Gleason's problem for harmonic Bergman and Bloch functions on half-spaces (Q1566819)

Let \(X\) be a function space on a domain \(\Omega\subset\mathbb{R}^n\). By Gleason's problem for \(X\) the following is meant: Given a point \(a\in\Omega\) and a function \(u\in X\), do there exist functions \(g_1,\dots,g_n\in X\) such that \(u(z)- u(a)=\sum_j (z_j-a_j) g_j(z)\), \(\forall z\in\Omega\)? The authors prove that Gleason's problem for \(X\) the harmonic \(L^p\)-Bergman space on the half-space \(H=\mathbb{R}^{n-1}\times\mathbb{R}_+\) is solvable if and only if \(p>n\), and the problem for \(X\) the harmonic Bloch space on \(H\) is always solvable.