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Kyungpook Mathematical Journal 2023; 63(3): 413-424

Published online September 30, 2023 https://doi.org/10.5666/KMJ.2023.63.3.413

Copyright © Kyungpook Mathematical Journal.

Lp-boundedness (1p) for Bergman Projection on a Class of Convex Domains of Infinite Type in 2

Ly Kim Ha

University of Science, Ho Chi Minh City, Vietnam
Vietnam National University, Ho Chi Minh City, Vietnam
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City, Vietnam
e-mail : lkha@hcmus.edu.vn

Received: December 14, 2020; Revised: June 11, 2021; Accepted: June 14, 2021

The main purpose of this paper is to show that over a large class of bounded domains Ω2, for 1<p<, the Bergman projection P is bounded from Lp(Ω,dV) to the Bergman space Ap(Ω); from L(Ω) to the holomorphic Bloch space BlHol(Ω); and from L1(Ω,P(z,z)dV) to the holomorphic Besov space Besov(Ω), where P(ζ,z) is the Bergman kernel for Ω.

Keywords: Bergman projection, Bloch functions, Besov functions, finite/infinite type

Let Ω be a bounded domain in 2 with smooth boundary bΩ. Let ρ be a defining function for Ω so that Ω={z2:ρ(z)<0} and bΩ={z2:ρ(z)=0}, ρ0 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 4, 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 such that

P[u](z)=Ωu(ζ)P(ζ,z)dV(ζ),

for all uL2(Ω), zΩ. Here, P(ζ,z) is the Bergman kernel on Ω, which is holomorphic with respect to zΩ, 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

uBl=supzΩ|ρ(z)||u(z)|+|ρ(z)||u(z)|<.

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 uBlHol(Ω)=uBl(Ω) for all uBlHol(Ω).

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

Definition 1.2. A function uA2(Ω,dV) is said to be a Besov function if and only if

uBesov(Ω)=Ω|3u(z)|(ρ(z))3P(z,z)dV(z)<,

where |3u(z)|= 1j+k3 j+ku z1jz2k(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

uB=Ω|u(z)|(ρ(z))P(z,z)dV(z)<.

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 2 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 LpAp boundedness when Ω is a strongly pseudoconvex domain. Then, this result was generalized to a certain class of convex domains in 2 (see [1]) and to finite type convex domains in n (see [15]). Even when Ω is the unit ball in n, for n2, 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 n, 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 2.

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 Ω2 be a bounded convex domain with smooth boundary bΩ with a defining function ρ. By the hypothesis that Ω is convex,

i,j=142ρxixj(x)aiaj0,

in which xbΩ, zj=x2j1+1x2j and a4 be a non-zero vector such that j=14ajρxj(x)=0 on bΩ. Let us define, for (ζ,z)bΩ×Ω:

Φ(ζ,z)= j=12ρζj(ζ)(ζjzj).

The convexity of Ω gives

j=12ρζj(ζ)(ζjzj)0,

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

Lemma 2.1. For any PbΩ, there are positive constants δ,c such that for all boundary points ζbΩB(P,δ), we have

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

  • 2. Φ(ζ,ζ)=0, and dzΦ|z=ζ0;

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

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

Now we set

C(ζ,z)=12πi j=12ρζj (ζ)dζj1Φ(ζ,z)for(ζ,z)bΩ×Ω

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

Ω0C(ζ,z)=C(ζ,z)(¯ζC(ζ,z))
= j0{1,2} A j 0 (ζ)Φ2(ζ,z)dζ1dζ2d ζ¯j0,

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

C˜(ζ,z)=12πi j=12ρζj (ζ)dζj1Φ(ζ,z)ρ(ζ).

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 Ω2 be a smooth bounded, convex domain. We say that Ω admitting F-type at a point PbΩ if there are positive constants c, c' satisfy that for all ζbΩB(P,c):

ρ(z)F(|zζ|2),

for all zB(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 2 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 2 admitting an F-type at PbΩ. Then there is a positive constant c such that the support function Φ(ζ,z) satisfies the following estimate

|Φ(ζ,z)ρ(ζ)||ρ(ζ)|+|ρ(z)|+|Φ(ζ,z)|+F(|zζ|2),

for every ζΩ¯B(P,c), and zΩ, |zζ|<c.

Definition 2.4. ([13, Definition 2.1]) We say that a smoothly bounded domain Ω2 has B-property if there is a positive constant CΩ such that the following holds:

(ρ(ζ))3Ω|z3P(ζ,z)|dV(z)+1P(ζ,ζ)Ω|z3P(ζ,z)|(ρ(z))3P(z,z)dV(z)CΩ

for all ζΩ.

