2 I. Theory of Distributions
Laplace equations the traditional approach is simpler, and we offer this
approach as problems (with some hints).
1. Spaces of infinitely differentiable functions
Denote by C0
∞(Rn)
the space of infinitely differentiable complex-valued func-
tions in
Rn
with compact supports, i.e., ϕ(x) C0
∞(Rn)
if ϕ(x) has contin-
uous partial derivatives of every order and ϕ(x) = 0 when |x| R for some
R depending of ϕ(x). For example,
(1.1) χ(x) =
e

1
1−|x|2 if |x| 1,
0 if |x| 1, |x| = x1
2
+ · · · + xn,2
is a C0

function.
Definition 1.1. Let f(x) be a continuous function in
Rn.
The support of
f(x) is the closure of the set where f(x) = 0.
We denote the support of f(x) by supp f.
Example 1.1.
a) The support of χ(x) is the closed ball of radius 1.
b) The support of f(x1) = x1
2
1 is
R1.
Definition 1.2. Let ϕ(x) be a measurable bounded and ψ(x) a Lebesgue
integrable functions in
Rn,
ψ
L1
=
Rn
|ψ(x)|dx +∞. Then the convo-
lution of ϕ(x) and ψ(x) (denoted by ψ)(x) ) is the following integral:
(1.2) ψ)(x) =
Rn
φ(x y)ψ(y)dy.
Proposition 1.1. If ϕ(x) L1 and ψ(x) L1, then the integral (1.2) exists
for almost all x
Rn,
ϕ ψ
L1(Rn)
and
(1.3) ϕ ψ
L1
ϕ
L1
ψ
L1
.
The proof of Proposition 1.1 is given at the end of this section.
1.1. Properties of the convolution.
a) ϕ ψ = ψ ϕ.
Proof: We have
(1.4) ψ)(x) =
Rn
ϕ(x y)ψ(y)dy.
Changing the variables x y = t, we get
Rn
ϕ(x y)ψ(y)dy =
Rn
ϕ(t)ψ(x t)dt = ψ ϕ.
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