The fractional $p$-Laplacian on hyperbolic spaces

TL;DR

The paper addresses defining and connecting nonlinear fractional Laplacians on hyperbolic spaces by presenting three equivalent representations: a singular-kernel formulation, a heat-semigroup representation, and a Caffarelli–Silvestre extension. It provides explicit normalizing constants to ensure consistency in the $s\to1^{-}$ limit and establishes the convergence of $(-\Delta_{\mathbb{H}^{n}})^{s}_{p}$ to the classical $p$-Laplacian $(-\Delta_{\mathbb{H}^{n}})_{p}$ for suitable functions. The work advances nonlinear analysis on curved spaces by furnishing a robust framework for fractional operators on hyperbolic geometry, with potential implications for PDEs and geometric analysis on manifolds. The combination of heat-kernel techniques, explicit kernel representations, and extension theory yields a versatile toolkit for studying nonlinear fractional diffusion in negatively curved spaces.

Abstract

We present three equivalent definitions of the fractional $p$-Laplacian $(-Δ_{\mathbb{H}^{n}})^{s}_{p}$, $0<s<1$, $p>1$, with normalizing constants, on hyperbolic spaces. The explicit values of the constants enable us to study the convergence of the fractional $p$-Laplacian to the $p$-Laplacian as $s \to 1^{-}$.

Theorems & Definitions (26)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3