26 I. Theory of Distributions

7.2. Convolution of f ∈ D and g ∈ E .

Definition 7.2. The convolution of f ∈ D and g ∈ E is defined by the

formula:

(7.9) (f ∗ g)(ψ) = f(g1 ∗ ψ), ∀ψ ∈ D,

where g1 is the distribution obtained from g by the change of variables

x → −x (cf. (3.8)):

(7.10) g1(ϕ) = g(ϕ(−x)).

Since g1 has a compact support, we have that g1 ∗ ψ = g1(ψ(x − ·)) ∈ D

(see Proposition 7.1). It follows from (7.3) and Proposition 6.1 that

∂k

∂xk

(g1 ∗ ψ) ≤ C|[ψ]|m+|k|,BR

for some m and C.

Consider a sequence ψn → ψ in D. We assume that supp ψ ⊂ BR and

supp ψn ⊂ BR. Then for any k,

max

x∈Rn

∂k

∂xk

(g1 ∗ ψn) −

∂k

∂xk

(g1 ∗ ψ) ≤ C|[ψn − ψ]|m+|k|,BR → 0

as n → ∞. By Proposition 7.1, supp(g∗ψn) ⊂ BR1+R, where supp g1 ⊂ BR1 .

Therefore g1 ∗ ψn → g1 ∗ ψ in D and f(g1 ∗ ψn) → f(g1 ∗ ψ).

Thus f ∗ g is a linear continuous functional on D.

It follows from Proposition 7.2 that Definition 7.2 agrees with Definition

7.1 for g ∈ C0

∞.

Proposition 7.3. Let f ∈ D and g ∈ E . Then for any k = (k1,...,kn),

(7.11)

∂k

∂xk

(f ∗ g) =

∂kf

∂xk

∗ g = f ∗

∂kg

∂xk

.

Proof: Indeed

∂k(f

∗ g)

∂xk

(ϕ) =

(−1)|k|(f

∗ g)

∂kϕ

∂xk

=

(−1)|k|f

g1 ∗

∂kϕ

∂xk

=

(−1)|k|f

∂k

∂xk

(g1 ∗ ϕ) =

∂kf

∂xk

(g1 ∗ ϕ) =

∂kf

∂xk

∗ g (ϕ)

and

∂k

∂xk

(f ∗ g)(ϕ) =

∂kf

∂xk

(g1 ∗ ϕ) =

(−1)|k|f

∂k

∂xk

(g1 ∗ ϕ)

=

(−1)|k|f

∂kg1

∂xk

∗ ϕ = f ∗

∂kg

∂xk

(ϕ),

where we used (7.4) and that

∂kg1

∂xk

=

(−1)k

∂kg

∂xk

.