HARDY TYPE INEQUALITIES
7
functions of compact support.
In Section 3 we derive our principal Hardy type inequalities. For
this we have to extend certain local inequalities of Section 2 to global
inequalities, i.e. to inequalities valid for functions that are not requi-
red to have compact support. The principal technical arguments are contai-
ned in the proof of Proposition 3.4, and the most important result of our
paper is Theorem 3.8.
In Section 4 we present some variants of the results of Section 3
(some of the inequalities proved in Section 2 are needed only here). In pa
particular we obtain Hardy type inequalities for weight functions cp that
may be less regular or grow more rapidly than those considered in Section
3. In Section 5 we discuss several applications to ordinary and partial
differential operators, and in an appendix we prove some inequalities on
fractional powers of positive self-adjoint operators.
D. Notes and related research. We end this introduction with some more
technical comments and an indication of additional related literature. We
begin by specifying and discussing a set of sufficient conditions for the
validity of the Hardy type inequality (1.8) given above.
Theorem 1.2 : Let ft be an open neighbourhood of infinity in ]Rn and, for
r 0, set ft = {x ft | |x| r}. Let A be a partial differential operator
of the form (1.1), defined for x ft, and assume that its coefficients ha-
ve the following properties:
(a) a., : ft - ]R are Lipschitz, a., (x) -* 5.. as |x| - °° and
J K J K J K
|grad a.,K (x)| ^ const.|x|"
"e
for some e 0,
J
Co) b.: ft - ( P is such that b.u G L? (ft) for each u H} (ft),
J J IOC lOC
2 2
(c) c: ft -* - ( D is such that cu G L, (ft) for each u E tf, (ft),
loc loc '
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