Convolution singular integral operators on lipschitz curves

Abstract

As the high-dimensional generalization of the boundedness of singular integrals on Lipschitz curves, the \(L^{p}(\Sigma )\)-boundedness of the Cauchy-type integral operators on the Lipschitz surfaces \(\Sigma \) is a meaningful question. The increase of the dimensions means that we need to apply a new method to solve the above question. In 1994, C. Li, A. McIntosh and S. Semmes embedded \(\mathbb {R}^{n+1}\) into Clifford algebra \(\mathbb {R}_{(n)}\) and considered the class of holomorphic functions on the sectors \(S_{w,\pm }\), see [1]. They proved that if the function \(\phi \) belongs to \(K(S_{w,\pm })\), then the singular integral operator \(T_{\phi }\) with the kernel \(\phi \) on Lipschitz surface is bounded on \(L^{p}(\Sigma )\).