<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {S}_p\big (A^2(\Omega )\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">S</mi> <mi>p</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msup> <mi>A</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the Schatten <i>p</i>-class of the Bergman space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A^2(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on a smoothly bounded strongly pseudoconvex domain <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega \subset \mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. This paper shows that if <Equation ID="Equ49"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll}\tau =0\ & \ \ \text {as}\ \ p\in \left( \frac{2n}{n+1},\infty \right) ;\\ \tau >\left( \frac{n+1}{2p}\right) \left( \frac{2n}{n+1}-p\right) \ & \ \ \text {as}\ \ p\in \left( 0,\frac{2n}{n+1}\right] , \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0</mn> <mspace width="4pt" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="4pt" /> <mspace width="4pt" /> <mtext>as</mtext> <mspace width="4pt" /> <mspace width="4pt" /> <mi>p</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>,</mo> <mi>∞</mi> </mfenced> <mo>;</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>τ</mi> <mo>></mo> <mfenced close=")" open="("> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </mfenced> <mfenced close=")" open="("> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>-</mo> <mi>p</mi> </mfenced> <mspace width="4pt" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="4pt" /> <mspace width="4pt" /> <mtext>as</mtext> <mspace width="4pt" /> <mspace width="4pt" /> <mi>p</mi> <mo>∈</mo> <mfenced close="]" open="("> <mn>0</mn> <mo>,</mo> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> belongs to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {S}_p\big (A^2(\Omega )\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">S</mi> <mi>p</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msup> <mi>A</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if and only if <Equation ID="Equ50"> <EquationSource Format="TEX">$\begin{aligned} \left( \int _\Omega \left( \left( \delta (z)\right) ^{n+1+2\tau }\int _\Omega \left| K_\tau \left( z,\varphi (w)\right) \right| ^2\textrm{d}v(w)\right) ^\frac{p}{2} K(z,z)\textrm{d}v(z)\right) ^\frac{1}{p}<\infty . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mfenced close=")" open="("> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <msup> <mfenced close=")" open="("> <msup> <mfenced close=")" open="("> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mfenced> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>τ</mi> </mrow> </msup> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <msup> <mfenced close="|" open="|"> <msub> <mi>K</mi> <mi>τ</mi> </msub> <mfenced close=")" open="("> <mi>z</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mfenced> </mfenced> <mn>2</mn> </msup> <mtext>d</mtext> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </msup> <mi>K</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mtext>d</mtext> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </msup> <mo><</mo> <mi>∞</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>This result extends [<CitationRef CitationID="CR13">13</CitationRef>, Theorem 1.1] from the range <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p > \frac{2n}{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (the previously known case) to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p \le \frac{2n}{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≤</mo> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (which was an open problem).</p>

Schatten class composition and Toeplitz operators on strongly pseudoconvex domains

期刊论文
发表日期：

2026年3月7日

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Jie Xiao,
Wenwan Yang,
Cheng Yuan

摘要

Let $\mathcal {S}_p\big (A^2(\Omega )\big )$ $S_{p} (A^{2} (Ω))$ denote the Schatten p-class of the Bergman space $A^2(\Omega )$ $A^{2} (Ω)$ on a smoothly bounded strongly pseudoconvex domain $\Omega \subset \mathbb {C}^n$ $Ω \subset C^{n}$ . This paper shows that if $\begin{aligned} {\left\{ \begin{array}{ll}\tau =0\ & \ \ \text {as}\ \ p\in \left( \frac{2n}{n+1},\infty \right) ;\\ \tau >\left( \frac{n+1}{2p}\right) \left( \frac{2n}{n+1}-p\right) \ & \ \ \text {as}\ \ p\in \left( 0,\frac{2n}{n+1}\right] , \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} τ = 0 & as p \in (\frac{2 n}{n + 1}, \infty); \\ τ > (\frac{n + 1}{2 p}) (\frac{2 n}{n + 1} - p) & as p \in (0, \frac{2 n}{n + 1}], \end{matrix}) \end{matrix}$ then $C_{\varphi }$ $C_{φ}$ belongs to $\mathcal {S}_p\big (A^2(\Omega )\big )$ $S_{p} (A^{2} (Ω))$ if and only if $\begin{aligned} \left( \int _\Omega \left( \left( \delta (z)\right) ^{n+1+2\tau }\int _\Omega \left| K_\tau \left( z,\varphi (w)\right) \right| ^2\textrm{d}v(w)\right) ^\frac{p}{2} K(z,z)\textrm{d}v(z)\right) ^\frac{1}{p}<\infty . \end{aligned}$ $\begin{matrix} {(\int_{Ω}, {({(δ (z))}^{n + 1 + 2 τ}, \int_{Ω}, {|K_{τ}, (z, φ (w))|}^{2}, d, v, (w))}^{\frac{p}{2}}, K, (z, z), d, v, (z))}^{\frac{1}{p}} < \infty . \end{matrix}$ This result extends [13, Theorem 1.1] from the range $p > \frac{2n}{n+1}$ $p > \frac{2 n}{n + 1}$ (the previously known case) to $p \le \frac{2n}{n+1}$ $p \leq \frac{2 n}{n + 1}$ (which was an open problem).