Kufner's conjecture for higher order Hardy inequalities
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 \textit{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 \textit{P. Drábek} and \textit{A. Kufner}, Bayreuther Math. Schr. 47, 99-104 (1994; Zbl 0818.26010)]. \textit{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).
scientific article; zbMATH DE number 1095851