Regularity of invariant distributions

Ulrich Bunke^{∗}and Martin Olbrich^{†}

September 7, 2001

Abstract

We consider a discrete subgroup Γ of the isometry group G of the real hyperbolic space X ≅ H^{n} of dimension n ≥ 2. We assume that Γ admits a fundamental domain with finitely many totally geodesic faces (i.e. Γ is geometrically finite). This domain may have finitely many cusps of rank varying in {1, …, n − 1}. Γ acts by conformal transformations on the sphere ∂X ≅ S^{n−1} at infinity. By Λ_{Γ} we denote its limit set which is a closed Γ-invariant subset of ∂X of Hausdorff dimension dim_{H}(Λ_{Γ}). We consider a G-equivariant irreducible complex vector bundle V → ∂X. Let Λ be the bundle of densities on ∂X. Then we define a real number s(V) by the condition that V ⊗ Λ^{s(V)} ≅ V^{♯} ⊗ Λ as G-equivariant bundles, where V^{♯} is the hermitian dual of V. The goal of the paper is to find lower bounds on the regularity of Γ-invariant distribution sections of V which are strongly supported on the limit set. In order to quantify this regularity we consider the spaces of sections of regularity H^{p,r}, p ∈ (1, ∞), r ∈ ℝ. For r ∈ ℕ_{0} the space H^{p,r} is the space of distribution sections which have all their derivatives up to order r in L^{p}. For r ∈ [0, ∞) we extend this scale by complex interpolation, while for negative r we define these spaces by duality. Our main result is the following. Let

r^{0} := (n − 1)\frac{s(V) − 1}{2} + \frac{n − 1 − dim_{H}(Λ_{Γ})}{p}
r := min (r^{0}, (n − 1)s(V) + \frac{n − 1}{p} − max_{cusps c} rank(c)) ,

bunke@uni-math.gwdg.de olbrich@.uni-math.gwdg.de

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, E-mail :

†Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, E-mail :

where the maximum over the empty set is, by convention, equal to −∞. Moreover, dim_{H}(Λ_{Γ})

has to be replaced by half of the rank of the cusp if Γ is elementary parabolic. If ϕ is a Γinvariant distribution section of V which is strongly supported on the limit set (and a cusp form), then ϕ ∈ H^{p,s} for all s < r (s < r^{0}).

If Γ is convex cocompact, then strongly supported on the limit set just means supported on the limit set as a distribution (and the cusp form condition holds true automatically). But if Γ has cusps, then this condition is stronger. Generically, boundary values of cusp forms and residues of Eisenstein series have this property. Actually, in order to prove our result, we also need some additional technical assumptions.

Contents

1 Statement of the result 3 1.1 Introduction and statement of the main theorem . . . . . . . . . . . . . . . . . . 3 1.2 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Related results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Elements of geometric scattering theory 16 2.1 The space B_{Γ}(σ_{λ}, φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Restriction, extension, and the scattering matrix . . . . . . . . . . . . . . . . . . 21

3 Some functional analytic preparations 25 3.1 Complex interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Regularity of intertwining operators . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Proof of Theorem 1.15 34 4.1 Compatibility with twisting and embedding . . . . . . . . . . . . . . . . . . . . . 35 4.2 Sobolev regularity for cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Sobolev regularity of invariant functions . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1 Statement of the result

1.1 Introduction and statement of the main theorem

Let G be one of the groups Spin(1, n), SO(1, n)_{0}, 2 ≤ n ∈ ℕ. Then the symmetric space X of G is the real hyperbolic space of dimension n. Furthermore, let Γ ⊂ G be a discrete subgroup. We assume that Γ is geometrically finite. One of many equivalent formulations of this condition is that Γ admits a fundamental domain in X which has finitely many totally geodesic faces. In the following we assume that Γ is torsion-free, but we shall see later that one can drop this assumption.

By ∂X we denote the geodesic boundary of X. It can be identified with the space of all minimal parabolic subgroups of G such that the minimal parabolic subgroup P ⊂ G corresponds to its unique fixed point ∞_{P} ∈ ∂X. Since G acts transitively on the set of minimal parabolic subgroups we can identify ∂X ≅ G/P for any fixed P.

The G-homogeneous complex vector bundles on ∂X are in one-to-one correspondence with finite-dimensional complex representations of P. On the one hand if (θ, V_{θ}) is a finite dimensional complex representation of P, then we can form the associated bundle V (θ) := G ×_{P,θ} V_{θ}. On the other hand, if V → ∂X is a G-homogeneous complex vector bundle, then we let V_{θ} be the representation of P on the fibre of V over ∞_{P}, and we have a natural isomorphism V (θ) ≅ V.

In order to parametrize the irreducible representations of P we consider the exact sequence

0 → N → P → L → 0 ,

where N is the unipotent radical of P. Any finite-dimensional irreducible representation of P factors over the quotient L. This group fits into a natural exact sequence

0 → M → L → A → 0 ,  (1)

where M is a compact group isomorphic to Spin(n − 1) or SO(n − 1), and A is isomorphic to the multiplicative group ℝ^{+}. Let 𝔞 denote the Lie algebra of A and 𝔞_{ℂ}^{∗} be the complexification of its dual.

For λ ∈ 𝔞_{ℂ}^{∗} we form the character a ↦ a^{λ} := exp(λ(log(a)). The group L acts on the Lie algebra 𝔫 of N. The induced representation on Λ^{dim(𝔫)}𝔫 factors over A and corresponds to the character 2ρ ∈ 𝔞^{∗}. Furthermore, we define the element α ∈ 𝔞^{∗} by

α := \frac{2ρ}{dim(𝔫)}

which plays the role of a unit.

If (θ, V_{θ}) is any irreducible representation of L, then its restriction to M is an irreducible unitary representation. Furthermore, the representation Λ^{dim(V_{θ})}θ factors over A and corresponds to an element of 𝔞_{ℂ}^{∗} (except in the case G = SL(2, ℝ). In this case M may act non-trivially on Λ^{dim(V_{θ})}θ and one employs the canonical split of (1) in order to define the character of A corresponding to θ).

Definition 1.1 Given (σ, V_{σ}) ∈ M̂ and λ ∈ 𝔞_{ℂ}^{∗} we let (σ_{λ}, V_{σ_{λ}}) be the unique irreducible representation of P such that (σ_{λ})_{|M} = σ and Λ^{dim V_{σ}}σ_{λ} corresponds to dim(V_{σ})(ρ − λ) ∈ 𝔞_{ℂ}^{∗}.

The bundle of densities Λ on ∂X is given in this parametrization by V (1_{−ρ}), where 1 ∈ M̂ denotes the trivial representation. The bundle V (σ_{λ})^{♯} ⊗ Λ is isomorphic to V (σ_{−λ̄}) as G-homogeneous bundle. Note that V (1_{−ρ})^{s} ⊗ V (σ_{λ}) = V (σ_{λ−2sρ}). We conclude that

s(V (σ_{λ})) = \frac{Re(λ)}{ρ} ,

where s(V (σ_{λ})) is the number introduced in the abstract.

Another important element of 𝔞^{∗} is the critical exponent δ_{Γ} of Γ which we now define. We fix any maximal compact subgroup K ⊂ G. This choice induces an embedding of L into P such that L is stable under the Cartan involution θ_{K} associated with K. Furthermore, we obtain a decomposition L = MA, where M = L^{θ_{K}} is the set of fixed points of θ_{K}, and if a ∈ A, then θ_{K}(a) = a^{−1}. In particular we now can consider A as a subgroup of G. The character ρ fixes a semigroup A_{+} := {a ∈ A | a^{ρ} ≥ 1}. By the Cartan decomposition any element g ∈ G can be written as g = k_{g}a_{g}h_{g}, where k_{g}, h_{g} ∈ K, and a_{g} ∈ A_{+} is uniquely determined.

Definition 1.2 The critical exponent δ_{Γ} ∈ 𝔞^{∗} of Γ is defined as the infimum of the set {μ ∈ 𝔞^{∗} | ∑_{γ∈Γ} a_{γ}^{−μ−ρ} < ∞}.

Let ∂X = Ω_{Γ} ∪ Λ_{Γ} be the Γ-invariant decomposition into the limit set Λ_{Γ} and the ordinary set Ω_{Γ}. The limit set Λ_{Γ} is closed and can be defined as the set of accumulation points of any Γ-orbit in X. By results of Patterson [10] in case n = 2 and Sullivan [14] for all n ≥ 2 the Hausdorff dimension of Λ_{Γ} is given by dim_{H}(Λ_{Γ}) = \frac{δ_{Γ}+ρ}{α} unless Γ is elementary parabolic.

Recall that a parabolic subgroup P ⊂ G is called Γ-cuspidal if Γ ∩ P is infinite and projects to a precompact subset l(Γ ∩ P) ⊂ L, where l : P → L is the projection. We form the compact group M_{Γ} := \overline{l(Γ ∩ P)} ⊂ M. Furthermore, there exists a subgroup N_{Γ} ⊂ N and a Langlands decomposition P = MAN (choice of K) such that Γ_{P} := Γ ∩ P is a cocompact lattice in the unimodular group P_{Γ} := N_{Γ}M_{Γ}.

A cusp of Γ is a Γ-conjugacy class [P]_{Γ} of Γ-cuspidal parabolic subgroups. The dimension dim(N_{Γ}) is called the rank of the cusp rank([P]_{Γ}) defined by P. We further define ρ_{Γ_{P}} := \frac{1}{2}rank([P]_{Γ})α and ρ^{Γ_{P}} := ρ − ρ_{Γ_{P}}. If dim(N_{Γ}) ∈ \{1, …, dim(𝔫) − 1\}, then we say that the cusp has smaller rank. If dim(N_{Γ}) = dim(𝔫), then the cusp has full rank.

A finite-dimensional representation (φ, V_{φ}) of Γ is called admissible twist (shortly twist) if for any Γ-cuspidal parabolic subgroup P the representation φ extends to AP_{Γ} such that A acts by semisimple endomorphisms. For example, if φ is a finite-dimensional representation of G, then its restriction to Γ is an admissible twist.

Given a representation (θ, V_{θ}) of P and an admissible twist (φ, V_{φ}) of Γ we form the bundle V (θ, φ) = V (θ) ⊗ V_{φ}. If π^{θ} denotes the left-regular representation of G on sections of V (θ), then we can form the representation π^{θ,φ} := π^{θ} ⊗ φ of Γ on sections of V (θ, φ). Note that π^{θ,φ} extends to AP_{Γ} for each Γ-cuspidal parabolic subgroup P.

Definition 1.3 The exponent δ_{φ} of the twist φ is defined as the infimum of the set {μ ∈ 𝔞^{∗} | sup_{γ∈Γ} ∥φ(γ)∥a_{γ}^{−μ} < ∞}, where ∥.∥ is any norm on End(V_{φ}).

Let σ̃ or φ̃ denote the ℂ-linear dual representations of σ or φ. The spaces

C^{−∞}(∂X, V (σ_{λ}, φ)) := C^{∞}(∂X, V (σ̃_{−λ}, φ̃))^{∗}

for varying λ ∈ 𝔞_{ℂ}^{∗} form locally trivial holomorphic bundles of dual Fréchet spaces so that it makes sense to speak of germs holomorphic families (ϕ_{λ})_{λ} such that ϕ_{λ} ∈ C^{−∞}(∂X, V (σ_{λ}, φ)). Let ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) be the subspace of Γ-invariant distributions.

We call ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) deformable, if there is a germ at λ of a holomorphic family (ϕ_{μ})_{μ} such that ϕ_{λ} = ϕ and ϕ_{μ} ∈ ^{Γ}C^{−∞}(∂X, V (σ_{μ}, φ)).

Definition 1.4 Let Fam_{Γ}(σ_{λ}, φ) ⊂ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) be the subspace of all deformable Γinvariant distributions.

If ϕ ∈ Fam_{Γ}(σ_{λ}, φ), then we say that ϕ is strongly supported on the limit set if res^{Γ}(ϕ) = 0. Here res^{Γ} is the restriction map introduced in [4] (we refer to Subsection 2.2 for details). More precisely, this condition means that there exists a germ at λ of a holomorphic family (ϕ_{μ})_{μ}, ϕ_{μ} ∈ ^{Γ}C^{−∞}(∂X, V (σ_{μ}, φ)), such that ϕ_{λ} = ϕ and (res^{Γ}ϕ_{μ})_{|μ=λ} = 0.

If ϕ is strongly supported on the limit set, then its support as a distribution supp(ϕ) is contained in Λ_{Γ}. (In fact it is equal to Λ_{Γ} if ♯Λ_{Γ} ≠ 2.) The converse may be false if Γ has cusps.

Definition 1.5 Let Fam_{Γ}(Λ_{Γ}, σ_{λ}, φ) ⊂ Fam_{Γ}(σ_{λ}, φ) be the subspace of all deformable invariant distributions which are strongly supported on the limit set.

Being deformable is a non-trivial condition. For example, if Γ has finite-covolume, then the boundary values of cusp forms are not deformable, while the boundary values of Eisenstein series are deformable. We also have examples of non-deformable invariant distributions in the case of convex cocompact Γ (see [3], [9], [5]).

Using embedding and twisting we can formulate a weaker condition. We first describe embedding. For a moment we write G_{n} for Spin(1, n) or SO(1, n)_{0}, and we attach the subscript n to all related objects. For n ≥ m have inclusions G_{m} ⊂ G_{n}, ∂X_{m} ⊂ ∂X_{n}, P_{m} ⊂ P_{n}, M_{m} ⊂ M_{n}. Note that A_{m} ≅ A_{n} =: A and thus 𝔞_{ℂ}^{∗}_{m} ≅ 𝔞_{ℂ}^{∗}_{n} =: 𝔞_{ℂ}^{∗} in a natural way.

Let σ_{n} ∈ M̂_{n}, σ_{m} ∈ M̂_{m} and assume that there is an M_{m}-invariant inclusion T : V_{σ_{m}} → V_{σ_{n}}. Then T^{∗} induces a projection V^{n}((σ̃_{n})_{−λ−ρ^{m}+ρ^{n}})_{|∂X^{n}} → V^{m}((σ̃_{m})_{−λ}). Let

i_{n,σ_{n},T}^{∗} : C^{∞}(∂X^{n}, V^{n}((σ̃_{n})_{−λ−ρ^{m}+ρ^{n}}, φ̃)) → C^{∞}(∂X^{m}, V^{m}((σ̃_{m})_{−λ}, φ̃))

be the restriction of functions and

i_{∗}^{n,σ_{n},T} : C^{−∞}(∂X^{m}, V^{m}((σ_{m})_{λ}, φ)) → C^{−∞}(∂X^{n}, V^{n}((σ_{n})_{λ+ρ^{m}−ρ^{n}}, φ))

be the dual map. Note that i_{∗}^{n,σ_{n},T} is injective and G_{m}-equivariant. For fixed m and σ_{m} the various choices of n, σ_{n} and T are are called embedding data. We will often abbreviate i_{∗}^{n,σ_{n},T} = i_{∗}.

Now we explain twisting. If (π, V_{π}) is a finite-dimensional representation of G, then there is an isomorphism of Γ-equivariant bundles R : V (θ⊗π_{|P} , φ) →^{∼} V (θ, φ⊗π) given by R([g, x⊗y]⊗z) := [g, x] ⊗ z ⊗ π(g)y. Fix σ ∈ M̂ and λ ∈ 𝔞_{ℂ}^{∗}. Assume that we are given σ′ ∈ M̂, μ ∈ 𝔞_{ℂ}^{∗}, a finitedimensional representation of (π, V_{π}) of G, and a P -equivariant inclusion T : V_{σ_{λ}} ↪ V_{σ′_{μ}} ⊗ V_{π}. Then we obtain a Γ-equivariant inclusion

i_{∗}^{σ′,μ,π,T} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(∂X, V (σ′_{μ} ⊗ π_{|P} , φ)) →^{R} C^{−∞}(∂X, V (σ′_{μ}, φ ⊗ π)) .

Note that i_{∗}^{σ′,μ,π,T} is P_{Γ}-equivariant for any Γ-cuspidal parabolic subgroup P. For simplicity we will from now on assume that (π, V_{π}) is irreducible. The various choices of σ′, μ, π and T are called twisting data. We often abbreviate i_{∗}^{σ′,μ,π,T} = i_{∗}.

We also allow iterated twisting and embedding, and we still denote the resulting inclusion by i_{∗}. We say that ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) is stably deformable if i_{∗}ϕ becomes deformable for suitable embedding or twisting data.

Definition 1.6 Let Fam_{Γ}^{st}(σ_{λ}, φ) denote the subspace of stably deformable invariant distributions.

Being stably deformable is a weaker condition than being deformable. E.g., if Γ has finite covolume, then the boundary values of cusp forms are not deformable, but generically they are stably deformable. In fact, for most λ ∈ 𝔞_{ℂ}^{∗} all invariant distributions are stably deformable. Let I_{𝔞} ⊂ 𝔞^{∗} denote the lattice of half-integral characters spanned by \frac{1}{2}α.

