<p>Consider weighted Bergman spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A^p_\alpha (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>A</mi> <mi>α</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded strongly pseudo-convex domain with smooth boundary in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. Our first main result demonstrates that if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\delta ^{(n+1+\alpha )}\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> be a Carleson measure for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A^p_\alpha (\Omega )(1<p<\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>A</mi> <mi>α</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the associated balayage function <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$B_\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>μ</mi> </msub> </math></EquationSource> </InlineEquation> possesses the property <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$1/(\delta ^{n+1+\alpha }B\mu ) \in L^\infty (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>δ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> </mrow> </msup> <mi>B</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if and only if the averaged measure <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$1/(\hat{\mu }_r\delta ^{n+1}) \in L^\infty (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> <mi>r</mi> </msub> <msup> <mi>δ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\widehat{\mu }_{r}\delta ^{n+1+\alpha }\,dA$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>μ</mi> <mo stretchy="false">^</mo> </mover> <mi>r</mi> </msub> <msup> <mi>δ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> </mrow> </msup> <mspace width="0.166667em" /> <mi>d</mi> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> constitutes a reverse Carleson measure. Furthermore, we provide a analysis of the intrinsic relationship between the asymptotic behavior of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$B_\mu (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the Carleson properties of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>. As a consequence, we generalize significantly Theorem 5 in Green-Wagner(Constructive Approximation, 2024) to a fully version.</p>

Balayage and Reverse Carleson Measures in Bergman Spaces on Strongly Pseudo-Convex Domains

期刊论文
发表日期：

2026年3月10日

查看全文

Zhengyuan Zhuo,
Senhua Zhu

摘要

Consider weighted Bergman spaces $A^p_\alpha (\Omega )$ $A_{α}^{p} (Ω)$ , where $\Omega $ $Ω$ is a bounded strongly pseudo-convex domain with smooth boundary in $\mathbb {C}^n$ $C^{n}$ . Our first main result demonstrates that if $\delta ^{(n+1+\alpha )}\mu $ $δ^{(n + 1 + α)} μ$ be a Carleson measure for $A^p_\alpha (\Omega )(1<p<\infty )$ $A_{α}^{p} (Ω) (1 < p < \infty)$ , the associated balayage function $B_\mu $ $B_{μ}$ possesses the property $1/(\delta ^{n+1+\alpha }B\mu ) \in L^\infty (\Omega )$ $1 / (δ^{n + 1 + α} B μ) \in L^{\infty} (Ω)$ if and only if the averaged measure $1/(\hat{\mu }_r\delta ^{n+1}) \in L^\infty (\Omega )$ $1 / ({\hat{μ}}_{r} δ^{n + 1}) \in L^{\infty} (Ω)$ if and only if $\widehat{\mu }_{r}\delta ^{n+1+\alpha }\,dA$ ${\hat{μ}}_{r} δ^{n + 1 + α} d A$ constitutes a reverse Carleson measure. Furthermore, we provide a analysis of the intrinsic relationship between the asymptotic behavior of $B_\mu (z)$ $B_{μ} (z)$ and the Carleson properties of $\mu $ $μ$ . As a consequence, we generalize significantly Theorem 5 in Green-Wagner(Constructive Approximation, 2024) to a fully version.