The investigation of the dimension of Bergman spaces has long been a central topic in several complex variables, uncovering profound connections with potential theory and function theory since the pioneering work of Carleson, Wiegerinck, and others in the 1960s. We investigate the dimension of p-Bergman spaces associated with pseudoconvex domains in <InlineEquation ID="IEq1"> <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>, particularly focusing on the case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p\in [1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. By constructing <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-versions of the extension theorems of Ohsawa and Ohsawa–Takegoshi, we establish several geometric and potential-theoretic criteria that ensure the spaces are infinite-dimensional. The range <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\in [1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is imposed because for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, classical tools like Ohsawa–Takegoshi and Skoda become unavailable or lack suitable analogs. Sufficient conditions for the infinite dimensionality of p-Bergman spaces of complete N-circled fibered Hartogs domains, weighted p-Fock spaces are obtained by applying the mentioned <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-analogs of extension theorems and generalizing a sufficient condition of Jucha. We also obtain sufficient conditions for triviality of p-Bergman spaces of balanced and complete N-circled fibered Hartogs domains with one-dimensional base.

On the Dimension of the p-Bergman Spaces

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发表日期：

2026年7月27日

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Shreedhar Bhat,
Achinta Kumar Nandi

摘要

The investigation of the dimension of Bergman spaces has long been a central topic in several complex variables, uncovering profound connections with potential theory and function theory since the pioneering work of Carleson, Wiegerinck, and others in the 1960s. We investigate the dimension of p-Bergman spaces associated with pseudoconvex domains in $\mathbb {C}^n$ $C^{n}$ , particularly focusing on the case $p\in [1,2)$ $p \in [1, 2)$ . By constructing $L^p$ $L^{p}$ -versions of the extension theorems of Ohsawa and Ohsawa–Takegoshi, we establish several geometric and potential-theoretic criteria that ensure the spaces are infinite-dimensional. The range $p\in [1,2)$ $p \in [1, 2)$ is imposed because for $p>2$ $p > 2$ , classical tools like Ohsawa–Takegoshi and Skoda become unavailable or lack suitable analogs. Sufficient conditions for the infinite dimensionality of p-Bergman spaces of complete N-circled fibered Hartogs domains, weighted p-Fock spaces are obtained by applying the mentioned $L^p$ $L^{p}$ -analogs of extension theorems and generalizing a sufficient condition of Jucha. We also obtain sufficient conditions for triviality of p-Bergman spaces of balanced and complete N-circled fibered Hartogs domains with one-dimensional base.