Sobolev spaces associated to singular and fractional Radon transforms

  • Brian Street

    University of Wisconsin, Madison, USA

Abstract

The purpose of this paper is to study the smoothing properties (in Sobolev spaces) of operators of the form , where is a function defined on a neighborhood of the origin in , satisfying , is a cut-off function supported on a small neighborhood of , and is a "multi-parameter fractional kernel" supported on a small neighborhood of . When is a Calderón–Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on were given under which the above operator is bounded on (). Under these same conditions, we introduce non-isotropic Sobolev spaces associated to . Furthermore, when 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 by Christ, Nagel, Stein, and Wainger, we prove optimal smoothing properties in isotropic Sobolev spaces for the above operator when is a fractional kernel which is smoothing of very low order.

Cite this article

Brian Street, Sobolev spaces associated to singular and fractional Radon transforms. Rev. Mat. Iberoam. 33 (2017), no. 2, pp. 633–748

DOI 10.4171/RMI/953