Sobolev spaces associated to singular and fractional Radon transforms

Abstract

The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) of operators of the form $f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt$, where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in\mathbb{R}^N\times \mathbb{R}^n$, satisfying $\gamma_0(x)\equiv x$, $\psi$ is a $C^\infty$ cut-off function supported on a small neighborhood of $0\in \mathbb{R}^n$, and $K$ is a "multi-parameter fractional kernel" supported on a small neighborhood of $0\in \mathbb{R}^N$. When $K$ is a Calder\'on-Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when $K$ is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on $\gamma$ were given under which the above operator is bounded on $L^p$ ($1<p<\infty$). Under these same conditions, we introduce non-isotropic $L^p$ Sobolev spaces associated to $\gamma$. Furthermore, when $K$ is a fractional kernel which is smoothing of an order which is close to $0$ (i.e., very close to a singular kernel) we prove mapping properties of the above operators on these non-isotropic Sobolev spaces. As a corollary, under the conditions introduced on $\gamma$ by Christ, Nagel, Stein, and Wainger, we prove optimal smoothing properties in isotropic $L^p$ Sobolev spaces for the above operator when $K$ is a fractional kernel which is smoothing of very low order.