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 d» =
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,/l9I//(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 L« 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 .
0a/?n 0a/3n
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(52p) 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(52Py1JeIP, (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)
0a/3n
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 + xr +1+ H^/(
X
)).
i/n+l X i/n *
//r
n
,
0
(/ , /) oo /or some a 3/2  p, then (/, /) G M£ and (/, /)Mp C
n
r
n
,
a
(/, / ) .