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Convolution with measures on hypersurfaces. (English) Zbl 0972.42009

The author considers the L p improving properties of convolution operators ff*dσ where dσ 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 L n . In this paper the author considers the slightly weaker restricted estimate L n+1/n,1 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 μ obeys the estimate μ(R)|R| (n-1)/(n+1) for all rectangles R. This is in particular achieved if μ is equal to surface measure times κ 1/(n+1) , where κ 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:
42B15Multipliers, several variables
44A12Radon transform
42A20Convergence and absolute convergence of Fourier and trigonometric series