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3.1
§3. Distributions. Examples and rules of calculus
3.1. Distributions.
The space C∞(Ω) is often denoted D(Ω) in the literature. The distribu-
0
tions are simply the elements of the dual space:
Definition 3.1. A distribution on Ω is a continuous linear functional on
C∞(Ω). The vector space of distributions on Ω is denoted D′(Ω). When
0
Λ∈D′(Ω), we denote the value of Λ on ϕ ∈ C∞(Ω) by Λ(ϕ) or hΛ;ϕi.
0
The tradition is here to take linear (rather than conjugate linear) func-
tionals. But it is easy to change to conjugate linear functionals if needed,
for ϕ 7→ Λ(ϕ) is a linear functional on C∞(Ω) if and only if ϕ 7→ Λ(ϕ) is a
0
conjugate linear functional.
See Theorem 2.5 (d) for how the continuity of a functional on C∞(Ω) is
0
checked.
The space D′(Ω) itself is provided with the weak∗-topology, i.e. the topol-
ogy defined by the system of seminorms p on D′(Ω):
ϕ
p : u 7→ |hu;ϕi|; (3.1)
ϕ
where ϕ runs through C∞(Ω). We here use Theorem B.5, noting that the
0
family of seminorms is separating (since u 6= 0 in D′(Ω) means that hu;ϕi 6= 0
for some ϕ).
Let us consider some examples. When f ∈ L1;loc(Ω), then the map
Λ : ϕ 7→ Z f(x)ϕ(x)dx; (3.2)
f
Ω
is a distribution. For we have on every Kj (cf. (2.4)), when ϕ ∈ C∞ (Ω),
Kj
Z Z
|Λ (ϕ)| = f(x)ϕ(x)dx ≤ sup|ϕ(x)| |f(x)|dx; (3.3)
f Kj Kj
so (2.15) is satisfied with N = 0 and c = kfk . Here one can in fact
j j L1(Kj)
identify Λ with f, in view of the following fact:
f
Lemma 3.2. When f ∈ L1;loc(Ω) with R f(x)ϕ(x)dx = 0 for all ϕ ∈
C∞(Ω), then f = 0.
0
Proof. Let ε > 0 and consider v (x) = (h ∗ f)(x) for j > 1=ε as in Lemma
j j
2.12. When x ∈ Ω , then h (x−y) ∈ C∞(Ω), so that v (x) = 0 in Ω . From
ε j 0 j ε
(2.46) we conclude that f = 0 in Ωε ∩ B(0;R). Since ε and R can take all
values in R+, it follows that f = 0 in Ω.
3.2
The lemma (and variants of it) is sometimes called “the fundamental
lemma of the calculus of variations” or “Du Bois-Reymond’s lemma”.
The lemma implies that when the distribution Λ defined from f ∈
f
L1;loc(Ω) by (3.2) gives 0 on all test functions, then the function f is equal
to 0 as an element of L (Ω). Then the map f 7→ Λ is injective from
1;loc f
L (Ω) to D′(Ω), so that we may identify f with Λ and write
1;loc f
L (Ω) ⊂ D′(Ω): (3.4)
1;loc
The element 0 of D′(Ω) will from now on be identified with the function 0
(where we as usual take the continuous representative).
Since Lp;loc(Ω) ⊂ L1;loc(Ω) for p > 1, these space are also naturally in-
jected in D′(Ω).
Remark 3.3. Let us also mention how Radon measures fit in here. The
space C0(Ω) of continuous functions with compact support in Ω is defined
0
in (C.7). In topological measure theory it is shown how the vector space
M(Ω) of complex Radon measures µ on Ω can be identified with the space
of continuous linear functionals Λ on C0(Ω) in such a way that
µ 0
Λ (ϕ) = Z ϕdµ for ϕ ∈ C0(Ω):
µ 0
suppϕ
Since one has that
|Λ (ϕ)| ≤ |µ|(suppϕ)·sup|ϕ(x)|; (3.5)
µ
∞ ′ ′
Λ is continuous on C (Ω), hence defines a distribution Λ ∈ D (Ω). Since
µ 0 µ
∞ 0 ◦ ′
C (Ω) is dense in C (Ω) (cf. Theorem 2.15 1 ), the map Λ 7→ Λ is injec-
0 0 µ µ
tive. Then the space of complex Radon measures identifies with a subset of
D′(Ω):
M(Ω)⊂D′(Ω): (3.6)
The inclusions (3.4) and (3.6) place L1;loc(Ω) and M(Ω) as subspaces of
D′(Ω). They are consistent with the usual injection of L1;loc(Ω) in M(Ω),
where a function f ∈ L (Ω) defines the Radon measure µ by the formula
1;loc f
µ (K)=Z fdx for K compact ⊂Ω: (3.7)
f
K
For, it is known from measure theory that
Z fϕdx=Z ϕdµf for allϕ∈C0(Ω) (3.8)
0
3.3
(hence in particular for ϕ ∈ C∞(Ω)), so the distributions Λ and Λ coin-
0 f µ
f
cide.
