Convolution singular integrals on Lipschitz surfaces

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 ${\left\Vert {\nabla g} \right\Vert _\infty } \leq {\text{tan}}\omega < \infty$. 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 $\left\vert {\phi (x)} \right\vert \leq C{\left\vert x \right\vert^{ - n}}$ on a sector

$$S_\mu ^o = \{ x = {x_0} + {\mathbf{x}} \in {\mathbb{R}^{n + 1}}:\... ...t {{x_0}} \right\vert < \left\vert {\mathbf{x}} \right\vert{\text{tan}} \mu \}$$

where $\mu > \omega$. Provided there exists an ${L_\infty }$ function $\underline \phi$ satisfying

$$\underline \phi (R) - \underline \phi (r) = \int_{\substack{\mathop r < \vert x\vert < R x \in {\mathbb{R}^{n}}}} \phi (x)\;dx$$

, then the related convolution singular integral operator

$$({T_{(\phi ,\underline {\phi )} }}u)(x) = \lim_{\varepsilon \to 0... ...\phi (x - y)n(y)u(y)\;d{S_y} + \underline \phi (\varepsilon n(x))u(x) \right\}$$

is bounded on ${L_p}(\Sigma )$ for $1 < p < \infty$.