Generalizations of Hardy-type integral inequalities for quasimonotone functions in weighted variable exponent Lebesgue spaces

Abstract

In 1992 V.I. Burenkov proved some Hardy's inequalities with sharp constants in Lebesgue spaces for monotone functions for \(0 < p < 1\). Later R.A. Bandaliev established analogous estimates in weighted variable exponent Lebesgue spaces for monotone functions for \(0 < p(x) \le q(x) < 1\). In 2020 A. Senouci and A. Zanou generalized the results of R.A. Bandaliev for quasi-monotone functions.

The aim of this paper is to obtain some generalizations of the previous results cited above for weighted Hardy operators by introducing a parameter \(\alpha \in \mathbb{R}\). Moreover, by using the quasi-norms \(\|f\|_{L^{BT}_{p(x)}(\Omega)}\) introduced by V.I. Burenkov and T.V. Tararykova, we obtain an improvement of constants in our previous estimates.