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.