On the Boundedness of Hypersingular Integrals Along Certain Radial Hypersurfaces

TL;DR

This paper investigates the boundedness of hypersingular oscillatory integral operators along specific radial hypersurfaces, establishing $L^p$ bounds and Sobolev estimates under curvature and monotonicity conditions.

Contribution

It provides new $L^p$ boundedness results and Sobolev estimates for hypersingular integrals associated with radial hypersurfaces, extending understanding of their analytical properties.

Findings

Proved $L^p$ boundedness under curvature and monotonicity conditions.

Established Sobolev estimates for the hypersingular operators.

Determined the range of $p$ depending on hypersingularity.

Abstract

We study a class of oscillatory hypersingular integral operators associated to a radial hypersurface of the form $\Gamma(t)=(t,\varphi(t)), t\in\R{n}$. When $\varphi$ satisfies suitable curvature and monotonicity conditions, we prove $L^p(\R{{n+1}})$ boundedness of the operator, where the range of $p$ depends on the hypersingularity of the operator. We also establish certain Sobolev estimates of the operator under consideration.