The fractional $p$-Laplacian on hyperbolic spaces

TL;DR

This paper introduces three equivalent definitions of the fractional p-Laplacian on hyperbolic spaces, providing explicit constants to analyze its convergence to the classical p-Laplacian as the fractional parameter approaches one.

Contribution

It offers explicit formulas and three equivalent definitions of the fractional p-Laplacian on hyperbolic spaces, facilitating convergence analysis.

Findings

Derived three equivalent definitions of the fractional p-Laplacian.

Provided explicit constants for the fractional p-Laplacian.

Studied the convergence to the p-Laplacian as s approaches 1.

Abstract

We present three equivalent definitions of the fractional $p$-Laplacian $(-\Delta_{\mathbb{H}^{n}})^{s}_{p}$, $0<s<1$, $p>1$, 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 $s \to 1^{-}$.