HARDY TYPE INEQUALITIES 13
n Co(I) k and f G.
We recall now some facts which will be used later on. A function
M : I -* - 8(H) (the Banach space of linear continuous operators in H) is
called locally Lipschitz if for each compact K = I there is a constant
C(K) « such that
|M(s) - M(t)lB(H) * C(K)|s-t| V s,t e K.
It is known that a distribution M on I with values in 8(H) is a locally
Lipschitz function if and only if its derivative Mr is a weakly measura-
ble locally bounded function Mf : I - B(H). In this case lMt(t)|g(H)= C(K)
for almost all t K .
We shall need the following easy consequence of the classical inequa-
lity of Hardy (see [15],Theorem 327) :
Lemma 2.1 : Let u : (0,») + H be an absolutely continuous function such
that ||u1 | | = (JQ^IU1 (t) |jj dt)1/2 o o and lim u(t) = 0 as t - 0. Then
I I t"1u|| S 2||uMI•
-1u
In particular, if u £
ff1
, and Pu H. then t " H (because if u
' loc ' o
is a constant, then t u £ H)
o
The following abstract form of Fatou's lemma will be used several ti-
mes below. Suppose that V is a linear subspace of H and let || | || | be a new
norm on V (here H can be any Hilbert space). Assume that || | || | is stronger
than | | | | , i.e. there is a finite constant c such that ||u|| ^ c|||u||| for
all u V. Let V be the completion of (P,|||-|||). P is a Banach space with
the norm || | || | extended by continuity. The inclusion V =. H extends by con-
tinuity to a linear, continuous mapping V -* - H which is not injective in ge-
neral. We say that || | * || | is closable (on V in H) if this mapping is injec-
tive, in which case we identify V c H through it (clearly || | || | is closable
if and only if for each sequence {u } of elements of V which is Cauchy for
HI• HI and convergent to zero for ||• | | , one has ||| u || | - 0). If V = V9
we say that || | || | is closed . The norm || | || | is called a reflexive norm
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