Kufner's conjecture for higher order Hardy inequalities
type: Item
instance of: scholarly article
title: Kufner's conjecture for higher order Hardy inequalities
zbMATH Open document ID: 0879.26052
published in: Real Analysis Exchange
publication date: 1997-12-10T00:00:00Z
review text: 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).
reviewed by: Petr Gurka
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 46N20
zbMATH DE Number: 1095851
zbMATH Keywords: Hardy's inequality
zbMATH Keywords: two-point boundary value problem
zbMATH Keywords: Green's function
zbMATH Keywords: weighted norm inequality
author: Gord Sinnamon
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1095851