Revista Matemática Iberoamericana


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Volume 33, Issue 2, 2017, pp. 633–748
DOI: 10.4171/RMI/953

Sobolev spaces associated to singular and fractional Radon transforms

Brian Street[1]

(1) University of Wisconsin, Madison, USA

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 Calderón–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.

Keywords: Sobolev spaces, singular Radon transforms, fractional Radon transforms, nonisotropic Sobolev spaces

Street Brian: Sobolev spaces associated to singular and fractional Radon transforms. Rev. Mat. Iberoamericana 33 (2017), 633-748. doi: 10.4171/RMI/953