MAXWELL - DIRAC EQUATIONS 31
We observe that
\\MpRi(Y)f\\h
= £ uivr^ion/iii-
1J3
We have djR^Y)/ = Rxj(Y)f + Pij(Y)f, where
fly(Y;s,0 = £:R1(Y,x,S),Plj(Y,x,S) = ^ ( y . a r . O ^ -
Since Rx satisfies (2.13b), the polynomials Rli and Pu satisfy condition (2.13a) of Qt.
This shows that
\\MPRi(X)f\\hCn
J2 IIM^M^/Hi,, YeIL',\Y\n. (2.36)
HMn
It follows from (2.13c) that R2(Y,x,£) =
Y^ij3rj
O ^ ^ O ^ ? where rj- satisfies condi-
tion (2.13a) of Q4. Then R,{Y)f = Ei*3 ^ f TO/ = Eii
3
djrf\Y)f +
R%]{Y)f,
where R2 (Y, x,£),r:- '(y,x,£), once more satisfies the homogeneity condition in (2.13c)
and deg i?2 1^1 + £(X) ~ 1 with R2 ' = 0 if this number is strictly smaller than one.
We now repeat this argument with R2 instead of R2 , R2 R2 until we have R2 ' = 0
for some L C(Y) + 1. The corresponding sequence r^\ ...,
r^L\
gives
L
/?a(r)/ = £ djrj(Y)f,rj(Y,x,0 = J2rf)(Y xV 2-37)
173 Z=l
where r^ satisfies (2.13a). Equalities (2.37) and (2.13a) show that
\\\V\'-lR2{Y)f\\L2Y,\\Mp-ldiri{Y)f\\L2 (2.38)
3
3
cn( E
IIM'M^/llk)4,
yen',|y|n.
HM*
It follows from expression (2.33) of \\u\\%n and inequalities (2.34), (2.35), (2.36) and (2.38)
that \\u\\E Cnqn{u), u G £"00» ^ 0, for some constants Cn. This proves the first
inequality of the theorem.
To prove the second inequality of the theorem, by induction we note that q0(u) =
\\u\\E = \\u\\E by definition and suppose that q^u) Ci\\u\\E for 0 i n. It follows
from definition (2.32a) of q™ that
(q™+i(v))2
=
(q™(v))2+
£
\\MvFv\\2M+
£ I I
M
/ ^ I |
2
M , (2-39)
\n\n |/z|=n+l
|i/|=n+l |i/|=n+l
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