Lacunary spherical maximal operators on hyperbolic spaces

TL;DR

This paper establishes the boundedness of lacunary spherical maximal operators on hyperbolic spaces, revealing how hyperbolic geometry influences harmonic analysis differently from Euclidean spaces.

Contribution

It proves the boundedness of lacunary spherical maximal operators on hyperbolic spaces for all dimensions and p-values, extending Euclidean results to non-Euclidean geometry.

Findings

Lacunary spherical maximal operator is bounded on L^p(H^n) for all n≥2 and 1<p≤∞.

The lacunary set in hyperbolic space is larger than in Euclidean space.

Hyperbolic geometry affects the behavior of maximal operators at infinity.

Abstract

We prove that the lacunary spherical maximal operator, defined on the $n$-dimensional real hyperbolic space, is bounded on $L^p(\mathbb{H}^n)$ for all $n\ge2$ and $1<p\le\infty$. In particular, the lacunary set is significantly larger than its Euclidean counterpart, reflecting the influence of the geometry at infinity of the hyperbolic space.