Variable Muckenhoupt weights with applications in approximation

TL;DR

This paper studies variable Muckenhoupt weights in variable exponent Lebesgue spaces, establishing boundedness of operators and deriving approximation inequalities with applications to polynomial approximation.

Contribution

It introduces new boundedness results for operators in variable exponent spaces and develops approximation inequalities using K-functionals and transference techniques.

Findings

Boundedness of averaging and Steklov operators in these spaces

Derivation of main inequalities of approximation

Application to polynomial approximation in variable exponent spaces

Abstract

Variable Muckenhoupt weights are considered in variable exponent Lebesgue spaces. Applications are given for polynomial approximation in these spaces. Boundedness of averaging operator is proved to gain a transference result. Almost all weighted norm inequalities are obtained using this transference result. Potential type approximate identities are considered. Translations of Steklov operator are proved to be bounded in these spaces. K-functional is a good apparatus for measuring smoothness properties of functions given in these function classes. Main inequalities of approximation are derived.