Hyperplane integrability conditions and smoothing for Radon transforms

TL;DR

This paper develops new $L^p$ to $L^q_s$ estimates for translation invariant Radon transforms using resolution of singularities, sometimes surpassing previous results and providing sharp endpoint estimates.

Contribution

It introduces a novel approach using $L^2$ Sobolev estimates from resolution of singularities to obtain sharp $L^p$ to $L^q_s$ estimates for Radon transforms, improving or complementing prior methods.

Findings

Derived new sharp $L^p$ to $L^q$ estimates up to endpoints.

Established global analogues with hyperplane integrability conditions.

Compared results with previous methods, showing sometimes stronger bounds.

Abstract

This paper may be viewed as a companion paper to [G1]. 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 $q \geq p$ 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 [G2] and combine with analogous interpolation arguments, again resulting in $L^p$ to $L^q_s$ estimates for translation invariant Radon transforms 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 [G1] are stronger. As in [G1], some of the sharp estimates of this paper occur when $s = 0$, thereby giving 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 [ISa1] on the $L^p$ boundedness of maximal averages over hypersurfaces.