On a weighted inequality of Hardy type for derivatives of higher order
The paper deals with the inequality \[ (1)\quad (\int^{\infty}_{0}| f(x)u(x)|^ q dx)^{1/q}\leq C(\int^{\infty}_{0}| f^{(k)}(x)v(x)|^ p dx)^{1/p} \] with \(q=p=2\), C is a positive constant independent of f, k is a positive integer. There are found necessary and sufficient conditions on weighted functions u(x), v(x) for the inequality to hold for all sufficiently smooth functions f(x) such that f together with all its derivatives up to the (k-1)th order including vanish at \(x=0.\) The inequality (1) has been completely investigated by many mathematicians for \(k=1.\) The case \(k>1\) is new. Nevertheless, the author has succeeded to improve his result presented in this paper to the case \(1\leq p\leq \infty,\quad 0<q\leq \infty.\) It will have appeared in a short time. The same problem was independently solved by another method by \textit{F. J. Martín-Reyes} and \textit{E. Sawyer} [Proc. Am. Math. Soc. 106, 727-733 (1989; Zbl 0704.42018)] with \(1<p\leq q<\infty.\) The same inequality (1) but on another class of functions f(x) (the function f and all its derivatives up to the mth order including vanish at \(x=0\) and all remaining derivatives up to the (k-1)th order including vanish at \(\infty)\) was investigated by \textit{A. Kufner} and \textit{H. P. Heinig} [Hardy's inequality for higher order derivatives (Russian) (to appear in Tr. Mat. Inst. Steklova)]. Some applications concerning continuity of the imbedding operator from certain weighted Sobolev space into certain weighted Lebesgue space are included.
scientific article; zbMATH DE number 4156825