Singular Integrals Associated to Hypersurfaces: $L^2$ Theory

Abstract

We consider singular integrals associated to a classical Calder\'on-Zygmund kernel $K$ and a hypersurface given by the graph of $\varphi(\psi(t))$ where $\varphi$ is an arbitrary $C^1$ function and $\psi$ is a smooth convex function of finite type. We give a characterization of those Calder\'on-Zygmund kernels $K$ and convex functions $\psi$ so that the associated singular integral operator is bounded on $L^2$ for all $C^1$ functions $\varphi$.