In this paper, we investigate the r-summing Carleson embeddings from generalized Fock spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$F_{\phi }^{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>F</mi> <mrow> <mi>ϕ</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_{\phi }^{p}(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi>ϕ</mi> </mrow> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\phi \in {\mathcal {C}}^{2}\left( {\mathbb {C}}^{n}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mn>2</mn> </msup> <mfenced close=")" open="("> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is real-valued and satisfies <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ m{\omega }_{0} \le d{d}^{c}\phi \le M{\omega }_{0} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <msub> <mi>ω</mi> <mn>0</mn> </msub> <mo>≤</mo> <mi>d</mi> <msup> <mrow> <mi>d</mi> </mrow> <mi>c</mi> </msup> <mi>ϕ</mi> <mo>≤</mo> <mi>M</mi> <msub> <mi>ω</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> for two positive constants <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m$</EquationSource> <EquationSource Format="MATHML"><math> <mi>m</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\omega }_{0} = d{d}^{c}{\left| z\right| }^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>d</mi> <msup> <mrow> <mi>d</mi> </mrow> <mi>c</mi> </msup> <msup> <mrow> <mfenced close="|" open="|"> <mi>z</mi> </mfenced> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is the Euclidean Kähler form on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\mathbb {C}}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${d}^{c} = \frac{\sqrt{-1}}{4}\left( {\bar{\partial } - \partial }\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi>d</mi> </mrow> <mi>c</mi> </msup> <mo>=</mo> <mfrac> <msqrt> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msqrt> <mn>4</mn> </mfrac> <mfenced close=")" open="("> <mrow> <mover accent="true"> <mrow> <mi>∂</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>-</mo> <mi>∂</mi> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is a positive Borel measure on <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. We characterize the r-summability of the embeddings <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\textrm{Id}: F_{\phi }^{p} \rightarrow L_{\phi }^{p}\left( \mu \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Id</mtext> <mo>:</mo> <msubsup> <mi>F</mi> <mrow> <mi>ϕ</mi> </mrow> <mi>p</mi> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>L</mi> <mrow> <mi>ϕ</mi> </mrow> <mi>p</mi> </msubsup> <mfenced close=")" open="("> <mi>μ</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$r \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$1< p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.

Absolutely summing Carleson embeddings on generalized Fock spaces

期刊论文
发表日期：

2026年6月15日

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Ermin Wang,
Yecheng Shi

摘要

In this paper, we investigate the r-summing Carleson embeddings from generalized Fock spaces $F_{\phi }^{p}$ $F_{ϕ}^{p}$ into $L_{\phi }^{p}(\mu )$ $L_{ϕ}^{p} (μ)$ , where $\phi \in {\mathcal {C}}^{2}\left( {\mathbb {C}}^{n}\right) $ $ϕ \in C^{2} (C^{n})$ is real-valued and satisfies $ m{\omega }_{0} \le d{d}^{c}\phi \le M{\omega }_{0} $ $m ω_{0} \leq d d^{c} ϕ \leq M ω_{0}$ for two positive constants $m$ $m$ and $M$ $M$ , ${\omega }_{0} = d{d}^{c}{\left| z\right| }^{2}$ $ω_{0} = d d^{c} {|z|}^{2}$ is the Euclidean Kähler form on ${\mathbb {C}}^{n}$ $C^{n}$ , ${d}^{c} = \frac{\sqrt{-1}}{4}\left( {\bar{\partial } - \partial }\right) $ $d^{c} = \frac{\sqrt{- 1}}{4} (\bar{\partial} - \partial)$ . $\mu $ $μ$ is a positive Borel measure on $\mathbb {C}^n$ $C^{n}$ . We characterize the r-summability of the embeddings $\textrm{Id}: F_{\phi }^{p} \rightarrow L_{\phi }^{p}\left( \mu \right) $ $Id : F_{ϕ}^{p} \to L_{ϕ}^{p} (μ)$ for any $r \ge 1$ $r \geq 1$ and $1< p<\infty $ $1 < p < \infty$ .