Sharp reverse Hölder property for A∞ weights on spaces of homogeneous type

Abstract

In this article we present a new proof of a sharp Reverse Hölder Inequality for A∞ weights. Then we derive two applications: a precise open property of Muckenhoupt classes and, as a consequence of this last result, we obtain a simple proof of a sharp weighted bound for the Hardy-Littlewood maximal function involving A∞ constants: kMkLp(w) ≤ c 1 p − 1 [w]Ap [σ]A∞ 1/p , where 1 < p < ∞, σ = w 1 1−p and c is a dimensional constant. Our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak Reverse H¨older Inequality which is still sufficient to prove the open property for Ap classes and the Lp boundedness of the maximal function. In this latter case, the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure µ and the geometric constant κ of the quasimetric.