Scholarlink

Abstract <p> We characterize the weighted bilinear Hardy inequality <Equation ID="Equi"> <EquationSource Format="TEX">\(\left(\int_{0}^{\infty}( \widetilde H_2(f,g)(x))^{q}u(x)\,dx\right)^{1/q} \leq C\left(\int_{0}^{\infty}f^{p_{1}}(x)v_1(x)\,dx\right)^{1/p_1} \left(\int_{0}^{\infty}g^{p_{2}}(x)v_2(x)\,dx\right)^{1/p_2}\)</EquationSource> </Equation> for all <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f,g\ge 0\)</EquationSource> </InlineEquation>, where <Equation ID="Equii"> <EquationSource Format="TEX">\(\widetilde H_{2}(f,g)(x)=Hf(x) \cdot H^{*}g(x)\)</EquationSource> </Equation> is the product of the Hardy operator and its adjoint. All cases <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(1&lt;p_1, p_2, q&lt;\infty\)</EquationSource> </InlineEquation> have been covered. We also point out that bilinear Hardy inequalities are equivalent to a pair of Hardy inequalities. </p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On Weighted Bilinear Inequalities with Mixed Hardy Operators

  • 期刊论文

  • 发表日期：

    2025年12月1日

查看全文
  • S. Mohanty,
  • S. Jain,
  • P. Jain

摘要

Abstract

We characterize the weighted bilinear Hardy inequality \(\left(\int_{0}^{\infty}( \widetilde H_2(f,g)(x))^{q}u(x)\,dx\right)^{1/q} \leq C\left(\int_{0}^{\infty}f^{p_{1}}(x)v_1(x)\,dx\right)^{1/p_1} \left(\int_{0}^{\infty}g^{p_{2}}(x)v_2(x)\,dx\right)^{1/p_2}\) for all \(f,g\ge 0\) , where \(\widetilde H_{2}(f,g)(x)=Hf(x) \cdot H^{*}g(x)\) is the product of the Hardy operator and its adjoint. All cases \(1<p_1, p_2, q<\infty\) have been covered. We also point out that bilinear Hardy inequalities are equivalent to a pair of Hardy inequalities.