Theorem 1.7 If λ ∉ I_{𝔞}, then Fam_{Γ}^{st}(σ_{λ}, φ) = ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)).

A proof will appear in a future publication. One of the main steps is to show that res^{Γ} is regular (i.e. res^{Γ}ϕ_{μ} is holomorphic for any germ at λ of a holomorphic family (ϕ_{μ})_{μ} of Γinvariant distributions) for λ ∉ I_{𝔞} for any Γ which is elementary parabolic, i.e. Γ = Γ_{P} for some Γ-cuspidal parabolic subgroup.

The following was shown in [5], proof of Cor. 6.12.

Theorem 1.8 If Γ is convex cocompact, then

Fam_{Γ}^{st}(σ_{λ}, φ) = ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))

holds for any λ ∈ 𝔞_{ℂ}^{∗}.

If Γ is convex cocompact and not cocompact, then twisting alone suffices to make an invariant distribution deformable. We do not know any example of an invariant distribution which is not stably deformable.

Conjecture 1.9 If Γ is geometrically finite, then

Fam_{Γ}^{st}(σ_{λ}, φ) = ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))

for all σ, λ and φ.

Note that if ϕ ∈ Fam_{Γ}^{st}(σ_{λ}, φ), then we can always find embedding and twisting data such that i_{∗}ϕ ∈ Fam_{Γ}(1_{λ′}, φ′). This will be employed later.

If ϕ ∈ Fam_{Γ}^{st}(σ_{λ}, φ), then we say that it is strongly supported on the limit set if i_{∗}ϕ is strongly supported on the limit set for some choice of embedding and twisting data with the property that i_{∗}ϕ is deformable. This definition makes sense since being strongly supported on the limit set is stable under twisting and embedding by Lemma 2.13.

Definition 1.10 We let Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ) be the space of stably deformable invariant distribution vectors which are strongly supported on the limit set.

An alternative way to sharpen the condition ”supported on the limit set” is to require a ”cusp form” condition. If λ ∉ I_{𝔞}, then ”cusp form” implies ”strongly supported on the limit set” (Theorem 1.14).

Let P be a Γ-cuspidal parabolic subgroup. We normalize the Haar measure dx on the unimodular group P_{Γ} such that vol(Γ_{P} \P_{Γ}) = 1. Note that π^{σ_{λ},φ} extends to P_{Γ}.

Definition 1.11 If ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)), then we define its constant term

ϕ_{P} ∈ ^{P_{Γ}}C^{−∞}(∂X, V (σ_{λ}, φ))

with respect to P by

ϕ_{P} := ∫_{Γ_{P} \P_{Γ}} π^{σ_{λ},φ}(x)ϕ dx .

Definition 1.12 We call ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) a cusp form if supp(ϕ) ⊂ Λ_{Γ} and ϕ_{P} = 0 for all Γ-cuspidal parabolic subgroups P.

If Γ has finite covolume and ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ})) is a cusp form, then the matrix coefficients G ∋ g ↦ ⟨ϕ, πσ̃_{−λ}(g)f⟩ for K-finite f ∈ C^{∞}(∂X, V (σ̃_{−λ})) are cusp forms in the classical sense. If Γ is convex cocompact, then any ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) with supp(ϕ) ⊂ Λ_{Γ} is a cusp form for trivial reasons. If ϕ is a cusp form, then i_{∗}ϕ is also a cusp form for any choice of embedding or twisting data.

Definition 1.13 Let Cusp_{Γ}(σ_{λ}, φ) denote the space of all cusp forms in ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)).

Theorem 1.14 If λ ∉ I_{𝔞}, then Cusp_{Γ}(σ_{λ}, φ) ⊂ Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ).

Note that the proof of this theorem uses Thm. 1.7 and the fact that res^{Γ} is regular outside I_{𝔞}. Assuming these facts in the present paper we show that a cusp form is strongly supported on the limit set.

Since ∂X is a closed manifold there are natural Sobolev spaces H^{p,r}(∂X, V (σ_{λ}, φ)), p ∈ (1, ∞), r ∈ ℝ. For r ∈ ℕ_{0} we let H^{p,r}(∂X, V (σ_{λ}, φ)) be the space of those distributions in C^{−∞}(∂X, V (σ_{λ}, φ)) which have L^{p}-integrable derivatives up to order r. For r ∈ −ℕ we define H^{p,r}(∂X, V (σ_{λ}, φ)) to be the dual of H^{q,−r}(∂X, V (σ̃_{−λ}, φ̃)), where \frac{1}{p} + \frac{1}{q} = 1. For non-integral r we define H^{p,r}(∂X, V (σ_{λ}, φ)) by complex interpolation (see Subsection 3.1). We also define

H^{p,<r}(∂X, V (σ_{λ}, φ)) := ⋂_{s<r} H^{p,s}(∂X, V (σ_{λ}, φ)) .

Now we can state our main result.

Theorem 1.15 Let r_{p,λ}(Γ), r_{p,λ}^{0}(Γ) ∈ ℝ be determined by

r_{p,λ}^{0}α := Re(λ) − δ_{φ} − ρ + \frac{ρ − δ_{Γ}}{p}
r_{p,λ}(Γ)α := min (r_{p,λ}^{0}(Γ)α, 2Re(λ) + \frac{2ρ}{p} − max_{cusps [P]_{Γ}} [2δ_{φ_{|Γ_{P}}} + α rank([P]_{Γ})])

(the maximum over the empty set is defined by convention as −∞). Then

Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ) ⊂ H^{p,<r_{p,λ}(Γ)}(∂X, V (σ_{λ}, φ))
Cusp_{Γ}(σ_{λ}, φ) ⊂ H^{p,<r_{p,λ}^{0}(Γ)}(∂X, V (σ_{λ}, φ))  if λ ∉ I_{𝔞} .

\underline{Remarks :}

1. If δ_{Γ} < ρ, then it is clear that ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) with supp(ϕ) ⊂ Λ_{Γ} cannot belong to H^{p,0}(∂X, V (σ_{λ}, φ)). Indeed, in this case the limit set Λ_{Γ} has trivial Lebesgue measure and cannot be the support of any non-trivial L^{p}-function. We conclude

Corollary 1.16 If r_{1,λ}(Γ) > 0 and δ_{Γ} < ρ, then

Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ) = {0} .

This is the vanishing result obtained in [4], Prop. 4.22, resp. [6]. Note that the Corollary does not follow logically from the present paper because it is used in the argument in order to deal with positive r_{p,λ}(Γ).

2. Assume that Γ has finite covolume. In this case ρ = δ_{Γ}. If we take σ = 1, φ = 1 and λ = ρ, then r_{p,λ}(Γ) = 0. However, the constant function on ∂X is a smooth Γ-invariant section of V (1_{ρ}) which is strongly supported on the limit set.

3. Assume that Γ is convex cocompact, σ = 1, φ = 1, and λ = δ_{Γ}. Then the Patterson-Sullivan measure generates Fam_{Γ}^{st}(Λ_{Γ}, 1_{λ}, 1). A measure belongs to H^{p,<− dim(∂X) \frac{1}{q}} for all p ∈ (1, ∞). In contrast, our estimate gives r_{p,λ}(Γ)α = −(dim(X) − dim_{H}(Λ))\frac{1}{q} > −\frac{1}{q} dim(∂X)α except if Γ is elementary hyperbolic. Thus the Patterson-Sullivan measure is more regular than a general measure. It has the regularity of a smooth measure on Λ if the latter would be a smooth submanifold.

4. It follows from lower regularity bounds obtained by W. Schmid in special cases that our estimate is essentially optimal. We refer to Subsection 1.4.1 for more details.

5. Probably, the condition λ ∉ I_{𝔞} in the cusp form case is not necessary.

Let us now explain the modifications in the case that Γ has torsion. In this case there is a torsion-free subgroup Γ′ ⊂ Γ of finite index which is still geometrically finite. Moreover, we have Λ_{Γ} = Λ_{Γ′} and δ_{Γ} = δ_{Γ′}. We define

Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ) := Fam_{Γ′}^{st}(Λ_{Γ′}, σ_{λ}, φ) ∩^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))

(this definition does not depend on the choice of Γ′). Cusp forms are characterized as in the torsion-free case. Then Theorem 1.15 holds true for Γ with torsion elements.

1.2 Idea of the proof

Let us demonstrate the main ideas leading to the proof of Theorem 1.15 in the special case that δ_{φ} = 0, σ = 1, Re(λ) < min(−δ_{Γ}, 0), λ ∉ I_{𝔞}, and that Γ has no cusps of full rank and is not cocompact.

Let ϕ ∈ Fam_{Γ}(Λ_{Γ}, 1_{λ}, φ). We apply the Knapp-Stein intertwining operator (see Subsection 2.2) and obtain ψ := Ĵ_{1_{λ},φ}^{w}ϕ ∈ ^{Γ}C^{−∞}(∂X, V (1_{−λ}, φ)). The fact that ϕ is strongly supported on the limit set implies that ψ is smooth on Ω_{Γ} and has a controlled growth along the cusps. This will be formalized by saying that ψ_{|Ω_{Γ}} belongs to the space B_{Γ}(1_{−λ}, φ) which will be introduced in Subsection 2.1. Using the convergence of the Poincaré series ∑_{γ∈Γ} a_{γ}^{−(δ_{Γ}+ρ+ϵ)}, 0 < ϵ ∈ 𝔞^{∗}, the Γ-invariance of ψ_{|Ω_{Γ}}, and the controlled growth along the cusps we show (Theorem 4.6) that ψ ∈ H^{p,<r_{p,λ}(Γ)−2 \frac{Re(λ)}{α}}. Since λ ∉ I_{𝔞} we recover ϕ from ψ by applying a multiple of Ĵ_{1_{−λ},φ}^{w^{−1}}. This operator decreases the regularity by −2\frac{Re(λ)}{α} (see Subsection 3.2) so that we obtain the desired result ϕ ∈ H^{p,<r_{p,λ}(Γ)}. If ϕ is a cusp form, then we get a stronger estimate of ψ along the cusps leading to ψ ∈ H^{p,<r_{p,λ}^{0}(Γ)−2 \frac{Re(λ)}{α}} and ϕ ∈ H^{p,<r_{p,λ}^{0}(Γ)}.

1.3 Specializations

In this subsection we discuss specializations of the main theorem. In our main theorem we quantify the regularity of distributions using the two-parameter family of spaces H^{p,r}. Let us indicate another way to measure the regularity of a distributions.

For 0 < s ∈ ℝ , s = [s] + s′, [s] ∈ ℕ_{0}, s′ ∈ [0, 1) let C^{s}(∂X, V (σ_{λ}, φ)) be the space of sections which have Hölder continuous (with exponent s′) derivatives up to order [s]. Furthermore, let C^{<s} := ⋂_{t<s} C^{t}. For negative s we define C^{s} (resp. C^{<s}) to be the space of distributions which can locally be written as N-fold derivatives of elements of C^{s+N} (resp. C^{<s+N}), where N is sufficiently large. For all s ∈ ℝ and p ∈ (1, ∞) have the embeddings H^{p,< \frac{n−1}{p} +s} ↪ C^{<s}. Since we deduce Hölder regularity from Sobolev regularity we cannot omit the <-sign. In some cases a direct approach to Hölder regularity may show C^{s} instead of C^{<s}-regularity (see Subsection 1.4.1).

12 1 STATEMENT OF THE RESULT 1.3.1 Γ cocompact Assume that Γ is cocompact. Then Λ_{Γ} = ∂X, δ_{Γ} = ρ. Any invariant distribution is stably deformable by Theorem 1.8 and strongly supported on the limit set. Theorem 1.17 Assume that Γ is cocompact, and let

rα := Re(λ) − δ_{φ} − ρ .

Then

ΓC^{−∞}(∂X, V (σ_{λ}, φ)) ⊂ H^{p,<r}(∂X, V (σ_{λ}, φ))

for any p ∈ (1, ∞). Using the freedom to choose p arbitrary large we obtain the following corollary. Corollary 1.18 If Γ is cocompact, then

ΓC^{−∞}(∂X, V (σ_{λ})) ⊂ C^{<− \frac{n−1}{2} + \frac{Re(λ)}{α}} .

1.3.2 Γ convex cocompact Assume now that Γ is convex cocompact. If 0 ≠ ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) satisfies supp(ϕ) ⊂ Λ_{Γ}, then it is deformable by Theorem 1.8 and strongly supported on the limit set.

We can thus apply Theorem 1.15 to the space ΓC^{−∞}(Λ_{Γ}, V (σ_{λ}, φ)) := {ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) | supp(ϕ) ⊂ Λ_{Γ}} .

Theorem 1.19 Let

rα := Re(λ) − δ_{φ} − ρ + \frac{ρ − δ_{Γ}}{p} .

Then

ΓC^{−∞}(Λ_{Γ}, V (σ_{λ}, φ)) ⊂ H^{p,<r}(∂X, V (σ_{λ}, φ)) .

Using the freedom to choose p arbitrary large we obtain Corollary 1.20 If Γ is convex cocompact, then

ΓC^{−∞}(Λ_{Γ}, V (σ_{λ}, φ)) ⊂ C^{<− \frac{n−1}{2} + \frac{Re(λ)−δφ}{α}} .

1.3.3 Γ has finite covolume

Now assume that Γ has finite covolume. Let ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ})) be a cusp form and λ ∉ I_{𝔞}. Applying Theorem 1.15 and using that δ_{Γ} = ρ we obtain the following result.

Theorem 1.21 Assume that Γ has finite covolume and λ ∉ I_{𝔞}, then

Cusp_{Γ}(σ_{λ}) ⊂ H^{p,<− \frac{n−1}{2} + \frac{Re(λ)}{α}} (∂X, V (σ_{λ}))

for any p ∈ (1, ∞).

Corollary 1.22 If Γ has finite covolume and λ ∉ I_{𝔞}, then

Cusp_{Γ}(σ_{λ}) ⊂ C^{<− \frac{n−1}{2} + \frac{Re(λ)}{α}} .

More general we have the following estimate.

Theorem 1.23 If Γ has finite covolume, then for p ∈ (1, ∞) we have

Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ) ⊂ H^{p,<− \frac{n−1}{q} +2 \frac{Re(λ)}{α}} (∂X, V (σ_{λ})) .

Corollary 1.24 If Γ has finite covolume, then

Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ) ⊂ C^{<−(n−1)+2 \frac{Re(λ)}{α}} (∂X, V (σ_{λ}, φ)) .

1.3.4 Residues of Eisenstein series

For general geometrically finite Γ a source of interesting invariant distributions are boundary values of residues of Eisenstein series. In the language of geometric scattering theory [4] these are invariant distributions obtained from the singular part of the extension map ext^{Γ}. We refer to Subsection 2.2 for more details.

Definition 1.25 Let Ext_{Γ}^{sing}(σ_{λ}, φ) ⊂ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) be the subspace of invariant distributions generated by the singular part of ext^{Γ}.

Essentially by definition the elements of Ext_{Γ}^{sing}(σ_{λ}, φ) belong to ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) and are deformable. Moreover, the relation res^{Γ}∘ext^{Γ} = id implies that the elements of Ext_{Γ}^{sing}(σ_{λ}, φ) are strongly supported on the limit set. We thus have Ext_{Γ}^{sing}(σ_{λ}, φ) ⊂ Fam_{Γ}(Λ_{Γ}, σ_{λ}, φ). We show in Section 2.2 that these spaces are in fact equal.

Note that for Re(λ) > δ_{Γ} + δ_{φ} the extension ext^{Γ} is regular and thus Ext_{Γ}^{sing}(σ_{λ}, φ) = 0. Theorem 1.15 gives the following result.

Theorem 1.26 We have

Ext_{Γ}^{sing}(σ_{λ}, φ) ⊂ H^{p,<r_{p,λ}(Γ)}(∂X, V (σ_{λ}, φ)) .

Choosing p arbitrary large we obtain the following consequence.

Corollary 1.27 If Γ is geometrically finite, then

Ext_{Γ}^{sing}(σ_{λ}, φ) ⊂ C^{<r} ,

where

r := min (−\frac{n − 1}{2} + \frac{Re(λ) − δ_{φ}}{α} , 2\frac{Re(λ)}{α} − max_{cusps [P]_{Γ}} [2\frac{δ_{φ_{|P_{Γ}}}}{α} + rank([P]_{Γ})]) .

1.4 Related results

1.4.1 Results of W. Schmid

In [13] W. Schmid considers a finitely generated discrete subgroup Γ of G = SL(2, ℝ). Note that G ≅ Spin(1, 2). If ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ})), then it defines a G-equivariant map

c_{ϕ} : C^{∞}(∂X, V (σ̃_{−λ})) → C^{∞}(Γ\G)

by c_{ϕ}(f)(g) := ⟨ϕ, πσ̃_{λ}(g)f⟩. Let us call ϕ square integrable, iff c_{ϕ}(f) ∈ L^{2}(Γ\G) for all f ∈ C^{∞}(∂X, V (σ̃_{−λ})). Schmid calls a square integrable ϕ (or rather the map c_{ϕ}) cuspidal, if c_{ϕ}(f) is a bounded function for all K-finite f, where K is some maximal compact subgroup of G.

