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arXiv · 2608.23335

Hardy-Littlewood type phenomena and the Girela-Pel\'aez conjecture for the M\"obius invariant Laplacian operator

Jiaolong Chen·Shaolin Chen·Hidetaka Hamada·Qianyun Li
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Abstract

The purpose of this paper is twofold. First, we investigate the Hardy-Littlewood type phenomena for Dirichlet solutions to the M\"obius invariant Laplace equation on the unit ball in $\mathbb{R}^n$. Our work extends and improves several key results due to Pavlov\'c [Rev. Mat. Iberoam. 23: 831-845, 2007] and Chen et al. [J. Geom. Anal. 34: 23 pp, 2024]. In particular, we give a complete answer to a question raised by Makoto Masumoto. Second, motivated by Aikawa's work, we study the boundedness of the operator norm of $P_{\alpha}$, where $P_{\alpha}[\varphi]$ is the Dirichlet solution of such equation for the boundary data $\varphi$. By using alternative proof techniques, we obtain an equivalent characterization of the boundedness of the operator norm of $P_{\alpha}$. Finally, we show that the Girela-Pel\'aez conjecture holds positively for more general classes of functions induced by the M\"obius invariant Laplacian operator.

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Jiaolong Chen, Shaolin Chen, Hidetaka Hamada, Qianyun Li. 2026-08-24. Hardy-Littlewood type phenomena and the Girela-Pel\'aez conjecture for the M\"obius invariant Laplacian operator. https://arxiv.org/abs/2608.23335

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