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Kufner’s conjecture for higher order Hardy inequalities. (English) Zbl 0879.26052

The author investigates the weighted Hardy inequality \[ \Big (\int _0^1|u(x)|^qw_0(x) dx\Big)^{1/q}\leq C\Big (\int _0^1|u^{(k)}(x)|^pw_k(x) dx\Big)^{1/p} \tag{1} \] where the order of the derivative \(k\) is a fixed positive integer; \(w_0\) and \(w_k\) are non-negative weight functions; \(1<p<\infty \), \(0<q<\infty \); the constant \(C\) is independent of the function \(u\) belonging to the certain class \(\mathcal M\). To define this class, put \(N_i=\{0,1,\dots ,i-1\}\) and fix subsets \(M_0, M_1\subset N_k\) such that \(|M_0|+|M_1|=k\) (where \(|M|\) denotes the number of elements of the set \(|M|\)). Then \(u\in \mathcal M=\mathcal M(M_0,M_1)\) if \(u\) is a solution of the following boundary value problem (BVP): \[ u^{(k)}=f;\quad u^{(i)}(0)=0 \text{for \(i\in M_0\)},\quad u^{(i)}(1)=0 \text{for \(i\in M_1\)} \tag{2} \] for some locally integrable function \(f\).
It has been known that (1) is meaningful if and only if the pair \((M_0,M_1)\) satisfies the Pólya condition, \[ |M_0\cap N_i|+|M_1\cap N_i|\geq i,\quad i=1,2,\dots ,k, \tag{3} \] moreover, in this case BVP (2) can be uniquely solved, i.e. there exists the Green function \(G(x,t)\) such that \(u(x)=\int _0^1G(x,t)f(t) dt\) [see P. Drábek and A. Kufner, Bayreuther Math. Schr. 47, 99-104 (1994; Zbl 0818.26010)]. A. Kufner [Real Anal. Exch. 21, No. 1, 380-381 (1996; Zbl 0869.34021); Bayreuther Math. Schr. 44, 105-146 (1993; Zbl 0785.26010)] conjectured a connection between the Green function \(G(x,t)\) and the weight characterization for the inequality (1).
The author verifies this conjecture and thereby completes the proof of the weight characterization for (1).
Reviewer: Petr Gurka (Praha)

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
34B05 Linear boundary value problems for ordinary differential equations
46N20 Applications of functional analysis to differential and integral equations
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