Factorizations for variable exponent Muckenhoupt weights

TL;DR

This paper establishes factorizations for variable exponent Muckenhoupt weights, extending known results and applying recent inequalities and transformations to improve understanding of weighted variable exponent spaces.

Contribution

It introduces new factorizations for variable exponent Muckenhoupt weights using recent reverse H"older inequalities and transformation formulas, advancing the theory of weighted variable exponent spaces.

Findings

Proves factorization of weights using reverse H"older inequalities.

Extends factorizations to restricted range contexts with transformation formulas.

Provides an alternative proof for extrapolation of compactness in weighted variable exponent spaces.

Abstract

Given two variable exponent Muckenhoupt weights $w\in A_{p(\cdot)}$ and $w_1\in A_{p_1(\cdot)}$, we prove that for all small enough $\theta>0,$ there holds that $w_0\in A_{p_0(\cdot)},$ where the weight is determined by $w = w_0^{1-\theta}w_1^{\theta}$ and exponent of the weight class by $1/p(\cdot) = (1-\theta)/p_0(\cdot) + \theta/p_1(\cdot).$ The proof is based on a recent reverse H\"older's inequality for variable exponent Muckenhoupt weights by Cruz-Uribe and Penrod. We upgrade these factorizations to the restricted range context by using a recent transformation formula due to Nieraeth. Then, following an extrapolation of compactness scheme by Hyt\"onen and Lappas, we provide an alternative proof of the recent extrapolation of compactness results of Lorist and Nieraeth in the context of weighted variable exponent Lebesgue spaces.