On weighted Hardy inequalities on semiaxis for functions vanishing at the endpoints (Q1383391)

Let \(u\) and \(v\) be weights on \((0,\infty)\) and let \(\|\cdot\|_2\) stand for the norm in the Lebesgue space \(L^2(0,\infty)\). The authors completely characterize the validity of the weighted Hardy inequality \(\| Fu\|_2\leq C\| F''v\|_2\) on the class of all functions \(F\) which have absolutely continuous derivates \(F'\) on \((0,\infty)\) and satisfy the boundary conditions \(F^{(i)}(0)= 0\), \(F^{(i)}(\infty)= 0\), \(i\in\{0, 1\}\).