<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1792_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$g\in H(\mathbb {D})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the Volterra type operator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1792_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$J_g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>J</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation> is defined by <InlineEquation ID="IEq1000"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1792_Article_IEq1000.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="337" /> </InlineMediaObject> <EquationSource Format="TEX">$\begin{aligned} (J_g f)(z) = \int _0^z f(\xi ) g'(\xi ) d \xi , \ \ \ f\in H(\mathbb {D}), z\in \mathbb {D}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>J</mi> <mi>g</mi> </msub> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>z</mi> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>ξ</mi> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <mspace width="4pt" /> <mi>f</mi> <mo>∈</mo> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">D</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </InlineEquation>In this paper, some properties of Volterra type operator <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40995_2025_1792_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$J_g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>J</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation> from Cauchy transform space into weighted Zygmund space will be investigated.</p>

Volterra Type Operator from Cauchy Transform Space into Weighted Zygmund Space

期刊论文
发表日期：

2025年4月21日

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Ebrahim Abbasi,
Mostafa Hassanlou,
Daryoush Molaei

摘要

Let $g\in H(\mathbb {D})$ $g \in H (D)$ , the Volterra type operator $J_g$ $J_{g}$ is defined by $\begin{aligned} (J_g f)(z) = \int _0^z f(\xi ) g'(\xi ) d \xi , \ \ \ f\in H(\mathbb {D}), z\in \mathbb {D}. \end{aligned}$ $\begin{matrix} (J_{g} f) (z) = \int_{0}^{z} f (ξ) g^{'} (ξ) d ξ, f \in H (D), z \in D . \end{matrix}$ In this paper, some properties of Volterra type operator $J_g$ $J_{g}$ from Cauchy transform space into weighted Zygmund space will be investigated.