Mathematical Research Letters

Volume 20 (2013)

Number 4

Restricted convolution inequalities, multilinear operators and applications

Pages: 675 – 694

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n4.a6

Authors

Dan-Andrei Geba (Department of Mathematics, University of Rochester, New York, U.S.A.)

Allan Greenleaf (Department of Mathematics, University of Rochester, New York, U.S.A.)

Alex Iosevich (Department of Mathematics, University of Rochester, New York, U.S.A.)

Eyvindur Palsson (Department of Mathematics, University of Rochester, New York, U.S.A.)

Eric Sawyer (Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada)

Abstract

For functions $F,G$ on $ {\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \subset {\Bbb R}^{n}$, $ 1\le k <n$, and exponents $p,q,r \ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, we prove the estimate\[{||(F*G)|_H||}_{L^{r}(H)} \leq {||F||}_{\Lambda^H_{2, p}({\Bbb R}^{n})} \cdot {||G||}_{\Lambda^H_{2, q}({\Bbb R}^{n})}.\]Here, the mixed norms on the right are defined in terms of the Fourier transform by\[{||F||}_{\Lambda^H_{2,p}({\Bbb R}^{n})}={\left( \int_{H^*} {\left( \int {|\widehat{F}|}^2 dH_{\xi}^{\perp} \right)}^{\frac{p}{2}} d\xi \right)}^{{1}/{p}},\]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 on affine subspaces. We use this, and a maximal variant, to prove results for a variety of multilinear convolution operators, including $L^p$-improving bounds for measures; bilinear variants of Stein’s spherical maximal theorem; estimates for $m$-linear oscillatory integral operators; Sobolev trace inequalities; and bilinear estimates for solutions to the wave equation.

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