Let Ω be a bounded domain in ℂ2 with smooth boundary bΩ. Let ρ be a defining function for Ω so that Ω={z∈ℂ2:ρ(z)<0} and bΩ={z∈ℂ2:ρ(z)=0}, ∇ρ≠0 on bΩ. Let O(Ω) be the space of functions that are holomorphic in Ω, with the topology of uniform convergence on compact subsets of Ω. For 1<p<∞, let Lp(Ω,dV) be the standard Lebesgue space over Ω with respect to the Lebesgue volume measure dV of ℝ4, and let the Bergman space Ap(Ω)=Lp(Ω,dV)∩O(Ω). The Bergman projection P is the orthogonal projection of L2(Ω) onto the Bergman space A2(Ω). The most important property of the Bergman projection is that there exists a function P:Ω×Ω→ℂ such that

for all u∈L2(Ω), z∈Ω. Here, P(ζ,z) is the Bergman kernel on Ω, which is holomorphic with respect to z∈Ω, and anti-holomorphic in ζ. In this paper, we investigate the Lp(Ω)-boundedness of the projection P. In the recent forty years, there have been many papers focused on studying Lp(Ω)-boundedness (see for example [16, 1, 14, 15, 2]) and its applications in studying commutator operators (see for example [10]), composition operators (see for example [4, 9]). Although there are many results on the Lp(Ω)-boundedness, the case p=1 and the case p=∞ are still open. In this paper, we provide an answer to solve these problems.

Definition 1.1. ([13, p. 478]) A differentiable function u on Ω is said to be a Bloch function if and only if

The space of all Bloch functions defined on Ω is denoted by Bl(Ω) and by BlHol(Ω)=Bl(Ω)∩O(Ω) the space of holomorphic Bloch functions on Ω. We also define ‖u‖BlHol(Ω)=‖u‖Bl(Ω) for all u∈BlHol(Ω).

Since P(z,z)>0 for all z∈Ω, P(z,z)dV(z) is a biholomorphically invariant measure of Ω.

Definition 1.2. A function u∈A2(Ω,dV) is said to be a Besov function if and only if

where |∇3u(z)|=∑ 1≤j+k≤3 ∂ j+ku ∂z1j∂z2k(z). The space of all holomorphic Besov functions defined on Ω is denoted by Besov(Ω). Here we have an explanation for this definition. Assume that Ω is a smoothly bounded, strongly pseudoconvex domains. The classical Besov space B(Ω) is a subspace of A2(Ω,dV) in which we equip the semi-norm

Since ∫Ω (−ρ(z)) −1dV(z)=∞, (B(Ω),‖⋅‖B) consists only constant functions on Ω. In order to make more natural, we use the semi-norm ‖⋅‖Besov(Ω) instead of ‖⋅‖B. This idea was used in [13] for strongly pseudoconvex domains.

The main result in this paper is following.

Main Theorem. Let Ω be a smoothly bounded convex domain in ℂ2 admitting a type F at all boundary points (see Definition 2.2) and satisfying the condition (B) (see Definition 2.4). Then the Bergman projection is bounded from:

• 1. Lp(Ω,dV) to Ap(Ω,dV) for all 1<p<∞.

• 2. L∞(Ω) to BlHol(Ω).

• 3. L1(Ω,P(z,z)) to Besov(Ω).

Phong and Stein in [16] established the Lp→Ap boundedness when Ω is a strongly pseudoconvex domain. Then, this result was generalized to a certain class of convex domains in ℂ2 (see [1]) and to finite type convex domains in ℂn (see [15]). Even when Ω is the unit ball in ℂn, for n≥2, the Bergman projection P can not be extended continuously from Lp(Ω) onto Ap(Ω) when p=1 or p=∞ (for example, see [20, Section 7.1]). In [14], using Cauchy-Fantappié integral theory, Ligocka obtained the L∞(Ω)→BlHol(Ω) boundedness on bounded strongly pseudoconvex domains. Recently, in studying Besov spaces on general domains in ℂn, Li and Luo (see [13]) have proved the L1(Ω,P(z,z))→Besov(Ω) boundedness also on bounded strongly pseudoconvex domains or convex domains of finite type in ℂ2.

The structure of the paper is as follows. Section 2 deals with preliminaries for the Bergman projection in terms of Cauchy-Fantappié forms on convex domains admitting the F-type condition. Section 3 deals with the proof of the Main Theorem.

