Restricted convolution inequalities, multilinear operators and applications

Abstract

For $ 1\le k <n$, we prove that for functions $F,G$ on $ {\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \subset {\Bbb R}^{n}$, and $p,q,r \ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, one has the estimate $$ {||(F*G)|_H||}_{L^{r}(H)} \leq {||F||}_{\Lambda^H_{2, p}({\Bbb R}^{n})} \cdot {||G||}_{\Lambda^H_{2, q}({\Bbb R}^{n})},$$ where the mixed norms on the right are defined by $$ {||F||}_{\Lambda^H_{2,p}({\Bbb R}^{n})}={(\int_{H^*} {(\int {|\hat{F}|}^2 dH_{\xi}^{\perp})}^{\frac{p}{2}} d\xi)}^{\frac{1}{p}},$$ with $dH_{\xi}^{\perp}$ the $(n-k)$-dimensional Lebesgue measure on the affine subspace $H_{\xi}^{\perp}:=\xi + H^\perp$. Dually, one obtains restriction theorems for the Fourier transform for affine subspaces. Applied to $F(x^{1},...,x^{m})=\prod_{j=1}^m f_j(x^{j})$ on $\R^{md}$, the diagonal $H_0={(x,...,x): x \in {\Bbb R}^d}$ and suitable kernels $G$, this implies new results for multilinear convolution operators, including $L^p$-improving bounds for measures, an $m$-linear variant of Stein's spherical maximal theorem, estimates for $m$-linear oscillatory integral operators, certain Sobolev trace inequalities, and bilinear estimates for solutions to the wave equation.