Regularly bounded functions and Hardy’s inequality

Abstract

We consider Hardy’s inequality

$$\int\limits_{0}^{\infty } {{{{\left( {\frac{1}{t}\int\limits_{0}^{t} F (s)ds} \right)}}^{p}}} W(t)dt \leqslant C\int\limits_{0}^{\infty } {{{F}^{P}}} (t)W(t)dt $$

where W, a positive function, is the weight and 1 ≤ p < ∞. In [6, Th. 330] this inequality is given with the weight W(t) = t ^α, for 0 < α < p - 1.