10 1. A CLASS O F FUNCTION SPACES

so for any z G R ,

(i + N)

A

" M»(i + \z\ni +

\v\)x+N+1

*cLn+

l"

+

* " dyCs»P I'M

|z + i,|)A(l +

IJ/D^+I

y

- ~ tf€I£ (1 + M)*'

which proves (1.1.13). Here we have used the inequality (l + |;z|)(l + |2/|) l + |z-fy|,

and the integrability o£ (1 +

\y\)~N~1.

Now, we use (1.1.13) to prove

(1.1.14) sup — J f ^ l CsuJ± f ' ^ L V

1 A

By the mean value theorem, for any 6 0,

(1.1.15)

Here, if $ 1,

(1.1.15 ) |/(z)| inf\f(y)\ + 6 sup |V/(y)|.

inf \f(y)\c(± I

\f(y)\rdy)

\v-z\S V 5 J\y-z\& )

,(1 + \z\)N/r+d / " ^

l/f-.M*-r

\ 1 / r

c-

i /• \f(y)\ . V

;i + kl)^4i1+iZi(i + M r V

Substituting this and (1.1.13) into (1.1.15) we obtain,

zel[N (i + i*D"/r+* -

0

^a^ yB(0iO) (i + iyi)^

y

C5 sup

z6R~(l + l*l)JV/r+""

The inequality (1.1.14) and the lemma follow if 5 is chosen small enough. •

We note the following corollary.

COROLLARY

1.1.8. Under the assumptions of Lemma 1.1.7 there is a constant

C = C(r, d, N, M) such that for 0 b 1/R

eM(f,B(0,b),L„)C(bR)MmJ±

[ - J M L d y ) .

a0 \ a JB(0,a) (1 + # M ) r d /

PROPOSITION

1.1.9. Let E satisfy the conditions in Definition 1.1.5. Then the

space YL{E) is continuously embedded in L

r

j

o c

and is complete.

PROOF.

The continuity of the embedding follows from Lemma 1.1.4.

In order to prove completeness it suffices to prove that every sum Y^dLi f^ °f

functions /W e YL(E) such that J X i ll/(0llyL(£7) °° i s convergent in YL(E).

Here n is the number defined in (1.1.2). By Definition 1.1.5 and the definition of

the norm in YL(E), each fW has a representation as a sum of entire functions,

f{l) = ET=ofil\ satisfying (1.1.9), so that

oo oo

£ii{/fh=oii£2£ii/(%L(£)°o.

1=1 1=1