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Smoothing theorems for Radon transforms over hypersurfaces and related operators

  • Michael Greenblatt ORCID logo EMAIL logo
Published/Copyright: August 11, 2020

Abstract

We extend the theorems of [M. Greenblatt, Lp Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron, J. Funct. Anal. 276 2019, 5, 1510–1527] on Lp to Lsp Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving Lp to Lsq boundedness results for such operators. Here qp but s can be positive, negative, or zero. For many such operators we will have a triangle Z(0,1)×(0,1)× such that one has Lp to Lsq boundedness for (1p,1q,s) beneath Z, and in the case of Radon transforms one does not have Lp to Lsq boundedness for (1p,1q,s) above the plane containing Z, thereby providing a Sobolev space improvement result which is sharp up to endpoints for (1p,1q) below Z. This triangle Z intersects the plane {(x1,x2,x3):x3=0}, and therefore we also have an Lp to Lq improvement result that is also sharp up to endpoints for certain ranges of p and q.

MSC 2010: 42B20; 44A12

Communicated by Christopher D. Sogge


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Received: 2020-06-04
Published Online: 2020-08-11
Published in Print: 2020-11-01

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