Theorem 1.28 (Schmid) Let ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ})) be square integrable.

1. If ϕ is cuspidal and \frac{λ}{α} ∉ ℤ + \frac{1}{2}, then ϕ is of Hölder regularity C^{\frac{Re(λ)−ρ}{α}}.

2. If \frac{λ}{α} ∈ ℤ + \frac{1}{2}, then ϕ is cuspidal and belongs to C^{< \frac{λ−ρ}{α}}.

3. If ϕ is not cuspidal, then ϕ ∈ C^{<2(λ−ρ)} if λ ∈ (0, ρ), and ϕ ∈ C^{<−1}, if λ ∈ (−ρ, 0).

Note that in the two-dimensional case a finitely generated discrete group is geometrically finite, so the classes of discrete subgroups considered by Schmid and in the present paper coincide. On the one hand, being a square integrable invariant distribution is a strong restriction on ϕ. On the other hand there are square integrable distributions which are not strongly supported on the limit set. If ϕ is cuspidal and λ ∉ I_{𝔞}, or if ϕ is stably deformable, then our estimate applies to ϕ and reproduces Schmid’s estimate. In fact, in the cuspidal case the result Schmid is slightly stronger than ours since he shows ϕ ∈ C^{r}, while we show ϕ ∈ C^{<r} for the appropriate r.

In the case λ ∈ (0, ρ) Schmid shows that his lower bounds of regularity are optimal in both, the cuspidal and the general case. This implies that our result is essentially optimal among estimates which can be stated using the same data and are compatible with embedding and twisting.

We thank W. Schmid for showing us a counterexample to an estimate claimed in a previous version of this paper.

1.4.2 Results of J. Lott

In [8] Lott considers discrete subgroups Γ of SO(1, n)_{0}. Let σ^{k} := Λ^{k}𝔫̄ ∈ M̂ and λ_{k} := ρ − kα. Note that V (σ_{λ_{k}}^{k}) = Λ^{k}T^{∗}∂X. Let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition of 𝔤. Then Λ^{k}𝔭 ∈ K̂ is the minimal K-type of π^{σ̃_{−λ_{k}}^{k}}. Fix 0 ≠ f_{k} ∈ C^{∞}(∂X, V (σ̃_{−λ_{k}}^{k}))(Λ^{k}𝔭). The main result of Lott is the following theorem.

Theorem 1.29 (Lott) If ϕ ∈ C^{−ω}(∂X, V (σ_{λ_{k}}^{k})) is exact (as a differential form) and such that c_{ϕ}(f_{k}) ∈ C^{∞}(G) is bounded, then ϕ ∈ H^{2,<−k}.

Among the variety of results proved in Lott’s paper the following can be compared with those of the present paper.

Theorem 1.30 (Lott) Let Γ be convex cocompact.

1. If k ∈ [1, \frac{n−1}{2}), and ϕ ∈ ^{Γ}C^{−∞}(Λ_{Γ}, V (σ_{λ_{k}}^{k})), then ϕ ∈ H^{2,<−k}.

2. If Γ is not cocompact and ϕ ∈ ^{Γ}C^{−∞}(Λ_{Γ}, V (σ_{λ_{1}}^{1})), then ϕ ∈ H^{2,−1}.

This can directly be compared with our result, which gives ϕ ∈ H^{2,<−k+ \frac{ρ−δ_{Γ}}{2}} . Thus if Γ is not cocompact, then our result improves Lott’s estimate.

1.4.3 The result of J. Bernstein and A. Reznikov

In [2] J. Bernstein and A. Reznikov consider G = SL(2, ℝ) ≅ Spin(1, 2), and a discrete subgroup of finite covolume. In Sec. 1.4 they ”prove” the following result.

Theorem 1.31 (Bernstein-Reznikov) If Re(λ) = 0 and ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ})) is square integrable, then ϕ ∈ (H^{1,1/2+ϵ})^{∗} for any ϵ > 0.

Our estimate applies if λ ≠ 0, because in this case ϕ is a cusp form and λ ∉ I_{𝔞}. We show ϕ ∈ (H^{p,1/2+ϵ})^{∗} for any ϵ > 0 and p ∈ (1, ∞). This implies Bernstein-Reznikov’s result.

2 Elements of geometric scattering theory

2.1 The space B_{Γ}(σ_{λ}, φ)

The spaces B_{Γ}(σ_{λ}, φ) were introduced in [4]. In this subsection we recall some of their basic properties. First we consider the case that all cusps of Γ have smaller rank.

Recall that Γ acts properly discontinuously on the complement Ω_{Γ} = ∂X \ Λ_{Γ} of the limit set Λ_{Γ} with quotient B_{Γ}. Furthermore, we have a quotient bundle V_{B_{Γ}}(σ_{λ}, φ) := Γ\V (σ_{λ}, φ)_{|Ω_{Γ}}. The bundles V_{B_{Γ}}(σ_{λ}, φ) depend holomorphically on λ ∈ 𝔞_{ℂ}^{∗}. We consider the family of spaces (C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)))_{λ∈𝔞_{ℂ}^{∗}} as a (trivial) bundle of Fréchet spaces over 𝔞_{ℂ}^{∗}.

2.1 The space B_{Γ}(σ_{λ}, φ) 17

For Re(λ) < −δ_{Γ} − δ_{φ} we can define the push-down map

π_{∗}^{Γ} : C^{∞}(∂X, V (σ_{λ}, φ)) → C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ))

by π_{∗}^{Γ}(f) := ∑_{γ∈Γ} π^{σ_{λ},φ}(γ)f_{|Ω_{Γ}}. Indeed, this sum converges and defines a smooth Γ-invariant section of V (σ_{λ}, φ)_{|Ω_{Γ}}, and thus an element of C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)). One of the main results of [4] is that π_{∗}^{Γ} has a meromorphic continuation to all of 𝔞_{ℂ}^{∗}. If Γ is not convex-cocompact, then π_{∗}^{Γ} is not surjective. Roughly speaking, the space B_{Γ}(σ_{λ}, φ) is defined to be the range of π_{∗}^{Γ}.

Definition 2.1 If Γ has no cusps of full rank, then we define (B_{Γ}(σ_{λ}, φ))_{λ∈𝔞_{ℂ}^{∗}} to be the minimal holomorphic subbundle of (C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)))_{λ∈𝔞_{ℂ}^{∗}} over which π_{∗}^{Γ} factors.

This defines B_{Γ}(σ_{λ}, φ) as a vector space. In fact, f ∈ B_{Γ}(σ_{λ}, φ) iff there exists a meromorphic family (F_{μ})_{μ}, F_{μ} ∈ C^{∞}(∂X, V (σ_{μ}, φ)), such that (π_{∗}^{Γ}F_{μ})_{μ=λ} = f.

As a topological vector space it will be equipped with a topology which is stronger than the topology induced from the embedding B_{Γ}(σ_{λ}, φ) → C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)). With this stronger topology we cannot show that (B_{Γ}(σ_{λ}, φ))_{λ∈𝔞_{ℂ}^{∗}} is a locally trivial bundle of Fréchet spaces. As explained in [4] we can consider it as a projective limit (see below) of locally trivial bundles of Fréchet spaces, and it therefore still makes sense to speak of holomorphic families (f_{μ})_{μ∈𝔞_{ℂ}^{∗}}, f_{μ} ∈ B_{Γ}(σ_{μ}, φ).

Note that C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) ⊂ B_{Γ}(σ_{λ}, φ), so that the elements of B_{Γ}(σ_{λ}, φ) are distinguished among all smooth sections by their behaviour at the ends of B_{Γ}. In order to describe this behaviour in detail we first assume that Γ ⊂ P for some Γ-cuspidal parabolic subgroup P ⊂ G. Then we have Ω_{Γ} = ∂X \ ∞_{P}. We choose a maximal compact subgroup K and a Langlands decomposition P = MAN, M ⊂ K. Furthermore, let w ∈ K be a representative of the non-trivial element of the Weyl group N_{K}(M)/M. Then we can parametrize Ω_{Γ} = NwP ⊂ G/P. A function f ∈ C^{∞}(Ω_{Γ}, V (σ_{λ}, φ)) gives rise to a smooth function on N with values in V_{σ} ⊗ V_{φ} by

N ∋ x → f(xw) ∈ V_{σ} ⊗ V_{φ} .

For X ∈ U(𝔫), W ⊂ N, and β ∈ 𝔞^{∗} we define

q_{W,X,β}^{Γ}(f) := sup_{x∈W} sup_{a∈A_{+}} a^{β−2(Re(λ)−ρ^{Γ})}|φ(a^{−1})f(x^{a}Xw)|  (2)
q_{W,X}(f) = sup_{x∈W} |f(xX)| .

Note that in this context we extend φ to AP_{Γ} (which is possible since we assume that φ is admissible) such that the minimal A-weight is 0 and the highest A-weight is not greater than 2δ_{φ}.

The family of seminorms q_{W,X} for all compact W ⊂ N and X ∈ U(𝔫) defines the topology of C^{∞}(Ω_{Γ}, V (σ_{λ}, φ)).

Definition 2.2 Let S_{Γ,β,κ}(σ_{λ}, φ) be the closure of C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) with respect to Banach space topology defined by the seminorms q_{W,X}, deg(X) ≤ κ, W ⊂ Ω_{Γ} compact, and q_{W,X,β}^{Γ}, deg(X) ≤ κ, W ⊂ N \ N_{Γ} compact. Furthermore, we define the Fréchet spaces S_{Γ,β}(σ_{λ}, φ) := ⋂_{κ∈𝔞^{∗}} S_{Γ,β,κ}(σ_{λ}, φ) and S_{Γ}(σ_{λ}, φ) := ⋂_{β∈𝔞^{∗}} S_{Γ,β}(σ_{λ}, φ).

The elements of B_{Γ}(σ_{λ}, φ) will now be characterized by the property that they have a certain asymptotic expansion near ∞_{P} with remainder in S_{Γ}(σ_{λ}, φ). In order to describe this asymptotic expansion we introduce the following space of homogeneous functions.

Definition 2.3 For β ∈ 𝔞^{∗} we define A_{P_{Γ},β}(σ_{λ}, φ) ⊂ C^{∞}(Ω_{Γ} \ N_{Γ}wP, V (σ_{λ}, φ)) to be the closed subspace of P_{Γ}-invariant sections satisfying

φ(a)^{−1}f(x^{a}w) = a^{2(λ−ρ^{Γ})−β}f(xw) .

In [4] we have shown that if f ∈ B_{Γ}(σ_{λ}, φ), then it has an asymptotic expansion

φ(a^{−1})f(x^{a}w) ∼^{a→∞} ∑_{k∈ℕ_{0}} a^{2(λ−ρ^{Γ})−kα}f_{k}(x) ,

where f_{k} ∈ A_{P_{Γ},kα}(σ_{λ}, φ).

Definition 2.4 We define R_{Γ}(σ_{λ}, φ)^{k} ⊂ A_{P_{Γ},kα}(σ_{λ}, φ) to be the subspace generated by the f_{k} for all f ∈ B_{Γ}(σ_{λ}, φ).

The space R_{Γ}(σ_{λ}, φ)^{k} is finite-dimensional. We further set R_{Γ,k}(σ_{λ}, φ) := ⨁_{l≤k} R_{Γ}(σ_{λ}, φ)^{l}, B_{Γ,k}(σ_{λ}, φ) := S_{Γ,kα}(σ_{λ}, φ) + B_{Γ}(σ_{λ}, φ), such that we have an exact sequence

0 → S_{Γ,kα}(σ_{λ}, φ) → B_{Γ,k}(σ_{λ}, φ) →^{AS} R_{Γ,k}(σ_{λ}, φ) → 0 .

We can construct a split L : R_{Γ,k}(σ_{λ}, φ) → B_{Γ,k}(σ_{λ}, φ) of this exact sequence using a suitable P_{Γ}-invariant cut-off function. So we have B_{Γ,k}(σ_{λ}, φ) = S_{Γ,kα}(σ_{λ}, φ) ⊕ L(R_{Γ,k}(σ_{λ}, φ)), and this decomposition defines a Fréchet space topology on B_{Γ,k}(σ_{λ}, φ). Note that B_{Γ}(σ_{λ}, φ) = ⋂_{k∈ℕ} B_{Γ,k}(σ_{λ}, φ).

We need the following seminorms on B_{Γ}(σ_{λ}, φ).

Definition 2.5 If W ⊂ N \ N_{Γ_{P}} is compact, k ∈ ℕ_{0}, and X ∈ U(𝔫) is homogeneous of degree deg(X) ∈ 𝔞^{∗}, then we define p_{W,X}^{Γ}(f) by

p_{W,X}^{Γ}(f) := sup_{a∈A_{+}} sup_{x∈W} a^{deg(X)−2(Re(λ)−ρ^{Γ})}|φ(a^{−1})f(x^{a}Xw)| .

Lemma 2.6 p_{W,X}^{Γ} is a continuous seminorm on B_{Γ}(σ_{λ}, φ).

Proof. Fix k ∈ ℕ such that kα > deg(X). Let f ∈ B_{Γ,k}(σ_{λ}, φ). We define f_{l} such that AS(f) = ⊕_{l≤k}f_{l}. Furthermore, we set g := f − LAS(f) ∈ S_{Γ,kα}(σ_{λ}, φ). Note that f_{l} and g depend continuously on f. The restriction of p_{W,X}^{Γ} to the space S_{Γ,kα}(σ_{λ}, φ) belongs to the family of seminorms which define the topology of S_{Γ,kα}(σ_{λ}, φ). So it remains to show that A_{P_{Γ},kα}(σ_{λ}, φ) ∋ f_{l} → p_{W,X}^{Γ}(Lf_{l}) is continuous.

Let χ_{Γ} be the P_{Γ}-invariant cut-off function used to define L. Furthermore, let Δ(X) = ∑_{j} Y_{j} ⊗ Z_{j} be the coproduct. Then we can write

φ(a^{−1})L(f_{l})(x^{a}Xw) = φ(a^{−1}) ∑_{j} χ(x^{a}Y_{j}w)f_{l}(x^{a}Z_{j}w)
= φ(a^{−1}) ∑_{j} χ(x^{a}Y_{j}w)f_{l}((xZ_{j}^{a^{−1}})^{a}w)
= ∑_{j} a^{2(λ−ρ^{Γ})−deg(Z_{j})−lα}χ(x^{a}Y_{j}w)f_{l}(xZ_{j}w)
= a^{2(λ−ρ^{Γ})−deg(X)−lα}χ(x^{a}w)f_{l}(xXw)

+ ∑_{j,deg(Y_{j})>0} χ(x^{a}wY_{j})f_{l}((xZ_{j}^{a^{−1}})^{a}w) .

The summands for deg(Y_{j}) > 0 vanish for large a ∈ A_{+} uniformly for x ∈ W. We therefore can majorize p_{W,X}^{Γ}(Lf_{l}) by a continuous seminorm of A_{P_{Γ},lα}(σ_{λ}, φ). □

We can now describe B_{Γ}(σ_{λ}, φ) for a general geometrically finite torsion-free group Γ without cusps of full rank. If P ⊂ G is a Γ-cuspidal parabolic subgroup, then there is a representative B_{P} ⊂ B_{Γ} of an end of B_{Γ} which is isomorphic to a representative of the end of B_{Γ_{P}}. Let χ_{P} ∈ C^{∞}(B_{Γ}) be supported on B_{P} such that it is identically one near infinity. If f ∈ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)), then we can consider χ_{P} f as a section T_{P} f ∈ C^{∞}(B_{Γ_{P}} , V_{B_{Γ_{P}}} (σ_{λ}, φ)). Then f ∈ B_{Γ}(σ_{λ}, φ) iff T_{P} f ∈ B_{Γ_{P}} (σ_{λ}, φ) for all Γ-cuspidal parabolic subgroups P ⊂ G.

Definition 2.7 As a topological vector space B_{Γ}(σ_{λ}, φ) is equipped with the smallest topology such that the inclusion B_{Γ}(σ_{λ}, φ) ⊂ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) and all the maps T_{P} are continuous.

