Fourier multipliers on Lipschitz curves

Abstract:

We develop the theory of Fourier multipliers acting on ${L_p}(\gamma )$ where $\gamma$ is a Lipschitz curve of the form $\gamma = \{ x + ig(x)\}$ with $\left \| g\right \| _\infty < \infty$ and $\left \| g\prime \right \| _\infty < \infty$ . The aim is to better understand convolution singular integrals $B$ defined naturally on such curves by \[ Bu(z) = {\text {p.v.}}\int _\gamma {\varphi (z - \zeta )u(\zeta )d\zeta } \] for almost all $z \in \gamma$ .