20 I. Theory of Distributions

Substituting (5.27) in (5.24) we get in the original system of coordinates

Rn

αj(x)ϕ(x)dx

S(x) ± i0

= p.v.

Rn

αj(x)ϕ(x)dx

S(x)

∓ iπ

S(x)=0

αj(x)ϕ(x)dσ

|∇S(x)|

.

Summing over j, we obtain

1

S(x) ± i0

= p.v.

1

S(x)

∓ iπδ(S),

where δ(S) is the delta-function of the surface S(x) = 0:

δ(S)(ϕ) =

S(x)=0

ϕ(x)

|∇S(x)|

dσ.

Note, in particular, that

δ(|x|2

−

k2)

=

1

2k

δ(|x| − k),

since

|∇x(|x|2

−

k2)|

= 2|x| = 2k for

|x|2

−

k2

= 0 and |∇x(|x| − k)| = 1 on

|x| = k.

6. Supports of distributions

Let f be a distribution. We say that f = 0 on an open set U ⊂

Rn

if

f(ϕ) = 0 for any ϕ ∈ C0

∞

with support in U. Let Umax be the largest open

set where f = 0. Then the complement of U is called the support of f:

(6.1) supp f =

Rn

\ Umax.

Example 6.1.

a) supp δ = {0} as the support of

∂kδ

∂xk

for any k.

b) supp δ(S) = {x : S(x) = 0}, where δ(S) is defined as in §5.

c) supp x+

λ

= [0, +∞).

For any ball BR and any m, we introduce the norms:

(6.2) |[ϕ]|m,BR =

m

|k|=0

max

x∈

¯

B

R

∂kϕ

∂xk

,

where ϕ ∈ C0

∞(BR),

i.e., supp ϕ ⊂ BR and

¯

B

R

is the closure of BR.

6.1. General form of a distribution with support at 0.

Proposition 6.1. For any distribution f ∈ D and for any ball BR, there

exist m and C depending on f and BR such that

(6.3) |f(ϕ)| ≤ C|[ϕ]|m,BR , ∀ϕ ∈ C0

∞(BR).