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.
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