On the Hardy inequality with measures
The author presents four theorems (without proofs) concerning necessary and sufficient conditions for the fulfillment of the general Hardy inequality of the form \[ \Biggl(\int_{[a,b]} v(x)\biggl(\int_{[a,x]} k(x,y)f(y)u(y)\,d\lambda(y)\biggr)^q \,d\mu(x)\Biggr)^{1/q} \leq C\Biggl(\int_{[a,b]}f^p w\,dv\Biggr)^{1/p} \] for all \(f\in\{\mathfrak{M}_{\lambda}\}^+\), where \(0<p<1\); \(0<q<+\infty\); \(\lambda,\mu\) and \(v\) are \(\sigma\)-finite measures on \([a,b]\subset\mathbb R\); \(\lambda\) and \(v\) are defined on the same \(\sigma\)-algebra \(\mathfrak{M}\) of subsets of the set \([a,b]\) containing all Borel subsets of \([a,b];\) \(u,v,w\) and \(k\) are the respective measurable functions. Those theorems generalize significantly previous results obtained by other mathematicians, including the author himself (seven papers cited in the list of references). Especially interesting in this paper is the substitution of Oinarov's condition for the kernel \(k\) by some monotonic-type conditions for \(k\).
scientific article; zbMATH DE number 5905453