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Restricted convolution inequalities, multilinear operators and applications
2012
For $ 1\le k 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.
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