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Convolution with measures on hypersurfaces. (English) Zbl 0972.42009

The author considers the ${L}^{p}$ improving properties of convolution operators $f↦f*d\sigma$ 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 $\mu \left(R\right)\lesssim {|R|}^{\left(n-1\right)/\left(n+1\right)}$ for all rectangles $R$. This is in particular achieved if $\mu$ is equal to surface measure times ${\kappa }^{1/\left(n+1\right)}$, 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.

##### MSC:
 42B15 Multipliers, several variables 44A12 Radon transform 42A20 Convergence and absolute convergence of Fourier and trigonometric series