Weighted Hardy modular inequalities in variable Lp spaces for decreasing functions

We study weighted modular inequalities with variable exponents for the Hardy operator restricted to non-increasing functions. We show that the exponents p(⋅) for which these modular inequalities hold must have a constant oscillation at zero, which implies that these exponents are either constant or extremely oscillating near the origin. Similarly to the constant case, we introduce the class of weights Bp(⋅), and prove some of the classical properties in this context.