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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 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.