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5.15. Lemma. Let p > 1 and α ∈Ê\(π/2)�. If u belongs to the kernel of
∂α : X 1,p (Σ) → X p (Σ),
then u is smooth on Σ \ {0}, it satisfies the boundary conditions pointwise, and the function
(Ru)(ζ) = u(ζ 2 ) is smooth on Cl(À+ ) ∩�1. In particular, u is continuous at 0, and Du(z) =
O(|z| −1/2 ) for z → 0.
Proof. The regularity theory for weak solutions of ∂ on�and on the half-planeÀ(Theorem 5.4)
implies - by a standard bootstrap argument - that u ∈ C ∞ (Σ \ {0}). We just need to check the
regularity of u at 0.
Consider the function f(ζ) := e αζ2 /2 u(ζ 2 ) onÀ+ ∩�1. Since
∂f(ζ) = 2ζe αζ2 /2 � ∂u(ζ 2 ) + αu(ζ 2 ) � = 0,
f is holomorphic onÀ+ ∩�1. Moreover, by (91) the function f belongs to W 1,p (À+ ∩�1), and in
particular it is square integrable. The function f is real onÊ+ and purely imaginary on iÊ+ , so a
double Schwarz reflection produces a holomorphic extension of f to�1 \ {0}. Such an extension
of f is still square integrable, so the singularity 0 is removable and the function is holomorphic on
the whole�1. It follows that
is smooth on Cl(À+ ) ∩�1, as claimed.
(Ru)(ζ) = u(ζ 2 ) = e −αζ2 /2 f(ζ)
The real Banach space X p (Σ) is the space of L p functions with respect to the measure defined
by the density
ρp(z) :=
� 1 if z ∈ Σ \�r,
|z| p/2−1 if z ∈ Σ ∩�r.
So the dual of Xp (Σ) can be identified with the real Banach space
�
v ∈ L 1 �
loc(Σ,�) | |v| q �
ρp(z)dsdt < +∞ , where 1 1
+ = 1, (100)
p q
Σ
by using the duality paring
�
� � p ∗ p
X (Σ) × X (Σ) →Ê, (v, u) ↦→ Re 〈v, u〉ρp(z)dsdt.
Σ
We prefer to use the standard duality pairing
�
� � p ∗ p
X (Σ) × X (Σ) →Ê, (w, u) ↦→ Re 〈w, u〉dsdt. (101)
With the latter choice, the dual of Xp (Σ) is identified with the space of functions w = ρp(z)v,
where v varies in the space (100). From 1/p + 1/q = 1 we get the identity
�w� q
�
Xq = |w|
Σ\�r
q �
dsdt + |w|
Σ∩�r
q |z| q/2−1 dsdt
�
= |v| q �
dsdt + |v| q |z| (p/2−1)q |z| q/2−1 dsdt
�
=
Σ\�r
Σ\�r
|v| q �
dsdt +
Σ∩�r
Σ∩�r
|v| q |z| p/2−1 �
dsdt =
Σ
Σ
|v| q ρp(z)dsdt,
which shows that the standard duality paring (101) produces the identification
� � p ∗
X (Σ) ∼= q 1 1
X (Σ), for + = 1.
p q
Therefore, we view the cokernel of ∂α : X 1,p (Σ) → X p (Σ) as a subspace of X q (Σ). Its elements
are a priori less regular at 0 than the elements of the kernel:
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