The \epsilon-Maximal Operator and Haar Multipliers on Variable Lebesgue Spaces

TL;DR

This paper extends the boundedness of the -maximal operator and Haar multipliers to variable Lebesgue spaces with broader exponent functions, also establishing their compactness on dyadic cubes.

Contribution

It proves the -maximal operator and Haar multipliers are bounded on variable Lebesgue spaces for more general exponents than previously known.

Findings

Boundedness of -maximal operator on broader variable Lebesgue spaces

Boundedness of Haar multipliers on these spaces

Haar multiplier is compact on dyadic cubes

Abstract

C. Stockdale, P. Villarroya, and B. Wick introduced the $\epsilon$-maximal operator to prove the Haar multiplier is bounded on the weighted spaces $L^p(w)$ for a class of weights larger than $A_p$. We prove the $\epsilon$-maximal operator and Haar multiplier are bounded on variable Lebesgue spaces $\Lpp(\R^n)$ for a larger collection of exponent functions than the log-Holder continuous functions used to prove the boundedness of the maximal operator on $\Lpp(\R^n)$. We also prove that the Haar multiplier is compact when restricted to a dyadic cube $Q_0$.