GLEASON’S PROBLEM ON FOCK-SOBOLEV SPACES∗

Jineng DAI(戴济能)Jingyun ZHOU(周静云)

Department of Mathematics,School of Science,Wuhan University of Technology,Wuhan 430070,China E-mail:jinengdai@whut.edu.cn;zhou19950614@163.com

Abstract In this article,we solve completely Gleason’s problem on Fock-Sobolev spaces Fp,mfor any non-negative integer m and 0

Key words Fock-Sobolev space;Gleason’s problem

1 Introduction

where dv is the normalized Lebesgue volume measure on Cso that the constant function 1 has norm 1 in L.Here we are abusing the term“norm”for 0

Let Fdenote the space of holomorphic functions in L.When m=0,the space Fis called the Fock space or the Segal-Bargmann space(see[1–7]).For a multi-index α=(α,···,α),where each α(1≤k≤n)is a non-negative integer,we write

where∂denotes partial differentiation with respect to the k-th component.An equivalent characterization for the space Fis as follows(see[4]):f∈Fif and only if∂f belongs to the Fock space for each multi-index α with|α|≤m.In this sense we call Fthe Fock-Sobolev space.The space Fis a closed subspace of the Hilbert space Lwith inner product

The orthogonal projection P :L→Fis given by

where K(z,w) is the reproducing kernel of the Fock-Sobolev space F.It is well known that

for all 0

In this article,we prefer to use the integral form of hto express the reproducing kernel of Fock-Sobolev spaces F.

Let X be a space of holomorphic functions on a domain Ω in C.Gleason’s problem for X is the following:if a ∈Ω and f ∈X,do there exist functions f,···,fin X such that

In this article,we solve Gleason’s problem on Fock-Sobolev spaces Fin a stronger form for the full range of p with 0

for all z ∈C(see Theorem 2.9 and 2.10).Because the form of the Bergman kernel of F(especially for m ≥1) is a bit complicated,some techniques are used for dealing with many integrals.

2 Gleason’s Problem on Fock-Sobolev Spaces

In this section,we begin with several useful lemmas,which are needed in the proof of the solvability of Gleason’s problem on Fock-Sobolev spaces F.

Lemma 2.1

Let 0

0.There exists a constant C only depending on p,α and β such that

for all holomorphic functions f and anti-holomorphic functions g on C.

Proof

It is known that g is anti-holomorphic if and only if g is holomorphic.By Lemma 4 in [4],we have that

Lemma 2.2

For fixed a in C,we have that

Remark 2.5

If we replace the quantity |z|(resp.|w|) by (1+|z|)(resp.(1+|w|)) in Lemma 2.4,then the inequality is also valid.

An important tool for tackling the boundedness of integral operators on L(1

Lemma 2.6

([15]) Let (X,µ) be a measure space and H be a non-negative measurable function on the product space X ×X.Let 1

Now we state our main results.We first solve Gleason’s problem on the Fock space,then we turn to generalized Fock-Sobolev spaces.

Theorem 2.9

For fixed a in Cand any 0

for all z in Cand f in F.