MAXWELL - DIRAC EQUATIONS
39
Proof. Let a be such that T
n 5 a
(/,/) is finite. Then the hypotheses of Theorem 2.13 are
satisfied and the right-hand side of the inequality in that theorem is bounded by CnTn^a,
after redefining the constants. This proves the corollary.
Later we shall need estimates of weighted supremum norms of solution of the ho-
mogeneous wave equation. These estimates follow directly from Kirchhoff's formula and
Theorem 2.12.
Proposition 2.15. Ifn 1, (/,/) G M^+2, 1/2 p 1, then the solution u of the wave
equation \2u = 0, with initial conditions u(0jx) f{x), J^(£,x)|t=o = f(x) satisfies
(l + \x\ +
\t\)3'2-"\u(t,x)\
+ (l + \x\ + \t\)
£ (I
+
| | *
M
Z | | )
1 / 2
- '
+ M +
' F ( | ) V M )
l|i/|+Kn
CnMfJ)\\M
Proof Give first fje 5(R3,E4). Then da(d/dt)mu(tJx) = uatm(t,x),a = (ai,a
2
,a
3
),
is the solution of the wave equation with initial conditions /ajTn, /a,m, where (/a,m fa,m)
Tffiif, /), w h e r e P = (m, *i, a2, a3), ^ = P^P^P^Pt is an element in the enveloping
algebra U(p) and where
Tjf1
denotes the restriction to the space
Mp
of the linear Lie
algebra representation Tx as defined by (1.5). As the initial data are in
5(R3,R4)
it is
sure that the solution is given by Kirchhoff's formula (cf. [18]):
up(t,x) = (47r)-1 I (fp{x + tv) +tV(Jdifpix + tu) +tfp(x + tu))duj. (2.70)
J\"\=i
v
V
J
Let r
n j a
be as in Corollary 2.14 and let
ja{t,x) =
(4:7r)-1
[ (l + \x +
ujt\2)-a!2duj,
aeR. (2.71)
J\u\ = l
It follows from (2.70) and (2.71) that
M*,X)| ro,a(fPiM0a(t,x) + \t\ja+!(t,x)). (2.72)
The easy explicit evaluation of the integral in (2.71) in polar coordinates leads to
ja{t,x) Ca(l + \x\ +
\t\)~a
for 0 a 2, (2.73a)
ja(t,x) Ca(l + \x\ +
\t\)-2(l
+ \\t\-
\x\\)2-a
for a 2. (2.73b)
We choose a = 3/2 - p + |/?| in (2.72), where 1/2 p 1. Then 1/2 + |/?| a 1 + |/3|
and it follows from (2.73) that
*.(*,*) + |*|j«+i(*,a;) CpM(l + \x\ + \t\)-W2-»\ \/3\ = 0, (2.74a)
3a(t,x) + \t\ja+1(t,x) CpM(l + \x\ + \t\)-\l + \\t\- IxWyM-P+W, |/3| l,(2.74b)
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