In 2, 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 2 admits type F(t)=t at all boundary points. Secondly, let m1,m2 be positive integers, and let

Ωm={(z1,z2)2:|z1|2m1+|z2|2m21<0}

be convex domain in 2. The family {Ωm} is the certain class of weakly convex domains in 2. Then, in [5], the author shows that Ωm admits type F(t)=tm 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 uC1(Ω¯)O(Ω) and u is holomorphic on Ω, by the Stoke Theorem, we get

u(z)=Ωu(ζ)¯ζΩ0(C˜(ζ,z)),   zΩ.

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),zbΩ}.

For 0<c<δ (c is the constant in Lemma 2.3), let us define Ωδ={z2:ρ(z)<δ} and let Pz be the Hörmander solution operator to the ¯-equation in the variables zΩδ (the existence of Pz can be found in [11]).

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

Q(ζ,z)=Pz(¯z¯ζΩ0((C˜(ζ,z))),G(ζ,z)=Q(ζ,z)+¯ζΩ0((C˜(ζ,z)),

where G(ζ,z) is holomorphic in z.

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

G(ζ,z)=1π21(Φ(ζ,z)ρ(ζ)3O(|ζz|)+det ρ(ζ) ρζ1 (ζ) ρζ2 (ζ) ρ ζ¯1 (ζ) 2 ρζ1 ζ¯1 (ζ) 2 ρζ2 ζ¯1 (ζ) ρ ζ¯2 (ζ) 2 ρζ1 ζ¯2 (ζ) 2 ρζ2 ζ¯2 (ζ)dζ1d ζ¯1dζ2d ζ¯2+non-singular terms.

Let u be a holomorphic function defined on Ωδ, since

Ωu(ζ)Pz(¯z¯ζΩ0((C˜(ζ,z)))=Ω P z (u(ζ)¯z¯ζΩ0((C˜(ζ,z)))            =Pz(Ωu(ζ)¯z¯ζΩ0((C˜(ζ,z)))            =Pz(Ωu(ζ)¯ζ¯zΩ0((C˜(ζ,z)))            =Pz(bΩu(ζ)¯zΩ0((C˜(ζ,z)))            =0(see[12, 1.4.2]),

we have the reproductive property of G(ζ,z) that u(z)=Ωu(ζ)G(ζ,z) for all zΩ. More generally, let uL2(Ω), and let us define

G[u](z)=Ωu(ζ)G(ζ,z)

and its dual

G*[u](z)=Ωu(ζ)G(ζ,z)¯.

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 Ω2 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(IB)1[u](z)=(I+B)1G*[u](z), where

B[u](z)=G*[u](z)G[u](z).

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

G[u]Lp(Ω,dV)uLp(Ω,dV)for all uLp(Ω,dV),1p,

and

G*[u]Lp(Ω,dV)uLp(Ω,dV)for alluLp(Ω,dV),1<p<.

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

G[u]L1(Ω,dV)uL1(Ω,dV)andG[u]L(Ω,dV)uL(Ω,dV).

Firstly, we recall the change of variables (α,w)=(α1,α2,w1,w2)=(ζ1,ζ2,z1ζ1,ρ(ζ)+iIm(Φ(ζ,z))) and let J be the Jacobian of this change. Then

det(J)=Im(Φ(ζ,z))x4ρ(z)x2Im(Φ(ζ,z))x2ρ(z)x4.

Since ρ(z)0, we can find a sufficiently small 0<δ<c so that ρx4 dominates others partial derivatives of ρ and |zζ|δ. As a consequence, we have det(J)0 on |ζz|δ.

Now let δ>0 depend on Ω,c,δ and ρ, and uL1(Ω,dV). Since {(z,ζ):|ζz|<c}, the kernel G(ζ,z) is bounded from above by

|ζz||Φ(ζ,z)ρ(ζ)|31(|ρ(z)|+|ρ(ζ)|+|Φ(ζ,z)|+F(|ζz|2))3.

and by Lemma 2.3, we have

(ζ,z)(ΩB(0,c/2))2|G(ζ,z)u(ζ)|dV(ζ,z)       (α,w)(ΩB(0,δ))×B(0,δ)|u(α)|(|w2|2+F2(|w1|2))|w1|dV(α,w)      uL1(Ω,dV)0 δ 0 δ r 1 r 2 (r 2 2 + F 2 (r 1 2 ))r 1 dr2dr1      uL1(Ω,dV)0 δ lnF(r12)dr1uL1(Ω,dV)      (by the property of F).

