Vol. 14, No. 7, 2021

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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
Abstract

This paper is concerned with the Lp integrability of N-harmonic functions with respect to the standard weights (1 |x|2)α on the unit ball 𝔹 of n , n 2. More precisely, our goal is to determine the real (negative) parameters α for which (1 |x|2)αpu(x) Lp(𝔹) implies that u 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 ΔN .

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 < . When n 3, we obtain an analogous characterization for n2 n1 p < 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.

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