A convolution estimate in Lorentz space for some surfaces of revolution

Abstract

The purpose of this thesis is to study a convolution operator associated with the affine surface measure supported on a surface of revolution. We prove a nearly sharp convolution estimates in Lorentz spaces by adapting a combinatorial argument developed by Christ and Stovall. With Kovanskiı̆’s theorem, an analog of Bézout’s theorem, we can handle not only hypersurfaces defined by polynomial functions, but also hypersurfaces defined by some non-polynomial functions.