In a similar manner we define the topological vector spaces B_{Γ,k}(σ_{λ}, φ) (S_{Γ,β}(σ_{λ}, φ)) to be the space of all f ∈ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) such that T_{P} f ∈ B_{Γ_{P} ,k}(σ_{λ}, φ) (T_{P} f ∈ S_{Γ_{P} ,β}(σ_{λ}, φ)) for all Γ-cuspidal parabolic subgroups P ⊂ G. Furthermore, we set R_{Γ,k}(σ_{λ}, φ) := ⨁_{P ∈P̃} R_{Γ_{P} ,k}(σ_{λ}, φ), where P̃ denotes a set of representatives of the Γ-conjugacy classes of Γ-cuspidal parabolic subgroups (P̃ parametrizes the ends of B_{Γ}). Then we have a split exact sequence

0 → S_{Γ,kα}(σ_{λ}, φ) → B_{Γ,k}(σ_{λ}, φ) →^{AS} R_{Γ,k}(σ_{λ}, φ) → 0 ,

and we can write

B_{Γ,k}(σ_{λ}, φ) = S_{Γ,kα}(σ_{λ}, φ) + L(R_{Γ,k}(σ_{λ}, φ)) .  (3)

We now describe the necessary modification if Γ has cusps of full rank. In this case the ends of B_{Γ} are parametrized by the set of Γ-conjugacy classes of Γ-cuspidal subgroups P^{<} corresponding to cusps of smaller rank. The same construction as above gives spaces B_{Γ}(σ_{λ}, φ)_{1}, B_{Γ,k}(σ_{λ}, φ)_{1}. The space B_{Γ,k}(σ_{λ}, φ) will be the direct sum of B_{Γ,k}(σ_{λ}, φ)_{1} and a finite-dimensional vector space B_{Γ}(σ_{λ}, φ)_{2} := ⊕_{P ∈P̃^{max}}R_{Γ_{P}} (σ_{λ}, φ), where P̃^{max} is a set of representatives of Γ-conjugacy classes Γ-cuspidal parabolic subgroups corresponding to cusps of full rank.

The space R_{Γ_{P}} (σ_{λ}, φ) is the fibre of a trivial holomorphic vector bundle as follows. Let Γ_{P} E_{∞_{P}} (σ, φ) denote the sheaf of holomorphic families f_{ν} ∈ ^{Γ_{P}} C^{−∞}(∂X, V (σ_{ν}, φ)) with supp(f) = ∞_{P}. Since ^{Γ_{P}} E_{∞_{P}} (σ, φ) is torsion-free it is the space of sections of a unique holomorphic vector bundle ^{Γ_{P}} E_{∞_{P}} (σ, φ) over 𝔞_{ℂ}^{∗}. By ^{Γ_{P}} E_{∞_{P}} (σ_{λ}, φ) we denote the fibre of ^{Γ_{P}} E_{∞_{P}} (σ, φ) at λ ∈ 𝔞_{ℂ}^{∗}. Then we define R_{Γ_{P}} (σ_{λ}, φ) := ^{Γ_{P}} E_{∞_{P}} (σ̃_{−λ}, φ̃)^{∗}.

The push-down π_{∗}^{Γ} now decomposes as (π_{∗}^{Γ})_{1}⊕(π_{∗}^{Γ})_{2}, where (π_{∗}^{Γ})_{1} is the average as in the case without cusps of full rank. The second component further decomposes as (π_{∗}^{Γ})_{2} = ⊕_{P ∈P̃^{max}}[π_{∗}^{Γ}]_{P}, where [π_{∗}^{Γ}]_{P} has values in R_{Γ_{P}} (σ_{λ}, φ). For Re(λ) < −δ_{Γ} − δ_{φ} it is defined by the condition

⟨ϕ, [π_{∗}^{Γ}]_{P} (f)⟩ = ∑_{[g]∈Γ/Γ_{P}} ⟨π^{σ̃_{−λ},φ̃}(g)ϕ, f⟩

for all ϕ ∈ ^{Γ_{P}} E_{∞_{P}} (σ̃_{−λ}, φ̃), and for general λ by meromorphic continuation.

The family of spaces (B_{Γ,k}(σ_{λ}, φ))_{λ∈𝔞_{ℂ}^{∗}} forms a trivial holomorphic bundle of Fréchet spaces over 𝔞_{ℂ}^{∗}. In the present paper we do not need an explicit trivialization, and we therefore omit its description, which can be found in [4].

2.2 Restriction, extension, and the scattering matrix

In the preceding subsection we have defined the Fréchet spaces B_{Γ}(σ_{λ}, φ).

Definition 2.8 We define D_{Γ}(σ_{λ}, φ) := B_{Γ}(σ̃_{−λ}, φ̃)^{∗}.

Furthermore, we introduce

D_{Γ,k}(σ_{λ}, φ) := B_{Γ,k}(σ̃_{−λ}, φ̃)^{∗} .

The family (D_{Γ,k}(σ_{λ}, φ))_{λ∈𝔞_{ℂ}^{∗}} is a trivial holomorphic bundle of dual Fréchet spaces. Moreover, we have D_{Γ}(σ_{λ}, φ) = lim_{k^{→}} D_{Γ,k}(σ_{λ}, φ). Therefore we can speak of holomorphic families (ϕ_{μ})_{μ∈𝔞_{ℂ}^{∗}}, ϕ_{μ} ∈ D_{Γ}(σ_{μ}, φ). Locally, ϕ_{μ} is a holomorphic family in D_{Γ,k}(σ_{μ}, φ) for some fixed k.

Definition 2.9 We define the extension map ext^{Γ} : D_{Γ}(σ_{λ}, φ) → C^{−∞}(∂X, V (σ_{λ}, φ)) to be the adjoint of π_{∗}^{Γ}.

If (ϕ_{μ})_{μ∈𝔞_{ℂ}^{∗}}, ϕ_{μ} ∈ D_{Γ}(σ_{μ}, φ), is a meromorphic family, then ext^{Γ}ϕ_{μ} ∈ C^{−∞}(∂X, V (σ_{λ}, φ)) is a meromorphic family of Γ-invariant distributions.

Definition 2.10 By Ext_{Γ}(σ_{λ}, φ) we denote the subspace of ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) consisting of evaluations at λ of germs families (ext^{Γ}ϕ_{μ})_{μ}, where (ϕ_{μ})_{μ}, ϕ_{μ} ∈ D_{Γ}(σ_{μ}, φ), is a germ of a meromorphic family. By Ext_{Γ}^{0}(σ_{λ}, φ) ⊂ Ext_{Γ}(σ_{λ}, φ) we denote the subspace of those evaluations, where ϕ_{λ} is regular.

In [4] we have shown that for generic λ ∈ 𝔞_{ℂ}^{∗} we have

Ext_{Γ}^{0}(σ_{λ}, φ) = Ext_{Γ}(σ_{λ}, φ) = ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) .

Definition 2.11 If (ψ_{μ})_{μ}, ψ_{μ} ∈ ^{Γ}C^{−∞}(∂X, V (σ_{μ}, φ)), is a germ of a meromorphic family at λ, then we define (res^{Γ}(ψ_{μ}))_{μ}, res^{Γ}(ψ_{μ}) ∈ D_{Γ}(σ_{μ}, φ), to be the unique germ of a meromorphic meromorphic family such that ext^{Γ}(res^{Γ}(ψ_{μ})) = ψ_{μ}.

Note that if ψ ∈ Ext_{Γ}^{0}(σ_{λ}, φ), then we can define res^{Γ}ψ ∈ D_{Γ}(σ_{λ}, φ).

Recall Definition 1.25 of Ext_{Γ}^{sing}(σ_{λ}, φ).

Lemma 2.12 We have the equality

Ext_{Γ}^{sing}(σ_{λ}, φ) = Fam_{Γ}(Λ_{Γ}, σ_{λ}, φ) .

Proof. The inclusion Ext_{Γ}^{sing}(σ_{λ}, φ) ⊂ Fam_{Γ}(Λ_{Γ}, σ_{λ}, φ) follows from the identity res^{Γ} ∘ext^{Γ} = id, which holds when both sides are applied to germs of meromorphic families. The inclusion Fam_{Γ}(Λ_{Γ}, σ_{λ}, φ) ⊂ Ext_{Γ}^{sing}(σ_{λ}, φ) follows from the similar identity ext^{Γ} ∘ res^{Γ} = id, which also holds after application to germs of meromorphic families. □

Next we show that the condition ”strongly supported on the limit set” is stable under twisting and embedding.

Lemma 2.13 If i_{∗} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(∂X′, V (σ′_{λ′}, φ′)) is induced by some twisting or embedding data, then

i_{∗}(Fam_{Γ}(Λ_{Γ}, σ_{λ}, φ)) ⊂ Fam_{Γ}(Λ_{Γ}, σ′_{λ′}, φ′) .

Proof. We show i_{∗}(Ext_{Γ}^{sing}(σ_{λ}, φ)) ⊂ Ext_{Γ}^{sing}(σ′_{λ′}, φ′) and then apply Lemma 2.12. But the latter inclusion follows immediately from the identity (see [4], Sec. 2.3) i_{∗} ∘ ext^{Γ} = ext^{Γ} ∘ i_{∗}^{Γ}, where the maps ext^{Γ} on the two sides of course act on different spaces, and i_{∗}^{Γ} : D_{Γ}(σ_{λ}, φ) → D_{Γ}(σ′_{λ′}, φ′).□

We can now define the scattering matrix. We fix a maximal compact subgroup K ⊂ G and a parabolic subgroup P ⊂ G. Let P = MAN be the associated Langlands decomposition and w ∈ N_{K}(M)/M be a representative of the non-trivial element of the Weyl group. Then we have the meromorphic family of Knapp-Stein intertwining operators Ĵ_{σ_{λ},φ}^{w} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(∂X, V (σ_{−λ}^{w}, φ)), where σ^{w}(m) = σ(m^{w^{−1}}). If Re(λ) < 0 and f ∈ C^{∞}(∂X, V (σ_{λ}, φ)), then it is given by

Ĵ_{σ_{λ},φ}^{w}(f)(g) = ∫N̄ f(gwn̄)dn̄ .

Let (ϕ_{μ})_{μ∈𝔞_{ℂ}^{∗}}, ϕ_{μ} ∈ D_{Γ}(σ_{μ}, φ), be a germ of a meromorphic family at λ. Then (J_{σ_{μ},φ}^{w}(ext^{Γ}ϕ_{μ}))_{μ}, J_{σ_{μ},φ}^{w}(ext^{Γ}ϕ_{μ}) ∈ ^{Γ}C^{−∞}(∂X, V (σ_{−μ}^{w}, φ)), is a germ of a meromorphic family.

Definition 2.14 We define the germ of a meromorphic family (Ŝ_{σ_{μ},φ}^{w}(ϕ_{μ}))_{μ} at λ, Ŝ_{σ_{μ},φ}^{w}(ϕ_{μ}) ∈ D_{Γ}(σ_{−μ}^{w}, φ), by Ŝ_{σ_{μ},φ}^{w}(ϕ_{μ}) := res^{Γ}(Ĵ_{σ_{μ},φ}^{w}(ext^{Γ}ϕ_{μ})).

Note that for generic λ we have a well-defined continuous map Ŝ_{σ_{λ},φ}^{w} : D_{Γ}(σ_{λ}, φ) → D_{Γ}(σ_{−λ}^{w}, φ). The identity

ext^{Γ} ∘ Ŝ_{σ_{μ},φ}^{w} = Ĵ_{σ_{μ},φ}^{w} ∘ ext^{Γ} .

is an immediate consequence of the definitions. Strictly speaking, this equation holds after applying both sides to a germ of a meromorphic family.

There is a natural pairing V_{B_{Γ}}(σ_{λ}, φ)⊗V_{B_{Γ}}(σ̃_{−λ}, φ̃) → V_{B_{Γ}}(−ρ). Since V_{B_{Γ}}(−ρ) is the bundle of densities on B_{Γ} its sections can be integrated. In [4] we have shown that if max_{P ∈P^{<}}(δ_{φ_{|P_{Γ}}} − ρ^{Γ_{P}}) < 0, then we have a non-degenerate paring B_{Γ}(σ_{λ}, φ) ⊗ B_{Γ}(σ̃_{−λ}, φ̃) → ℂ and thus an inclusion B_{Γ}(σ_{λ}, φ) ↪ D_{Γ}(σ_{λ}, φ). Furthermore, we have shown the following result

Lemma 2.15 If (f_{μ})_{μ} is a germ at λ of a meromorphic family, f_{μ} ∈ B_{Γ}(σ_{μ}, φ), then (Ŝ_{σ_{μ},φ}^{w}(f_{μ}))_{μ} is a germ at λ of a meromorphic family, Ŝ_{σ_{μ},φ}^{w}(f_{μ}) ∈ B_{Γ}(σ_{−μ}^{w}, φ).

We now prove Theorem 1.14 assuming that res^{Γ} is regular outside I_{𝔞}. Let ϕ ∈ Cusp(σ_{λ}, φ). By Theorem 1.7 we know that ϕ is stably deformable. After suitable twisting and embedding we can assume that ϕ ∈ Cusp(σ_{λ}, φ) ∩ Fam_{Γ}(σ_{λ}, φ) and there are no cusps of full rank. Let (ϕ_{μ})_{μ} be a germ of a holomorphic family such that ϕ_{λ} = ϕ. Since supp(ϕ) ⊂ Λ_{Γ} we conclude that {res^{Γ}ϕ_{μ}}_{|μ=λ} = 0, where {ψ} denotes the restriction of ψ ∈ D_{Γ}(σ_{λ}, φ) to S_{Γ}(σ̃_{−λ}, φ̃) ⊂ B_{Γ}(σ̃_{−λ}, φ̃) (here we use that fact that C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ̃_{−λ}, φ̃) is dense in S_{Γ}(σ̃_{−λ}, φ̃)). Using the decomposition [4], (22), we can write

(res^{Γ}ϕ_{μ})_{μ=λ} = ∑_{P ∈P̃}(T_{P}^{∗}res^{Γ_{P}} ϕ_{μ})_{μ=λ} .

It suffices to show that (T_{P}^{∗}res^{Γ_{P}} ϕ_{μ})_{μ=λ} = 0 for all P ∈ P̃. Let χ ∈ C_{c}^{∞}(P_{Γ}) be such that ∑_{γ∈Γ_{P}} γ^{∗}χ ≡ 1. Then we can write

ϕ_{P} := ∫_{P_{Γ}} χ(x)π^{σ_{λ},φ}(x)ϕ dx .

Since we assume that λ ∉ I_{𝔞} the push-down π_{∗}^{Γ_{P}} is regular and surjective at λ. In fact, the poles of π_{∗}^{Γ_{P}} are the same as the poles of π_{∗}^{P_{Γ}} (see the definition below). The location of the latter is determined in [4], Lemma 3.14. Furthermore, regularity of res^{Γ} is equivalent to surjectivity of π_{∗}^{Γ_{P}}. Thus in order to check that T_{P}^{∗}(res^{Γ}ϕ_{μ})_{μ=λ} = 0 it suffices to show that ⟨χ_{P} res^{Γ_{P}} ϕ_{μ}, π_{∗}^{Γ_{P}} f_{−μ}⟩_{μ=λ} = 0 for all germs (f_{μ})_{μ} at −λ of holomorphic families f_{μ} ∈ C^{∞}(∂X, V (σ̃_{μ}, φ̃)). In fact, since {χ_{P} res^{Γ_{P}} ϕ_{μ}}_{μ=λ} = 0 and B_{Γ_{P}} (σ̃_{μ}, φ̃) is a C_{c}^{∞}(B_{Γ_{P}})-module it suffices to show

⟨res^{Γ_{P}} ϕ_{μ}, π_{∗}^{Γ_{P}} f_{−μ}⟩_{μ=λ} = 0

for all families with supp(π_{∗}^{Γ_{P}} f_{μ}) ⊂ {χ_{P} = 1}. We now consider

π_{∗}^{P_{Γ}}(f_{μ}) := ∫_{P_{Γ}} π^{σ̃_{μ},φ̃}(x)(f_{μ})_{|Ω_{Γ_{P}}} dx .

In [4] we have shown that π_{∗}^{P_{Γ}}(f_{μ}) ∈ B_{Γ_{P}} (σ̃_{μ}, φ̃), and that (π^{Γ_{P}} (f_{μ}) − π_{∗}^{P_{Γ}}(f_{μ}))_{μ} is a germ of a holomorphic family in S_{Γ}(σ̃_{μ}, φ̃). Since π_{∗}^{P_{Γ}} factorizes over π_{∗}^{Γ_{P}} we also have supp(π_{∗}^{P_{Γ}}f_{μ}) ⊂ {χ_{P} = 1}. Because of {χ_{P} res^{Γ_{P}} ϕ_{μ}}_{μ=λ} = 0 we get

⟨res^{Γ_{P}} ϕ_{μ}, π_{∗}^{Γ_{P}} f_{−μ}⟩_{μ=λ} = ⟨res^{Γ_{P}} ϕ_{μ}, π_{∗}^{P_{Γ}}f_{−μ}⟩_{μ=λ} .  (4)

We compute

⟨res^{Γ_{P}} ϕ_{μ}, π_{∗}^{P_{Γ}}f_{−μ}⟩_{μ=λ} = ⟨res^{Γ_{P}} ϕ_{μ}, π_{∗}^{Γ_{P}} ∫_{P_{Γ}} χ(x)π^{σ_{−μ},φ̃}(x)f_{−μ}dx⟩_{μ=λ}
= ⟨ext^{Γ_{P}} res^{Γ_{P}} ϕ_{μ}, ∫_{P_{Γ}} χ(x)π^{σ_{−μ},φ̃}(x)f_{−μ}dx⟩_{μ=λ}
= (∫_{P_{Γ}} χ(x)⟨ϕ_{μ}, π^{σ_{−μ},φ̃}(x)f_{−μ}⟩dx)_{μ=λ}
= ⟨(ϕ_{μ})_{P} , f_{−μ}⟩_{μ=λ}
= 0 .

