2. PRELIMINARY INEQUALITIES

In this section we present the general framework in which our estima-

tes will be made . Then we establish an important identity ( Lemma 2.3

and Corollary 2.4 ) and derive some useful inequalities from it .

Throughout this paper I denotes an open interval of the form

I = (a,+°°) c ]R with a n , H is a complex , separable Hilbert space and

o

H - L (I;H) . | | . | | denotes the norm in H, obtained from the scalar

product

(u,v) = L (u(t),v(t)) dt .

.m,

The notation |.| will be used for the norm in H . We denote by H (I;H)

= tfm the Sobolev space of order m =1,2 of H-valued functions on I . If

P = -id/dt ( the derivatives are always in the sense of distributions ) ,

then H1 is explicitly given as

Hm = { u € H | Pmu E H }.

with norm

u||Hm = (||u||2 + ||Pmu||2)1/2

We define tfm as the set of functions u £ Hm with compact support in I and

Ho m as the closure of tfm in tfm . H is similarly defined .

c c J

We now introduce a notion which will considerably simplify the appli-

cation of our results . The terminology is borrowed from Hormander ([18],

\r

Chapter 10.1). Let k € {0,1,2, ,«},let C (I) be the space of complex-

valued functions of class C on I and C (I) the subspace of C (I) of

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