The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Abstract

Let Ω ⊆ ℝn be a bounded convex domain with C2 boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space Hkp,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, Hkp,q,φ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.