The aim of this paper is to explore the equivalent characterizations for the boundedness and compactness of Cφ − Cψ acting from classical (little) Zygmund space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3224_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">${\cal{Z}}({\cal{Z}}_{0})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">Z</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mrow> <mrow> <mi mathvariant="script">Z</mi> </mrow> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> to (little) Bloch-type space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3224_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">${\cal{B}}^{\alpha}\;({\cal{B}}_{0}^{\alpha})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mrow> <mi>α</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <msubsup> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mrow> <mn>0</mn> </mrow> <mrow> <mi>α</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>. Especially, we creatively develop a useful lemma, which not only plays a crucial role in the estimations but also offers a sufficient condition for the bounded below property of composition operators.

Differences of Composition Operators from Classical Zygmund Space to Bloch-type Space

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发表日期：

2025年5月15日

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Jinhao Liu,
Yuxia Liang,
Zicong Yang

摘要

The aim of this paper is to explore the equivalent characterizations for the boundedness and compactness of C_φ − C_ψ acting from classical (little) Zygmund space ${\cal{Z}}({\cal{Z}}_{0})$ $Z (Z_{0})$ to (little) Bloch-type space ${\cal{B}}^{\alpha}\;({\cal{B}}_{0}^{\alpha})$ $B^{α} (B_{0}^{α})$ . Especially, we creatively develop a useful lemma, which not only plays a crucial role in the estimations but also offers a sufficient condition for the bounded below property of composition operators.