arXiv subjects

Zhijian Qiu

Publications and source records attributed to Zhijian Qiu.

6 recordsLinked to original sources

arXiv2018

The Riemann Mapping Problem

In this article we investigate the century-old continuous extension problem of the Riemann map. Let $G$ be a simply connected domain. We call $λ$ in $\partial G$ a multiple point if there are simply connected subdomains $ U$ and $V$ such that $λ\in\partial U \cap\partial V$ and $ dist (\partial U\cap G , \partial V\cap G )>0$. We show that the Riemann map of $G$ has a continuous extension to $\overline G$ if and only if $\partial G$has no multiple points. All of the results in this paper, together with the Riemann mapping theorem, give a complete and desirable solution to the mapping problem that was originally raised by Riemann in 1851 and intensively investigated by many famous mathematicians throughout history.

math.CA↗

arXiv2018

Continuous extension of conformal maps

For a simply connected domain $G$, let $\partial_{a}G$ be the set of accessible points in $\partial G$ and let $\partial_{n} G=\partial G-\partial_{a}G$. A point $a\in\partial G$ is called semi-unreachable if there is a crosscut $J$ of $G$ and domains $U$ and $V$ such that $G-J=U\cup V$ and $a\in(\partial_{n} U\cup\partial_{n} V)-J$. We use $\partial_{sn}G$ to denote the set of semi-unreachable points. In this article we show that a univalent analytic function $ψ$ from the unit disk $D$ onto $G$ extends continuously to $\overline D$ if and only if $\partial_{sn}G=\emptyset$. As a consequence, we provide a very short and elementary proof for the Osgood conjecture: if $G$ is a Jordan domain, then $ψ^{-1}$, the Riemann map, extends to be a homeomorphism from $\overline G$ to $\overline D$.

math.CA↗

arXiv2013

Continuous boundary values of conformal maps

Let $G$ be a bounded simply connected domain in the complex plane. A point $a\in \partial G$ is said to be accessible from inside of $G$ if there is a Jordan arc $J$ such that $J\subset \bar G$ and $J\cap\partial G=\{a\}$. In this paper the author shows that a univalent analytic function $ψ$ from the unit disk $D$ onto $G$ extends continuously to $\bar D$ if and only if every $a\in\partial G$ is accessible. The main result covers a famous theorem proved by C. Caratheödory, which says that if $G$ is a Jordan domain, then $ψ$ extends to be a homeomorphism from $\bar D$ onto to $\bar G$.

math.AP↗

arXiv2013

Beurling's Theorem And Invariant Subspaces For The Shift On Hardy Spaces

Let $G$ be a bounded open subset in the complex plane and let $H^{2}(G)$ denote the Hardy space on $G$. We call a bounded simply connected domain $W$ perfectly connected if the boundary value function of the inverse of the Riemann map from $W$ onto the unit disk $D$ is almost 1-1 rwith respect to the Lebesgure on $\partial D$ and if the Riemann map belongs to the weak-star closure of the polynomials in $H^{\infty}(W)$. Our main theorem states: In order that for each $M\in Lat(M_{z})$, there exist $u\in H^{\infty}(G)$ such that $ M = \vee\{u H^{2}(G)\}$, it is necessary and sufficient that the following hold: 1) Each component of $G$ is a perfectly connected domain. 2) The harmonic measures of the components of $G$ are mutually singular. 3) % $P^{\infty}(ω) The set of polynomials is weak-star dense in $ H^{\infty}(G)$. \noindent Moreover, if $G$ satisfies these conditions, then every $M\in Lat(M_{z})$ is of the form $u H^{2}(G)$, where %$u\in H^{\infty}(G)$ and the restriction of $u$ to each of the components of $G$ is either an inner function or zero.

math.FA↗

arXiv2013

The Structure of the Closure of the Rational Functions in $L^{q}$($μ$)$

Let $K$ be a compact subset in the complex plane and let $A(K)$ be the uniform closure of the functions continuous on $K$ and analytic on $K^{\circ}$. Let $μ$ be a positive finite measure with its support contained in $K$. For $1 \leq q < \infty$, let $A^{q}(K,μ)$ denote the closure of $A(K)$ in $L^{q}(μ)$. The aim of this work is to study the structure of the space $A^{q}(K,μ)$. We seek a necessary and sufficient condition on $K$ so that a Thomson-type structure theorem for $A^{q}(K,μ)$ can be established. Our results essentially give perfect solutions to the major open problem in the research filed of theory of subnormal operators and aproximation by analytic functions in the mean .

math.FA↗

arXiv2013

Carleson measures on planar sets

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 $\partial W$. We characterize all Carleson measures for those open subsets so that each of their components is multi-nicely connected and harmonic measures of the components are mutually singular. Our results suggest the extend of Carleson measures probably is up to this class of open subsets.

math.CA↗