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
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