<p>We study weighted composition operators acting between Zygmund- and Bloch-type spaces on the unit ball in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}^N.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>N</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Their boundedness, compactness, and norm estimates are characterized via properties of the symbols <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varphi ,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> with special emphasis on a distinguished component of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varphi .\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Under standard integrability conditions, these spaces are shown to be automorphism-invariant and boundary-regular. The results provide essential tools for clarifying the relationship between boundedness and compactness of weighted composition operators.</p>

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On the Smallness of Zygmund-Type, Bloch-Type Spaces and a New Approach to Weighted Composition Operators

  • Thai Thuan Quang

摘要

We study weighted composition operators acting between Zygmund- and Bloch-type spaces on the unit ball in \(\mathbb {C}^N.\) C N . Their boundedness, compactness, and norm estimates are characterized via properties of the symbols \(\psi \) ψ and \(\varphi ,\) φ , with special emphasis on a distinguished component of \(\varphi .\) φ . Under standard integrability conditions, these spaces are shown to be automorphism-invariant and boundary-regular. The results provide essential tools for clarifying the relationship between boundedness and compactness of weighted composition operators.