THE ε-MAXIMAL OPERATOR AND HAAR MULTIPLIERS ON VARIABLE LEBESGUE SPACES

Recently, Stockdale, Villarroya, and Wick introduced the & epsilon;-maximal operator to prove the Haar multiplier is bounded on the weighted spaces Lp(w) for a class of weights larger than Ap . We prove the & epsilon;-maximal operator and Haar multiplier are bounded on variable Lebesgue spaces Lp(& BULL;)(I8n) for a larger collection of exponent functions than the log-Ho & BULL;lder continuous functions used to prove the boundedness of the maximal operator on Lp(& BULL;)(I8n). We also prove that the Haar multiplier is compact when restricted to a dyadic cube Q0 .