Weighted integrability of polyharmonic functions in the higher-dimensional case

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 ${\mathrm{\Delta }}^N$.

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 . When $n\ge 3$, we obtain an analogous characterization for 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.