MAXWELL - DIRAC EQUATIONS 43
If we now take \vi\ |z/21, we obtain the same estimate (2.88), so (2.88) is true for
I^ll + N N,N1. Prom (2.85) and (2.88) it follows that
/(M,^1,^2,^1,^2) C'N(||uiH^\\U2\\EN + K b J M b J , (2.89)
with the range of multi-indices defined in (2.85).
Let \v\\ |v21 in (2.86). We can choose \i\ and fj,2, without changing the value of
J(/i,Mi,^2,^1,^2), such that |/xi| |i/i|, |/X21 N \i/i\. Inequality (2.83) then gives
||M
M
|M
Ml
^/il|M
M2
^a
2
|||
L 2
q^ilKM^^/i.OJII^HM^^aall^ (2.90)
where we have used Theorem 2.9. An estimate for the second term in (2.86) is obtained
by permuting u\ and 1x2 in (2.90). This gives
J(fi,m,fi2l 1/1,i/2) CAT(||WI||E J M * , , , + IMIi? , . M * ), \v\\ |^| .
Since |i/i| + I1/2I N and iV + 1 \v\\ N in this inequality, Corollary 2.6 gives
J(M,/xi,M2,^i,z/2) CN(Wm\\Ei\\u2\\EN + I M J E J M J S J , N W2V (2.91)
Let \ui\ 11/21
m
(2.86). As above we can choose |//i| \i/i\ and |/X21 N \i/i\.
Application of the inequality (2.84) to the two terms in (2.86) gives
j(M,Ml,M2,i,1^2)cw(||(/1,o)||M
JMJT*'2,
J M 2 C 7 ,
\ l^ll l + N-\u1\ N-\u1\
+
ll(/2,0)||Mki|||a1||^2-l„llII^II^Xll).
N * l«*l-
It now follows from the definition of the norms that
J ( M , / ^ 2 , ^
2
) C
n
( | M
E
IMir
1 7 2
,
,\M\EI~'
(2-92)
\ l"ll JV + l - | ^ l l J V - k i l
+ IMU
JKIir1/',
,IMIBW27,)'
MN -
Application of Corollary 2.6 to the terms (||^i|U \\u2\\E
)P~1/2
and
l"ll A T + I - I J / J I
(||txi||E \\u2\\E )ZI2~P and the corresponding terms with u\ and u2 permuted in
\vx\ iV-|i/2|
(2.92), gives
J{n,m,ti2,vuv2)
C W ( N I W M J S
+ 1
+ \\UI\\EN+1\\ME)P~1/2 (2-93)
(Iltiibjltiall^ + KII^NII^)
372
^, M N -
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