DOI QR코드

DOI QR Code

INEQUALITIES FOR THE INTEGRAL MEANS OF HOLOMORPHIC FUNCTIONS IN THE STRONGLY PSEUDOCONVEX DOMAIN

CHO, HONG-RAE;LEE, JIN-KEE

  • Published : 2005.04.01

Abstract

We obtain the following two inequalities on a strongly pseudoconvex domain $\Omega\;in\;\mathbb{C}^n\;:\;for\;f\;{\in}\;O(\Omega)$ $$\int_{0}^{{\delta}0}t^{a{\mid}a{\mid}+b}M_p^a(t, D^{a}f)dt\lesssim\int_{0}^{{\delta}0}t^{b}M_p^a(t,\;f)dt\;\int_{O}^{{\delta}O}t_{b}M_p^a(t,\;f)dt\lesssim\sum_{j=0}^{m}\int_{O}^{{\delta}O}t^{am+b}M_{p}^{a}\(t,\;\aleph^{i}f\)dt$$. In [9], Shi proved these results for the unit ball in $\mathbb{C}^n$. These are generalizations of some classical results of Hardy and Littlewood.

Keywords

strongly pseudo convex domain;integral means;Levi polynomial

References

  1. F. Beatrous, Estimates for Derivatives of Holomorphic Functions in Pseudoconvex Domains, Math. Z. 191 (1986), 91-116 https://doi.org/10.1007/BF01163612
  2. H. R. Cho, Estimates on the mean growth of Hp functions in convex domains of finite type, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2393-2398
  3. H. R. Cho and E. G. Kwon, Sobolev-type embedding theorems for harmonic and holomorphic Sobolev spaces, J. Korean Math. Soc. 40 (2003), no. 3, 435-445 https://doi.org/10.4134/JKMS.2003.40.3.435
  4. H. R. Cho and E. G. Kwon, Growth rate of the functions in Bergman type spaces, J. Math. Anal. Appl. 285 (2003), 275-281 https://doi.org/10.1016/S0022-247X(03)00416-5
  5. P. L. Duren, Theory of $H^p$ spaces, Academic Press, New York, 1970
  6. M. M. Peloso, Hankel operators on weighted Bergman spaces on strongly domains, Illinois J. Math. 38 (1994), no. 2, 223-249
  7. R. M. Range, Holomorphic functions and integral representations in several complex variables, Springer-Verlag, Berlin, 1986
  8. J. H. Shi, Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of Cn, Trans. Amer. Math. Soc. 328 (1991), no. 2, 619-637 https://doi.org/10.2307/2001797
  9. F. Beatrous, $L^p$ estimates for extensions of holomorphic functions, Michigan Math. J. 32 (1985), 361-380 https://doi.org/10.1307/mmj/1029003244

Cited by

  1. On Traces in Some Analytic Spaces in Bounded Strictly Pseudoconvex Domains vol.2015, 2015, https://doi.org/10.1155/2015/265245