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