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Carleson measures on Planar Sets

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Acta Mathematica Sinica,English Series 2009年第11期25卷 1881-1892页

作者： Zhijian QIU School of Mathematics,Southwestern University of Finance and Economics,Chengdu 610074,P. R. China School of Mathematics Southwestern University of Finance and Economics Chengdu 610074 P. R. China

In this paper, we investigate what are Carleson measures on open subsets in the complex plane. A circular domain is a connected open subset whose boundary consists of finitely many disjoint circles. We call a domain G multi-nicely connected if there exists a circular domain W and a conformal map ψ from W onto G such that ψ is almost univalent with respect the arclength on δW. We characterize all Carleson measures for those open subsets so that each of their components is multinicely connected and harmonic measures of the components are mutually singular. Our results suggest the extension of Carleson measures probably is up to this class of open subsets

关键词： Carleson measure circular domain harmonic measure multi-nicely connected domain

On Pointwise Bounded Approximation

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Acta Mathematica Sinica,English Series 2009年第7期25卷 1217-1222页

作者： Zhijian QIU School of Mathematics Southwestern University of Finance and Economics Chengdu 610074 P. R. China

Let G be an open subset in the extended complex plane and let A（G） denote the algebra of all functions analytic on G and continuous on G. We call a domain multi-nicely connected if there is a circular domain W and a conformal map ~ from W onto G such that the boundary value function of φ is univalent almost everywhere with respect to the arclength on aW. Suppose that every component of G is finitely connected and none of the components of G have single point boundary components. We show that for every bounded analytic function on G to be the pointwise limit of a bounded sequence of functions in A（G）, it is necessary and sufficient that each component of G is multi-nicely connected and the harmonic measures of G are mutually singular. This generalizes the corresponding result of Davie for the case when the components of G are simply connected.

关键词： pointwise approximation harmonic measure Dirichlet algebra

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Science China Mathematics 2007年第3期50卷 305-312页

作者： Zhi-jian QIU Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China 1. Department of Economic Mathematics Southwestern University of Finance and Economics Chengdu 610074 China

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.

关键词： subnormal operator quasi-similarity harmonic measure Hardy space 47B20 30H05 30E10 46E15