Kyungpook Mathematical Journal 2023; 63(3): 413-424
Published online September 30, 2023
Copyright © Kyungpook Mathematical Journal.
-boundedness ( ) for Bergman Projection on a Class of Convex Domains of Infinite Type in
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 : email@example.com
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
Keywords: Bergman projection, Bloch functions, Besov functions, finite/infinite type
Definition 1.1. ([13, p. 478]) A differentiable function
The space of all Bloch functions defined on
Definition 1.2. A function
The main result in this paper is following.
Main Theorem. Let
to for all .
Phong and Stein in  established the
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
The convexity of
Lemma 2.1. For any
is holomorphic in ;
, and ;
3. There exists a constant
such that for all and ;
for all zwith and .
Now we set
which is a
which is a Cauchy-Fantappié
Definition 2.2. Let
for some which is small enough,
is non-decreasing function.
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
Definition 2.4. ([13, Definition 2.1]) We say that a smoothly bounded domain
be convex domain in
By the smoothness of each component in
Definition 2.5. For
we have the reproductive property of
and its dual
Theorem 2.6. ([6, Theorem 3.4][Ligocka's decomposition]) Let
3. Proof of the Main Theorem
3.1. Proof of the assertion (1)
This fact has been proved in . For convenience, we briefly sketch its proof here. The
Lemma 3.1. The operators
Due to the strong duality and the Marcinkiewicz Interpolation Theorem from harmonic analysis (see Theorem B.7, Appendix B in  for more details), it is sufficient to show that
Firstly, we recall the change of variables
and by Lemma 2.3, we have
Therefore, we obtain the
3.2. Proof of the assertion (2)
Since the continuity of
We consider the first term in (3.1). Since the integral
To estimate the last integral in the above inequality, we use the following Henkin coordinates on
Lemma 3.2 (Henkin's coordinates). There exist positive constants
Therefore, for some
Next, for the second term in (3.1), we have the note that
Therefore, we conclude that for all
3.3. Proof of the assertion (3)
Therefore the proof of the assertion (3) is complete.
- D. Bekollé, Inégalités
Lppour les projecteurs de Bergman de certains domaines de , C. R. Acad. Sci. Paris Sér. I Math., 294(12)(1982), 395-397.
- P. Charpentier and Y. Dupain,
Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form, Publ. Mat., 50(2)(2006), 413-446.
- S. C. Chen and M. C. Shaw,
Partial Differential Equations in Several Complex Variables, AMS/IP, Studies in Advanced Mathematics, AMS, (2001).
- Z. Cuckovic and R. Zhao,
Essential norm estimates of weighted composition operators between Bergman spaces on strongly pseudoconvex domains, Math. Proc. Camb. Phil. Soc., 145(2007), 525-533.
- L. K. Ha, Tangential Cauchy-Riemann equations on pseudoconvex boundaries of finite and infinite type in
, Results in Math., 72(2017), 105-124.
- L. K. Ha, On the global Lipschitz continuity of the Bergman projection on a class of convex domains of infinite type in
, Collo. Math., 150(2017), 187-205.
- L. K. Ha,
-estimates for -equation on certain convex domains of infinite type in , J. Geom. Anal., 31(2021), 2058-2087.
- L. K. Ha and L. H. Khoi, Composition Operators Between Hardy Spaces on Linearly Convex Domains in
, Complex Anal. Oper. Theory, 13(2019), 2589-2603.
- L. K. Ha and L. H. Khoi, On boundedness and compactness of composition operators between Bergman spaces on infinite type convex domains in
, (2023), DOI: 10.1080/17476933.2023.2221642.
- F. Haslinger,
The, Czechoslovak Math. J., -Neumann operator and commutators of the Bergman projection and multiplication operators 58(133)(2008), 1247-1256.
- L. H¨ormander,
L2 estimates and existence theorems for the -operator, Acta Math., 113(1965), 89-152.
- N. Kerzman and E. Stein,
The Szegö kernel in terms of Cauchy-Fantappié kernels, Duke Math. J., 45(2)(1978), DOI: 10.1215/S0012-7094-78-04513-1, 197-224 pp.
- S. Y. Li and W. Luo,
On characterization of Besov space and application, Part I, J. of Math. Analysis and Applications, 310(2005), 477-491.
- E. Ligocka,
The Bergman projection on harmonic functions, Studia Math., 85(1987), 229-246.
- J. D McNeal and E. M. Stein,
Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J., 73(1)(1994), 177-199.
- D. H. Phong and E. M. Stein,
Estimates for the Bergman and Szeg, Duke Math. J., rojections on strongly pseudo-convex domains 44(3)(1977), 695-704.
- M. Range,
The Carathéodory metric and holomorphic maps on a class of weakly pseudoconvex domains, Pacific J. Math., 78(1)(1978), 173-189.
- M. Range,
On the Hölder estimates for, Proc. Inter. Conf. Cortona, Italy 1976-1977. Scoula. Norm. Sup. Pisa, (1978), 247-267. on weakly pseudoconvex domains
- M. Range, Holomorphic Functions and Integral Representation in Several Complex Variables, (1986).
- W. Rudin. Function theory in the unit ball of
, Springer, Berlin(1980).