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This paper is concerned with the $L^p$ integrability of $N$-harmonicfunctions with respect to the standard weights $(1-|x|^2)^{\alpha}$ on the unitball $\mathbb{B}$ of $\mathbb{R}^n$, $n\geq 2$. More precisely, our goal is todetermine the real (negative) parameters $\alpha$, for which$(1-|x|^2)^{\alpha/p} u(x) \in L^p(\mathbb{B})$ implies that $u\equiv 0$,whenever $u$ is a solution of the $N$-Laplace equation on $\mathbb{B}$. Thisquestion is motivated by the uniqueness considerations of the Dirichlet problemfor the $N$-Laplacian $\Delta^N$.  Our study is inspired by a recent work of Borichev and Hedenmalm [Adv. Math.,264(2014), pp. 464-505], where a complete answer to the above question in thecase $n=2$ is given for the full scale $0<p<\infty$. When $n\geq 3$, we obtainan analogous characterization for $\frac{n-2}{n-1}\leq p<\infty$, and remarkthat the remaining case can be genuinely more difficult. Also, we extend theremarkable cellular decomposition theorem of Borichev and Hedenmalm to alldimensions.

Weighted integrability of polyharmonic functions in the higher dimensional case