Potential theory Proofs of Theorem 3.1.5, Lemma 3.1.6
Lemma 3.4.4 For i, j
∈ and α, β = 1, . . . , d, let b
i,α
and a
i j,αβ
be functions in
C
K satisfying uniform bounds
b
i,α ∞
≤ M
1
∀i ∈ , α = 1, . . . , d a
i j,αβ ∞
≤ M
2
∀i, j ∈ , α, β = 1, . . . , d. 3.4.17
Then for each f ∈
C
2 sum
K and for all finite
n
⊂ with
n
↑ , the limit lim
n →∞
i ∈
n
, α
b
i,α
x
∂ ∂
x
α i
+
i j ∈
n
, αβ
a
i j,αβ
x
∂
2
∂ x
α i
∂ x
β j
f x 3.4.18
exists in the topology on
C
K and does not depend on the choice of the
n
.
Proof of Lemma 3.4.4: We treat only the convergence of the first order derivatives; the argument for second order derivatives is then the same. Define operators A
n
by A
n
f x : =
i ∈
n
,α
b
i,α
x
∂ ∂
x
α i
f x. 3.4.19
For each n ≤ m and for each f ∈
C
2 sum
K we have
A
n
f − A
m
f
∞
= sup
x ∈K
i ∈
m
\
n
, α
b
i,α
x
∂ ∂
x
α i
f x ≤ M
1
sup
x ∈K
i ∈\
n
, α ∂
∂ x
α i
f x .
3.4.20
The functions g
n
: K → l
1
{1, . . . , d} × given by g
n
x : =
i ∈\
n
, α ∂
∂ x
α i
f x 3.4.21
are continuous functions decreasing to zero as n → ∞, and hence by Dini’s the-
orem g
n
→ 0 uniformly. Thus we see by 3.4.20 that the sequence A
n
f is a Cauchy sequence in the Banach space
C
K .
To see that the limit does not depend on the summation order, observe that for each x
∈ K
i,α
b
i,α
x
∂ ∂
x
α i
f x ≤
M
1 i,α
∂ ∂
x
α i
f x ∞.
3.4.22 This means that we can write
lim
n →∞
A
n
f x =
i,α
b
i,α
x
∂ ∂
x
α i
f x, 3.4.23
where the sums are pointwise absolutely convergent so that the result does not depend on the summation order.
For each i ∈ and α, β ∈ {1, . . . , d} the maps
x →
j
a j − ix
α j
− x
α i
x → w
αβ
x
i
3.4.24 are continuous on K
and satisfy the uniform bounds sup
x ∈K
j
a j − ix
α j
− x
α i
≤
k
ak sup
α
′
sup
y,z ∈K
y
α
′
− z
α
′
sup
x ∈K
w
αβ
x
i
≤ sup
α
′
β
′
w
α
′
β
′
∞
. 3.4.25
Therefore, by Lemma 3.4.4, the operator A
′
f x : =
i,α j
a j − ix
α j
− x
α i
∂ ∂
x
α i
+
i,αβ
w
αβ
x
i ∂
2
∂ x
α i
∂ x
β i
f x 3.4.26 is well-defined on
C
2 sum
K , where the infinite sums are convergent in
C
K and
the result is independent of the summation order. We now show that A is a core for A
′
.
Lemma 3.4.5 Let A be the closure of the operator A in 3.1.10, and let
D
A be its domain. Then
C
2 sum
K ⊂
D
A and A f x
=
i,α j
a j − ix
α j
− x
α i
∂ ∂
x
α i
+
i,αβ
w
αβ
x
i ∂
2
∂ x
α i
∂ x
β i
f x 3.4.27 for each f
∈
C
2 sum
K .
Proof of Lemma 3.4.5: We have to show that for each f ∈
C
2 sum
K there exist
f
n
∈
D
A such that f
n
→ f and A f
n
→ A
′
f , with A
′
as in 3.4.26. Fix f
∈
C
2 sum
K and define π
n
, f
n
as in 3.2.1 and 3.2.2, so that f
n
→ f . It is easy to see that there exists a constant C such that
A f
n
x − A
′
f x
∞
≤ C sup
x ∈K
i,α ∂
∂ x
α i
f
n
x −
∂ ∂
x
α i
f x +
i,αβ ∂
2
∂ x
α i
∂ x
β i
f
n
x −
∂
2
∂ x
α i
∂ x
β i
f x ≤ C sup
x ∈K
i ≤n,α
∂ ∂
x
α i
f π
n
x −
∂ ∂
x
α i
f x +
i n,α ∂
∂ x
α i
f x +
i ≤n,αβ
∂
2
∂ x
α i
∂ x
β i
f π
n
x −
∂
2
∂ x
α i
∂ x
β i
f x +
i n,αβ ∂
2
∂ x
α i
∂ x
β i
f x .
3.4.28
By the compactness of K , the maps
x →
∂ ∂
x
α i
f x
α =1,...,d
i ∈
x →
∂
2
∂ x
α i
∂ x
β j
f x
α,β =1,...,d
i, j ∈
3.4.29 are uniformly continous with respect to the norm on the spaces l
1
{1, . . . , d} × and l
1
{1, . . . , d}
2
×
2
. This implies that lim
n →∞
sup
x ∈K
i,α ∂
∂ x
α i
f π
n
x −
∂ ∂
x
α i
f x =
0, 3.4.30
and similarly for second derivatives. Finally, by Dini’s theorem see 3.4.21 lim
n →∞
sup
x ∈K
i n,α ∂
∂ x
α i
f x =
0, 3.4.31
and similarly for second derivatives, so A f
n
→ A
′
f .