MAXWELL - DIRAC EQUATIONS 37
Proof. Let p = 6(3
2p)_1.
Then 3 p 6 for 1/2 p 1. Suppose for the moment
that / C£°(R3). The inequality (cf. Theorem 4.5.3. of [18])
II/IIL»(R3)
CpHIVI'/ll^,), p = 6(3 - 2p)~\ (2.61)
where the constant Cp is independent of the support of / , and inequality (2.57) with
n |/x| 1 3/p of lemma 2.10 then give
(l + M) 3 / 2 -^'|^/Cr)| ?,,,, J ] IIIVI'M^/U^ (2.62a)
I«II/3|IMI+I
^4.(/,o),
where qM was defined in (2.32a). It follows from this inequality and Theorem 2.9 that
(1 + \x\f/2-»+M\d»f(x)\ CMfP||(/,0)||Mi+|M| (2.62b)
for (/, 0) e Mi+JM|. As a matter of fact,
C£°(R3, R4)
0
Cg°(R3, R4)
is dense in
S(R3, R4)
0
5(R3,R4),
which by definition is dense in M£. In particular, with \x = 0, this proves
(2.60a).
According to (2.62a) we have
(1 + \x\fl2-"+^\d^Xif{x)\ C^q^iQjiO), (2.63)
where (Qif) = Xif{x). By the definition of qM
C|+1(Qi/0)2= £ IIM'M^QJIli, (2.64)
Wl0lM+i
l«ll/J|M+i
1J3
But
djMad0Qi
= Qjai(x, -iV), where
Q0jal{x,£)
is a polynomial of degree \a\ + |/?| + 2
2|/i| + 4 and
Q^al{a~lx,ai)
= a^-^Q^al{x,0- Therefore
QM+x(Qj,0)CMq^+2(0j),
which together with (2.63) and (2.64) shows that
(l + \x\)3/2-"+M\d^xJ\ C7,,M«ffi+2(0,/) CpJ(0,f)\\Kf+a, (2.65)
where the last inequality follows from Theorem 2.9.
In a similar way it follows from (2.62a) that
|a*/(aO|CM,,||(0,/)||„, . (2.66)
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