Multiparameter maximal functions along dilation-invariant hypersurfaces

Abstract:

Consider the hypersurface ${x_{n + 1}} = \Pi _1^nx_i^{{\alpha _i}}$ in ${{\mathbf {R}}^{n + 1}}$. The associated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles $\{ 0 \leqslant {x_i} \leqslant {h_i},\;i = 1, \ldots ,n\}$. We prove that this operator is bounded on ${L^p}$ for $p > 1$. An analogous result is proved for a quadratic surface in ${{\mathbf {R}}^3}$.