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A weak-type inequality of subharmonic functions

Authors:	Changsun Choi

Institution:	Department of Mathematics, KAIST, Taejon 305-701, Korea

Abstract:	We prove the weak-type inequality $\lambda \mu(u+\|v\|\ge \lambda )\le(\alpha +2) \int _{\partial D}u\,d\mu$ , $\lambda >0$ , between a non-negative subharmonic function and an $\mathbb H$ -valued smooth function , defined on an open set containing the closure of a bounded domain 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.

Keywords:	Subharmonic function smooth function harmonic measure weak-type inequality

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