Hyperplane Integrability Conditions and Smoothing for Radon Transforms

Abstract

This paper may be viewed as a companion paper to Greenblatt ( arXiv:1910.04547 ). In that paper, $$L^2$$ Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove $$L^p$$ to $$L^q_s$$ estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here $$q \ge p$$ and s can be positive, negative, or zero. In this paper, we instead use $$L^2$$ Sobolev estimates derived from the resolution of singularities methods of Greenblatt ( arXiv:1810.10507 ) and combine with analogous interpolation arguments, again resulting in $$L^p$$ to $$L^q_s$$ estimates for translation invariant Radon transforms for $$q \ge p$$ 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 ( arXiv:1910.04547 ) are stronger. As in Greenblatt (submitted), some of the sharp estimates of this paper occur when $$s = 0$$ , thereby giving some new sharp $$L^p$$ to $$L^q$$ 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 $$L^p$$ boundedness of maximal averages over hypersurfaces.