Lacunary spherical maximal operators on hyperbolic spaces

TL;DR

This work studies lacunary spherical maximal operators on hyperbolic spaces H^n and establishes L^p-boundedness for all p in (1, ∞], leveraging the geometry at infinity of hyperbolic space. The authors construct an analytic family of lacunary operators L_*^alpha with corresponding Fourier multipliers m_t^alpha(D) and decompose the problem into nonlocal and local parts, applying Kunze-Stein estimates to the nonlocal component and Calderón-Zygmund-style kernel analysis to the local part. Key technical inputs are precise multiplier and kernel bounds for m_t^alpha and K_t^alpha, together with an interpolation argument across alpha, which yields boundedness for the lacunary operator at alpha = 0. The results reveal that hyperbolic geometry enables broader L^p-boundedness for lacunary maximal operators than for the full spherical operator, highlighting the influence of the geometry at infinity on harmonic analysis in non-doubling spaces.

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.

Figures (1)

• Figure 1: The admissible range of $(\alpha,p)$ for which the operators $\mathcal{M}^\alpha_*$ and $\mathcal{L}_{*}^{\alpha}$ are respectively $L^p$-bounded.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Theorem 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Proposition 3.4