A Bilinear Rubio de Francia Inequality for Arbitrary Rectangles

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.

A BILINEAR RUBIO DE FRANCIA INEQUALITY FOR ARBITRARY SQUARES

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 $$\begin{eqnarray}(f,g)\mapsto \biggl(\mathop{\sum }_{\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}}\biggl|\int _{\mathbb{R}^{2}}\hat{f}(\unicode[STIX]{x1D709}){\hat{g}}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})e^{2\unicode[STIX]{x1D70B}ix(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702})}\,d\unicode[STIX]{x1D709}\,d\unicode[STIX]{x1D702}\biggr|^{r}\biggr)^{1/r},\end{eqnarray}$$ provided $r>2$. More exactly, we show that the above operator maps $L^{p}\times L^{q}\rightarrow L^{s}$ whenever $p,q,s^{\prime }$ are in the ‘local $L^{r^{\prime }}$’ range, that is, $$\begin{eqnarray}\frac{1}{p}+\frac{1}{q}+\frac{1}{s^{\prime }}=1,\quad 0\leqslant \frac{1}{p},\frac{1}{q}<\frac{1}{r^{\prime }},\quad \text{and}\quad \frac{1}{s^{\prime }}<\frac{1}{r^{\prime }}.\end{eqnarray}$$ Note that we allow for negative values of $s^{\prime }$, which correspond to quasi-Banach spaces $L^{s}$.