We now apply (4). □

3 Some functional analytic preparations

3.1 Complex interpolation

For p ∈ (1, ∞) and r ∈ ℕ_{0} the space H^{p,r}(∂X, V (σ_{λ}, φ)) is the usual Sobolev space of sections of V (σ_{λ}, φ) which have distributional derivatives in L^{p} up to order r. This space is well-defined, but there is no prefered norm. There are several slightly different ways to define Sobolev spaces of non-integral order. For our purpose a natural choice is complex interpolation. If r_{0}, r_{1} ∈ ℕ_{0}, θ ∈ (0, 1), and r := r_{0} + θ(r_{1} − r_{0}), then we define (see [11], IX.4, [1], Ch. 6)

H^{p,r}(∂X, V (σ_{λ}, φ)) := [H^{p,r_{0}}(∂X, V (σ_{λ}, φ)), H^{p,r_{1}}(∂X, V (σ_{λ}, φ))]_{θ} .

If r ∈ ℕ_{0}, then this definition coincides with the former. We extend the scale of Sobolev spaces to negative orders by duality

H^{p,−r}(∂X, V (σ_{λ}, φ)) := H^{q,r}(∂X, V (σ̃_{−λ}, φ̃))^{∗} ,

where r ≥ 0 and \frac{1}{p} + \frac{1}{q} = 1. Then for r_{0}, r, r_{1} ∈ ℝ with r_{0} < r < r_{1}, r = r_{0} + θ(r_{1} − r_{0}) and p_{0}, p, p_{1} ∈ (1, ∞), \frac{1}{p} = \frac{1}{p_{0}} + θ(\frac{1}{p_{1}} − \frac{1}{p_{0}} ), we have

H^{p,r}(∂X, V (σ_{λ}, φ)) := [H^{p_{0},r_{0}}(∂X, V (σ_{λ}, φ)), H^{p_{1},r_{1}}(∂X, V (σ_{λ}, φ))]_{θ} .  (5)

We further will consider the Fréchet spaces

H^{p,<r}(∂X, V ((σ_{λ}, φ)) := ⋂_{s<r} H^{p,s}(∂X, V ((σ_{λ}, φ)) .

For the convenience of the reader we recall the definition of the interpolation space (5). The pair of spaces

(H^{p_{0},r_{0}}(∂X, V (σ_{λ}, φ)), H^{p_{1},r_{1}}(∂X, V (σ_{λ}, φ)))

considered as subspaces of C^{−∞}(∂X, V (σ_{λ}, φ)) forms a couple in the sense of [1], Sec. 2.3. There is a natural Banach norm on the sum Σ := H^{p_{0},r_{0}}(∂X, V (σ_{λ}, φ)) + H^{p_{1},r_{1}}(∂X, V (σ_{λ}, φ)). Let F be the space of all bounded, continuous functions f on the strip {Re(z) ∈ [0, 1]} with values in Σ which are holomorphic on the interior of this strip, such that for t ∈ ℝ we have that f(it) ∈ H^{p_{0},r_{0}}(∂X, V (σ_{λ}, φ)) and f(it + 1) ∈ H^{p_{1},r_{1}}(∂X, V (σ_{λ}, φ)) are continuous and

sup_{t∈ℝ}∥f(it)∥_{H^{p_{0},r_{0}}} < ∞ , sup_{t∈ℝ}∥f(it + 1)∥_{H^{p_{1},r_{1}}} < ∞ .

Then sup_{t∈ℝ}{∥f(it)∥_{H^{p_{0},r_{0}}} , ∥f(it + 1)∥_{H^{p_{1},r_{1}}} } is a norm making F into a Banach space. Let F_{θ} be the closed subspace of all f ∈ F with f(θ) = 0. Then

[H^{p_{0},r_{0}}(∂X, V (σ_{λ}, φ)), H^{p_{1},r_{1}}(∂X, V (σ_{λ}, φ))]_{θ} := F/F_{θ} .

We employ complex interpolation in the following way. We will show for a fixed distribution f ∈ C^{−∞}(∂X, V (σ_{λ}, φ)) that f ∈ H^{p_{0},r_{0}} ∩ H^{p_{1},r_{1}}. If θ ∈ (0, 1), r = r_{0} + θ(r_{1} − r_{0}) and \frac{1}{p} = \frac{1}{p_{0}} + θ(\frac{1}{p_{1}} − \frac{1}{p_{0}} ), then we conclude that f ∈ H^{p,r}. In order to see this we consider f as a constant function from {Re(z) ∈ [0, 1]} to Σ and apply the interpolation result (5) above.

3.2 Regularity of intertwining operators

In this subsection we consider the Sobolev spaces H^{p,r} := H^{p,r}(ℝ^{n}) for p ∈ (1, ∞) and r ∈ ℝ and the regularity properties of the operator A_{s} : C_{c}^{∞}(ℝ^{n}) → C^{−∞}(ℝ^{n}) which is given by the convolution kernel a_{s}(x) := ∥x∥^{−n−2s} for s ∈ ℂ, where ∥x∥^{2} := ∑_{i=1}^{n} x_{i}^{2}. To be precise, a_{s} is a regular distribution for Re(s) < 0, and it is defined by meromorphic continuation for all s ∈ ℂ. Let Δ := ∑_{i=1}^{n}(\frac{∂}{∂x_{i}})^{2} be the Laplace operator. Then we have Δ∥x∥^{−n−2s} = (2s+n)(2s+ 2)∥x∥^{−n−2−2s}. Iterating the equation

A_{s+1} = \frac{1}{(2s + n)(2s + 2)}ΔA_{s} .  (6)

we obtain the required meromorphic continuation.

For Re(s) = 0, s ≠ 0 the operator A_{s} is a singular integral operator. It is well-known that A_{s} ∈ B(H^{p,r}) for p ∈ (1, ∞), r ∈ ℝ. If Re(s) ∈ ℕ_{0}, then we can study the mapping properties of A_{s} by reduction to the case Re(s) = 0 using (6). If Re(s) ≥ 0 is not integral, then we can reduce to the case Re(s) ∈ (−1, 0). In this case A_{s} is given by convolution by a regular distribution. But A_{s} does not act nicely on H^{p,r} since this distribution grows at infinity. So we must restrict the domain and extend the range.

Let H_{c}^{p,r} ⊂ H^{p,r} be the subspace of all elements of compact support, and let H_{loc}^{p,r} ⊂ C^{−∞}(ℝ^{n}) be the space of all distributions f such that χf ∈ H^{p,r} for all χ ∈ C_{c}^{∞}(ℝ^{n}). We define H_{loc}^{p,<r} := ⋂_{s<r} H_{loc}^{p,s}.

If m ∈ ℕ_{0}, then let A_{m} = (s − m)^{−1}B_{m,−1} + B_{m,0} + … be the Laurent expansion of A_{s} at s = m. Our main result is the following theorem.

Theorem 3.1 If Re(s_{0}) ≥ 0 and s_{0} ∉ ℕ_{0}, then we have

A_{s} : H_{c}^{p,r} → H_{loc}^{p,r−2Re(s_{0})} .

For m ∈ ℕ_{0} we have

B_{m,−1} : H^{p,r} → H^{p,r−2m} ,
B_{m,k} : H_{c}^{p,r} → H_{loc}^{p,<r−2m}, k ∈ ℕ_{0} .

Proof. The space H_{c}^{p,r} is a limit of Banach spaces, while H_{loc}^{p,r} is a Fréchet space in a natural way. For each compact K ⊂ ℝ^{n} let H_{K}^{p,r} := {f ∈ H^{p,r} | supp(f) ⊂ K}. Then H_{K}^{p,r} is a closed subspace of H^{p,r}, and H_{c}^{p,r} = lim_{K^{→}} H^{p,r}. For each χ ∈ C_{c}^{∞}(ℝ^{n}) we define the seminorm q_{χ,p,r}(f) := ∥χf∥_{H^{p,r}} on H_{loc}^{p,r}. The family of these seminorms defines the topology of H_{loc}^{p,r}. If A : H_{c}^{p,r} → H_{loc}^{p,s} is continuous, then for each K and χ, we define

q(A)_{K,χ,p,r,s} := sup_{f∈H_{K}^{p,r},∥f∥_{H^{p,r}} =1} q_{χ,p,s}(Af) .

First we assume that Re(s) ∈ [−1, 0]. In order to fix the pole at s = 0 we set Â := sA_{s}.

Proposition 3.2 For each compact subset K ⊂ ℝ^{n} and χ ∈ C_{c}^{∞}(ℝ^{n}) there exists C < ∞ such that for all s ∈ ℂ with Re(s) ∈ [−1, 0] we have

q_{K,χ,p,0,0}(Â_{s}) ≤ C(1 + |s|)^{3} .

Proof. Fix a compact K ⊂ ℝ^{n} and χ ∈ C_{c}^{∞}(ℝ^{n}). Let

R := sup_{x∈K,y∈supp(χ)} dist(x, y) .  (7)

Let κ ∈ C_{c}^{∞}(ℝ^{n}) be a cut-off function such that κ(x) = 1 for ∥x∥ ≤ R and κ(x) = 0 for ∥x∥ ≥ 2R. We define a_{s}^{κ} := κa_{s} and let A_{s}^{κ} be the convolution operator defined by a_{s}^{κ}. If supp(f) ⊂ K, then we have χA_{s}f = χA_{s}^{κ}(f). It suffices to show

∥sA_{s}^{κ}∥_{B(L^{p})} ≤ C(1 + |s|)^{3} ,

where C is independent of s ∈ {Re(s) ∈ [−1, 0]}. If Re(s) < 0, then a_{s}^{κ} ∈ L^{1}(ℝ^{n}), so that by Yang’s inequality

∥A_{s}^{κ}∥_{B(L^{p})} ≤ ∥a_{s}^{κ}∥_{L^{1}} .

There is a constant C < ∞ such that ∥a_{s}^{κ}∥_{L^{1}} ≤ C uniformly for {Re(s) ∈ [−1, −1/8]}, and hence

∥A_{s}^{κ}∥_{B(L^{p})} ≤ C .  (8)

We cannot use this estimate close to the imaginary axis, since in that region ∥a_{s}^{κ}∥_{L^{1}} explodes like Re(s)^{−1}, and we need a uniform estimate. The point is that we must take into account the oscillating behaviour of a_{s}^{κ}(x) near x = 0. Then the divergence above only occurs at s = 0, and this is fixed because we consider Â_{s}^{κ} = sA_{s}^{κ}.

Lemma 3.3 There exists C < ∞ such that on {Re(s) ∈ [−1/4, 0]}

∥Â_{s}^{κ}∥_{B(L^{2})} ≤ C(1 + |s|) .

Proof. We first estimate ∥A_{s}^{κ}∥_{B(L^{2})} using the Fourier transform. The Fourier transform F(a_{s}^{κ}) is the convolution of F(a_{s}) and F(κ) ∈ S(ℝ^{n}). We have

F(a_{s})(ξ) = c 2^{−2s} \frac{Γ(−s)}{Γ(\frac{1}{2}n + s)}∥ξ∥^{2s} ,

where c is independent of s. Note that (see [12], 6.328)

lim_{|y|→∞} |Γ(x + iy)|e^{\frac{π}{2} |y|}|y|^{\frac{1}{2} −x} = \sqrt{2π} ,

locally uniformly for x ∈ ℝ. Using this property of the Γ-function we see that there is a constant C < ∞ such that for Re(s) ∈ [−1/4, 0] we have |s\frac{Γ(−s)}{Γ( \frac{1}{2} n+s)}| ≤ C(1 + Im(s))^{−2Re(s)−n/2+1}. Let χ_{1} ∈ C_{c}^{∞}(ℝ^{n}) be a cut-off function such that χ_{1}(ξ) = 1 for ∥ξ∥ ≤ 1. Then uniformly on {Re(s) ∈ [−1/4, 0]} we have ∥(1 − χ_{1})F(â_{s})∥_{L^{∞}} ≤ C(1 + |s|). Furthermore, on {Re(s) ∈ [−1/4, 0]} we obtain a uniform estimate ∥χ_{1}F(â_{s})∥_{L^{1}} ≤ C(1 + |s|). Writing F(â_{s}^{κ}) = F(κ) ∗ χ_{1}F(â_{s}) + F(κ) ∗ (1 − χ_{1})F(â_{s}) we can estimate

∥F(â_{s}^{κ})∥_{L^{∞}} ≤ ∥F(κ)∥_{L^{∞}}∥χ_{1}F(â_{s})∥_{L^{1}} + ∥F(κ)∥_{L^{1}}∥(1 − χ_{1})F(â_{s})∥_{L^{∞}} ≤ C(1 + |s|)

uniformly on {Re(s) ∈ [1/4, 0]}. This gives

∥Â_{s}^{κ}∥_{B(L^{2})} ≤ C(1 + |s|)

uniformly on {Re(s) ∈ [1/4, 0]}. □

In order to extend the assertion of Lemma 3.3 from L^{2} to L^{p} we need the following estimate.

Lemma 3.4 There exists a constant C < ∞ such that if {Re(s) ∈ [−1/4, 0]}, then

∫_{dist(y,z)≥4dist(x,z)} |â_{s}^{κ}(y − x) − â_{s}^{κ}(y − z)|dy ≤ C(1 + |s|^{3})  (9)

for all x, z ∈ ℝ^{n}.

Proof. On the domain of integration we have

\frac{1}{2}dist(z, y) ≤ dist(x, y) ≤ 2dist(z, y) .

Recall the definition (7) of R. We decompose the domain of integration in (9) into the regions {dist(z, y) ≤ R/2}, {R/2 ≤ dist(z, y) ≤ 2R}, {dist(z, y) ≥ 2R}, and let I_{1}, I_{2}, I_{3} denote the corresponding integrals. Since the integrand vanishes on the third region we have I_{3} = 0. Next we consider I_{1}. In this region we can replace â_{s}^{κ} by â_{s}. For 0 ≠ x ∈ ℝ^{n} let x^{0} := \frac{x}{∥x∥}. We compute

|â_{s}(y − z) − â_{s}(y − x)|  (10)
= |s| |∥y − z∥^{−n−2s} − ∥y − x∥^{−n−2s}|
= |s|∥y − z∥^{−n−2Re(s)} |1 − \frac{∥y − z + z − x∥^{−n−2s}}{∥y − z∥^{−n−2s}} |
= |s|∥y − z∥^{−n−2Re(s)} |1 − ∥(y − z)^{0} + \frac{z − x}{∥y − z∥}∥^{−n−2s}|
= |s|∥y − z∥^{−n−2Re(s)} |1 − (1 + 2 ⟨(y − z)^{0}, \frac{z − x}{∥y − z∥}⟩ + \frac{∥z − x∥^{2}}{∥y − z∥^{2}} )^{\frac{−n−2s}{2}} | . (11)

Note that b := 2⟨(y − z)^{0}, \frac{z−x}{∥y−z∥}⟩ + \frac{∥z−x∥^{2}}{∥y−z∥^{2}} satisfies |b| ≤ 1/2 + 1/16 < 3/4. For b ∈ ℝ, |b| < 3/4 we write (1 + b)^{−n−2s} = exp((−n − 2s) log(1 + b)). Furthermore, log(1 + b) = b(1 + r_{1}), where sup_{|b|≤3/4} |r_{1}| < ∞. Then we expand exp((−n − 2s)b(1 + r_{1})) = 1 + (−n − 2s)b(1 + r_{1}) + br_{2}, where |r_{2}| ≤ \frac{1}{2}|b|[(−n − 2s)(1 + r_{1})]^{2} sup_{ξ∈[0,1]} | exp(ξ(−n − 2s)b(1 + r_{1}))|. We conclude that |r_{2}| < C(1 + |s|)^{2}, where C is independent of s and b. We further get (1 − (1 + b)^{−n−2s}) = (−n − 2s)b + br_{3}, where |r_{3}| ≤ C(1 + |s|)^{2} uniformly in s. Inserting this estimate into (11) we obtain

|â_{s}(y, z) − â_{s}(y, x)|

= |s|∥y − z∥^{−n−2Re(s)} |(2 ⟨(y − z)^{0}, \frac{z − x}{∥y − z∥}⟩ + \frac{∥z − x∥^{2}}{∥y − z∥^{2}} )((−n − 2s) + r_{3})|
= |s|∥y − z∥^{−n−2Re(s)} \frac{∥z − x∥}{∥y − z∥} |(2 ⟨(y − z)^{0}, (z − x)^{0}⟩ + \frac{∥z − x∥}{∥y − z∥})((−n − 2s) + r_{3})|
≤ (1 + |s|)^{3}C∥y − z∥^{−n} \frac{∥z − x∥}{∥y − z∥} .

