Annulus Maximal Averages on Variable Hyperplanes

TL;DR

This paper investigates maximal averages over variable hyperplanes in 3D, revealing how the matrix rank influences operator norms and establishing boundedness results in higher dimensions based on eigenvalue types.

Contribution

It introduces a new analysis of maximal averages on variable hyperplanes, linking matrix properties to operator bounds and extending results to higher dimensions with complex eigenvalues.

Findings

Operator norms depend on the rank of specific matrix combinations.

Boundedness in higher dimensions occurs when matrix A has only complex eigenvalues.

The model hyperplane relates to the horizontal plane in the Heisenberg group.

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 $\mathcal{M}^A_\delta$ and $\mathcal{N}^A_\delta$ over those annuli and tubes now imbedded on the variable hyperplanes $(x,x_3)+\left\{\left(y, \langle A(x), y\rangle\right): y\in\mathbb{R}^2\right\}\subset \mathbb{R}^3$ 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 $\|\mathcal{M}^A_\delta\|_{op} $ or $\|\mathcal{N}^A_\delta\|_{op}$ respectively. In the higher dimension, the corresponding spherical maximal means is bounded in $L^p$ if $A$ has only complex eigenvalues.