  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 $Q_0$.
