Weighted integrability of polyharmonic functions in the higher-dimensional case

Liu, Congwen; Perälä, Antti; Si, Jiajia

doi:10.2140/apde.2021.14.2047

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Weighted integrability of polyharmonic functions in the higher-dimensional case

Congwen Liu, Antti Perälä and Jiajia Si

Vol. 14 (2021), No. 7, 2047–2068

DOI: 10.2140/apde.2021.14.2047

Abstract

This paper is concerned with the $L^{p}$ integrability of $N$ -harmonic functions with respect to the standard weights ${(1 - | x |^{2})}^{α}$ on the unit ball $𝔹$ of $ℝ^{n}$ , $n \geq 2$ . More precisely, our goal is to determine the real (negative) parameters $α$ for which ${(1 - | x |^{2})}^{α ∕ p} u (x) \in L^{p} (𝔹)$ implies that $u \equiv 0$ whenever $u$ is a solution of the $N$ -Laplace equation on $𝔹$ . This question is motivated by the uniqueness considerations of the Dirichlet problem for the $N$ -Laplacian $Δ^{N}$ .

Our study is inspired by a recent work of Borichev and Hedenmalm (Adv. Math. 264 (2014), 464–505), where a complete answer to the above question in the case $n = 2$ is given for the full scale $0 < p < \infty$ . When $n \geq 3$ , we obtain an analogous characterization for $\frac{n - 2}{n - 1} \leq p < \infty$ and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.

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Keywords

polyharmonic functions, weighted integrability, boundary behavior, cellular decomposition

Mathematical Subject Classification 2010

Primary: 31B30

Secondary: 35J40

Milestones

Received: 3 August 2018

Revised: 13 May 2020

Accepted: 31 July 2020

Published: 10 November 2021

Authors

Congwen Liu
CAS Wu Wen-Tsun Key Laboratory of Mathematics School of Mathematical Sciences University of Science and Technology of China Hefei China

Antti Perälä
Department of Mathematics and Mathematical Statistics Umeå University Umeå Sweden

Jiajia Si
School of Science Hainan University Haikou China

Vol. 14, No. 7, 2021