## Kufner’s conjecture for higher order Hardy inequalities.(English)Zbl 0879.26052

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 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 P. Drábek and A. Kufner, Bayreuther Math. Schr. 47, 99-104 (1994; Zbl 0818.26010)]. 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).
Reviewer: Petr Gurka (Praha)

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 34B05 Linear boundary value problems for ordinary differential equations 46N20 Applications of functional analysis to differential and integral equations

### Citations:

Zbl 0818.26010; Zbl 0785.26010; Zbl 0869.34021