<p>We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A^p_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>A</mi> <mi>α</mi> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. To be more precise, we give a sufficient condition for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \left\| \int u_tC_{\phi _t}\, dt\right\| _e = \int \left\| u_tC_{\phi _t}\right\| _e \, dt $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mfenced close="∥" open="∥"> <mo>∫</mo> <msub> <mi>u</mi> <mi>t</mi> </msub> <msub> <mi>C</mi> <msub> <mi>ϕ</mi> <mi>t</mi> </msub> </msub> <mspace width="0.166667em" /> <mi>d</mi> <mi>t</mi> </mfenced> <mi>e</mi> </msub> <mo>=</mo> <mo>∫</mo> <msub> <mfenced close="∥" open="∥"> <msub> <mi>u</mi> <mi>t</mi> </msub> <msub> <mi>C</mi> <msub> <mi>ϕ</mi> <mi>t</mi> </msub> </msub> </mfenced> <mi>e</mi> </msub> <mspace width="0.166667em" /> <mi>d</mi> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation> to hold in terms of geometric properties of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$u_t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\phi _t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ϕ</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation>. We also provide some necessary conditions for the equality to hold and calculate the essential norm of some integral operators such as some Volterra operators.</p>

Essential norm and Integration of a Family of Weighted Composition Operators

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2026年1月30日

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David Norrbo

摘要

We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces $A^p_\alpha $ $A_{α}^{p}$ , where $p>1$ $p > 1$ and $\alpha \ge 0$ $α \geq 0$ . To be more precise, we give a sufficient condition for $ \left\| \int u_tC_{\phi _t}\, dt\right\| _e = \int \left\| u_tC_{\phi _t}\right\| _e \, dt $ ${∥\int u_{t} C_{ϕ_{t}} d t∥}_{e} = \int {∥u_{t}, C_{ϕ_{t}}∥}_{e} d t$ to hold in terms of geometric properties of $u_t$ $u_{t}$ and $\phi _t$ $ϕ_{t}$ . We also provide some necessary conditions for the equality to hold and calculate the essential norm of some integral operators such as some Volterra operators.