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

Next, let uL(Ω). The Hölder's Inequality and Lemma 2.3 imply

ΩB(0,c/2)|G(ζ,z)u(ζ)|dV(ζ)uL(Ω)ΩB(0,c/2) |ζz| |Φ(ζ,z)ρ(ζ)|3 dV(ζ)uL(Ω)ΩB(0,δ) dV(w1 ,w2 ) (|ρ(z)|+|w2|+F(|w1 |2))2 |w1 |uL(Ω)|(t1,t2,t3,t4)|δ d t1 d t2 d t3 d t 4 (|ρ(z)|+| t3|+| t 4|+F( t12+ t22))2 |( t1 , t2 )|(where w1=t1+-1t2,w2=t3+-1t4)uL(Ω)|(t1,t2,t3)|δ d t1 d t2 d t3 (|ρ(z)|+| t3 |+F( t12+ t22))|( t1 , t2 )|uL(Ω)0δ|lnF(r2)|druL(Ω)(by the property of F).

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[IB]={0}, IB is a Fredholm isomorphism of L(Ω). Thus, it is sufficient to prove that G maps continuously L(Ω) into BlHol(Ω).

Let uL(Ω), we must show that

|ρ(z)||Gu(z)|+|ρ(z)||zGu(z)|u

for all zΩ.

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

|ρ(z)||Gu(z)|u1+|ρ(z)|Ω ¯ζ Ω0 ((C˜(ζ,z))dV(ζ).

For 0<c<σ (c is the constant in Lemma 2.3), let hC(2) be a cutoff function such that h=1 on {ζ2:|ρ(ζ)|+|ρ(z)|+|(Φ(ζ,z))|+F(|ζz|2)<σ/2} and h=0 on {ζ2:|ρ(ζ)|+|ρ(z)|+|(Φ(ζ,z))|+F(|ζz|2)>σ}. Then,

Ω¯ζΩ0((C˜(ζ,z))dV(ζ)1+ |ρ(ζ)|+|ρ(z)|+|(Φ(ζ,z))|+F(|ζz|2)<σ¯ζΩ0((C˜(ζ,z))dV(ζ) |ρ(ζ)|+|ρ(z)|+|(Φ(ζ,z))|+F(|ζz|2)<σ¯ζΩ0((C˜(ζ,z))dV(ζ).

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

|ρ(z)||ρ(ζ)|+|ρ(z)|+|(Φ(ζ,z))|+F(|ζz|2)<σ ¯ζ Ω0 ((C˜(ζ,z))dV(ζ)  |ρ(ζ)|+|ρ(z)|+|(Φ(ζ,z))|+F(|ζz|2)<σ |ζz| |Φ(ζ,z)ρ(ζ) | 2 dV(ζ).

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 ηc, 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

t(z,z)=0,t1(ζ)=ρ(ζ)ρ(z),t2(ζ)=(Φ(ζ,z)),|t|<δfor ζB(z,c),|J(t)|Mand|detJ(t)|1M,

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

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

|ρ(ζ)|+|ρ(z)|+|(Φ(ζ,z))|+F(|ζz|2)<σ 1 |Φ(ζ,z)ρ(ζ) | 2 dV(ζ)|(t 1,,t4)|σ 1 (| t1|+| t 2|+F(|( t3, t 4) | 2 )) 2 dt1dt4(r 1,r2)(0,σ)2 r1 r 2 r1 2+F 2 ( r 2 2)dr1dr2(using the polar coordinates r1=|(t1,t2)| and r2=|(t3,t4)|)0σ|lnF(r2)|dr<.

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 zΩ, using the Henkin coordinates and the cutoff function h again, we have

|ρ(z)|zGu(z)u1+Ωh(ζ)dV(ζ)|Φ(ζ,z)ρ(ζ)|3       u1+|(t1,,t4)|σ d t 1d t 4 (| t 1|+| t2|+F(|( t 3, t 4) |2 ))2 |( t 1 ,, t 4)|       u1+0σ |lnF(r2)|dr<.

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

3.3. Proof of the assertion (3)

Let uL1(Ω,P(,)dV). Firstly we have

P[u]Besov(Ω)=Ω|z3P[u](z)|(ρ(z))3P(z,z)dV(z)    =Ω z 3 ΩP (ζ,z)u(ζ)dV(ζ)(ρ(z))3P(z,z)dV(z)    =Ω Ω z3 P(ζ,z)u(ζ)dV(ζ)(ρ(z))3P(z,z)dV(z)    Ω Ω | z3P(ζ,z)u(ζ)|dV(ζ)(ρ(z))3P(z,z)dV(z)    Ω Ω | z3P(ζ,z)|(ρ(z))3P(z,z)dV(z)|u(ζ)|dV(ζ).

Secondly the (B)-property implies

Ω|z3P(ζ,z)|(ρ(z))3P(z,z)dV(z)CΩP(ζ,ζ)

for all ζΩ. Hence, combining these facts yields that

P[u]Besov(Ω)CΩΩ|u(ζ)||P(ζ,ζ)|dV(ζ)=CΩuL1(Ω,P(,)dV).

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

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