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

Abstract

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

Strengthened the conclusion by removing the log-loss in the constant entirely