Geometric-arithmetic averaging of dyadic weights∗

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 Ap weights from a measurably varying family of dyadic Ap weights. This averaging process is suggested by the relationship between the Ap weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Hölder (RHp) conditions from families of dyadic RHp weights, and extends to the polydisc as well.