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## Colloquium Mathematicum

1996 | 69 | 2 | 213-238
Tytuł artykułu

### Estimates for the integral means of holomorphic functions on bounded domains in $ℂ^{n}$

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let 𝓓 = {z ∈ ℂ^{n} : λ(z) < 0} be a bounded domain with $C^{∞}$ boundary. For f holomorphic in 𝓓, let $M_{p}(f,r)$ be the pth integral mean of f on $∂𝓓_{r}= {z ∈ 𝓓 : λ(z)=-r}$. In this paper we prove that $∫_{0}^{ε} r^{s+|α|q} M_{p}^{q}(D^{α}f,r)dr ≤ C ∫_{0}^{ε} r^{s}M_{p}^{q}(f,r)dr$ and $∫_{0}^{ε} r^{s} M_{p}^{q}(f,r)dr ≤ C { ∑_{|α|<m} |(D^{α}f)(z_{0})|^{q} + ∑_{|α|=m} ∫_{0}^{ε} r^{s+mq} M_{p}^{q}(D^{α}f,r)dr }$, where $z_{0} ∈ 𝓓$ is fixed, 0 < p ≤ ∞, 0 < q < ∞, s>-1, m ∈ ℕ, α =(α_{1},...,α_{n}) is a multi-index, and ε > 0 is small enough. These inequalities generalize the known results in [9,10] on the unit ball of $ℂ^{n}$. Two applications are given. The methods used in the proof of the inequalities also enable us to obtain some theorems about pluriharmonic functions on 𝓓.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
213-238
Opis fizyczny
Daty
wydano
1996
otrzymano
1994-10-26
poprawiono
1995-01-19
Twórcy
autor
• Department of Mathematics, Huzhou Teachers' College, Huzhou, Zhejiang, 313000 P.R. China
Bibliografia
• [1] P. L. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
• [2] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746-765.
• [3] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, II, Math. Z. 34 (1932), 403-439.
• [4] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.
• [5] Z. J. Hu, Mean value properties of pluriharmonic functions, Chinese J. Math. 13 (1993), 299-303.
• [6] S. G. Krantz, Function Theory of Several Complex Variables, Wiley, New York, 1982.
• [7] S. G. Krantz and D. W. Ma, Bloch functions on strongly pseudoconvex domains, Indiana Univ. Math. J. 37 (1988), 145-163.
• [8] J. H. Shi, On the rate of growth of the means $M_p$ of holomorphic and pluriharmonic functions on bounded symmetric domains of $ℂ^{n}$, J. Math. Anal. Appl. 126 (1987), 161-175.
• [9] J. H. Shi, Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $ℂ^{n}$, Trans. Amer. Math. Soc. 328 (1991), 619-637.
• [10] J. H. Shi, Some results on singular integrals and function spaces in several complex variables, in: Contemp. Math. 142, Amer. Math. Soc., 1993, 75-101.
• [11] E. M. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton Univ. Press, Princeton, N.J., 1972.
• [12] M. Stoll, On the rate of growth of the means $M_p$ of holomorphic and pluriharmonic functions on the ball, J. Math. Anal. Appl. 93 (1983), 109-127.
• [13] K. Stroethoff, Besov-type characterizations for the Bloch space, Bull. Austral. Math. Soc. 39 (1989), 405-420.
• [14] V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, M.I.T. Press, Cambridge, Mass., 1966.
• [15] K. H. Zhu, The Bergman spaces, the Bloch space and Gleason's problem, Trans. Amer. Math. Soc. 309 (1988), 253-265.
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Bibliografia
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