Hardy inequalities for overdetermined classes of functions (Q1363051)

Let \(1<p\leq q<\infty\), \(k\in\mathbb{N}\), and let \(w_0\), \(w_k\) be weights on \((0,1)\). The aim of the paper is to find sufficient conditions for the \(k\)th order Hardy inequality \[ \Biggl( \int^1_0| u(t)|^q w_0(t)dt\Biggr)^{1/q}\leq c\Biggl( \int^1_0| u^{(k)}(t)|^p w_k(t)dt\Biggr)^{1/p} \] to hold on classes of functions satisfying boundary conditions \(u^{(i)}(0)= 0\), \(i\in M_0\), and \(u^{(j)}(1)= 0\), \(j\in M_1\), provided that \(M_0= M_1=\{0, 1,\dots,k-1\}\) (Theorem 1) or \(M_0= \{0,1,\dots, k-1\}\) and \(M_1= \{k-1\}\) (Theorem 2).