Decompositions of \(L^p\) and Hardy spaces of polyharmonic functions
Denote by \(B\) the open Euclidean unit ball in \(\mathbb{R}^n\). For \(0< p, q\leq\infty\), \(\alpha\in \mathbb{R}\) and \(k\in\mathbb{N}\) denote by \(H^{p,q,\alpha}_k\) the space of all \(f\in C^\infty(B)\) satisfying \(\Delta^k f=0\) for which \[ \| f\|_{p,q,\alpha}:= \Biggl(\int_B (M_p(f,| x|))^q(1-| x|^2)^{q\alpha- 1}dx\Biggr)^{1/q}< \infty, \] where \[ M_p(f, r):= \Biggl(\int_{\partial B}| f(ry)|^p d\sigma(y)\Biggr)^{1/p},\quad 0\leq r< 1, \] and \(d\sigma\) the normalized surface measure of \(\partial B\). The author shows that for \(k\geq 2\) and \(\alpha> -1\) the map \[ Q: H^{p,q,\alpha}_k\to H^{p,q,\alpha}_1\oplus H^{p,q,\alpha+ 2}_{k- 1},\quad Q(f)= (Pf,\Delta f) \] is an isomorphism, for \(Pf\) being the Poisson integral of \(f\). From this it follows that \(H^{p,q,\alpha}_k\) is isomorphic to the direct sum of the spaces \(H^{p,q,\alpha+ 2j}_1\), \(j= 0,\dots,k- 1\), of harmonic functions. Furthermore, another decomposition result of this type is given which uses the Almansi representation theorem.
scientific article; zbMATH DE number 1115583