We now estimate

I_{1} ≤ (1 + |s|)^{3}C∥z − x∥ ∫_{R/2≥dist(y,z)≥4dist(x,z)} ∥y − z∥^{−n−1}dy
≤ (1 + |s|)^{3}C′ ,

where C′ can be chosen uniformly for Re(s) ∈ [−1/4, 0]. Finally, on the compact region {R/2 ≤ dist(z, y) ≤ 2R} and for Re(s) ∈ [−1/4, 0] the integrand is uniformly bounded with respect to x, y, z. Thus

I_{2} ≤ C

uniformly on {Re(s) ∈ [−1/4, 0]}. □

Lemma 3.5 For p ∈ (1, ∞) there exists C < ∞ such that on {Re(s) ∈ [−1/4, 0)}

∥Â_{s}^{κ}∥_{B(L^{p})} ≤ C(1 + |s|)^{3} .

Proof. We follow the argument of [7], Thm. III.2.4. There the following fact was shown. Let K be an integral operator given by an integral kernel k(x, y). Assume that K extends to a bounded operator on L^{2} and ∥K∥_{B(L^{2})} ≤ C_{1}. Furthermore, assume that there are constants C_{2}, C_{3} such that for all z, x ∈ ℝ^{n}

∫_{dist(y,z)≥C_{2}dist(x,z)} |k(y, x) − k(y, z)|dy ≤ C_{3} .

Then K extends to a bounded operator on L^{p}, p ∈ (1, 2], and ∥K∥_{B(L^{p})} ≤ C(p, n, C_{2})(1 + C_{1} + C_{3}).

For Re(s) < 0 the operator Â_{s}^{κ} is given by an integral kernel. We set C_{2} = 4. Then we obtain the assertion of the lemma by combining Lemma 3.4 (which estimates C_{3}) and Lemma 3.3 (which provides an estimate for C_{1}). For p ∈ [2, ∞) we argue by duality using (Â_{s}^{κ})^{∗} = Âκ ; s̄. □

We now finish the proof of Proposition 3.2. We must discuss the case Re(s) = 0. Let f ∈ S(ℝ^{n}). Then we know that in S(ℝ^{n}) and thus in L^{p}(ℝ^{n})

lim_{ϵ↓0} Â_{s−ϵ}^{κ}f = Â_{s}^{κ}f .

Since we have the bound ∥Â_{s−ϵ}^{κ}∥_{B(L^{p})} ≤ C(1 + |s|)^{3} uniformly for ϵ ∈ (0, 1) we conclude that ∥Â_{s}^{κ}∥_{B(L^{p})} ≤ C(1 + |s|)^{3} uniformly if Re(s) ∈ [−1, 0]. □

Lemma 3.6 Let r ∈ ℝ. For each compact subset K ⊂ ℝ^{n} and χ ∈ C_{c}^{∞}(ℝ^{n}) there exists C < ∞ such that for all s ∈ ℂ with Re(s) ∈ [−1, 0] we have

q_{K,χ,p,r,r}(Â_{s}) ≤ C(1 + |s|)^{3} .

Proof. It again suffices to show that

∥Â_{s}^{κ}∥_{B(H^{p,r})} ≤ C(1 + |s|)^{3} .

Let J_{r} be the operator on S′(ℝ^{n}) given by F(J_{r})f(ξ) = (1 + |ξ|^{2})^{r/2}F(f)(ξ). Then it is wellknown that H^{p,r} = {f ∈ S′(ℝ^{n}) | J_{r}f ∈ L^{p}}, and ∥f∥_{H^{p,r}} := ∥J_{r}f∥_{L^{p}} is a norm on the Banach space H^{p,r}. Since Â_{s}^{κ} commutes with translation, it also commutes with J_{r} (when restricted to the Schwartz space). So we can write Â_{s}^{κ} = J_{−r}Â_{s}^{κ}J_{r}. The right hand side obviously acts on H^{p,r}, and we have the estimate

∥Â_{s}^{κ}∥_{B(H^{p,r})} ≤ ∥Â_{s}^{κ}∥_{B(L^{p})} ≤ C(1 + |s|)^{3} .

This proves the lemma. □

We now consider the case that Re(s) ∈ [m − 1, m], m ∈ ℕ. In order to fix the poles of A_{s} at s = m − 1, m we consider Ã_{m,s} := s(s + 1)A_{s+m}, Re(s) ∈ [−1, 0].

Lemma 3.7 Let r ∈ ℝ. For each compact subset K ⊂ ℝ^{n} and χ ∈ C_{c}^{∞}(ℝ^{n}) there exists C < ∞ such that for all s ∈ ℂ with Re(s) ∈ [−1, 0] we have

q_{K,χ,p,r,r−2m}(Ã_{s}) ≤ C(1 + |s|)^{2} .

Proof. We use (6) in order to reduce to the case Re(s) ∈ [−1, 0]. We have

Ã_{m,s} = q_{m}(s)^{−1}Δ^{m}Â_{s} ,

where q_{m} is some polynomial of degree 2m − 1 which does not vanish if Re(s) ∈ [−1, 0]. Let χ′ ∈ C_{c}^{∞} be such that χ′χ = χ. Since ∥χΔ^{m}∥_{B(H^{p,r},H^{p,r−2m})} ≤ C we get for f ∈ H_{K}^{p,r} with ∥f∥_{H^{p,r}=1}

∥χÃ_{m,s}(f)∥_{H^{p,r−2m}} = |q_{m}(s)^{−1}|∥χΔ^{m}Â_{s}(f)∥_{H^{p,r−2m}}
= |q_{m}(s)^{−1}|∥χΔ^{m}χ′Â_{s}(f)∥_{H^{p,r−2m}}
≤ |q_{m}(s)^{−1}|Cq_{K,χ′,p,r,r}(Â_{s})
≤ C(1 + |s|)^{2}

using Lemma 3.6 in the last step. □

We now prove the theorem. First we consider s_{0} ∈ ℂ \ ℕ_{0} such that Re(s_{0}) ≥ 0. Then there exists m ∈ ℕ such that Re(s_{0}) ∈ [m − 1, m]. Let K ⊂ ℝ^{n} be compact and χ ∈ C_{c}^{∞}(ℝ^{n}). Let f ∈ H_{K}^{p,r} be given. We must show that χA_{s}f ∈ H^{p,r−2Re(s_{0})}.

Assume first that f ∈ C_{c}^{∞}(ℝ^{n})∩H_{K}^{p,r}. We consider the holomorphic family F_{s} with values in S(ℝ^{n}) given by F_{s} := (s + 2)^{−3}f for Re(s) ∈ [−1, 0]. Then χÃ_{m,s}F_{s} is a holomorphic family in C_{c}^{∞}(ℝ^{n}). Thus it is holomorphic with values in H^{p,r−2m} and sup_{Re(s)=[−1,0]} ∥χÃ_{m,s}F_{s}∥_{H^{p,r−2m}} ≤ C∥f∥_{H^{p,r}} (see Lemma 3.7). Furthermore, we know that if Re(s) = 0, then χÃ_{m−1,s}F_{s−1} ∈ H^{p,r−m+2} is uniformly bounded by ∥f∥_{H^{p,r}}. Thus sup_{Re(s)=−1} ∥χÃ_{m,s}F_{s}∥_{H^{p,r−2m+2}} ≤ C∥f∥_{H^{p,r}}. By complex interpolation we now conclude that

χÃ_{m,s_{0}−m}F_{s_{0}} ∈ H^{p,r−2Re(s_{0})}
∥χÃ_{m,s_{0}−m}F_{s_{0}}∥_{H^{p,r−2Re(s_{0})}} ≤ C∥f∥_{H^{p,r}}

Note that any element of f ∈ H_{K}^{p,r} can be approximated by smooth compactly supported functions. Since Ã_{m,s_{0}−m} and F_{s_{0}} are multiples of A_{s_{0}} and f we conclude that

χA_{s_{0}}f ∈ H^{p,r−2Re(s_{0})} .

It remains to consider the integral points m ∈ ℕ_{0}. First of all note that B_{m,−1} is a multiple of Δ^{m}. For k ∈ ℕ_{0} and f ∈ S′(ℝ^{n}) we write

B_{m,k}f = \frac{1}{2πi} ∫_{S(m,ϵ)} \frac{1}{(s − m)^{k+1}} A_{s}fds ,

where S(m, ϵ) is the circle with radius ϵ > 0 centered at m, and the integral is taken in S′(ℝ^{n}). If f ∈ H^{p,r} and χ ∈ C_{c}^{∞}(ℝ^{n}), then we have ∥χA_{s}f∥_{H^{p,r−2m−2ϵ}} ≤ C uniformly on S(m, ϵ). For any ϕ ∈ C_{c}^{∞}(ℝ^{n}) we can estimate

|⟨χB_{m,k}f, ϕ⟩| = |\frac{1}{2πi} ∫_{S(m,ϵ)} \frac{1}{(s − m)^{k+1}} ⟨χA_{s}f, ϕ⟩ds|
≤ \frac{1}{2π} ∫_{S(m,ϵ)} |\frac{1}{(s − m)^{k+1}} ⟨χA_{s}f, ϕ⟩|ds
≤ \frac{1}{ϵ^{k}} sup_{s∈S(m,ϵ)} ∥χA_{s}f∥_{H^{p,r−2m−2ϵ}}∥ϕ∥_{H^{q,−r+2m+2ϵ}}ds
≤ C∥ϕ∥_{H^{q,−r+2m+2ϵ}} .

We conclude that χB_{m,k}f ∈ (H^{q,−r+2m+2ϵ})^{∗} = H^{p,r−2m−2ϵ}. Since ϵ > 0 can be chosen arbitrary small we have χB_{m,k}f ∈ H^{p,<(r−2m)} as asserted. □

We now apply this result to the spherical intertwining operator Ĵ_{1_{μ},φ}^{w}. Let λ ∈ 𝔞_{ℂ}^{∗} be such that Re(λ) ≥ 0 and

Ĵ_{1_{μ},φ}^{w} = ∑_{l}(\frac{μ − λ}{α})^{l}B_{l}

be the Laurent expansion of the meromorphic family (Ĵ_{1_{μ},φ}^{w})_{μ} at λ. The coefficients have the following mapping property.

Corollary 3.8 For all p ∈ (1, ∞) and r ∈ ℝ

B_{l} : H^{p,<r} → H^{p,<r−2Re(λ)} .

Proof. We fix a parabolic subgroup P, a maximal compact subgroup K, a Langlands decomposition P = MAN with M ⊂ K, and a representative w ∈ N_{K}(M) of the non-trivial element of the Weyl group N_{K}(M)/M. Let G \ wP = N̄MAN, g = n̄(g)m(g)α(g)ñ(g), be the Bruhat decomposition. Furthermore, we define F : N \ {1} → N̄ \ {1} such that F (x)P = xwP.

If Re(λ) < 0 and f ∈ C^{∞}(∂X, V (1_{λ}, φ)), then by definition

Ĵ_{1_{λ},φ}^{w}f(g) = ∫N̄ f(gwx̄)dx̄ .

We now assume that supp(f) ⊂ N̄P. We compute

(Ĵ_{1_{λ},φ}^{w}f)(1) = ∫N̄ f(wx̄)dx̄

= ∫N̄ f(x̄^{w}w)dx̄
= ∫_{N} f(xw)dx
= ∫_{N\{1}} α(xw)^{λ−ρ}f(F (x))dx.

We now identify the group N with its Lie algebra 𝔫 using the exponential map, and further we identify 𝔫 with ℝ^{n−1} such that the action of M is orthogonal (this fixes the identification up to scale) and such that the measure dx identifies with the standard Lebesgue measure (this fixes the scale). The function α(xw)^{α} is M-invariant and satisfies α(x^{a}w)^{α} = a^{2α}α(xw), a ∈ A. Thus in the coordinates it is given by α(xw)^{α} = c_{1}∥x∥^{2} for some constant c_{1} > 0. We identify N̄ with 𝔫̄ using the exponential map, and furthermore, 𝔫̄ with ℝ^{n−1} using the isomorphism 𝔫̄ ∋ X̄ ↦ X^{w} ∈ 𝔫. The map F : N \{0} → N̄ \{0} is M-equivariant and satisfies F (x^{a}) = F (x)^{a}. Therefore, in our coordinates F (x) = c_{2} \frac{x}{∥x∥^{2}} for some 0 ≠ c_{2} ∈ ℝ. So we can further compute

∫_{N\\{1\}} α(xw)^{λ−ρ}f(F (x))dx = c_{1}^{2λ/α−(n−1)} ∫_{ℝ^{n−1}} ∥x∥^{2λ/α−(n−1)}f(c_{2} \frac{x}{∥x∥^{2}} )dx
= (c_{1}c_{2})^{2λ/α−(n−1)} ∫_{ℝ^{n−1}} ∥ȳ∥^{−2λ/α−(n−1)}f(ȳ)dȳ

Let now z̄ ∈ N̄. Then we define fz̄(g) = f(z̄g). Since Ĵ_{1_{λ},φ}^{w} is G-equivariant, we have (Ĵ_{1_{λ},φ}^{w}f)(z̄) = (Ĵ_{1_{λ},φ}^{w}fz̄)(1). In our coordinates we thus have

(Ĵ_{1_{λ},φ}^{w}f)(z̄) = (c_{1}c_{2})^{2λ/α−(n−1)} ∫_{ℝ^{n−1}} ∥ȳ∥^{−2λ/α−(n−1)}f(z̄ + ȳ)dȳ
= (c_{1}c_{2})^{2λ/α−(n−1)} ∫_{ℝ^{n−1}} ∥z̄ − ȳ∥^{−2λ/α−(n−1)}f(ȳ)dȳ .

Thus up to the harmless factor (c_{1}c_{2})^{2λ/α−(n−1)} the intertwining operator coincides with A_{s} introduced at the beginning of the present subsection, where s = λ/α. The assertion of the corollary now follows from Theorem 3.1 since we can cover ∂X by finitely many charts of the form N̄P (with varying P). □

4 Proof of Theorem 1.15

4.1 Compatibility with twisting and embedding

We fix σ ∈ M̂, λ ∈ 𝔞_{ℂ}^{∗}, and a twist φ. If σ′, μ, π and T is some twisting data, then we have the map

i_{∗} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(∂X, V (σ′_{μ}, φ ⊗ π)) .

Lemma 4.1 Let r ∈ ℝ and p ∈ (1, ∞). If ϕ ∈ C^{−∞}(∂X, V (σ_{λ}, φ)) satisfies i_{∗}ϕ ∈ H^{p,r}(∂X, V (σ′_{μ}, φ ⊗ π)), then ϕ ∈ H^{p,r}(∂X, V (σ_{λ}, φ)).

Proof. The map i_{∗} is induced by an inclusion V (σ_{λ}, φ) ↪ V (σ′_{μ}, φ ⊗ π) of vector bundles. Since we can find a complementary bundle of V (σ_{λ}, φ) inside V (σ′_{μ}, φ ⊗ π) there is also a projection from V (σ′_{μ}, φ ⊗ π) to V (σ_{λ}, φ). In local trivializations this projection is given by a matrix with smooth entries. Since smooth functions act continuously on the Sobolev space H^{p,r} the projection provides a left-inverse of i_{∗}. Now the lemma follows. □

Now we discuss embedding. Let G = G_{m}, σ_{m} ∈ M̂_{m} and λ ∈ 𝔞_{ℂ}^{∗}. If n > m, σ_{n} and T is some embedding data, then we have the map

i_{∗} : C^{−∞}(∂X^{m}, V^{m}((σ_{m})_{λ}, φ)) → C^{−∞}(∂X^{n}, V^{n}((σ_{n})_{λ+ρ^{m}−ρ^{n}}, φ)) .

Lemma 4.2 Let r ∈ (−∞, 0], p ∈ (1, ∞), and q be given by \frac{1}{p} + \frac{1}{q} = 1. If ϕ ∈ C^{−∞}(∂X^{m}, V^{m}((σ_{m})_{λ}, φ)) satisfies i_{∗}ϕ ∈ H^{p,<r− \frac{n−m}{q}} (∂X^{n}, V^{n}((σ_{n})_{λ+ρ^{m}−ρ^{n}}, φ)), then ϕ ∈ H^{p,<r}(∂X^{m}, V^{m}((σ_{m})_{λ}, φ)).

Proof. The map T induces an inclusion of bundles

V^{m}((σ_{m})_{λ}, φ) ↪ V^{n}((σ_{n})_{λ+ρ^{m}−ρ^{n}}, φ)_{|∂X^{m}} .