Let Ω⊂ℂ2 be a bounded convex domain with smooth boundary bΩ with a defining function ρ. By the hypothesis that Ω is convex,

in which x∈bΩ, zj=x2j−1+−1x2j and a∈ℝ4 be a non-zero vector such that ∑ j=14aj∂ρ∂xj(x)=0 on bΩ. Let us define, for (ζ,z)∈bΩ×Ω:

The convexity of Ω gives

so that Φ(ζ,z)≠0 for all ζ∈bΩ,z∈Ω. Moreover, the following lemma proved in [17] is a consequence of the definition of Φ(ζ,z).

Lemma 2.1. For any P∈bΩ, there are positive constants δ,c such that for all boundary points ζ∈bΩ∩B(P,δ), we have

• 1. Φ(ζ,z) is holomorphic in z∈B(ζ,δ);

• 2. Φ(ζ,ζ)=0, and dzΦ|z=ζ≠0;

• 3. There exists a constant A>0 such that |Φ(ζ,z)|≥A for all z∈Ω and |z−ζ|≥c;

• 4. ρ(z)>0 for all z with Φ(ζ,z)=0 and 0<|z−ζ|<c.

Now we set

which is a (1,0)-form of ζ-variables. The Cauchy-Leray kernel for the convex domain Ω is

which is a Cauchy-Fantappié (2,1)-form on bΩ×Ω, where Aj0(ζ) is a polynomial involving first and second derivatives in ζ of ρ.

For each z∈Ω we extend C(.,z) smoothly to the interior of Ω as follows

Definition 2.2. Let F:[0,∞)→[0,∞) be a smooth, strictly increasing function such that

• 1. F(0)=0,

• 2. ∫0σ lnF(r2)dr<∞ for some σ>0 which is small enough,

• 3. F(t)t is non-decreasing function.

Let Ω⊂ℂ2 be a smooth bounded, convex domain. We say that Ω admitting F-type at a point P∈bΩ if there are positive constants c, c' satisfy that for all ζ∈bΩ∩B(P,c′):

for all z∈B(ζ,c) with Φ(ζ,z)=0.

If Ω admits the same F-type at every point on bΩ, we simply call that Ω admitting F-type. In case F(t)=tm, for m=1,2,…, the F-type notion agrees with the finite type condition in the sense of Range in []. Here the notation B(ζ,r) means the Euclidean ball centered at ζ of radius r>0. Also the notations ≲ and ≳ denote inequalities up to a positive constant, and ≈ means the combination of ≲ and ≳.

Some examples to illustrate that the F-type condition consists a large class of convex domains of finite and infinite type in ℂ2 can be found in [8, 9].

The following lemma provides the important lower estimate for the Cauchy-Fantappié form. Its proof is rather similar to the proof of [5, Lemma 3.3] with a negligible modification and can be found in [7, before Corollary 2.6].

Lemma 2.3. Let Ω be a smoothly bounded, convex domain in ℂ2 admitting an F-type at P∈bΩ. Then there is a positive constant c such that the support function Φ(ζ,z) satisfies the following estimate

for every ζ∈Ω¯∩B(P,c), and z∈Ω, |z−ζ|<c.

Definition 2.4. ([13, Definition 2.1]) We say that a smoothly bounded domain Ω⊂ℂ2 has B-property if there is a positive constant CΩ such that the following holds:

for all ζ∈Ω.

In ℂ2, there are many bounded domains which admitting a type F at all boundary points and satisfying the condition (B). Firstly, all strictly convex domains in ℂ2 admits type F(t)=t at all boundary points. Secondly, let m1,m2 be positive integers, and let

be convex domain in ℂ2. The family {Ωm} is the certain class of weakly convex domains in ℂ2. Then, in [5], the author shows that Ωm admits type F(t)=t^m at all boundary points. In [13, p. 480-p. 481], it is proved that any strictly convex domain or any Ωm satisfies B-property.

For u∈C1(Ω¯)∩O(Ω) and u is holomorphic on Ω, by the Stoke Theorem, we get

By the smoothness of each component in Ω0((C˜(ζ,z)) then the form ∂¯ζΩ0((C˜(ζ,z)) also is a smooth form on (Ω¯×Ω¯)∖{(z,z),z∈bΩ}.

For 0<c<δ (c is the constant in Lemma 2.3), let us define Ωδ={z∈ℂ2:ρ(z)<δ} and let P_z be the Hörmander solution operator to the ∂¯-equation in the variables z∈Ωδ (the existence of P_z can be found in [11]).

Definition 2.5. For (ζ,z)∈Ω¯× Ω ¯ δ, let us define

where G(ζ,z) is holomorphic in z.

