On the Boundedness of Hypersingular Integrals Along Certain Radial Hypersurfaces

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.