Some remarks on the Hardy inequality for higher order derivatives

Abstract

The paper deals with necessary and sufficient conditions on p, q and nonnegative weights ω₀, ω _k for the validity of the inequality

$${\left( {\int_0^1 {{{\left| {u\left( x \right)} \right|}^q}{\omega _0}\left( x \right)dx} } \right)^{1/q}} \leqslant C{\left( {\int_0^1 {{{\left| {{u^{\left( k \right)}}\left( x \right)} \right|}^p}{\omega _k}\left( x \right)dx} } \right)^{1/p}}$$

for some classes of functions u satisfying certain conditions at the endpoints 0, 1.