A Bilinear Rubio de Francia Inequality for Arbitrary Rectangles

Abstract

Abstract Let $\mathscr{R}$ be a collection of disjoint dyadic rectangles $R$, let $\pi _R$ denote the non-smooth bilinear projection onto $R$ and let $r>2$. We show that the bilinear Rubio de Francia operator associated with $\mathscr{R}$ given by $$\begin{equation*} f,g \mapsto \left(\sum_{R\in\mathscr{R}} |\pi_{R} (f,g)|^r \right)^{1/r} \end{equation*}$$ is $L^p \times L^q \rightarrow L^s$ bounded whenever $1/p + 1/q = 1/s$, $r^{\prime}<p,q<r$. This extends from squares to rectangles a previous result by the same authors in [7], and as a corollary extends in the same way a previous result from [2] for smooth projections, albeit in a reduced range.