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On overdetermined Hardy inequalities. (English) Zbl 0936.26009

Let \(M_0\) and \(M_1\) be subsets of \(\{0,1,\ldots ,k-1\}\) such that \(M_0 +M_1 >k\), \(k\in \mathbb N\). The paper is devoted to a characterization of the \(k\)-th order Hardy inequality \[ \Big (\int ^1_0 |u(x)|^q w_0(x)dx\Big)^{1/q} \leq C\Big (\int ^1_0 |u^{(k)} (x)|^pw(x)dx\Big)^{1/p}, \] where \(w_0\) and \(w\) are weights, \(1<p<\infty \), \(0<q<\infty \), and \(u\in AC^{(k-1)} (0,1)\) satisfies the “boundary” conditions \(u^{(i)} (0) = 0\), \(i\in M_0\), \(u^{(j)} (1) = 0\), \(j\in M_1\).
Reviewer: B.Opic (Praha)

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
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