Higher order Hardy inequalities. (English) Zbl 0785.26010

The present well-written paper is devoted to the study of the \(k\)-th order Hardy-type integral inequality \[ \left[ \int^ 1_ 0| u(x)|^ qw_ 0(x)dx\right]^{1/q}\leq C\left[\int^ 1_ 0 \bigl| u^{(k)}(x)\bigr|^ pw_ k(x)dx\right]^{1/p}, \tag{1} \] where \(k\geq 1\) is an integer, \(p>1\) and \(q>0\) are constants, the weight functions \(w_ 0\), \(w_ k\) are measurable and positive a.e. in (0,1), \(C\) is a suitable positive number.
Denote by \(AC^{(k-1)}(I_ 0,I_ 1)\) the class of all functions \(u\) absolutely continuous on [0,1] with derivatives of order \(\leq k-1\) and satisfying the boundary conditions: (2) \(u^{(i)}(0)=0\) for \(i\in I_ 0\), \(u^{(j)}(1)=0\) for \(j\in I_ 1\), where the sets \(I_ 0,I_ 1\subset\{0,1,\ldots,k-1\}\), \(\text{Card} I_ 0+\text{Card} I_ 1=k\).
The problem under consideration is to find necessary and sufficient conditions on \(p,q,w_ 0,w_ k\) which ensure that (1) holds for all \(u\in AC^{(k-1)}(I_ 0,I_ 1)\) for any integer \(k\geq 1\) fixed. For \(k=1\) this problem is completely solved [cf. B. Opic and the author, “Hardy-type inequalities” (1990; Zbl 0698.26007)]. In the case \(k>1\), an exhaustive answer to the problem is only known for some special choice of the sets \(I_ 0\), \(I_ 1\) [see the author and A. Wannebo, General Inequalities VI, Proc. 6th Int. Conf., Oberwolfach/Ger. 1990, ISNM 103, 33-48 (1992; Zbl 0766.26014), V. D. Stepanov, Preprint No. 39, Math. Inst. Czech. Acad. Sci., Prague (1988), and the author and H. P. Heinig, Tr. Mat. Inst. Steklova 192, 105-113 (1990; Zbl 0716.26008)].
In this article, by using Hardy-operator approach, some ideas due to V. D. Stepanov, and the equivalence of functions defined in the paper, the author establishes a general approach for the case when \(k>1\). A large number of particular cases and examples are also discussed in detail.


26D10 Inequalities involving derivatives and differential and integral operators
34B05 Linear boundary value problems for ordinary differential equations