MAXWELL - DIRAC EQUATIONS 45
Since, according to the definition of || H^ and Corollary 2.6
I K - x l U l ^ d b , ^ + liri««n-iiyi«„lljsm+„
c'iWu^ \\E\\uin I I
V|+i+]v
+ lk„-L b
m + 1 +
I K ||B),
and since the case n = 1 is trivial, we obtain
\\TrP+\lq
®T?®
J _ , _
P
) ( T
B
®y=1 ^ ) | |
£
(2.95)
n - 1
^ E J I W ^ k w i
1
nl,p=l,2,n-p0.
i j=l
Formula (2.94) and inequality (2.95) prove, after changing the constant C, that the first
statement of the lemma is true for \Y'\ = L + 1. So, by induction it is true for all \Y\ 0.
According to Theorem 2.4 of [28] we have, for X e U and Z e II',
Txz=
Yl
T!Tx(Tzl®T^)rn+T^,
n2, (2.96)
ni-\-n2—n Z,2
where Y^zi
*s a s u m o v e r a sutset
of couples ( Z i , ^ ) with Z$ G
II7
and 0 \Z{\ |Z|,
\Z\\ + \Zz\ = \Z\ and where rn is the normalized symmetrization operator on g E. Let
Y = XZ,X e U,Z e IT, let /, e £00,1 I m,gj E^,! j n2, and let
/ = fx (g) •. g /
n i
, g = px (g). (g) pn2. According to statement ii) of Lemma 2.17, we obtain
\\T$(T5l{f)®i?MK
(2-97)
CN(\\T2l(f)\\BJT%(g)\\Ei +
||T^(/)||
B I
||T^(
9
)||
£ N
)
3 / 2
-''
(\\T%(f)\\ENJT%(9)\\E +
||T^(/)||
£
||T^(
P
)||
£ N + I
)''-
1 / 2
,
J V 0.
Let N 1. Then it follows from the first inequality of Lemma 2.19 which is already proved
and from the second inequality of Corollary 2.6, that
II^WIkPzJGrtlk + l l ^ / ) I M
r
£ G ? ) l k (2-98)
CN,n,lZ\ Y, ll/bb H/'-i
WE\\9J2\\E
\\9u2
WE
1,3
(WfiAEN+fJ\9jA\E1+IZ2l + Wh\\El+lzj9jAEN+lJ
CNn ,Z|
Y,\\^E---\\K\\E\\9^\\E---\\9^\\E
1,3
(\\flA\EN+j9h\\El + \\h\\El\\93A\EN+J, N1.
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