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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A weak-type inequality
of subharmonic functions

Author: Changsun Choi
Journal: Proc. Amer. Math. Soc. 126 (1998), 1149-1153
MSC (1991): Primary 31B05
MathSciNet review: 1425115
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the weak-type inequality $\lambda \mu(u+|v|\ge \lambda )\le(\alpha +2)\break \int _{\partial D}u\,d\mu$, $\lambda >0$, between a non-negative subharmonic function $u$ and an $\mathbb H$-valued smooth function $v$, defined on an open set containing the closure of a bounded domain $D$ in a Euclidean space $\mathbb R^n$, satisfying $|v(0)|\le u(0)$, $|\nabla v|\le|\nabla u|$ and $|\Delta v|\le \alpha \Delta u$, where $\alpha \ge 0$ is a constant. Here $\mu$ is the harmonic measure on $\partial D$ with respect to 0. This inequality extends Burkholder's inequality in which $\alpha =1$ and $\mathbb H=\mathbb R^\nu$, a Euclidean space.

References [Enhancements On Off] (What's this?)

  • 1. Donald L. Burkholder, Strong differential subordination and stochastic integration, Ann. Probab. 22 (1994), no. 2, 995–1025. MR 1288140
  • 2. W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. London Mathematical Society Monographs, No. 9. MR 0460672
  • 3. S. Lang, Analysis I, Addison-Wesley, Reading, Mass. (1968).

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Additional Information

Changsun Choi
Affiliation: Department of Mathematics, KAIST, Taejon 305-701, Korea

Keywords: Subharmonic function, smooth function, harmonic measure, weak-type inequality
Received by editor(s): May 9, 1996
Received by editor(s) in revised form: October 1, 1996
Additional Notes: This work was partially supported by GARC-KOSEF
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society