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 (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
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