MAXWELL - DIRAC EQUATIONS 35

The inequality q^\{ot) Cn^a\\D is proved in a similar way, but one has only to

consider the degree of the corresponding polynomials r(x,£) and not their homogeneity

properties. Since the proof of this part is very similar, we omit it. Together with (2.56)

we then have

Qn+l(u) Cn\\u\\En+i,

which according to the induction hypothesis proves the second inequality of the theorem.

To prove that Ec is dense in E^ we first observe that C^(R3, C) is dense in 5(E 3 , C)

which after inverse Fourier transform shows that Dc is dense in D^. We therefore only

have to prove that Mc is dense in M ^ . Let (p G Co°(M3), 0 ip(k) 1 for k G M3,

(p(k) = 1 for |A;| 1, tp(k) = 0 for \k\ 2. Let pn(k) = ip{n~lk){l - p(nk)) for k G R3,

n G N, n 2, and let i\)n = 1 - pn. Then 0 ^n(fc) 1 for k G R 3 and i/n(k) = 0

for 2/n \k\ n. Moreover if |a| 1 then \k\^\daijjn{k)\ 2 l a l q

a

| for \k\ 2/n

and \daipn(k)\ n~^C\a\ for |A;| n, where the C\ are constants independent of n and

k. For ( / , / ) G Moo we define / » ( f c ) = Pn(k)f(k) and /W(fe) = Vn{k)f(k). Then

(/ ( n ) , / ( n ) ) e Mc and / » -f = ^

n

/ , fW - / = ipnf. The first inequality of the theorem

and the Plancherel theorem now show that

ll(/

( n )

-/,/

( n )

-/)ll^

C'N Y. J {\k\2p\d»{k^n(k)f{k))\2 + |^|2^-2|^(A;n(^)/(A:))|2) dfc,

n 2, where £

n

= {A: G M 3 |2/n |fc| n}. The estimates of VVi and commutation of 9M

and kv give that

||(/»-/

J

/V)-_/)||^

C'N

E /

(l*l2pl^(*V))l2

+

l*|2p-2l^(*"/(*))l2)

dfc.

The last inequality converges to zero when n — ex) since #JV ((//)) *s finite. This proves

the theorem.

The elements of the space have important asymptotic decrease properties:

Lemm a 2.10. Letlpoo and let f G LP(R3) be such that Mad^f G L P (R 3 ), Ma(x) =

xa, for 0 |a| |/?| n. Ifv is a multi-index and (n — \v\)p 3, then (after a change

on a set of measure zero)

(l + \x\f/r+\»\\d»f(x)\Cu,p E

llM«^/||LP.

(2.57)

\*\\P\n

Proof A Sobolev embedding gives at once with Ip 3 that

II^/IIL-

^ E U^/IL* C Y, WfWis, (2-58)

Mt |/3|n