy inequality

TL;DR

This paper investigates weighted variable exponent Lebesgue spaces with exponents less than one, establishing their quasi-Banach space structure, non-local convexity, embedding theorems, and two-weight Hardy operator estimates.

Contribution

It introduces and analyzes the properties of $L_{p(x), ext{omega}}$ spaces for $0<p(x)<1$, including their quasi-Banach structure and embedding theorems, which are novel contributions.

Findings

Spaces are quasi-Banach but not locally convex.

Embedding theorems between weighted variable Lebesgue spaces are proved.

Two-weight estimates for Hardy operators are established.

Abstract

In this paper a weighted variable exponent Lebesgue spaces $L_{p(x), \omega}$ for $0< p(x)< 1$ is investigated. We show that this spaces is a quasi-Banach spaces. Note that embedding theorem between weight variable Lebesgue spaces is proved. In particular, we show that $L_{p(x), \omega}(\Omega)$ for $0< p(x)< 1$ isn't locally convex. Also, in this paper a some two-weight estimates for Hardy operator are proved.