HARDY TYPE INEQUALITIES
15
(2.3)
£ . , -i
1 / 2
« i ! . , } " « S •.
"2-
Now we introduce the object of our study. Throughout this paper we
assume that the following are given :
2
1) A 2-semilocal subspace P of H such that P c H
c
2) A function M : I - 8(H) which is locally Lipschitz and such that
M(t) is a positive symmetric operator for each t £ I and satisfies
lim |M(t)-l|
t-K»
8(H)
lim IWCt)!,,,^ = 0
t-M»
8(H)
3) A 2-ultralocal mapping Q : P H which is symmetric as an opera-
tor in H .
4) A 1-ultralocal mapping S : V •* - H , where P is the 1-semilocal
closure of the linear space V + PP.
5) A 2-ultralocal mapping R : P - H .
We define then the "unperturbed" operator L H. . and the
loc
perturbed operator L
by the formulas :
loc
K,
o " loc
( which is the object we want to study)
L = PMP + Q
L = L + SP + R
o
In fact, these operators are initially defined only on P. Since they
are local , they extend uniquely to local operators on P and we denote
the extensions by the same letter (we use the same convention for Q,S,R).
2
Remark that P, c H . We provide P with the Hilbertian norm
loc loc ^
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