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5.16. Lemma. Let p > 1 and α ∈Ê\(π/2)�. If v ∈ Xq (Σ), 1/p+1/q = 1, belongs to the cokernel
of
∂α : X 1,p (Σ) → X p ∈Ê
(Σ),
then v is smooth on Σ \ {0}, it solves the equation ∂v − αv = 0 with boundary conditions
v(s) ∈Ê, v(−s) ∈ iÊ∀s > 0,
(102)
v(s + i) ∀s ∈Ê,
and the function (T v)(ζ) = 2ζv(ζ2 ) is smooth on Cl(À+ ) ∩�1. In particular, v(z) = O(|z| −1/2 )
and Dv(z) = O(|z| −3/2 ) for z → 0.
Proof. Since v ∈ Xq (Σ) annihilates the image of ∂α, there holds
�
Re 〈v(z), ∂u(z) + αu(z)〉ds dt = 0, (103)
Σ
for every u ∈ X1,p (Σ). By letting u vary among all smooth functions in X1,p (Σ) which are
compactly supported in Σ \ {0}, the regularity theory for weak solutions of ∂ (the analogue of
Theorem 5.4) and a bootstrap argument show that v is smooth on Σ\{0} and it solves the equation
∂v − αv = 0 with boundary conditions (102). There remains to study the regularity of v at 0.
Set w(ζ) := (T v)(ζ) = 2ζv(ζ2 ). By (89), the function w is in Lq (À+ ∩�1). Let ϕ ∈
C∞ c (Cl(À+ ) ∩�1) be real onÊ+ and purely imaginary on iÊ+ . Then the function u defined by
u(ζ2 ) = ϕ(ζ) belongs to X1,p (Σ), and by (103) we have
�
0 = Re 〈v, ∂u + αu〉dsdt = 4Re
Σ
The above identity can be rewritten as
Re
�À+ ∩�1
�À+ ∩�1
= Re
|ζ 2 |〈 1 1
w(ζ), ∂ϕ(ζ) + αϕ(ζ)〉dσdτ
2ζ 2ζ
�À+ ∩�1
〈w(ζ), ∂ϕ(ζ)〉dσdτ = −Re
〈w(ζ), ∂ϕ(ζ) + 2αζϕ(ζ)〉dσdτ.
�À+ ∩�1
〈2αζw(ζ), ϕ(ζ)〉dσdτ,
so w is a weak solution of ∂w = 2αζw on Cl(À+ )∩�1 with real boundary conditions. Since w is in
L q (À+ ∩�1), Lemma 5.7 implies that w is in W 1,q (À+ ∩�1). In particular, w is square integrable
onÀ+ ∩�1, and so is the function
f(ζ) := e −αζ2 /2 w(ζ).
The function f is anti-holomorphic, it takes real values onÊ+ and on iÊ+ , so by a double Schwarz
reflection it can be extended to an anti-holomorhic function on�1\{0}. Since f is square integrable,
the singularity 0 is removable and f is anti-holomorphic on�1. Therefore
is smooth on Cl(À+ ) ∩�1, as claimed.
(T v)(ζ) = w(ζ) = e αζ2 /2 f(ζ)
We can finally prove the following Liouville type result:
5.17. Proposition. If 0 < α < π/2, the operator
is an isomorphism, for every 1 < p < ∞.
∂α : X 1,p (Σ) → X p (Σ)
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