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

• Autorzy: Zhangjian Hu
• 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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1996
• Tom: 69
• Numer: 2
• Strony: 213-238

Daty

wydano

1996

otrzymano

1994-10-26

poprawiono

1995-01-19

Twórcy

autor

Zhangjian Hu

• Department of Mathematics, Huzhou Teachers' College, Huzhou, Zhejiang, 313000 P.R. China

Bibliografia

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