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commutes. By the elliptic estimates of Lemma 5.6, suitable domain and codomain for the operator
on the lower horizontal arrow are the standard W 1,p and Lp spaces, for 1 < p < ∞. Moreover, if
u ∈ Map(À,�n ) then
�Ru� p
Lp (À+ ) = 1
�À1
4 |z| |u(z)|p dsdt, (87)
�D(Ru)� p
Lp (À+ ) = 2 p−2
�À|Du(z)| p |z| p/2−1 dsdt, (88)
�T u� p
Lp (À+ ) = 2 p−2
�À|u(z)| p |z| p/2−1 dsdt. (89)
Note also that by the generalized Poincaré inequality, the W 1,p norm onÀ+ ∩�r, where�r denotes
the open disk of radius r, is equivalent to the norm
and the ˜ W 1,p norm of Ru is
�v� p
˜W 1,p := �Dv�p
(À+ ∩�r) Lp (À+ ∩�r) +
�À+
|v(ζ)|
∩�r
p |ζ| p dσdτ, (90)
�Ru� p
˜W 1,p �À∩�r
1
=
(À+ ∩�r) 4 2
|u(z)| p |z| p/2−1 dsdt + 2 p−2
�À∩�r2 |Du(z)| p |z| p/2−1 dsdt. (91)
So when dealing with bounded domains, both the transformations R and T involve the appearance
of the weight |z| p/2−1 in the L p norms. Note also that when p = 2, this weight is just 1, reflecting
the fact that the L 2 norm of the differential is a conformal invariant.
By the commutativity of diagram (86) and by the identities (88), (89), Lemma 5.6 applied to
Ru implies the following:
5.8. Lemma. Let V and W be partially orthogonal linear subspaces ofÊn . For every p ∈]1, +∞[,
there exists a constant c = c(p, n) such that
�À|∇u(z)| p |z| p/2−1 ds dt ≤ c p
�À|∂u(z)| p |z| p/2−1 ds dt
for every compactly supported map u : Cl(À) →�n such that ζ ↦→ u(ζ 2 ) is smooth on Cl(À+ ), and
u(s) ∈ N ∗ V ∀s ∈] − ∞, 0], u(s) ∈ N ∗ W ∀s ∈ [0, +∞[.
5.3 Strips with jumping conormal boundary conditions
Let us consider the following data: two integers k, k ′ ≥ 0, k + 1 linear subspaces V0, . . .,Vk of
Ên such that Vj−1 and Vj are partially orthogonal, for every j = 1, . . .,k, k ′ + 1 linear subspaces
′ , . . . , V k ofÊn ′ such that V j−1 and V ′
j are partially orthogonal, for every j = 1, . . .,k′ , and real
numbers
V ′
0
−∞ = s0 < s1 < · · · < sk < sk+1 = +∞, −∞ = s ′ 0 < s′ 1 < · · · < s′ k ′ < −s′ k ′ +1
Denote by V the (k + 1)-uple (V0, . . . , Vk), by V ′ the (k ′ + 1)-uple (V ′ ′
0 , . . . , V ), and set
Let Σ be the closed strip
S := {s1, . . .,sk, s ′ 1 + i, . . .,s′ k ′ + i}.
Σ := {z ∈�|0≤Im z ≤ 1}.
k
= +∞.
The space C ∞ S (Σ,�n ) is the space of maps u : Σ →�n which are smooth on Σ \S , and such that
the maps ζ ↦→ u(sj + ζ 2 ) and ζ ↦→ u(s ′ j + i − ζ2 ) are smooth in a neighborhood of 0 in the closed
upper-right quadrant
Cl(À+ ) = {ζ ∈�|Re ζ ≥ 0, Im ζ ≥ 0}.
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