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Abstract

We prove weighted Littlewood–Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on ${{{\mathbb R}}^d}$ with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg.