On the Riesz-transforms along surfaces in $\textbf {R}^ 3$

Abstract:

For $x = ({x_1},{x_2},{x_3})$ in ${R^3}$, $t = ({t_1},{t_2})$ in ${R^2}$, $|t| = {(t_1^2 + t_2^2)^{1/2}}$ and $a > 0$, we define \[ {R_a}f(x) = \int _{{R^2}} {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - |t{|^a})\frac {{{t_1}}}{{|t{|^3}}}dt.} \] The transformation ${R_a}$ can be thought of as a Riesz-transform along the surface $({t_1},{t_2},|t{|^a})$ in ${R^3}$. Our purpose here is to show that, for all $a > 0$, the operator ${R_a}$ is bounded on ${L^p}$ for $6/5 < p < 6$. Singular integrals along lower dimensional nonaffine varieties have been studied by several authors. See for example [3, 5, 6]. In the proof of our result, we use techniques similar to the ones used in [1, 2].