<p>In this article, we investigate the connection between certain real variable methods and the Bergman theory. We first use Hardy-type inequalities to give an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> Hartogs-type extension theorem and an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> integrability theorem for the Bergman kernel <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K_\Omega (\cdot ,w)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mi mathvariant="normal">Ω</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We then use the Sobolev-Morrey inequality to show the absolute continuity of Bergman kernels on planar domains with respect to logarithmic capacities. Finally, we give lower bounds of the minimum <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\kappa (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the Bergman kernel <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$K_\Omega (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mi mathvariant="normal">Ω</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of the interior capacity radius for planar domains and the volume density for bounded pseudoconvex domains in <InlineEquation ID="IEq6"> <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>. As a consequence, we show that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\kappa (\Omega )\ge c_0 \lambda _1(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> holds on planar domains, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$c_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is a numerical constant and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda _1(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the first Dirichlet eigenvalue of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$-\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Real Variable Methods in Bergman Theory

期刊论文
发表日期：

2026年1月30日

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Bo-Yong Chen,
Yuanpu Xiong

摘要

In this article, we investigate the connection between certain real variable methods and the Bergman theory. We first use Hardy-type inequalities to give an $L^2$ $L^{2}$ Hartogs-type extension theorem and an $L^p$ $L^{p}$ integrability theorem for the Bergman kernel $K_\Omega (\cdot ,w)$ $K_{Ω} (\cdot, w)$ . We then use the Sobolev-Morrey inequality to show the absolute continuity of Bergman kernels on planar domains with respect to logarithmic capacities. Finally, we give lower bounds of the minimum $\kappa (\Omega )$ $κ (Ω)$ of the Bergman kernel $K_\Omega (z)$ $K_{Ω} (z)$ in terms of the interior capacity radius for planar domains and the volume density for bounded pseudoconvex domains in $\mathbb {C}^n$ $C^{n}$ . As a consequence, we show that $\kappa (\Omega )\ge c_0 \lambda _1(\Omega )$ $κ (Ω) \geq c_{0} λ_{1} (Ω)$ holds on planar domains, where $c_0$ $c_{0}$ is a numerical constant and $\lambda _1(\Omega )$ $λ_{1} (Ω)$ is the first Dirichlet eigenvalue of $-\Delta $ $- Δ$ .