On the Boundedness of Hypersingular Integrals Along Certain Radial Hypersurfaces

We study a class of oscillatory hypersingular integral operators associated to a radial hypersurface of the form $\mathrm{\Gamma }(t)=(t,\phi (t)),t\in \mathbb{R}n$. When $\phi$ satisfies suitable curvature and monotonicity conditions, we prove $L^p(\mathbb{R}n+1)$ 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.