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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.

MSC:

42B15 Multipliers for harmonic analysis in several variables
44A12 Radon transform
42A20 Convergence and absolute convergence of Fourier and trigonometric series
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