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
H - L (I;H) . | | . | | denotes the norm in H, obtained from the scalar
(u,v) = L (u(t),v(t)) dt .
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],
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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