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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-3-1,
author = {J. M. Wilson},
title = {A semi-discrete Littlewood-Paley inequality},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {207-233},
zbl = {1013.42014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-3-1}
}
J. M. Wilson. A semi-discrete Littlewood-Paley inequality. Studia Mathematica, Tome 151 (2002) pp. 207-233. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm153-3-1/