When f ∈ L (Ω), we shall usually write f instead of Λ ; then we also
1;loc f
write
Λ (ϕ) = hΛ ;ϕi = hf;ϕi = Z f(x)ϕ(x)dx: (3.9)
f f
Ω
Moreover, one often writes µ instead of Λ when µ ∈ M(Ω). In the following
µ
we shall even use the notation f or u (resembling a function) to indicate an
arbitrary distribution!
In the systematical theory we will in particular be concerned with the
inclusions
C∞(Ω)⊂L (Ω)⊂D′(Ω) (3.10)
0 2
(and other L2-inclusions of importance in Hilbert space theory). We shall
show how the large gaps between C∞(Ω) and L2(Ω), and between L2(Ω) and
0
D′(Ω), are filled out by Sobolev spaces.
Here is another important example.
Let x0 be a point in Ω. The map
δ : ϕ 7→ ϕ(x ) (3.11)
x 0
0
sending a testfunction into its value at x0 is a distribution, for it is clearly
a linear map from C∞(Ω) to C, and one has for any j, when suppϕ ⊂ Kj
0
(where Kj is as in (2.4)),
|hδ ; ϕi| = |ϕ(x )| ≤ sup{|ϕ(x)| | x ∈ K } (3.12)
x 0 j
0
(note that ϕ(x ) = 0 when x ∈= K ). Here (2.15) is satisfied with c = 1,
0 0 j j
Nj =0, for all j. In a similar way one finds that the maps
Λ : ϕ 7→ (Dαϕ)(x ) (3.13)
α 0
are distributions, now with c = 1 and N = |α| for each j. The distribution
j j
(3.11) is the famous “Dirac’s δ-function” or “δ-measure ”. The notation
measure is correct, for we can write
hδ ; ϕi = Z ϕdµ ; (3.14)
x x
0 0
where µ is the point measure that has the value 1 on the set {x } and the
x 0
0
value 0 on compact sets disjoint from x0. The notation δ-function is a wild
“abuse of notation” (see also (3.22)ff. later). Maybe it has survived because
3.4
it is so bad that the motivation for introducing the concept of distributions
becomes clear.
The distribution δ is often just denoted δ.
0
Still other distributions are obtained in the following way: Let f ∈
L1;loc(Ω) and let α ∈ Nn. Then the map
0
Λ :ϕ7→Z f(x)(Dαϕ)(x)dx; ϕ∈C∞(Ω); (3.15)
f;α 0
is a distribution, since we have for any ϕ ∈ C∞ (Ω):
Kj
Z Z
|hΛ ; ϕi| = f Dαϕdx ≤ |f(x)|dx· sup |Dαϕ(x)|; (3.16)
f;α Kj Kj x∈Kj
here (2.15) is satisfied with c = kfk and N =|α| for each j.
j L1(Kj) j
Onecanshowthatthemostgeneraldistributionsarenotmuchworsethan
this last example. One has in fact that when Λ is an arbitrary distribution,
then for any fixed compact set K ⊂ Ω there is an N (depending on K) and
a system of functions fα ∈ C0(Ω) for |α| ≤ N such that
hΛ;ϕi = X hfα;Dαϕi for ϕ∈C∞(Ω) (3.17)
K
|α|≤N
(the Structure Theorem). We shall show this later in connection with the
theorem of Sobolev in Chapter 6.
In the fulfillment of (2.15) one cannot always find an N that works for all
Kj ⊂ Ω (only one Nj for each Kj); another way of expressing this is to say
that a distribution does not necessarily have a finite order, where the concept
of order is defined as follows:
Definition 3.4. We say that Λ ∈ D′(Ω) is of order N ∈ N0 when the
inequalities (2.15) hold for Λ with N ≤ N for all j (but the constants c
j j
may very well depend on j). Λ is said to be of infinite order if it is not of
order N for any N; otherwise it is said to be of finite order. The order of
Λ is the smallest N that can be used, resp. ∞.
In all the examples we have given, the order is finite. Namely, L1;loc(Ω)
andM(Ω)definedistributionsoforder0(cf. (3.3), (3.5) and (3.12)), whereas
Λ and Λ in (3.13) and (3.15) are of order |α|. To see an example of a
α f;α
distribution of infinte order we consider the distribution Λ ∈ D′(R) defined
by
∞
X (N)
hΛ;ϕi = h1[N;2N];ϕ (x)i; (3.18)
N=1
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