## 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.

### MSC:

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

### Citations:

Zbl 0698.26007; Zbl 0766.26014; Zbl 0716.26008