The Gleason's problem on normal weight general function spaces in the unit ball of C^(n)

In this paper,we first discuss the boundedness of certain integral operator T_(t) on the normal weight general function space F(p,μ,s)in the unit ball Bnof C^(n).As an application of this operator,we prove that the Gleason’s problem is solvable on F(p,μ,s).

GLEASON'S PROBLEM ON THE SPACE F^(p,q,s)(B) IN C^(n)

Let Ω be a domain in C^(n) and let Y be a function space on Ω.If a∈Ω and g∈Y with g(a)=0,do there exist functions f_(1),f_(2),…,f_(n)∈Y such that g(z)=∑_(l=1)^(n)(z_(l)−a_(l))f_(l)(z)for all z=(z_(1),z_(2),…,z_(n))∈Ω?This is Gleason’s problem.In this paper,we prove that Gleason’s problem is solvable on the boundary general function space F^(p,q,s)(B)in the unit ball B of C^(n).

The Gleason's problem on mixed norm spaces in convex domains

Let Ω be a bounded convex domain with C2 boundary in Cn and for given 0 ＜ p, q ≤∞ and normal weight function ψ(r) let Hp,q,ψ be the mixed norm space on Ω. In this paper we prove that the Gleason's problem (Ω, a, Hp,q,ψ) is solvable for any fixed point a ∈Ω. While solving the Gleason's problem we obtain the boundedness of certain integral operator on H-p,q,ψ.

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

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