MAXWELL - DIRAC EQUATIONS 41

Proof. It follows from (1.8a) that

\\n0{ui®u2)\\\N (2.78)

= ||(0,i(ai7a2 +

a27ai))||2Mr,

+ l l ^ ( E ( ^ 7 V «

2

+

/2M7Vai))l|2D„.

/x=0

It follows from Theorem 2.13 and Leibniz rule that

||(0,5l7aj)||MB Cn J2 \\M0\dr*ai\\8r'aj\\\LP, (2.79)

\0\\vi\+Mn

where p = 6(5 -

2p)_1.

By definition (2.32b) of q% and by the first part of Theorem 2.9:

£ Mrf-faj K C'nJ2

Vnifinl0^)

(2.80)

\P\n

The triangle inequality, inequalities (2.78), (2.79) and (2.80) prove the first inequality in

the lemma. The proof of the second inequality is so similar that we omit it.

Lemma 2.17. If Ui G Eoo, i = 1,2, and X G II, then

i)

\\T2x{Ul

®U2)\\EN

C V O K H ^ J K H ^ + | k b

2

| |

£ j v + i

) , N 0,

ii) ||2$(«i®«2)IU„ CN(\\Ul\\EJu2\\Ei +\\u1\\Ei\\u2\\ENf2-p

{\WI\\ENJME0 +

\\UI\\E0\\U2\\EN+X~1/2,

N0,

iii) ||r^(^

a

cS ) ^2)11^ Cmin

(M^1l|jEJ|-«2||^7a/2!|«2M^22-'', ||^iil^7a/:2ll^1|1^22-'

il^2||jE;i)-

Proof Since T\ = 0 if X G II and X ^ P0, X ^ M0j, j = 1,2,3, we have according to

Lemma 2.16 for X G II, m = (/*, /^, a^) G I ^ , z = 1,2,

I|T| K

®«2)||B c(J2

II^MMMIIL-

(2-81)

IMII

+ J2

(WMMI\\*2\\\L*

+ l|MM|/2||ai|||L2)), p = 6(5 - 2p)-\

IMII

Since||M

M

|ai||a

2

|||

iP

||a

1

||

L2

||M

M

a

2

||

Lg

,p = 6(5-2p)-

1

,

9

= 3(l-p)-

1

and||M

M

a

2

||

L

,

C E

M

i II^Af^aall^, we obtain

£ HM^axllaallli, q K L * £ ||MM5"a2||La (2.82)

IMII

CH^Hcllaall^, P = 6(5 - 2p)- \ 1/2 p 1.