Convolution singular integrals on Lipschitz surfaces

Abstract:

We prove the ${L_p}$-boundedness of convolution singular integral operators on a Lipschitz surface \[ \Sigma = \{ g({\mathbf {x}}){e_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:{\mathbf {x}} \in {\mathbb {R}^n}\} \] where $g$ is a Lipschitz function which satisfies ${\left \| {\nabla g} \right \|_\infty } \leq {\text {tan}}\omega < \infty$. Here we have embedded ${\mathbb {R}^{n + 1}}$ in the Clifford algebra ${\mathbb {R}_{(n)}}$ with identity ${e_0}$, and are considering convolution with right-monogenic functions $\phi$ which satisfy $\left | {\phi (x)} \right | \leq C{\left | x \right |^{ - n}}$ on a sector \[ S_\mu ^o = \{ x = {x_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:\left | {{x_0}} \right | < \left | {\mathbf {x}} \right |{\text {tan}} \mu \} \] where $\mu > \omega$. Provided there exists an ${L_\infty }$ function $\underline \phi$ satisfying \[ \underline \phi (R) - \underline \phi (r) = \int _{\substack {r < |x| < R\\x \in {\mathbb {R}^{n}}}} \phi (x)\;dx \], then the related convolution singular integral operator \[ ({T_{(\phi ,\underline {\phi )} }}u)(x) = \lim _{\varepsilon \to 0+}\left \{\int _{\substack {y \in \Sigma \\|x - y| \geq \varepsilon }} \phi (x - y)n(y)u(y)\;d{S_y} + \underline \phi (\varepsilon n(x))u(x) \right \} \] is bounded on ${L_p}(\Sigma )$ for $1 < p < \infty$.