The fact Q(ζ,z)∈C∞(Ω¯)×C1(Ω¯) implies that

Let u be a holomorphic function defined on Ωδ, since

we have the reproductive property of G(ζ,z) that u(z)=∫Ωu(ζ)G(ζ,z) for all z∈Ω. More generally, let u∈L2(Ω), and let us define

and its dual

Then G:L2(Ω)→A2(Ω) is a well-defined, continuous operator. Moreover, we also have:

Theorem 2.6. ([6, Theorem 3.4][Ligocka's decomposition]) Let Ω⊂ℂ2 be a smoothly bounded, convex domain. Assume that Ω admits a F-type at all boundary points for some function F. Then P[u](z)=G(I−B)−1[u](z)=(I+B)−1G*[u](z), where

3.1. Proof of the assertion (1)

This fact has been proved in [9]. For convenience, we briefly sketch its proof here. The Lp(Ω,dV)-boundedness (for p∈(1,∞)) is a consequence of the following lemma.

Lemma 3.1. The operators G and G* are bounded on Lp(Ω,dV). In particular, we have

and

Due to the strong duality and the Marcinkiewicz Interpolation Theorem from harmonic analysis (see Theorem B.7, Appendix B in [3] for more details), it is sufficient to show that

Firstly, we recall the change of variables (α,w)=(α1,α2,w1,w2)=(ζ1,ζ2,z1−ζ1,ρ(ζ)+iIm(Φ(ζ,z))) and let J be the Jacobian of this change. Then

Since ρ(z)≠0, we can find a sufficiently small 0<δ<c so that ∂ρ∂x4 dominates others partial derivatives of ρ and |z−ζ|≤δ. As a consequence, we have det(J)≠0 on |ζ−z|≤δ.

Now let δ′>0 depend on Ω,c,δ and ρ, and u∈L1(Ω,dV). Since {(z,ζ):|ζ−z|<c}, the kernel G(ζ,z) is bounded from above by

and by Lemma 2.3, we have

Therefore, we obtain the L1(Ω,dV)-boundedness.

Next, let u∈L∞(Ω). The Hölder's Inequality and Lemma 2.3 imply

Hence the L∞-boundedness is established and the proof of Lp(Ω,dV), for p∈(1,∞), is complete.

3.2. Proof of the assertion (2)

Since the continuity of B in Theorem 2.6 and the fact that Ker[I−B]={0}, I−B is a Fredholm isomorphism of L∞(Ω). Thus, it is sufficient to prove that G maps continuously L∞(Ω) into BlHol(Ω).

Let u∈L∞(Ω), we must show that

for all z∈Ω.

We consider the first term in (3.1). Since the integral ∫Ω Q(ζ,z)dV(ζ) is non-singular, we have

For 0<c<σ (c is the constant in Lemma 2.3), let h∈C∞(ℂ2) be a cutoff function such that h=1 on {ζ∈ℂ2:|ρ(ζ)|+|ρ(z)|+|ℑ(Φ(ζ,z))|+F(|ζ−z|2)<σ/2} and h=0 on {ζ∈ℂ2:|ρ(ζ)|+|ρ(z)|+|ℑ(Φ(ζ,z))|+F(|ζ−z|2)>σ}. Then,

Since |∂¯ζΩ0((C˜(ζ,z))| is dominated by |ζ−z||Φ(ζ,z)−ρ(ζ)|3 when ζ near to z, we obtain

To estimate the last integral in the above inequality, we use the following Henkin coordinates on Ω (see [19, Lemma V3.4]). These coordinates do exist since ∇ρ(ζ)|ζ=z and ∇ImΦ(ζ,z)|ζ=z are nonzero and are not proportial.

Lemma 3.2 (Henkin's coordinates). There exist positive constants M,a and η≤c, and, for each z with dist(z,bΩ)≤a, there is a smooth local coordinate system (t1,t2,t3,t4)=t=t(ζ,z) on the ball B(z,c) such that we have

where Jℝ(t) is the Jacobian of the transformation t.

Therefore, for some 0<σ′<σ small enough,

Next, for the second term in (3.1), we have the note that ∂∂zj ∂¯ζΩ0(C˜(ζ,z)) is dominated by 1|Φ(ζ,z)−ρ(ζ)|4. Thus, for all z∈Ω, using the Henkin coordinates and the cutoff function h again, we have

Therefore, we conclude that for all u∈L∞(Ω), G[u]∈BlHol(Ω). So P is bounded from L∞(Ω) to BlHol(Ω).

3.3. Proof of the assertion (3)

Let u∈L1(Ω,P(⋅,⋅)dV). Firstly we have

Secondly the (B)-property implies

for all ζ∈Ω. Hence, combining these facts yields that

Therefore the proof of the assertion (3) is complete.