#### 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 ${L}^{p}$ integrability of $N$-harmonic functions with respect to the standard weights ${\left(1-|x{|}^{2}\right)}^{\alpha }$ on the unit ball $\mathbb{𝔹}$ of ${ℝ}^{n}$, $n\ge 2$. More precisely, our goal is to determine the real (negative) parameters $\alpha$ for which ${\left(1-|x{|}^{2}\right)}^{\alpha ∕p}u\left(x\right)\in {L}^{p}\left(\mathbb{𝔹}\right)$ implies that $u\equiv 0$ whenever $u$ is a solution of the $N$-Laplace equation on $\mathbb{𝔹}$. This question is motivated by the uniqueness considerations of the Dirichlet problem for the $N$-Laplacian ${\Delta }^{\phantom{\rule{-0.17em}{0ex}}N}$.

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. When $n\ge 3$, we obtain an analogous characterization for $\frac{n-2}{n-1}\le p<\infty$ 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.

##### Keywords
polyharmonic functions, weighted integrability, boundary behavior, cellular decomposition
Primary: 31B30
Secondary: 35J40