6. Supports of distributions 23

Therefore, by the Cauchy-Schwartz inequality,

(6.14) |[ϕ]|m1,BR

2

≤ C1 ϕ

2

m1+n, ∀ϕ ∈ C0

∞(BR),

i.e.,

(6.15) |f(ϕ)| ≤ C2 ϕ m1+n, ∀ϕ ∈ C0

∞(BR).

Denote by L2(BR) the space of vector-valued functions g(x) = {gk(x) ∈

L2(BR), k = (k1,...,kn), 0 ≤ |k| ≤ m1 + n}, with the norm g

2

0

=

∑

m1+n

|k|=0

gk

2.

0

By the Riesz theorem (see, for example, [R]), any bounded

linear functional Φ on L2(BR) has the form Φ(g) =

∑m1+n(gk,hk),

|k|=0

where

hk ∈ L2(BR) and (gk,hk) is the scalar product in L2(BR). Let L be a linear

(not closed) subspace of L2(BR) consisting of all vector-valued functions of

the form {

∂kϕ

∂xk

, ϕ ∈ C0

∞(Bk),

0 ≤ |k| ≤ m1 + n}. It follows from (6.15) that

f(ϕ) defines a bounded linear functional Ψ on L ⊂ L(BR). Extending Ψ as

a bounded linear functional to L2(BR) (the Hahn-Banach theorem; see, for

example, [R]) we get {ψk(x), ψk(x) ∈ L2(BR), 0 ≤ |k| ≤ m1 + n} so that

Ψ(g) =

∑m1+n(gk,ψk).

|k|=0

In particular,

(6.16) f(ϕ) =

m1+n

|k|=0

∂kϕ

∂xk

, ψk =

m1+n

|k|=0

BR

∂kϕ(x)

∂xk

ψk(x)dx.

Let

ψk1(x) =

x1

a−1

· · ·

xn

an

ψk(y)dy1 · · · dyn,

where (a1,...,an) is a point in BR. Then ψk1(x) are continuous in BR and

ψk(x) =

∂nψk1(x)

∂x1 · · · ∂xn

in BR.

Therefore we have

(6.17) f(ϕ) =

m

|k|=0

BR

∂kϕ(x)

∂xk

fk1(x)dx, ∀ϕ ∈ C0

∞(BR),

where fk1(x) are continuous in BR, m = m1 +2n. Since supp f ⊂ BR, there

exists χ(x) ∈ C0

∞(BR)

equal to 1 in a neighborhood of supp f. Therefore,

for any ϕ ∈ C0

∞(Rn),

we have from (6.17):

(6.18)

f(ϕ) = f(χϕ) =

m

|k|=0

fk1

∂k(χϕ)

∂xk

=

m

|k|=0

∂fk

∂xk

(ϕ), ∀ϕ ∈ C0

∞(Rn),

where fk are regular functionals corresponding to continuous functions fk(x)

in

Rn

with supp fk ⊂ BR.