22 I. Theory of Distributions

Since |Rm(x)| ≤

C|x|m+1

and β(

x

ε

) = 0 for |x| ε, we have |β(

x

ε

)Rm(x)| ≤

Cεm+1.

Analogously, for any derivative of β(

x

ε

)Rm(x) we get

∂k

∂xk

β

x

ε

Rm(x) ≤

Cεm+1−|k|,

0 |k| ≤ m.

Thus

β

x

ε

Rm(x)

m,B1

≤ Cε.

Since f(β(

x)Rm(x))

ε

does not depend on ε, we have that f(β(

x)Rm(x))

ε

= 0.

Therefore

f(ϕ) =

m

|k|=0

dk

∂kϕ(0)

∂xk

=

m

|k|=0

(−1)|k|dk

∂kδ

∂xk

(ϕ).

6.2. Distributions with compact supports.

Denote by E ⊂ D the space of distributions with compact supports.

Proposition 6.3. Any distribution f can be represented as a limit in D of

distributions fn ∈ E .

Proof: Let β(x) be the same as in the proof of Theorem 6.2, and let

fn = β(

x

n

)f. Then supp fn ⊂ Bn and fn(ϕ) = f(β(

x

n

)ϕ) → f(ϕ), ∀ϕ, as

n → ∞.

Proposition 6.1 combined with the Hahn-Banach theorem allows one to

prove the following theorem:

Theorem 6.4. Any distribution with a compact support can be represented

in the form

f =

m

|k|=0

∂kfk

∂xk

,

where fk are regular functionals corresponding to the continuous functions

fk(x) with compact supports.

Proof: Let supp f ⊂ BR. By Proposition 6.1,

(6.12) |f(ϕ)| ≤ C|[ϕ]|m1,BR , ∀ϕ ∈ C0

∞(BR)

for some m1 ≥ 0. Denote by ϕ

p

the following norm:

(6.13) ϕ

2

p

=

p

|k|=0

BR

∂kϕ

∂xk

2

dx.

Note that

∂kϕ(x)

∂xk

=

x1

−∞

· · ·

xn

−∞

∂n

∂y1∂y2 · · · ∂yn

∂kϕ(y)

∂yk

dy.