Some new characterizations of weights in dynamic inequalities involving monotonic functions
In this paper, the authors derive and prove some new characterizations of weighted functions for dynamic inequalities of Hardy type involving monotonic functions on a time scale \(\mathbb{T}\) in spaces \(L^{P}(\mathbb{T})\) and \(L^{q}(\mathbb{T})\) where \(0 < p < q <\infty\) and \(p\leq 1\). The main results of this paper are given in Theorems 3.1 and 3.2 and the authors employ the reverse Hölder inequality, integration by parts, and the Fubini theorem on time scales to prove them. Several consequences of the results obtained in the cases of time scales \(\mathbb{T}= \mathbb{R}\) and \(\mathbb{T}= \mathbb{N}\) are derived and discussed. In particular, the case \(p < 1\) appears to be new. The results in this paper unify and extend continuous and discrete Hardy-type inequalities in the literature.
scientific article; zbMATH DE number 7365499