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Abstract

We apply a decomposition lemma of Uchiyama and results of the author to obtain good weighted Littlewood–Paley estimates for linear sums of functions satisfying reasonable decay, smoothness, and cancellation conditions. The heart of our application is a combinatorial trick treating $m$-fold dilates of dyadic cubes. We use our estimates to obtain new weighted inequalities for Bergman-type spaces defined on upper half-spaces in one and two parameters, extending earlier work of R. L. Wheeden and the author.