Hardy-Littlewood型现象与Möbius不变Laplacian算子的Girela-Peláez猜想
Hardy-Littlewood type phenomena and the Girela-Peláez conjecture for the Möbius invariant Laplacian operator
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中文总结 AI 辅助
本文研究单位球上Möbius不变Laplace方程的Hardy-Littlewood型现象与相关算子有界性，改进已有结果、解答公开问题，并证明Girela-Peláez猜想对更一般函数类成立。
中文摘要 AI 辅助
本文的研究目的分为两部分。首先，我们研究了$\n\bbr^n$中单位球上Möbius不变Laplace方程Dirichlet解的Hardy-Littlewood型现象。我们的工作推广并改进了Pavlovć[《Rev. Mat. Iberoam.》23卷: 831-845页, 2007年]以及Chen等人[《J. Geom. Anal.》34卷: 23页, 2024年]的若干关键结果，特别地，我们完整解答了Makoto Masumoto提出的一个问题。其次，受Aikawa工作的启发，我们研究了算子$P_α$的算子范数有界性，其中$P_α[φ]$是边界数据为$φ$的上述方程的Dirichlet解。通过采用替代证明技巧，我们得到了$P_α$算子范数有界性的等价刻画。最后，我们证明了Girela-Peláez猜想对于由Möbius不变Laplacian算子诱导的更一般函数类是成立的。
英文摘要
The purpose of this paper is twofold. First, we investigate the Hardy-Littlewood type phenomena for Dirichlet solutions to the Möbius invariant Laplace equation on the unit ball in $\mathbb{R}^n$. Our work extends and improves several key results due to Pavlovć [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 $P_α$, where $P_α[φ]$ is the Dirichlet solution of such equation for the boundary data $φ$. By using alternative proof techniques, we obtain an equivalent characterization of the boundedness of the operator norm of $P_α$. Finally, we show that the Girela-Peláez conjecture holds positively for more general classes of functions induced by the Möbius invariant Laplacian operator.
