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)

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).
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