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Convolution with measures on hypersurfaces. (English) Zbl 0972.42009
The author considers the $L^p$ improving properties of convolution operators $f \mapsto f * d\sigma$ where $d\sigma$ is a compactly supported measure on a $C^2$ hypersurface $S$. For surfaces of non-zero curvature the sharp estimate is $L^{n+1/n} \to L^n$. In this paper the author considers the slightly weaker restricted estimate $L^{n+1/n,1} \to L^n$. Under very mild conditions on $S$ (namely that the Gauss map generically has bounded multiplicity, plus another technical condition of a similar flavor) the author shows that one can obtain the above restricted estimate if and only if $\mu$ obeys the estimate $\mu(R) \lesssim |R|^{(n-1)/(n+1)}$ for all rectangles $R$. This is in particular achieved if $\mu$ is equal to surface measure times $\kappa^{1/(n+1)}$, where $\kappa$ is the Gaussian curvature. The heart of the argument is a certain $L^n$ estimate which, after multiplying everything out and changing variables, hinges on the estimation of various Jacobians and on certain multilinear estimates with these Jacobians as kernels.

42B15Multipliers, several variables
44A12Radon transform
42A20Convergence and absolute convergence of Fourier and trigonometric series
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