Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> be the defining function of a bounded strongly pseudoconvex domain D with smooth boundary in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {C}}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In this paper, we study the essential norm of Hankel operators <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq3.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$H^\beta _f$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mi>f</mi> <mi>β</mi> </msubsup> </math></EquationSource> </InlineEquation> which are considered as operators from weighted Bergman spaces <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$A^p(D,|\rho |^\alpha \,dV)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>p</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi>ρ</mi> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> </msup> <mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">$L^q(D,|\rho |^\beta \,dV)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>q</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi>ρ</mi> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> </msup> <mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$1<p\le q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$-1<\alpha ,\beta <\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in L^1(D,|\rho |^\beta \,dV)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi>ρ</mi> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> </msup> <mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we obtain some quantities in terms of the symbol function f, which are comparable to the essential norm of the Hankel operator <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq9.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$H^\beta _f$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mi>f</mi> <mi>β</mi> </msubsup> </math></EquationSource> </InlineEquation>. Furthermore, it is shown that the essential norm of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2024_403_Article_IEq10.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$H^\beta _f$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>H</mi> <mi>f</mi> <mi>β</mi> </msubsup> </math></EquationSource> </InlineEquation> is equivalent to the distance norm from itself to compact Hankel operators.

Essential norm of Hankel operators on weighted Bergman spaces of strongly pseudoconvex domains

期刊论文
发表日期：

2025年1月8日

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Zhicheng Zeng,
Xiaofeng Wang,
Jin Xia

摘要

Let $\rho $ $ρ$ be the defining function of a bounded strongly pseudoconvex domain D with smooth boundary in ${\mathbb {C}}^n$ $C^{n}$ . In this paper, we study the essential norm of Hankel operators $H^\beta _f$ $H_{f}^{β}$ which are considered as operators from weighted Bergman spaces $A^p(D,|\rho |^\alpha \,dV)$ $A^{p} {(D, | ρ |}^{α} d V)$ to $L^q(D,|\rho |^\beta \,dV)$ $L^{q} {(D, | ρ |}^{β} d V)$ with $1<p\le q<\infty $ $1 < p \leq q < \infty$ and $-1<\alpha ,\beta <\infty $ $- 1 < α, β < \infty$ . For $f\in L^1(D,|\rho |^\beta \,dV)$ $f \in L^{1} {(D, | ρ |}^{β} d V)$ , we obtain some quantities in terms of the symbol function f, which are comparable to the essential norm of the Hankel operator $H^\beta _f$ $H_{f}^{β}$ . Furthermore, it is shown that the essential norm of $H^\beta _f$ $H_{f}^{β}$ is equivalent to the distance norm from itself to compact Hankel operators.