24 I. Theory of Distributions

Remark 6.1. Let g ∈ D (BR ), where BR ⊂ BR (cf. the definition of D (Ω)

in §2). Then, as in (6.17), the restriction of g to BR can be represented in

the form

g(ϕ) =

m

|k|=0

∂kgk

∂xk

(ϕ), ∀ϕ ∈ C0

∞(BR),

where the gk are regular functionals in D (BR) corresponding to the func-

tions gk(x) continuous in BR.

7. The convolution of distributions

7.1. Convolution of f ∈ D and ϕ ∈ C0

∞.

If ϕ ∈ C0

∞(Rn),

then, obviously, ϕ(x − y) ∈ C0

∞(Rn)

as a function of y for

any fixed x ∈

Rn.

Definition 7.1. We define the convolution of a distribution f ∈ D and a

function ϕ ∈ D by the formula:

(7.1) (f ∗ ϕ)(x) = f(ϕ(x − ·)),

where f acts on ϕ(x − y) as a function of y.

Since ϕ(xm −y) → ϕ(x0 −y) in D as xm → x0 and since f is a continuous

functional, we have that (f ∗ ϕ)(x) is a continuous function of x. Also we

have that

(7.2)

ϕ(x − y , xn + hn − yn) − ϕ(x − y , xn − yn)

hn

=

∂ϕ(x − y , xn + θhn − yn)

∂xn

→

∂ϕ(x − y , xn − yn)

∂xn

in D,

as hn → 0 and (x , xn) is fixed. Here x = (x1,...,xn−1),y = (y1,...,yn−1).

Therefore, the limit

∂(f ∗ ϕ)

∂xn

= lim

Δxn→0

(f ∗ ϕ)(x , xn + Δxn) − (f ∗ ϕ)(x , xn)

Δxn

= lim

Δxn→0

f

∂ϕ(x − ·,xn + θΔxn − ·)

∂xn

exists and

∂(f ∗ ϕ)

∂xn

= f

∂ϕ(x − ·)

∂xn

.

Analogously, one can show that each partial derivative

∂k(f∗ϕ)

∂xk

exists and is

continuous, and

(7.3)

∂k(f

∗ ϕ)

∂xk

= f

∂kϕ(x

− ·)

∂xk

= f ∗

∂kϕ

∂xk

.