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Abstract <p> Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((X,d,\mu)\)</EquationSource> </InlineEquation> be a set with quasimetric <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\sigma\)</EquationSource> </InlineEquation>-finite measure <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mu\)</EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(I=(0,t_0)\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(0&lt;t_0\le\infty\)</EquationSource> </InlineEquation>. We consider the classes <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal H^{p,r}\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(0&lt;p,r\le\infty\)</EquationSource> </InlineEquation>, consisting of complex-valued measurable functions <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(u\)</EquationSource> </InlineEquation> on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(X\times I\)</EquationSource> </InlineEquation> for which the maximal function <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\mathcal N u(x):=\sup\{|u(y,t)|\colon d(x,y)&lt;t\}\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(x\in X\)</EquationSource> </InlineEquation>, belongs to the Lorentz space <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(L^{p,r}(X)\)</EquationSource> </InlineEquation>. In special cases, these classes are extensions of certain Hardy and Hardy–Lorentz spaces (of analytic, harmonic, etc., functions). For functions in the classes <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(\mathcal H^{p,r}\)</EquationSource> </InlineEquation>, we consider generalizations of the classical Hardy–Littlewood inequalities as well as fractional integration operators. </p>

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Hardy–Littlewood Inequalities for Functions in Hardy–Lorentz Type Spaces

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • V. G. Krotov,
  • M. M. Loginovskaya

摘要

Abstract

Let \((X,d,\mu)\) be a set with quasimetric \(d\) and \(\sigma\) -finite measure \(\mu\) , and let \(I=(0,t_0)\) , \(0<t_0\le\infty\) . We consider the classes \(\mathcal H^{p,r}\) , \(0<p,r\le\infty\) , consisting of complex-valued measurable functions \(u\) on \(X\times I\) for which the maximal function \(\mathcal N u(x):=\sup\{|u(y,t)|\colon d(x,y)<t\}\) , \(x\in X\) , belongs to the Lorentz space \(L^{p,r}(X)\) . In special cases, these classes are extensions of certain Hardy and Hardy–Lorentz spaces (of analytic, harmonic, etc., functions). For functions in the classes \(\mathcal H^{p,r}\) , we consider generalizations of the classical Hardy–Littlewood inequalities as well as fractional integration operators.