On the maximal Riesz-transforms along surfaces

Abstract:

Let $b(t)$ be an arbitrary bounded radial function. For $x = ({x_1},{x_2}),t = ({t_1},{t_2})$ in ${R^2},\left | t \right | = {({t_1} + {t_2})^{1/2}}$, we establish that the following maximal Riesz-transforms along the surfaces $({t_1},{t_2},|t{|^a}),a > 0$: \[ {T^*}f(x) = \sup \limits _{\varepsilon > 0} \left | {\int _{|t| > \varepsilon } {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - |t{|^a})b(t)\left . {\frac {{{t_1}}}{{|t{|^3}}}dt} \right |} } \right .\] are bounded in ${L^p}({R^3})$ for all $1 < p < \infty$. The $n$-dimensional result can be found at the end of this paper.