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<p>We study the structural “smallness” of weighted high-order growth spaces on the unit ball—through automorphism-invariance and boundary continuity—and its effect on when the boundedness and compactness of weighted composition operators <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( W_{\psi ,\varphi }: f \mapsto \psi \cdot (f \circ \varphi ) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> <mo>:</mo> <mi>f</mi> <mo>↦</mo> <mi>ψ</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> coincide. Using a new approach based on the function-theoretic properties of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\psi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> and a component <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varphi _p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>φ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>, we provide new characterizations of boundedness, compactness, and asymptotic norm estimates for these operators.</p>

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Weighted-type high-order growth spaces: the smallness and a new approach to characterizing weighted composition operators

  • 期刊论文

  • 发表日期：

    2026年2月6日

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  • Thai Thuan Quang

摘要

We study the structural “smallness” of weighted high-order growth spaces on the unit ball—through automorphism-invariance and boundary continuity—and its effect on when the boundedness and compactness of weighted composition operators \( W_{\psi ,\varphi }: f \mapsto \psi \cdot (f \circ \varphi ) \) W ψ , φ : f ↦ ψ · ( f ∘ φ ) coincide. Using a new approach based on the function-theoretic properties of \(\psi \) ψ and a component \(\varphi _p\) φ p of \(\varphi \) φ , we provide new characterizations of boundedness, compactness, and asymptotic norm estimates for these operators.