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 ^