Hardy Inequalities for Overdetermined Classes of Functions

Abstract

Conditions on weights $w_0$ and $w_k$ are given for the $k$-th order Hardy inequality $({\int }_0^1|u(t){|}^q{w}_0(t)d{t}^{1/q}\le c({\int }_0^1|u^{(k)}(t){|}^p{w}_k(t)dt{)}^{1/p}$ to hold for two special classes of functions $u$ satisfying $2k$ and $k+1$ boundary conditions, respectively. The conditions are sufficient and partially also necessary. For one class, a hypothesis is formulated describing necessary and sufficient conditions on $w_0$ and $w_k$.