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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
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