Let U be a bounded domain in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb C^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">C</mi> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p_a(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>a</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le p < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, denote the space of functions that are analytic on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{U}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>U</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> and bounded in the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> norm on U. A point <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$x \in \overline{U}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mover> <mi>U</mi> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> is said to be a bounded point evaluation for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p_a(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>a</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if the linear functional <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$f \rightarrow f(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">→</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is bounded in <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p_a(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>a</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we provide a purely geometric condition given in terms of the Sobolev q-capacity for a point to be a bounded point evaluation for <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1132_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p_a(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>a</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This extends results known only for the single variable case to several complex variables.

Geometric conditions for bounded point evaluations in spaces of several complex variables

期刊论文
发表日期：

2025年9月29日

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Stephen Deterding

摘要

Let U be a bounded domain in $\mathbb C^d$ $C^{d}$ and let $L^p_a(U)$ $L_{a}^{p} (U)$ , $1 \le p < \infty $ $1 \leq p < \infty$ , denote the space of functions that are analytic on $\overline{U}$ $\bar{U}$ and bounded in the $L^p$ $L^{p}$ norm on U. A point $x \in \overline{U}$ $x \in \bar{U}$ is said to be a bounded point evaluation for $L^p_a(U)$ $L_{a}^{p} (U)$ if the linear functional $f \rightarrow f(x)$ $f \to f (x)$ is bounded in $L^p_a(U)$ $L_{a}^{p} (U)$ . In this paper, we provide a purely geometric condition given in terms of the Sobolev q-capacity for a point to be a bounded point evaluation for $L^p_a(U)$ $L_{a}^{p} (U)$ . This extends results known only for the single variable case to several complex variables.