38 NONLINEAR REPRESENTATION AND SPACES
Inequalities (2.65) and (2.66) with p, 0 prove (2.60b) with v = 0 and / = 0. Since [9M, Xi]
is a monomial of degree \p\ 1 in V, it follows from (2.65) and (2.66) by induction that
(2.60b) is true for \v\ 0, with / = 0.
Let =
dvdi
in (2.62b). Then (2.60b), with / = 0, follows from (2.62b). Since
(2.60b) is true for the particular cases (/,0) G M?^2 and (0,/) G M£,+2, it is also true
for (/,/) G MfH+2, because ||(/,0)||Mn + ||(0,/)||Mn v^||(/,/||
M n
-
T h i s
P
r o v e s t h e
theorem.
We shall prove that
Mp
contains the long range potentials which it is expected to
contain. It follows from the next theorem that (/,/) G E^ if (1 +
\x\)a+W\duf
(x)\ and
(1 + |x|)a+1+l,/l|9I//(x)| are uniformly bounded in x for some a 3/2 p.
Theorem 2.13. Let 1/2 p l,p = 6(5 - 2p)~1,q = 6(3 - 2p)~\ f G and f G Lp.
IfMad^dif G
L^(R3,M4)
and Mad?f G
LP(R3,R4)
for 0 |a| |/?|,1 i 3, then
(/, /) G Mn and
\\(fJ)\\MnCn( £ | | M
a
a ^ / | |
L P
+ £ | | M
a
^ / | |
L P
) , n 0 .
0|a||/?|n 0|a||/3|n
1«3
Proof Suppose for the moment that / G
Co°(R3).
The inequality (cf. Theorem 4.5.3. of
[WD
I l l V r V W ^ C p l l / W a ) , P = 6(5-2p)- 1 , (2.67)
where Cp is independent of the support of / , gives ||(0,/)||MP Cp\\f\\LP1f G CQ°. Since
CQ° is dense Lp, for our given p, and dense in 5(R3), it follows by continuity and by the
definition of the space Mp that
I I ( 0 , / ) | | M P C
P
| | / | |
L
P , p = 6(5-2Py1JeIP, (2.68)
and that (0, / ) G M? for / G IP.
According to inequality (2.68), Theorem 2.9 and definition (2.32a) of q%* we have
K0,f)\\M,Cp,ng(0j)Cpn YI Wad^fW^, (2.69)
0|a||/3|n
p = 6(5- 2p)"\ n 0, if Mad^f G LP for 0 \a\ \(3\ n. Since ||(/,0)||Mp
Cp,n^(/,0) according to Theorem 2.9 and q™(f,0) C n E i ^ C M t / ) using definition
(2.32a) of qjf it follows from (2.69) and the triangle inequality that the inequality of the
theorem is true.
Corollary 2.14. Let n 0, 1/2 p 1, / G Cn+1(R3,R4), / Cn(R3,R4) and let
r
n
,a(/,/)= £ 8Up((l + |x|)a+W|^/(x)|)+ EsU
P
(( l + |x|r +1+ H|^/(
X
)|).
|i/|n+l X |i/|n *
//r
n
,
0
(/ , /) oo /or some a 3/2 - p, then (/, /) G and ||(/, /)||Mp C
n
r
n
,
a
(/, / ) .
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