Smoothing theorems for Radon transforms over hypersurfaces and related operators

TL;DR

This paper extends Sobolev space boundedness results for Radon transforms over hypersurfaces, establishing sharp $L^p$ to $L^q_s$ estimates and geometric regions where these bounds hold, improving understanding of such operators.

Contribution

The paper provides new sharp $L^p$ to $L^q_s$ boundedness results for Radon transforms over hypersurfaces, including fractional singular variants, with a geometric characterization of the boundedness region.

Findings

Established $L^p$ to $L^q_s$ boundedness within a specific geometric triangle $Z$

Proved sharpness of Sobolev space improvements up to endpoints

Derived $L^p$ to $L^q$ improvements for certain $p,q$ ranges

Abstract

We extend the theorems of [G1] on $L^p$ to $L^p_s$ Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving $L^p$ to $L^q_s$ boundedness results for such operators. Here $q \geq p$ but $s$ can be positive, negative, or zero. For many such operators we will have a triangle $Z \subset (0,1) \times (0,1) \times {\mathbb R} $ such that one has $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q}, s)$ beneath $Z$, and in the case of Radon transforms one does not have $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q}, s)$ above the plane containing $Z$, thereby providing a Sobolev space improvement result which is sharp up to endpoints for $({1 \over p}, {1 \over q})$ below $Z$. This triangle $Z$ intersects the plane $\{(x_1,x_2,x_3): x_3 = 0\}$, and therefore we also have an $L^p$ to $L^q$ improvement result that is also sharp up to endpoints for certain ranges of $p$ and $q$.