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 $(\int^1_0 |u(t)|^q w_0(t)dt^{1/q} ≤ c(\int^1_0 |u^{(k)}(t)|^p w_k(t) dt)^{1/p}$ 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 $w_k$.