Oberlin, Daniel M. Convolution with measures on hypersurfaces. (English) Zbl 0972.42009 Math. Proc. Camb. Philos. Soc. 129, No. 3, 517-526 (2000). 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. Reviewer: Terence Tao (Los Angeles) Cited in 16 Documents MSC: 42B15 Multipliers for harmonic analysis in several variables 44A12 Radon transform 42A20 Convergence and absolute convergence of Fourier and trigonometric series Keywords:Radon transforms; convolution operator; \( L^p\) improving estimates; compactly supported measure; hypersurface; Jacobians PDFBibTeX XMLCite \textit{D. M. Oberlin}, Math. Proc. Camb. Philos. Soc. 129, No. 3, 517--526 (2000; Zbl 0972.42009) Full Text: DOI