Let E → U be an extension of V^{m}((σ_{m})_{λ}, φ) to a tubular neighbourhood U of ∂X^{m}. It is now clear that i_{∗} = i_{∗}^{2} ∘ i_{∗}^{1} is a composition of the push-forward i_{∗}^{1} of distributions induced by the inclusion ∂X^{m} ↪ ∂X^{n} of a regular submanifold, and the inclusion i_{∗}^{2} induced by the inclusion of bundles. If i_{∗}ϕ ∈ H^{p,<r− \frac{n−m}{q}} (∂X^{n}, V^{n}((σ_{n})_{λ+ρ^{m}−ρ^{n}}, φ)), then i_{∗}^{1}(ϕ) ∈ H^{p,<r− \frac{n−m}{q}} (U, E) by Lemma 4.1. It remains to conclude that ϕ ∈ H^{p,<r}(∂X^{m}, V^{m}((σ_{m})_{λ}, φ)).

Fix ϵ > 0. We first assume that n = m + 1. Let l := [−r + ϵ + \frac{1}{q}] be the integral part of −r + ϵ + \frac{1}{q}. We define a restriction operator R : C^{∞}(U, E^{+}) → ⨁_{i=0}^{l} C^{∞}(∂X^{m}, E_{|∂X^{m}}^{+}) by

R(f) := (f_{|∂X^{m}}, (∂_{n}f)_{|∂X^{m}}, …, (∂_{n}^{l}f)_{|∂X^{m}}) ,

where ∂_{n} denotes the normal derivative, and E^{+} := E^{∗} ⊗ Λ, Λ being the bundle of densities. We have an exact sequence

0 → H_{0}^{q,−r+ϵ+ \frac{1}{q}} (U, E^{+}) → H^{q,−r+ϵ+ \frac{1}{q}} (U, E^{+}) →^{R} ⨁_{i=0}^{l} B_{q,q}^{−r+ϵ−i}(∂X^{m}, E_{|∂X^{m}}^{+}) → 0 ,

(see [15] for the definition of the Besov spaces and [15], 4.7.1, for the characterization of the range and the kernel of R). In particular, the space of smooth sections of E^{+} vanishing to infinite order along ∂X^{m} is dense in ker R.

Since i_{∗}^{1}(ϕ) ∈ H^{p,r−ϵ− \frac{1}{q}} (U, E) we know that i_{∗}^{1}ϕ extends continuously to H^{q,−r+ϵ+ \frac{1}{q}} (U, E^{+}). Furthermore, since i_{∗}^{1}ϕ vanishes on the space of smooth sections of E^{+} vanishing to infinite order along ∂X^{m}, we conclude that i_{∗}^{1}ϕ vanishes on H_{0}^{q,−r+ϵ+ \frac{1}{q}} (U, E^{+}) and thus factors over the quotient ⨁_{i=0}^{l} B_{q,q}^{−r+ϵ−i}(∂X^{m}, E_{|∂X^{m}}^{+}). This means that ϕ extends continuously to B_{q,q}^{−r+ϵ}(∂X^{m}, E_{|∂X^{m}}^{+}). We now use the continuous embedding ([15], 4.6.1)

H^{q,−r+2ϵ} ↪ B_{q,max(2,q)}^{−r+2ϵ} ↪ B_{q,q}^{−r+ϵ}

in order to conclude that ϕ restricts to a continuous functional on H^{q,−r+2ϵ}. Since this holds for all ϵ > 0 we have ϕ ∈ H^{p,<r}(∂X^{m}, V^{m}((σ_{m})_{λ}, φ)). If n − m > 1, then we argue by induction. □

We now state the following stable version of our main theorem.

Theorem 4.3 1. If ϕ ∈ Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ), then for p in a generic subset of (1, ∞) there exist suitable twisting and embedding data such that i_{∗}ϕ ∈ H^{p,<r_{p,μ}(Γ)′}(∂X′, V (σ′_{μ}, φ′)), where we attach a ”′ ” to objects which are associated to ∂X′ and φ′.

2. If ϕ ∈ Cusp_{Γ}(σ_{λ}, φ) and λ ∉ I_{𝔞}, then for all p ∈ (1, ∞) there exist suitable twisting and embedding data such that i_{∗}ϕ ∈ H^{p,<r_{p,μ}^{0}(Γ)′}(∂X′, V (σ′_{μ}, φ′)).

The contents of this theorem is that given a stably deformable invariant distribution which is strongly supported on the limit set, or given a cusp form and λ ∉ I_{𝔞}, then after suitable embedding and twisting it belongs to the Sobolev space as asserted by Theorem 1.15 (for generic p).

Proposition 4.4 Theorem 4.3 implies Theorem 1.15.

Proof. Assume that i_{∗} is induced by some embedding data. In this case φ = φ′, μ = λ + ρ − ρ′, δ′_{Γ} = δ_{Γ} + ρ − ρ′. We observe that

r_{p,λ}(Γ)′ = r_{p,μ}(Γ) − \frac{dim(∂X′) − dim(∂X)}{q}
( r_{p,λ}^{0}(Γ)′ = r_{p,μ}^{0}(Γ) − \frac{dim(∂X′) − dim(∂X)}{q} ) .

If r_{p,λ}(Γ) > 0 ( r_{p,λ}^{0}(Γ) > 0 ), then we employ the vanishing result [6] in order to conclude that ϕ = 0. Else we combine the regularity of i_{∗}ϕ asserted in Theorem 4.3 with Lemma 4.2 in order to see that ϕ has the regularity claimed in Theorem 1.15 provided p is generic or ϕ is a cusp form and λ ∉ I_{𝔞}.

Now assume that i_{∗} is induced by some twisting data. In this case ∂X = ∂X′, ρ = ρ′, δ_{Γ} = δ′_{Γ}. Moreover, φ′ = φ ⊗ π. Let ν be the highest A-weight of π. Then μ − ν = λ and δ_{φ} + ν = δ_{φ′}. It is at this point where we use the assumption that π is irreducible. We observe that

r_{p,λ}(Γ)′ = r_{p,μ}(Γ)
( r_{p,λ}^{0}(Γ)′ = r_{p,μ}^{0}(Γ) ) .

Combining the regularity of i_{∗}ϕ asserted in Theorem 4.3 with Lemma 4.1 we see that ϕ has the regularity claimed in Theorem 1.15 again provided that p is generic or ϕ is a cusp form and λ ∉ I_{𝔞}.

In the general case, where i_{∗} is the composition of several embeddings and twistings, we argue by induction.

In order to drop the assumption that p is generic we argue as follows. Let s < r_{p,λ}(Γ). Then we find p̃ > p such that Theorem 4.3 applies and s < r_{p̃,λ}(Γ). The argument above gives ϕ ∈ H^{p̃,<r_{p̃,λ}(Γ)} By the embedding theorem H^{p̃,<r_{p̃,λ}(Γ)} ⊂ H^{p,s}. Since s < r_{p,λ}(Γ) was arbitrary, we conclude that ϕ ∈ H^{p,<r_{p,λ}(Γ)}. □

4.2 Sobolev regularity for cusps

In the present subsection we assume that Γ = Γ_{P} for some (uniquely determined) Γ-cuspidal parabolic subgroup P. We assume that the cusp defined by Γ has smaller rank. We have a natural embedding

S_{Γ}(σ_{λ}, φ) ⊂ B_{Γ}(σ_{λ}, φ) ⊂ ^{Γ}C^{∞}(Ω_{Γ}, V (σ_{λ}, φ)) .

Under certain conditions f ∈ B_{Γ}(σ_{λ}, φ) ( f ∈ S_{Γ}(σ_{λ}, φ) ) defines a regular distribution f̂ ∈ C^{−∞}(∂X, V (σ_{λ}, φ)). Our goal in the present subsection is to determine these conditions and to estimate the regularity of f̂.

Let N ⊂ P be the unipotent radical of P. Then we have the connected subgroup N_{Γ} ⊂ N which is AP_{Γ}-invariant and such that Γ\N_{Γ} is compact. Here P acts on N by (p, x) ↦ p.x such that pxwP = p.xwP. Let 2ρ_{Γ} ∈ 𝔞^{∗} denote the character of the action of A on Λ^{max}𝔫_{Γ}, where 𝔫_{Γ} denotes the Lie-algebra of N_{Γ}. This is consistent with the definition given earlier in the present paper. Furthermore, recall that ρ^{Γ} := ρ − ρ_{Γ}. Note that δ_{Γ} = −ρ^{Γ}.

Let S(N/N_{Γ}) be the unit sphere with respect to some choice of an euclidean M_{Γ}-invariant metric on the vector space N/N_{Γ}. Furthermore, we let U be the preimage of S(N/N_{Γ})^{A_{+}} under the projection N → N/N_{Γ}. Then Uw ⊂ Ω_{Γ} is a Γ-invariant subset which projects onto a neighbourhood B_{Γ}^{+} of the cusp of B_{Γ}.

We fix a maximal compact subgroup K and a corresponding Langlands decomposition P = MAN. Let w ∈ K be a representative of the non-trivial element of the Weyl group N_{K}(M)/M. Then we define the group N̄ := N^{w} with Lie algebra 𝔫̄. For r ∈ ℕ_{0} let (X̄_{j})_{j} be a base of U(𝔫̄)^{≥−rα} (note that X ∈ 𝔫̄ has degree −α). If supp(f) ⊂ B_{Γ}^{+}, then f̂ is supported near ∞_{P}. It therefore makes sense to consider the norm

N̄∥f̂∥_{H^{p,r}}^{p} := ∑_{j} ∫N̄ |f̂(x̄X̄_{j})|^{p}dx̄ ,

where we have fixed a norm on V_{φ}.

Theorem 4.5 Let f ∈ B_{Γ}(σ_{λ}, φ) ( f ∈ S_{Γ}(σ_{λ}, φ) ) be supported in B_{Γ}^{+}.

1. If ρ^{Γ} > δ_{φ} ( no condition ) and Re(λ) > −ρ^{Γ} + δ_{φ}, then f determines a regular distribution f̂.

2. If p ∈ (1, ∞) and r, r^{0} ∈ ℕ_{0} satisfy

rα < min (Re(λ) − ρ − δ_{φ} + \frac{ρ + ρ^{Γ}}{p} , −2δ_{φ} − 2ρ_{Γ} + 2\frac{ρ}{p})

( r^{0}α < Re(λ) − ρ − δ_{φ} + \frac{ρ + ρ^{Γ}}{p} ) ,

then we have f̂ ∈ H^{p,r}(∂X, V (σ_{λ}, φ)) ( f̂ ∈ H^{p,r^{0}}(∂X, V (σ_{λ}, φ)) ).

Proof. Note that f̂ is smooth on Ω_{Γ}. In order to prove 1.) me must show that f̂ is locally integrable near ∞_{P}. Since N̄ ∋ x̄ ↦ x̄P defines coordinates of ∂X near ∞_{P} it therefore suffices to show that

N̄∥f̂∥_{H^{1,0}} := ∫_{N̄\{1}} |f(x̄)|dx̄ < ∞ .  (12)

Let F : N \ {1} → N̄ \ {1} be the diffeomorphism such that F (x)m(xw)α(xw)ñ(xw) = xw with m(xw)α(xw)ñ(xw) ∈ MAN. We have

f(xw) = f(F (x)m(xw)α(xw)ñ(xw)) = σ(m(xw))^{−1}α(xw)^{λ−ρ}f(F (x)) .

Thus |f(F (x))| = |f(xw)|α(xw)^{ρ−Re(λ)}. Furthermore F^{∗}dx̄ = | det DF (x)|dx. Note that F (x^{a}) = F (x)^{a}. From this we conclude that

| det DF (x^{a})| = | det DF (x)|a^{−4ρ} .  (13)

Now (12) is equivalent to

∫_{N} |f(xw)|α(xw)^{ρ−Re(λ)}| det DF (x)|dx < ∞ .

Taking into account that supp(f) ∈ B_{Γ}^{+} we can write this integral in double polar coordinates as

∫_{A_{+}} ∫_{S(N/N_{Γ})} ∫_{A} ∫_{S(N_{Γ})} |f(η^{b}ξ^{a}w)|α(η^{b}ξ^{a}w)^{ρ−Re(λ)}| det DF (η^{b}ξ^{a})|a^{2ρ^{Γ}}b^{2ρ_{Γ}}dηdbdξda . (14)

We choose some compact preimage Ŝ of S(N/N_{Γ}) inside N.

Recall that φ extends to a semisimple representation of A. We can and will assume that V_{φ} decomposes into eigenspaces V_{φ}(β), β ∈ 𝔞_{+}^{∗} ∪ {0}, w.r.t. the action of A so that a ∈ A acts on V_{φ}(β) by a^{β}, and such that max_{V_{φ}(β)≠{0}} β = 2δ_{φ}. Let f_{β} be the component of f in V_{φ}(β). Furthermore, let φ_{β_{1},β_{2}} be the component of φ mapping V_{φ}(β_{2}) to V_{φ}(β_{1}). Note that φ_{β_{1},β_{2}}(p) = 0 if β_{2} > β_{1} and p ∈ AP_{Γ}. Since f is Γ-invariant we have

f(γxw) = φ(γ)f(xw) = ∑_{β_{1}≥β_{2}} φ_{β_{1},β_{2}}(γ)f_{β_{2}}(xw) .

We obtain the estimate

sup_{ξ∈Ŝ,η∈S(N_{Γ})} |f(η^{b}v^{a}w)| ≤ C ∑_{β_{1}≥β_{2}} sup_{v∈V} |f_{β_{2}}(ξ^{a}w)| sup_{η∈S(N_{Γ})} ∥φ_{β_{1},β_{2}}(η^{b})∥ ,

where V ⊂ N \ N_{Γ} is a sufficiently large compact subset, and C is independent of a, b. Furthermore, using 2.5, α(x^{a}w) = a^{2}α(xw), and (13), we get

sup_{η∈S(N_{Γ})} ∥φ_{β_{1},β_{2}}(η^{b})∥ ≤ Cb^{β_{1}−β_{2}}
sup_{v∈V} |f_{β_{2}}(v^{a}w)| ≤ p_{V,1}^{Γ}(f)a^{2(Re(λ)−ρ^{Γ})+β_{2}}  (15)
sup_{η∈S(N_{Γ}),ξ∈Ŝ} |α(η^{b}ξ^{a}w)^{ρ−Re(λ)}| ≤ C max(a, b)^{2(ρ−Re(λ))}
sup_{η∈S(N_{Γ}),ξ∈Ŝ} | det DF (η^{b}ξ^{a})| ≤ C max(a, b)^{−4ρ} .

Thus for a ≤ b we have uniformly in (η, ξ)

|f(η^{b}ξ^{a}w)|α(η^{b}ξ^{a}w)^{ρ−Re(λ)}| det DF (η^{b}ξ^{a})|a^{2ρ^{Γ}}b^{2ρ_{Γ}}
≤ C ∑_{β_{1}≥β_{2}} b^{β_{1}−β_{2}}a^{2(Re(λ)−ρ^{Γ})+β_{2}}b^{2(ρ−Re(λ))}b^{−4ρ}a^{2ρ^{Γ}}b^{2ρ_{Γ}} .

Similarly, for b ≤ a we have uniformly in (η, ξ)

|f(η^{b}ξ^{a}w)|α(η^{b}ξ^{a}w)^{ρ−Re(λ)}| det DF (η^{b}ξ^{a})|a^{2ρ^{Γ}}b^{2ρ_{Γ}}
≤ C ∑_{β_{1}≥β_{2}} b^{β_{1}−β_{2}}a^{2(Re(λ)−ρ^{Γ})+β_{2}}a^{2(ρ−Re(λ))}a^{−4ρ}a^{2ρ^{Γ}}b^{2ρ_{Γ}}

We now easily conclude that (14) converges if ρ^{Γ} > δ_{φ} and Re(λ) > −ρ^{Γ} + δ_{φ}.

If f ∈ S_{Γ}(σ_{λ}, φ), then (15) can be improved to

sup_{v∈V} |f_{β_{2}}(v^{a}w)| ≤ q_{W,1,kα}^{Γ}(f)a^{2(Re(λ)−ρ^{Γ})+β_{2}−kα}

for any k ∈ ℕ_{0}. We see that in this case (14) converges if Re(λ) > −ρ^{Γ} + δ_{φ}.

In order to prove assertions 2.) and 3.) of the theorem we extend the argument above. Let X̄ ∈ U(𝔫) be homogeneous of degree −ν ∈ 𝔞^{∗}, i.e. X̄^{a} = a^{−ν}X̄ for a ∈ A. We then estimate

∫_{N̄\{1}} |f(x̄X̄)|^{p}dx̄ .  (16)

We again transform this integral into an integral over N. We define the function X : N \ {1} → U(𝔫) so that the the equality F (xX(x)) = F (x)X̄ of differential operators holds true. Let X(x) = ∑_{μ≤ν} X_{μ}(x) be its decomposition into homogeneous components. Because of

F (x^{a}X(x^{a})) = F (x^{a})X̄ = F (x)^{a}X̄ = a^{ν}(F (x)X̄)^{a} = a^{ν}F (xX(x))^{a}
= a^{ν}F (x^{a}X(x)^{a}) = ∑_{μ≤ν} a^{ν+μ}F (x^{a}X_{μ}(x))

we conclude that

X_{μ}(x^{a}) = a^{μ+ν}X_{μ}(x) .

