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<p>The distance from the identity operator <i>I</i> to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^*\)</EquationSource> </InlineEquation>, the dual of the Hardy averaging operator, is studied on the cone of nonnegative, nonincreasing functions in Lebesgue space. The exact value is obtained. Optimal lower bounds are also given for difference, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^*-I\)</EquationSource> </InlineEquation>, of these two operators acting on the same cone. A positive answer is given to a conjecture made in “The norm of Hardy-type oscillation operators in the discrete and continuous settings” by A. Ben Said, S. Boza, and J. Soria. Preprint, 2024. In addition, a direct comparison, with optimal constants, is given between the operators <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H-I\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(H^*-I\)</EquationSource> </InlineEquation> acting on the cone.</p>

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Best bounds for the dual Hardy operator minus identity on decreasing functions

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Achraf Ben Said,
  • Gord Sinnamon

摘要

The distance from the identity operator I to \(H^*\) , the dual of the Hardy averaging operator, is studied on the cone of nonnegative, nonincreasing functions in Lebesgue space. The exact value is obtained. Optimal lower bounds are also given for difference, \(H^*-I\) , of these two operators acting on the same cone. A positive answer is given to a conjecture made in “The norm of Hardy-type oscillation operators in the discrete and continuous settings” by A. Ben Said, S. Boza, and J. Soria. Preprint, 2024. In addition, a direct comparison, with optimal constants, is given between the operators \(H-I\) and \(H^*-I\) acting on the cone.