Bilinear Rubio de Francia inequalities for collections of non-smooth squares

TL;DR

This paper proves boundedness of a bilinear Rubio de Francia operator involving non-smooth projections onto dyadic squares, extending classical inequalities to a bilinear setting with specific integrability conditions.

Contribution

It establishes $L^p imes L^q o L^s$ bounds for a bilinear operator with non-smooth projections, generalizing Rubio de Francia inequalities to a bilinear context.

Findings

Boundedness of the bilinear Rubio de Francia operator for certain $p,q,s$ ranges.

Independence of the bounds from the collection of dyadic squares $\Omega$.

Extension of classical linear inequalities to a bilinear, non-smooth setting.

Abstract

Let $\Omega$ be a collection of disjoint dyadic squares $\omega$, let $\pi_\omega$ denote the non-smooth bilinear projection onto $\omega$ \[ \pi_\omega (f,g)(x):=\int\int \mathbf{1}_{\omega}(\xi,\eta) \widehat{f}(\xi) \widehat{g}(\eta) e^{2\pi i (\xi + \eta) x} d \xi d\eta \] and let $r>2$. We show that the bilinear Rubio de Francia operator \[ \Big(\sum_{\omega\in\Omega} |\pi_{\omega} (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded with constant independent of $\Omega$ whenever $1/p + 1/q = 1/s$, $r'<p,q<r$, $r'/2 < s < r/2$.