Hyperplane Integrability Conditions and Smoothing for Radon Transforms

This paper may be viewed as a companion paper to Greenblatt (). In that paper, L2 Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove Lp to Lsq estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here q >= p and s can be positive, negative, or zero. In this paper, we instead use L2 Sobolev estimates derived from the resolution of singularities methods of Greenblatt () and combine with analogous interpolation arguments, again resulting in Lp to Lsq\documentclass[12pt] estimates for translation invariant Radon transforms for q >= p which can be sharp up to endpoints. It will turn out that sometimes the results of this paper are stronger, and sometimes the results of Greenblatt () are stronger. As in Greenblatt (submitted), some of the sharp estimates of this paper occur when s=0, thereby giving some new sharp Lp to Lq estimates for such operators, again up to endpoints. Our results lead to natural global analogues whose statements can be recast in terms of a hyperplane integrability condition analogous to that of Iosevich and Sawyer in their work (Adv Math 132(1):46-119, 1997) on the Lp boundedness of maximal averages over hypersurfaces.