The author considers the improving properties of convolution operators where is a compactly supported measure on a hypersurface . For surfaces of non-zero curvature the sharp estimate is . In this paper the author considers the slightly weaker restricted estimate .
Under very mild conditions on (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 obeys the estimate for all rectangles . This is in particular achieved if is equal to surface measure times , where is the Gaussian curvature.
The heart of the argument is a certain 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.