10 I. Theory of Distributions

Definition 3.2. For any f ∈ D and any a(x) ∈

C∞,

af is the distribution

defined by the formula:

(3.5) af(ϕ) = f(aϕ), ∀ϕ ∈ D.

Note that the right hand side of (3.5) defines a linear continuous func-

tional on D. Indeed, aϕ ∈ D, supp(aϕ) ⊂ supp ϕ, and the Leibniz formula

gives that ϕm → ϕ in D implies that aϕm → aϕ in D.

Example 3.2. x1δ = 0. Indeed, x1δ(ϕ) = δ(x1ϕ) = 0ϕ(0) = 0 for ∀ϕ ∈ D.

3.3. Change of variables for distributions.

Let x = s(y) be a one-to-one

C∞

map of

Rn

onto

Rn:

(3.6) xk = sk(y1,...,yn), 1 ≤ k ≤ n.

We assume that the inverse map y =

s−1(x)

is also

C∞.

Denote by J(x) = det

∂s−1(x)

i

∂xj

n

i,j=1

the Jacobian of the inverse map

y =

s−1(x).

For a regular functional, after changing the variable y =

s−1(x)

we get:

(3.7)

Rn

f(s(y))ϕ(y)dy =

Rn

f(x)ϕ(s−1(x))|J(x)|dx.

Let (f ◦ s)(y) = f(s(y)). Then we can rewrite (3.7) in the following form:

(3.8) f ◦ s (ϕ) =

f(ϕ(s−1(x))|J(x)|).

Note that if ϕ ∈ D, then ψ(x) =

ϕ(s−1(x))J(x)

is also in D. Moreover, if

ϕn → ϕ in D, then ψn → ψ in D. Therefore the right hand side of (3.8) is

a linear continuous functional on D.

Hence for any f ∈ D we can define the change of variable x = s(y) in f

by formula (3.8).

Example 3.3. If x1 = s(y1) = ky1 + b is a linear map in

R1,

we get

f ◦ s (ϕ) = f ϕ

x1 − b

k

1

|k|

.

4. Convergence of distributions

Definition 4.1. We say that a sequence of distributions fn ∈ D converges

to a distribution f if

(4.1) fn(ϕ) → f(ϕ) for any ϕ ∈ D,

i.e., fn → f in D if (4.1) holds.