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.


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