The Gleason's problem for some polyharmonic and hyperbolic harmonic function spaces
Let \(\Omega\subseteq{\mathbb R}^n\) be a bounded convex domain with \({\mathcal C}^2\) boundary and let \(\lambda\) be the defining function of \(\Omega,\) that is, \(\Omega=\{x: \lambda(x)<0\}.\) A \({\mathcal C}^\infty\) function \(f\) is said to be polyharmonic of order \(k\) if \(\Delta^k f\equiv 0\) for the \(k\)-th formal power of the Laplacian \(\Delta.\) For \(0<p,q \leq \infty \) and a normal weight \(\varphi,\) the mixed norm \(\| f\| _{p,q,\varphi}\) is the \(L_q (0,\varepsilon)\) norm of the weighted function \(M_p(f,r){\varphi(r) \over r^{1/q}},\) where \(M_p(f,r)=\| f\| _{L_p(\partial\Omega_r,d\sigma_r)}\) for the surface measure \(\sigma_r\) of the boundary \(\partial\Omega_r=\{x: \lambda(x)=-r \}.\) The mixed norm space \(H^{p,q,\varphi}_k(\Omega)\) consists of all polyharmonic functions \(f\) of order \(k\) for which the mixed norm is finite. The authors prove that the Gleason problem \((\Omega,a,H^{p,q,\varphi}_k)\) is always solvable for any reference point \(a \in \Omega.\) Also, the Gleason problem for the polyharmonic \(\varphi\)-Bloch (little \(\varphi\)-Bloch) space is solvable. Parallel results for the hyperbolic harmonic mixed norm space are also obtained.
scientific article; zbMATH DE number 5118318
scientific article