The ε-Maximal Operator and Haar Multipliers on Variable Lebesgue Spaces

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 $\text{Lpp}({\mathbb{R}}^n)$ for a larger collection of exponent functions than the log-Holder continuous functions used to prove the boundedness of the maximal operator on $\text{Lpp}({\mathbb{R}}^n)$. We also prove that the Haar multiplier is compact when restricted to a dyadic cube $Q_0$.