Norm of a Volterra Integral Operator on Some Analytic Function Spaces

Hao Li, Songxiao Li

Type: Article

Publication Date: 2013-09-03

Citations: 0

DOI: https://doi.org/10.1155/2013/492052

Abstract

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:math> be an analytic function in the unit disc <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow></mml:math>. The Volterra integral operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> is defined as follows: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy="false">∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>h</mml:mi><mml:mi mathvariant="normal">'</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>w</mml:mi><mml:mo>,</mml:mo><mml:mi>h</mml:mi><mml:mo>∈</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>𝔻</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi>𝔻</mml:mi><mml:mo>.</mml:mo></mml:math> In this paper, we compute the norm of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> on some analytic function spaces.

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