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 ).