A transference result for Lebesgue spaces with $A_{\infty }$ weights and its applications

TL;DR

This paper establishes a transference theorem for Lebesgue spaces with $A_{ olinebreak}{}_{ olinebreak}{}_{ olinebreak}$ weights, enabling easier derivation of weighted norm inequalities and applications to fractional difference operators and approximation inequalities.

Contribution

It introduces a novel transference technique for $A_{ olinebreak}{}_{ olinebreak}{}_{ olinebreak}$-weighted Lebesgue spaces, simplifying the derivation of weighted inequalities and applications.

Findings

Derived weighted norm inequalities using transference.

Applied transference to fractional difference operators.

Provided new proofs for approximation inequalities.

Abstract

In this work we obtain a transference theorem for Lebesgue spaces with $A_{\infty }$ weights, namely, starting from some uniform-norm inequalities it is possible to obtain similar inequalities in Lebesgue spaces with $A_{\infty }$ weights. This transference technic allows us to obtain some weighted norm inequalities easily. Also transference result gives possibility to use fractional difference operators in weighted Lebesgue spaces easier than the classical known one. We can obtain some norm-like inequalities easily as a consequence. Some important approximation inequalities of approximation by integral functions of finite degree can be obtained with a different proof.