The fractional $ p $-Laplacian on hyperbolic spaces

Jongmyeong Kim; Minhyun Kim; Ki-Ahm Lee; Jongmyeong Kim; Minhyun Kim; Ki-Ahm Lee

doi:10.3934/mine.2026003

Mathematics in Engineering

2026, Volume 8, Issue 1: 70-97. doi: 10.3934/mine.2026003

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The fractional $ p $-Laplacian on hyperbolic spaces

1.
Institute of Mathematics, Academia Sinica, Taipei 106319, Taiwan
2.
Department of Mathematics & Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea
3.
Department of Mathematical Sciences & Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

Received: 16 September 2025 Revised: 21 January 2026 Accepted: 21 January 2026 Published: 27 January 2026

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^{-} $.
- fractional $ p $-Laplacian,
- hyperbolic space,
- pointwise convergence
Citation: Jongmyeong Kim, Minhyun Kim, Ki-Ahm Lee. The fractional $ p $-Laplacian on hyperbolic spaces[J]. Mathematics in Engineering, 2026, 8(1): 70-97. doi: 10.3934/mine.2026003

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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^{-} $.

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