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.