1.3 Unitary representations of Fuchsian groups
2.1.6 Poisson operator
i.e. theH1-closure of the space of smooth, compactly supported in the interior ofM, sections.
2.1.6 Poisson operator
The last kind of operator we will need to define in this section is the so-calledPoisson operator.
Their aim is to provide a way to extend sections of the restricted bundle E|∂M defined over the compact boundary∂M of a (possibly noncompact) Riemann surfaceM.
Poisson operator for the Chern Laplacian. The idea behind this first type of Poisson oper-ator is to take a sectionsofE|∂M over∂M and to solve the followingDirichlet problem
(
∆g,hE +z
v = 0
γ∂Mv = s
,
for sectionsv ofE overM which are regular enough, where the first equality is to be understood in a distributional sense. The resolution of this problem will be done in a way similar to the one presented by Burghelea, Friedlander, and Kappeler in [20, Sec. 2.7].
Definition 2.1.54. Letzbe a complex number lying inC\R−. We define the applicationΦz by Φz : H2 (M, g),(E, h),∆g,hE
−→ L2((M, g),(E, h))⊕ H3/2 (∂M, g), E|∂M, h
v 7−→ ∆(M,g),(E,h)+z
v, γ∂Mv
.
As can be gathered from the statement of the Dirichlet problem we aim to solve, the first step will be to prove that everyΦz is bijective.
Proposition 2.1.55. For any complex numberz∈C\R−, the applicationΦz is an isomorphism.
Proof. The fact that Φz is linear directly stems from the definition. We now note that Φz is injective, as anH2-sectionv sent to 0byΦzsolves the problem
( ∆g,hE +z
v = 0
γ∂Mv = 0 .
Such a section then belongs to the intersection of Sobolev spaces D = H2((M, g),(E, h),∆g,hE
∩H01 (M, g),(E, h),∇(E,h) ,
which is the domain of the Chern Laplacian with Dirichlet boundary conditions. The operator
∆g,hE +z : D −→ L2((M, g),(E, h))
being invertible, the sectionv considered above has to be zero, which yields the injectivity of Φz. We now move to prove the surjectivity of this application. To do that, we consider an element
(u, w) ∈ L2((M, g),(E, h))⊕H3/2 (∂M, g), E|∂M, h .
In order to prove that this pair is reached byΦz, we first need to consider an extensionweofwto the whole Riemann surfaceM, meaning a sectionwe∈H2 (M, g),(E, h),∆g,hE
such that we have γ∂Mwe = w .
This is possible, since the boundary trace operator is surjective. We can then consider the image ofwe by∆ +z, which yields a section
(∆ +z)we ∈ L2((M, g),(E, h)) .
The distributional Laplacian considered here was denoted by∆instead of∆g,hE so as to differentiate it from the Chern Laplacian with Dirichlet boundary condition. We can then take the inverse image of the section written above by∆g,hE +z, which yields a section
∆g,hE +z−1
(∆ +z)we ∈ H2((M, g),(E, h),∆g,hE
∩H01 (M, g),(E, h),∇(E,h) .
We then set
ev = we− ∆g,hE +z−1
(∆ +z)we ∈ H2 (M, g),(E, h),∆g,hE . This section satisfies
(∆ +z)ev = (∆ +z)we−(∆ +z) ∆g,hE +z−1
(∆ +z)we = 0,
and its image by the boundary trace operatorγ∂M yieldsw, which means going from we to ev has not caused the boundary value to be changed. We further note that no loss of regularity has occured, and that we have actually gained information on the image by∆ +z, which is relevant here. However, this elementveis not the one whose image Φz will give (u, w). To remedy that problem, we set
v = ev+ ∆g,hE +z−1
u ∈ H2((M, g),(E, h),∆g,hE .
This modification of ev has still not modified the boundary value, as the inverse image is taken relatively to the Chern Laplacian with Dirichlet boundary condition, which means that the resulting section vanishes on∂M. The difference is that we now have
(∆ +z)v = (∆ +z)ev+ (∆ +z)
∆g,hE +z−1
u = u .
The image ofv byΦz now yields the pair(u, w), which is exactly what remained to be proved.
Definition 2.1.56. Letz ∈C\R−. The Poisson operator P(z)is defined to be the restriction of the isomorphism Φ−1z to the subspace {0} ⊕H3/2 (∂M, g), E|∂M, h
. This yields a linear operator defined between Sobolev spaces
P(z) : H3/2 (∂M, g), E|∂M, h
−→ H2((M, g),(E, h),∆g,hE .
