Restricted convolution inequalities, multilinear operators and applications

Abstract

For $1\le k<n$, we prove that for functions $F,G$ on ${\mathbb{R}}^n$, any $k$-dimensional affine subspace $H\subset {\mathbb{R}}^n$, and $p,q,r\ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, one has the estimate $${||(F\ast G){|}_H||}_{L^r(H)}\le {||F||}_{{\Lambda }_{2,p}^H({\mathbb{R}}^n)}\cdot {||G||}_{{\Lambda }_{2,q}^H({\mathbb{R}}^n)},$$ where the mixed norms on the right are defined by $${||F||}_{{\Lambda }_{2,p}^H({\mathbb{R}}^n)}={({\int }_{H^{\ast }}{(\int {|\hat{F}|}^{2}d{H}_{\xi }^{\perp })}^{\frac{p}{2}}d\xi )}^{\frac{1}{p}},$$ with $d{H}_{\xi }^{\perp }$ 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 ${\mathbb{R}}^{md}$, the diagonal $H_0=(x,...,x):x\in {\mathbb{R}}^d$ 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.

20 pages