HARDY TYPE INEQUALITIES
£ . , -i
1 / 2
« i ! . , • } " « • S „ •.
Now we introduce the object of our study. Throughout this paper we
assume that the following are given :
1) A 2-semilocal subspace P of H such that P c H
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 IWCt)!,,,^ = 0
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
perturbed operator L
by the formulas :
o " loc
( which is the object we want to study)
L = PMP + Q
L = L + SP + R
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).
Remark that P, c H . We provide P with the Hilbertian norm
loc loc ^