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Weighted Hardy’s inequalities for negative indices. (English) Zbl 1068.26022

The aim of the paper is to characterize the validity of the reverse Hardy inequality \[ \Big(\int^{\infty}_0 f(x)^p \,dx\Big)^{1/p} \leq C \Big(\int^{\infty}_0 u(x) \Big(\int^x_0 f(y)v(y)\,dy\Big)^q\,dx\Big)^{1/q}, \;f \geq 0, \] provided that either \(p,q < 0\) or \(p,q \in (0,1)\). (Here \(u\) and \(v\) are weight functions on \((0,\infty)\) and \(C\) is a positive constant independent of \(f\).) Moreover, the author establishes a duality between these two cases.

MSC:

26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
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