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)