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C. Stockdale, P. Villarroya, and B. Wick introduced the $\epsilon$-maximaloperator 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 variableLebesgue spaces $\Lpp(\R^n)$ for a larger collection of exponent functions thanthe log-Holder continuous functions used to prove the boundedness of themaximal operator on $\Lpp(\R^n)$. We also prove that the Haar multiplier iscompact when restricted to a dyadic cube $Q_0$.

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