A bilinear Rubio de Francia inequality for arbitrary squares

Abstract

We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane\[\left(f, g \right)\mapsto \left( \sum\_{ω\in Ω}\left| \int\_{\mathbb{R}^2} \hat{f}(ξ) \hat{g}(η) Φ\_ω(ξ, η) e^{2 πi x\left(ξ+η\right)} d ξd η\right|^r \right)^{1/r},\] provided $r\textgreater{}2$. More exactly, we show that the above operator maps $L^p\times L^q\to L^s$ whenever $p,q,s^{\prime }$ are in the "local $L^{r^{\prime }}$" range, i.e. $\frac{1}{p}+\frac{1}{q}+\frac{1}{s^{\prime }}=1$, $\displaystyle0 \leq \frac{1}{p}, \frac{1}{q} \textless{}\frac{1}{r'}$, and $\displaystyle\frac{1}{s'}\textless{}\frac{1}{r'}$. Note that we allow for negative values of $s^{\prime }$, which correspond to quasi-Banach spaces $L^s$.

References in corpus (1)

• Multiple Vector Valued Inequalities via the Helicoidal Method