Convolution Singular Integral Operators on Lipschitz Surfaces

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 $$L^{p}(\Sigma )$$ .