Littlewood–Paley–Rubio De Francia Inequality for the Two-Parameter Walsh System

Abstract

A version of Littlewood–Paley–Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles $$ {I}_k={I}_k^1\times {I}_k^2 $$ in ℤ+ × ℤ+ and a family of functions fk with Walsh spectrum inside Ik the following is true: $$ {\left\Vert \sum_k{f}_k\right\Vert}_{L^p}\le {C}_p{\left\Vert {\left(\sum_k{\left|{f}_k\right|}^2\right)}^{1/2}\right\Vert}_{L^p},\kern2em 1<p\le 2, $$ where Cp does not depend on the choice of rectangles {Ik} or functions {fk}. The arguments are based on the atomic theory of two-parameter martingale Hardy spaces. In the course of the proof, a two-parameter version of Gundy’s theorem on the boundedness of operators taking martingales to measurable functions is formulated, which might be of independent interest.