7. The convolution of distributions 29
∂wr
∂xn
=
∑r−1
k=0
pk(x,

∂x
)wk+1 + f,
∂wk
∂xn
= wk+1, 1 k r 1, and the
equation Pu = f can be written as a system
(7.17)
∂w
∂xn
= A x,

∂x
w + Φ,
where w = (w1,...,wr),A(x,

∂x
) is an r × r matrix of differential operators
in

∂x
, and Φ = (f,..., 0).
It follows from Remark 6.1 that the restriction of the distribution w to
V × (a, b) can be represented in the form
(7.18) w =
m
|k|=0
∂k
∂xk
vk,
where vk are regular functionals corresponding to vector-valued functions
vk(x) continuous in V × [a, b]. We rewrite (7.18) in the following form:
(7.19) w =
m
j=0
∂j
∂xnj
hj,
where hj are linear functionals continuous in xn [a, b] with values in D (V ).
We call m the order of w. A negative order of w means that w is
continuously differentiable in xn up to the order |m| with values in D (V )).
Note that A(x,

∂x
)w has the same order as w since A(x,

∂x
) does not
contain derivatives with respect to xn. We denote the right hand side of
(7.17) by b:
∂w
∂xn
= b.
Note that the order of b is m, i.e., b =
∑m
j=0
∂j
bj
∂xnj
. Let
b(1)
=
m
j=0
∂j
∂xn
j
xn
a
bjdyn.
Then
b(1)
has order m 1 and
∂b(1)
∂xn
= b since

∂xn
xn
a
bjdyn = bj. Hence
∂(w−b(1))
∂xn
= 0 in V × (a, b). By Proposition 7.5, w
b(1)
=
b(0)
does not
depend on xn, i.e., the order of
b(0)
is −∞. Therefore w =
b(1)
+
b(0)
has
order m 1. Repeating the same argument with w of order m 1, we get
that the order of w is m 2. After N steps we get that the order of w is
m N, ∀N. Therefore w
C∞
in xn (a, b) with values in D (V ).
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