Hardy Inequalities for p-Weakly Monotone Functions

Abstract

We prove Hardy-type inequalities

\[ \left( \int_d^\infty \left| \int_d^s f(x) \, dx \right|^p s^{\beta} \, ds \right)^{1/p} \leq C \left( \int_d^\infty \left| f(s) \right|^q s^{\alpha} \, ds \right)^{1/q} \]

for the class of \(p\)-weakly monotone functions with \(q\) or \(p\) smaller than 1 and \(d \geq 0\).