If <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is a positive Borel measure on the interval [0, 1), we denote the n-th moment of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, that is, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu _{n}=\int _{[0,1)}t^{n}d\mu (t).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>μ</mi> <mi>n</mi> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <msup> <mi>t</mi> <mi>n</mi> </msup> <mi>d</mi> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> This matrix formally induces the operator <Equation ID="Equ19"> <EquationSource Format="TEX">$ \mathcal {C}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left( \mu _{n} \sum _{k=0}^{n}a_{k}\right) z^{n} $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="script">C</mi> <mi>μ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mfenced close=")" open="("> <msub> <mi>μ</mi> <mi>n</mi> </msub> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>a</mi> <mi>k</mi> </msub> </mfenced> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </Equation>on the space of all analytic functions <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f(z)=\sum _{k=0}^{\infty } a_k z^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>a</mi> <mi>k</mi> </msub> <msup> <mi>z</mi> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, in the unit disk <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation>. In this paper, we characterize the measures <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> for which the operator <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {C}_{\mu }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">C</mi> <mi>μ</mi> </msub> </math></EquationSource> </InlineEquation> is bounded (resp., compact) from the F(p, q, s) spaces to the Dirichlet-type spaces and the Bloch-type spaces.

Cesàro-Type Operators from $\varvec{F(p,q,s)}$ Spaces to some Spaces of Analytic Functions

期刊论文
发表日期：

2025年12月17日

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Yuting Guo,
Pengcheng Tang

摘要

If $\mu $ $μ$ is a positive Borel measure on the interval [0, 1), we denote the n-th moment of $\mu $ $μ$ as $\mu _{n}$ $μ_{n}$ , that is, $\mu _{n}=\int _{[0,1)}t^{n}d\mu (t).$ $μ_{n} = \int_{[0, 1)} t^{n} d μ (t) .$ This matrix formally induces the operator $ \mathcal {C}_{\mu }(f)(z)=\sum _{n=0}^{\infty }\left( \mu _{n} \sum _{k=0}^{n}a_{k}\right) z^{n} $ $C_{μ} (f) (z) = \sum_{n = 0}^{\infty} (μ_{n}, \sum_{k = 0}^{n}, a_{k}) z^{n}$ on the space of all analytic functions $f(z)=\sum _{k=0}^{\infty } a_k z^k$ $f (z) = \sum_{k = 0}^{\infty} a_{k} z^{k}$ , in the unit disk $\mathbb {D}$ $D$ . In this paper, we characterize the measures $\mu $ $μ$ for which the operator $\mathcal {C}_{\mu }$ $C_{μ}$ is bounded (resp., compact) from the F(p, q, s) spaces to the Dirichlet-type spaces and the Bloch-type spaces.

Cesàro-Type Operators from \(\varvec{F(p,q,s)}\) Spaces to some Spaces of Analytic Functions

摘要