Proposition 2.1.57. The family of Poisson operatorsP(z), depending onz∈C\R−, is weakly continuous for theL2-norms.
Proof. Letf andgbe elements ofH3/2 (∂M, g), E|∂M, h
andL2((M, g),(E, h)), respectively.
Now, letz be a complex number inC\R−, andhbe another complex number of modulus small enough so that we havez+h∈C\R−. We have
h(∆ + (z+h)) (P(z+h)−P(z))f, giL2((M,g),(E,h))
= h(∆ + (z+h))P(z+h)f, giL2((M,g),(E,h))− h(∆ +z)P(z)f, giL2((M,g),(E,h))
−hhP(z)f, giL2((M,g),(E,h)).
Taking instead ofg the particular function(∆ +z) (P(z+h)−P(z))f, we get
k(∆ + (z+h)) (P(z+h)−P(z))fk2L2((M,g),(E,h)) = |h|2kP(z)fk2L2((M,g),(E,h)) . Noting that P(z+h)−P(z) is H2 and satisfies the Dirichlet boundary condition, we see that the image of this section by the distributional Laplacian∆ + (z+h) is the same as its image by the Laplacian with Dirichlet boundary condition∆g,hE + (z+h). The latter is invertible, and its bounded inverse induces a holomorphic family of operators, in the sense of [63, Sec. 7.11]. We get
k(P(z+h)−P(z))fk2L2((M,g),(E,h))
≤ C
∆g,hE + (z+h)
(P(z+h)−P(z))f
2
L2((M,g),(E,h))
≤ C|h|2kP(z)fk2L2((M,g),(E,h)),
whereC >0 is a constant, independant ofh. This completes the proof, as we have
h→0lim k(P(z+h)−P(z))fkL2((M,g),(E,h)) = 0 .
Proposition 2.1.58. The family of Poisson operators
P(z) : H3/2 (∂M, g), E|∂M, h
−→ L2((M, g),(E, h)) depending onz∈C\R− is holomorphic, with respect to the L2-norms.
Proof. Using the same notations as in the proof of the last proposition, we have
1
hh(∆ +z) (P(z+h)−P(z))f, giL2((M,g),(E,h))
= 1hh
h(∆ + (z+h))P(z+h)f, giL2((M,g),(E,h))
−hhP(z+h)f, giL2((M,g),(E,h))− h(∆ +z)P(z)f, giL2((M,g),(E,h))
i
= − hP(z+h)f, giL2((M,g),(E,h)) −→
h→0 − hP(z)f, giL2((M,g),(E,h))
by a previously shown continuity. Using the fact that we have
1 h
D
∆g,hE +z
(P(z+h)−P(z))f, ∆g,hE +z−1
gE
L2((M,g),(E,h))
= h1D
∆g,hE +z−1
∆g,hE +z
(P(z+h)−P(z))f, gE
L2((M,g),(E,h))
= h1h(P(z+h)−P(z))f, giL2((M,g),(E,h)),
we now deduce that we have lim
h→0 1
hh(P(z+h)−P(z))f, giL2((M,g),(E,h))
= lim
h→0 1 h
D
∆g,hE +z
(P(z+h)−P(z))f, ∆g,hE +z−1 gE
L2((M,g),(E,h))
= −D
P(z)f, ∆g,hE +z−1
gE
L2((M,g),(E,h))
= −D
∆g,hE +z−1
P(z)f, gE
L2((M,g),(E,h)). This proves the proposition, and further gives the formula
dP
dz = −
∆g,hE +z−1
P(z) .
Remark 2.1.59. Each and every one of the results given above can be extended to complex numberszwith−znot in the spectrum of the Laplacian with Dirichlet boundary condition∆g,hE . Proposition 2.1.60. The Poisson operatorP(z)takes a smooth section ofEover∂M to a smooth section ofE overM.
Proof. This result stems from the elliptic regularity of the Chern Laplacian, and the bijectivity of the applicationΦz, which must then induce an application between the spaces of smooth sections.
Poisson operator for the Dolbeault Laplacian. We will need another Poisson operator, this time related to the Dolbeault Laplacian. Every statement made in the last paragraph can be adapted, using the Dolbeault Laplacian instead of the Chern Laplacian. We get an operator
P∂
E(z) : H3/2 (∂M, g), E|∂M, h
−→ H2((M, g),(E, h),∆g,h∂
E
which solves the same kind of Dirichlet problem as before, with the Dolbeault Laplacian being used, and not the Chern Laplacian.