An equivalence theorem for some integral conditions with general measures related to Hardy's inequality
Let \(M(x):= \int_{[x,\infty)} d\mu<\infty\) and \(\Lambda(x):= \int_{(-\infty,x]} d\lambda<\infty\), where \(\lambda\) and \(\mu\) are measures on \(\mathbb{R}\). The authors define five functions as follows \[ \begin{aligned} A_1(x;\alpha,\beta)&= M(x)^\alpha\Lambda(x)^\beta,\\ A_2(x;\alpha,\beta,s)&= \Biggl(\int_{[x,\infty)}\Lambda^{(\beta- s)/\alpha} \,d\mu\Biggr)^\alpha\Lambda(x)^s,\\ A_3(x;\alpha,\beta, s)&= \Biggl(\int_{(-\infty, x]} M^{(\alpha- s)/\beta} \,d\lambda\Biggr)^\beta M(x)^s,\\ A_4(x;\alpha,\beta, s)&= \Biggl(\int_{(-\infty, x]}\Lambda^{(\beta+ s)/\alpha} \,d\mu\Biggr)^\alpha\Lambda(x)^{-s},\\ A_5(x;\alpha,\beta, s)&= \Biggl(\int_{[-x,\infty)} M^{(\alpha+ s)/\beta} \,d\lambda\Biggr)^\beta M(x)^{-s}, \end{aligned} \] where \(\alpha\), \(\beta\) and \(s\) are positive fixed numbers. They prove as their main result that the numbers \(\sup_{x\in \mathbb{R}} A_s(x;\alpha,\beta)\) and \(\sup_{x\in \mathbb{R}} A_i(x;\alpha, \beta,s)\), \(i= 2,3,4,5\), are mutually equivalent. This is a general equivalence theorem of independent interest. In particular it can be applied for characterizing the Hardy type inequality with general measures for the case \(1< p\leq q<\infty\) (see the Hardy inequality).
Hardy's inequality with general measures
scientific article; zbMATH DE number 5083574