Weighted Hardy's inequalities for negative indices
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.
scientific article; zbMATH DE number 2145404