Let E be a closed subset of the extended real line containing <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$[-\infty ,-1]\cup [1,+\infty ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mi>∞</mi> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∪</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>: we say that E is Carleson-homogeneous if there exists <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> such that for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x\in E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> we have <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$|E\cap [x-t,x+t]|\ge Ct$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>E</mi> <mo>∩</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>-</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo stretchy="false">|</mo> <mo>≥</mo> <mi>C</mi> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation>, where |A| stands for the Lebesgue measure of A. If E is such a set, we say that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega =\bar{\mathbb {C}}\backslash E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mover accent="true"> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="true">\</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> is a Carleson-homogeneous Denjoy domain. Such a domain is in particular a hyperbolic Riemann surface, meaning that there is a conformal bijection <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Phi : \mathbb {D}\rightarrow \Omega ^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Ω</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, the universal cover of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. Let now <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f=\Pi \circ \Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">Π</mi> <mo>∘</mo> <mi mathvariant="normal">Φ</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Π</mi> </math></EquationSource> </InlineEquation> is the natural projection from <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Omega ^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Ω</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> onto <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. It is a holomorphic function defined on <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$f'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> does not vanish. The main result of this paper is that <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\log f'\in BMOA(\mathbb {D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>B</mi> <mi>M</mi> <mi>O</mi> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a Carleson homogeneous Denjoy domain.

Denjoy Domains and BMOA

期刊论文
发表日期：

2026年7月6日

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Shengjin Huo,
Michel Zinsmeister

摘要

Let E be a closed subset of the extended real line containing $[-\infty ,-1]\cup [1,+\infty ]$ $[- \infty, - 1] \cup [1, + \infty]$ : we say that E is Carleson-homogeneous if there exists $C>0$ $C > 0$ such that for every $x\in E$ $x \in E$ and $t>0$ $t > 0$ we have $|E\cap [x-t,x+t]|\ge Ct$ $| E \cap [x - t, x + t] | \geq C t$ , where |A| stands for the Lebesgue measure of A. If E is such a set, we say that $\Omega =\bar{\mathbb {C}}\backslash E$ $Ω = \bar{C} \ E$ is a Carleson-homogeneous Denjoy domain. Such a domain is in particular a hyperbolic Riemann surface, meaning that there is a conformal bijection $\Phi : \mathbb {D}\rightarrow \Omega ^*$ $Φ : D \to Ω^{*}$ , the universal cover of $\Omega $ $Ω$ . Let now $f=\Pi \circ \Phi $ $f = Π \circ Φ$ , where $\Pi $ $Π$ is the natural projection from $\Omega ^*$ $Ω^{*}$ onto $\Omega $ $Ω$ . It is a holomorphic function defined on $\mathbb {D}$ $D$ such that $f'$ $f^{'}$ does not vanish. The main result of this paper is that $\log f'\in BMOA(\mathbb {D})$ $log f^{'} \in B M O A (D)$ if $\Omega $ $Ω$ is a Carleson homogeneous Denjoy domain.