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)