## Weighted inequalities of Hardy type for higher order derivatives, and their applications.(English. Russian original)Zbl 0674.26008

Sov. Math., Dokl. 38, No. 2, 389-393 (1989); translation from Dokl. Akad. Nauk SSSR 302, No. 5, 1059-1062 (1988).
In this note we find necessary and sufficient conditions on the weight functions u(x) and v(x) for inequalities of the form $(\int^{\infty}_{0}| f(x)u(x)|^ pdx)^{1/p}\leq C(\int^{\infty}_{0}| f^{(k)}(x)v(x)|^ pdx)^{1/p},\quad k\geq 1,$ to hold for any f(x) that vanish together with all their derivatives up to order k-1 at $$x=0$$ or at infinity. For $$k=1$$, our conditions are the same as the well-known criterion for the generalized Hardy inequality. In particular, we obtain as applications a number of complete results on the boundedness and compactness of certain embedding operators, and on the discreteness of the spectrum of certain classes of differential operators.

### MSC:

 26D15 Inequalities for sums, series and integrals 26A33 Fractional derivatives and integrals 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators