Annulus Maximal Averages on Variable Hyperplanes

Abstract

By giving a thin width of $0<\delta \ll 1$ to both a unit circle and a unit line, we set an annulus and a tube on the Euclidean plane ${\mathbb{R}}^2$. Consider the maximal means $M_{\delta }$ over dilations of the annulus, and $N_{\delta }$ over rotations of the tube. It is known that their operator norms on $L^2({\mathbb{R}}^2)$ are $O(|\log 1/\delta {|}^{1/2})$. In this paper, we study the maximal averages ${\mathcal{M}}_{\delta }^A$ and ${\mathcal{N}}_{\delta }^A$ over those annuli and tubes now imbedded on the variable hyperplanes $(x,x_3)+\left\{\left(y,⟨A(x),y⟩\right):y\in {\mathbb{R}}^2\right\}\subset {\mathbb{R}}^3$ where $A$ is a $2\times 2$ matrix. The model hyperplane is the horizontal plane of the Heisenberg group when $A$ is the skew--symmetric matrix denoted by $E$. It turns out that a rank of matrix $EA+(EA{)}^T$ or $A+A^T$ determines $\Vert {\mathcal{M}}_{\delta }^A{\Vert }_{op}$ or $\Vert {\mathcal{N}}_{\delta }^A{\Vert }_{op}$ respectively. In the higher dimension, the corresponding spherical maximal means is bounded in $L^p$ if $A$ has only complex eigenvalues.

59 pages. We revised the whole structure of the previous version, providing corrections and organizations

Cited by in corpus (3)

• Lebesgue space estimates for spherical maximal functions on Heisenberg groups
• $L^p-L^q$ estimates for the circular maximal operators on Heisenberg radial functions
• Circular maximal functions on the Heisenberg group