Geometric-arithmetic averaging of dyadic weights

Abstract: The theory of (Muckenhoupt) weights arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing A_p weights from a measurably varying family of dyadic A_p weights. This averaging process is suggested by the relationship between the A_p weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Holder (RH_p) conditions from families of dyadic RH_p weights, and extends to the polydisc as well.