A semi-discrete Littlewood–Paley inequality
Volume 153 / 2002
Studia Mathematica 153 (2002), 207-233 MSC: Primary 42B25; Secondary 42B35. DOI: 10.4064/sm153-3-1
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.