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Up to multiplying u by U, we can replace the boundary conditions (82) by
u(s) ∈Ên ∀s ∈ [0, +∞[, u(it) ∈ Y ∀t ∈ [0, +∞[, (83)
where Y is a totally real n-dimensional subspace of�n such that Y = Y . Define a�n-valued map
v on the right half-plane {Re z ≥ 0} by Schwarz reflection,
�
u(z) if Imz ≥ 0,
v(z) :=
u(z) if Imz ≤ 0.
By (83) and by the fact that Y is self-conjugate, v belongs to W 1,p ({Re z > 0},�n ), and satisfies
Moreover,
and since ∂v(z) = ∂u(z) for Imz ≤ 0,
v(it) ∈ Y ∀t ∈Ê.
�∇v� p
Lp ({Re z>0}) = 2�∇v�p Lp ({Re z>0, Im z>0}) ,
�∂v� p
Lp ({Re z>0}) = 2�∂v�p Lp ({Re z>0, Im z>0}) .
Then (81) follows from the Calderon-Zygmund estimate on the half plane with totally real boundary
conditions (Theorem 5.3).
Similarly, Theorem 5.4 has the following consequence about regularity of weak solutions of ∂
on the upper right quadrantÀ+ :
5.7. Lemma. Let V and W be partially orthogonal linear subspaces ofÊn . Let u ∈ L p (À+ ,�n ),
f ∈ L p (À+ ,�n ), 1 < p < ∞, be such that
Re
�À+
�À+
〈u, ∂ϕ〉dsdt = −Re 〈f, ϕ〉dsdt,
for every ϕ ∈ C ∞ c (�,�n ) such that ϕ(Ê) ⊂ N ∗ V , ϕ(iÊ) ⊂ N ∗ W. Then u ∈ W 1,p (À+ ,�n ),
∂u = f, the trace of u onÊtakes values into N ∗ V , and the trace of u on iÊtakes values into
(N ∗ W) ⊥ = N ∗ (W ⊥ ) = iN ∗ W.
Proof. By means of a linear unitary transformation, as in the proof of Lemma 5.6, we may assume
that V = N ∗ V =Ên . A Schwarz reflection then allows to extend u to a map v on the right
half-plane {Re z > 0} which is in L p and is a weak solution of ∂v = g ∈ L p , with boundary
condition in iN ∗ W on iÊ. The thesis follows from Theorem 5.4.
We are now interested in studying the operator ∂ on the half-planeÀ, with boundary conditions
u(s) ∈ N ∗ V, u(−s) ∈ N ∗ W ∀s > 0,
where V and W are partially orthogonal linear subspaces ofÊn . Taking Lemmas 5.6 and 5.7 into
account, the natural idea is to obtain the required estimates by applying a conformal change of
variable mapping the half-planeÀonto the the upper right quadrantÀ+ . More precisely, let R
and T be the transformations
R : Map(À,�n ) → Map(À+ ,�n ), (Ru)(ζ) = u(ζ 2 ), (84)
T : Map(À,�n ) → Map(À+ ,�n ), (T u)(ζ) = 2ζu(ζ 2 ), (85)
where Map denotes some space of maps. Then the diagram
Map(À,�n )
⏐
∂
−−−−→ Map(À,�n )
⏐
�
�R T
Map(À+ ,�n )
∂
−−−−→ Map(À+ ,�n )
58
(86)