Three weights higher order Hardy type inequalities (Q998165)

The authors investigate the following Hardy type inequality \[ \| g \| _{q,u} \, \leq \, C \, \| D^k_{\rho}g \| _{p, v} ,\tag{1} \] where \(1 < q < p < \infty\), where \(u\) and \(v\) are positive and in \(L^q_{loc}(0, \, +\infty)\), \(L^p_{loc}(0, \, +\infty)\) respectively and where \(D_{\rho}^i\) denotes the weighted differential operator: \[ D^i_{\rho} g(t) = \begin{cases} \frac{d^ig(t)}{dt^i}, & i=0, \, 1, \, \dots, \, m-1, \\ \frac{d^{i-m}}{dt^{i-m}}\left( \rho (t) \frac{d^m g(t)}{dt^m}\right) & i=m, \, m+1, \, \dots, \, k, \end{cases} \] for a convenient weight function \(\rho\). The authors give a complete characterization of the weights \(u, \, v\) and \(\rho\) so that (1) holds for the case \(1 < q < p < \infty\), while \textit{A. A. Kalybaj} proved the corresponding characterization in [Sib. Mat. Zh. 45, No. 1, 119--133 (2004); translation in Sib. Math. J. 45, No. 1, 100--111 (2004; Zbl 1054.26011)] for the case \(1 < p \leq q < \infty.\) The authors introduced the following ``boundary conditions'' that must be satisfied by all functions \(g\), \[ \begin{aligned}\lim_{t\rightarrow 0^+} D^i_{\rho}g(t) & = 0, \quad i=0, \, 1, \, \dots, \, l-1, \tag{i} \\ \lim_{t \rightarrow \infty}D^i_{\rho}g(t) & = 0, \quad j=l, \, l+1, \, \dots, \, k-1, \tag{ii}\end{aligned} \] where \(1 \leq l \leq k-1\). In the fourth section, the authors presented their main result that can be stated as follows. Under the assumptions: \(1 < q < p < \infty\), \(k\geq 3\) and \(g\) satisfies the conditions (i) and (ii), the inequality (1) holds with a real constant \(C>0\) independent of \(g\) if and only if four conditions depending on \(u,\, v, \, \rho, p, \, q, \, k\) and on two intermediate exponents \( 1\leq m, \, l < k \), are fulfilled. They divided the problem into three cases, namely: \(m< l\), \(m>l\) and \(m=l\) and in each of these cases the proofs depend mainly on a Hardy type inequality for a Volterra type operator and on various estimates.