Because of α(xw)^{ρ−λ}σ(m(xw))f(xw) = f(F (x)) we must apply the Leibniz rule in order to express f(F (x)X̄). We obtain a sum of terms of the form

h(x)α(xY_{1}w)^{ρ−λ}σ(m(xY_{2}w))f(xY_{3}w) ,

where Y_{i} ∈ U(𝔫) is of degree d_{i}, d_{1} + d_{2} + d_{3} =: μ ≤ ν, and h(x^{a}) = a^{μ+ν}h(x) for a ∈ A. We must estimate

∫_{N} |h(x)α(xY_{1}w)^{ρ−λ}σ(m(xY_{2}w))f(xY_{3}w)|^{p}| det DF (x)|dx .

We again write this integral in the form

∫_{A_{+}} ∫_{S(N/N_{Γ})} ∫_{A} ∫_{S(N_{Γ})} |h(η^{b}ξ^{a})α(η^{b}ξ^{a}Y_{1}w)^{ρ−λ}σ(m(η^{b}ξ^{a}Y_{2}w))f(η^{b}ξ^{a}Y_{3}w)|^{p}
| det DF (η^{b}ξ^{a})|a^{2ρ^{Γ}}b^{2ρ_{Γ}}dηdbdξda .  (17)

Using the Γ-equivariance of f, d_{3} ≤ ν ≤ kα we can estimate

|f(η^{b}ξ^{a}Y_{3}w)| ≤ C ∑_{β_{1}≥β_{2}} sup_{v∈V} |f_{β_{2}}(v^{a}Y_{3}w)| sup_{η∈S(N_{Γ})} ∥φ_{β_{1},β_{2}}(η^{b})∥
sup_{v∈V} |f_{β_{2}}(v^{a}Y_{3}w)| ≤ p_{V,Y_{3}}^{Γ}(f)a^{2(Re(λ)−ρ^{Γ})+β_{2}−d_{3}}  (18)
|h(η^{b}ξ^{a})| ≤ C max(a, b)^{μ+ν}
|α(η^{b}ξ^{a}Y_{1}w)^{ρ−λ}| ≤ C max(a, b)^{2(ρ−Re(λ))−d_{1}}
|σ(m(η^{b}ξ^{a}Y_{2}w))| ≤ C max(a, b)^{−d_{2}} .

For a ≤ b we have uniformly in (η, ξ)

|h(η^{b}ξ^{a})α(η^{b}ξ^{a}Y_{1}w)^{ρ−λ}σ(m(η^{b}ξ^{a}Y_{2}w))f(η^{b}ξ^{a}Y_{3}w)|^{p}| det DF (η^{b}ξ^{a})|a^{2ρ^{Γ}}b^{2ρ_{Γ}}

≤ Cp_{V,Y_{3}}^{Γ}(f)^{p} ∑_{β_{1}≥β_{2}} b^{p(μ+ν)}b^{p[2(ρ−Re(λ))−d_{1}]}b^{−pd_{2}}a^{p[2(Re(λ)−ρ^{Γ})+β_{2}−d_{3}]}b^{p(β_{1}−β_{2})}b^{−4ρ}a^{2ρ^{Γ}}b^{2ρ_{Γ}} .

Similarly for b ≤ a we have uniformly in (η, ξ) |h(η^{b}ξ^{a})α(η^{b}ξ^{a}Y_{1}w)^{ρ−λ}σ(m(η^{b}ξ^{a}Y_{2}w))f(η^{b}ξ^{a}Y_{3}w)|^{p}| det DF (η^{b}ξ^{a})|a^{2ρ^{Γ}}b^{2ρ_{Γ}}

≤ Cp_{V,Y_{3}}^{Γ}(f)^{p} ∑_{β_{1}≥β_{2}} a^{p(μ+ν)}a^{p[2(ρ−Re(λ))−d_{1}]}a^{−pd_{2}}a^{p[2(Re(λ)−ρ^{Γ})+β_{2}−d_{3}]}b^{p(β_{1}−β_{2})}a^{−4ρ}a^{2ρ^{Γ}}b^{2ρ_{Γ}} .

¿From these two estimates we conclude that (17) converges for ν < min(Re(λ) − ρ − δ_{φ} + \frac{ρ+ρ^{Γ}}{p} , −2δ_{φ} + 2ρ^{Γ} − 2\frac{ρ}{q}). This implies assertion 2.) provided that f̂ belongs to H^{p,r}.

If f ∈ S_{Γ}(σ_{λ}, φ), then (18) can be improved to

sup_{v∈V} |f_{β_{2}}(v^{a}Y_{3}w)| ≤ q_{V,Y_{3},kα}^{Γ}(f)a^{2(Re(λ)−ρ^{Γ})+β_{2}−kα}

for any k ∈ ℕ_{0}. We see that (17) now converges for ν < Re(λ) − ρ − δ_{φ} + \frac{ρ+ρ^{Γ}}{p}. We thus obtain assertion 2.) in the case that f belongs to the Schwartz space again provided that f̂ belongs to H^{p,r^{0}}.

In order to show that f̂ ∈ H^{p,r} ( resp. f̂ ∈ H^{p,r^{0}} ) we must show that its distributional derivatives up to order r are indeed regular distributions. We proceed as follows. We construct a family of cut-off functions κ_{n} ∈ C_{c}^{∞}(Ω_{Γ}), n ∈ ℕ, such that κ_{n} → 1 locally uniformly on Ω_{Γ} as n → ∞. We then have lim_{n→∞} κ_{n}f̂ = f̂ in the sense of distributions. For X̄ ∈ U(𝔫̄) of degree −ν, ν ≤ rα we consider (κ_{n}f̂)(x̄X̄) = κ_{n}(x)f̂(x̄X̄) + r_{n}(x̄). Thus r_{n} subsumes all terms involving at least one derivative of κ_{n}. The estimate above already shows that κ_{n}(x)f̂(x̄X̄) converges in H^{p,0} to { \atop f(.X̄)}. We will then show that r_{n} tends to zero in H^{p,0}. This proves that the distributional derivatives up to order r of f̂ belong to H^{p,0}.

Let χ ∈ C^{∞}(A) be a cut-off function such that χ(a) ≡ 1 for a ≤ 1 and χ(a) ≡ 0 for a^{α} ≥ 2. Furthermore, fix 1 < a_{0} ∈ A. Then we define κ̃_{n}(x) ∈ C^{∞}(N \ {1}) by κ̃_{n}(x) := χ(α(xw)a_{0}^{−n}). Furthermore, we set κ_{n}(xw) := κ̃_{n}(x). In order to estimate ∥r_{n}∥_{H^{p,r}} we proceed as above but we replace f(xw) by κ̃_{n}(x)f(xw). Then

(κ_{n}f)(η^{b}ξ^{a}Y_{3}w) = ∑_{k} κ̃_{n}(η^{b}ξ^{a}Z′_{k})f(η^{b}ξ^{a}Z_{k}w)

where Z_{k}, Z′_{k} ∈ U(𝔫), deg(Z_{k}) + deg(Z′_{k}) = deg(Y_{3}) are defined by the decomposition of the coproduct Δ(Y_{3}) = ∑_{k} Z_{k} ⊗ Z′_{k}. The terms which contribute to r_{n} have deg(Z′_{k}) > 0. In this case we have for 0 < ϵ ∈ 𝔞^{∗}

|κ̃_{n}(η^{b}ξ^{a}Z′_{k})| = χ(α(η^{b}ξ^{a}Z′_{k}w)a_{0}^{−n})
≤ C max(a, b)^{− deg(Z′_{k})+ϵ}a_{0}^{−nϵ/2}  (19)
|f(η^{b}ξ^{a}Z_{k}w)| ≤ C ∑_{β_{1}≥β_{2}} a^{2(Re(λ)−ρ^{Γ})+β_{2}−deg(Z_{k})}b^{β_{1}−β_{2}} .

In order to see (19) note that χ(α(η^{b}ξ^{a}Z′_{k}w)a_{0}) ≠ 0 implies that a_{0}α(η^{b}ξ^{a}w) ∈ [1, a_{1}], where a_{1}^{α} = 2. We can choose ϵ so small that (ν + ϵ) < min(Re(λ) − ρ − δ_{φ} + \frac{ρ+ρ^{Γ}}{p} , −2δ_{φ} + 2ρ^{Γ} − 2\frac{ρ}{q}). Then we get a bound

∥r_{n}∥_{H^{p,r}} ≤ Ca_{0}^{−nϵ/2} .

In the case that f ∈ S_{Γ}(σ_{λ}, φ) the argument is similar. □

4.3 Sobolev regularity of invariant functions

In this subsection Γ is a general geometrically finite torsion-free subgroup of G. We assume that all cusps have smaller rank. Again, we have inclusions

S_{Γ}(σ_{λ}, φ) ⊂ B_{Γ}(σ_{λ}, φ) ⊂ ^{Γ}C^{∞}(Ω_{Γ}, V (σ_{λ}, φ)) .

Under certain conditions f ∈ B_{Γ}(σ_{λ}, φ) ( f ∈ S_{Γ}(σ_{λ}, φ) ) determines a regular distribution f̂ ∈ C^{−∞}(∂X, V (σ_{λ}, φ)). Our goal ist to determine sufficient conditions for this to happen and to estimate the regularity of f̂.

Let P denote the set of Γ-conjugacy classes of Γ-cuspidal parabolic subgroups. It parametrizes the set of cusps of Γ. For P ∈ P we define Γ_{P} := Γ ∩ P. Then the results of Subsection 4.2 can be applied to Γ_{P}.

The main result of the present subsection is the following theorem.

Theorem 4.6 Let f ∈ B_{Γ}(σ_{λ}, φ) ( f ∈ S_{Γ}(σ_{λ}, φ) ).

1. If max_{[P]_{Γ}∈P} (δ_{φ_{|Γ_{P}}} − ρ^{Γ_{P}} ) < 0 ( no condition ) and Re(λ) > δ_{Γ}+δ_{φ}, then f determines a regular distribution f̂.

2. If p ∈ (1, ∞) and r, r^{0} ∈ ℕ_{0}, r ≤ r^{0},

rα < min (Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} , −2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p})

( r^{0}α < Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} ) ,

then we have f̂ ∈ H^{p,r}(∂X, V (σ_{λ}, φ)) ( f̂ ∈ H^{p,r^{0}}(∂X, V (σ_{λ}, φ)) ).

Proof. We first prove 1.). It suffices to show that

∫_{Ω_{Γ}M} |f(k)|dk < ∞ .  (20)

Let Y_{Γ} := Γ\X be the locally symmetric space of Γ and Ȳ_{Γ} := Γ\(X ∪ Ω_{Γ}) be its geodesic compactification. Let P ∈ P and Ȳ_{P} be a representative of the end of Ȳ_{Γ} corresponding to the cusp labeled by P such that we have an embedding e_{P} : Ȳ_{P} ↪ Ȳ_{Γ_{P}}. We choose a maximal compact subgroup K of G such that wP ∈ Ω_{Γ}. Let O ∈ X denote the origin of X determined by this choice of K. We let Γ^{P} ⊂ Γ be a set of representatives of Γ_{P} \Γ, such that if γ ∈ Γ^{P}, then dist_{X}(O, γO) ≤ dist(O, γ′O) for all γ′ ∈ [γ] (comp. [4], Sec. 4.2). Let B_{P} := B_{Γ} ∩ Ȳ_{P}. We first assume that supp(f) ⊂ B_{P}. Since f̂ is smooth in a neighbourhood of wP in order to prove (20) we must verify that

∫V̄ |f(x̄)|dx̄ < ∞ ,

where V̄ ⊂ N̄ is a sufficiently large compact subset.

We consider the lift (e_{P}^{−1})^{∗}f =: f_{P} ∈ B_{Γ_{P}} (σ_{λ}, φ). Then we have

f = ∑_{γ∈Γ^{P}} π^{σ_{λ},φ}(γ^{−1})(f_{P})_{|Ω_{Γ}}

as smooth functions on Ω_{Γ}, where the sum is locally finite. Since δ_{φ_{|Γ_{P}}} − ρ^{Γ_{P}} < 0 ( or f belongs to the Schwartz space ) and Re(λ) > δ_{Γ} + δ_{φ} ≥ δ_{φ_{|Γ_{P}}} − ρ^{Γ_{P}} we already know by Theorem 4.5 that f_{P} determines a regular distribution ˆ(f_{P}) ∈ L^{1}. Let G \ wP ∋ g = n̄(g)m(g)α(g)ñ(g) ∈ N̄MAN be the Bruhat decomposition of g. We now compute using \frac{dn̄(γ^{−1}x̄)}{dx̄} = α(γ^{−1}x̄)^{−2ρ} and α(γn̄(γ^{−1}x̄)) = α(γ^{−1}x̄)^{−1}

∫V̄ |f(x̄)|dx̄ ≤ ∑_{γ∈Γ^{P}} ∫V̄ |φ(γ)^{−1}f_{P} (γx̄)|dx̄
= ∑_{γ∈Γ^{P}} ∫V̄ |φ(γ)^{−1}α(γx̄)^{λ−ρ}σ(m(γx̄)^{−1})f_{P} (n̄(γx̄))|dx̄
= ∑_{γ∈Γ^{P}} ∫V̄ |φ(γ)^{−1}f_{P} (n̄(γx̄))|α(γx̄)^{Re(λ)−ρ}dx̄
= ∑_{γ∈Γ^{P}} ∫_{n̄(γV̄)} |φ(γ)^{−1}f_{P} (x̄)|α(γn̄(γ^{−1}x̄))^{Re(λ)−ρ} \frac{dn̄(γ^{−1}x̄)}{dx̄}dx̄
= ∑_{γ∈Γ^{P}} ∫_{n̄(γV̄)} |φ(γ)^{−1}f_{P} (x̄)|α(γ^{−1}x̄)^{−Re(λ)−ρ}dx̄
≤ ∑_{γ∈Γ^{P}} ∥φ(γ^{−1}∥ ∫N̄ |f_{P} (x̄)|α(γ^{−1}x̄)^{−Re(λ)−ρ}dx̄ .

In order to proceed further we employ the following estimate. Let g = k_{g}a_{g}h_{g} ∈ KA_{+}K denote the Cartan decomposition of g.

Lemma 4.7 There is c ∈ A_{+} such that c^{−1}a_{γ} ≤ α(γ^{−1}x̄) ≤ ca_{γ} uniformly for all x̄ ∈ supp(f_{P}) and γ ∈ Γ^{P}.

Assuming the lemma and using

∥φ(γ)^{−1}∥ ≤ C_{ϵ}a_{γ}^{δ_{φ}+ϵ}
∫N̄ |f_{P} (x̄)|dx̄ < ∞

for arbitrary small 𝔞^{∗} ∋ ϵ > 0 we obtain

∫V̄ |f(x̄)|dx̄ ≤ C ∑_{γ∈Γ^{P}} a_{γ}^{−Re(λ)−ρ+δ_{φ}+ϵ} .

The sum on the right hand side is finite if Re(λ) > δ_{Γ} + δ_{φ} + ϵ.

This argument for all P ∈ P shows that f̂ is a regular distribution if supp(f) is concentrated near the cusps of B_{Γ}.

Assume now that f ∈ C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)). We choose some cut-off function χ ∈ C_{c}^{∞}(Ω_{Γ}) such that ∑_{γ∈Γ} γ^{∗}χ ≡ 1 on supp(f). Then we can write

f = ∑_{γ∈Γ} π^{σ_{λ},φ}(γ^{−1})(χf) .

We further choose any parabolic subgroup P ⊂ G and a maximal compact subgroup K such that

wP ∈ Ω_{Γ} \ supp(f) .  (21)

This subgroup will be denoted by P_{1} below. In order to prove that f defines a regular distribution f̂ is suffices to show that

∫V̄ |f(x̄)|dx̄ < ∞

for a sufficiently large compact subset V̄ ⊂ N̄. We proceed in a similar manner as above. Define χ̃ by χ̃(n̄(g)) = χ(g).

∫V̄ |f(x̄)|dx̄ ≤ ∑_{γ∈Γ} ∫V̄ |φ(γ)^{−1}χ(γx̄)f(γx̄)|dx̄

= ∑_{γ∈Γ} ∫V̄ χ̃(n̄(γx̄))|φ(γ)^{−1}α(γx̄)^{λ−ρ}σ(m(γx̄)^{−1})f(n̄(γx̄))|dx̄
= ∑_{γ∈Γ} ∫V̄ χ̃(n̄(γx̄))|φ(γ)^{−1}f(n̄(γx̄))|α(γx̄)^{Re(λ)−ρ}dx̄
= ∑_{γ∈Γ} ∫_{n̄(γV̄)} χ̃(x̄)|φ(γ)^{−1}f(x̄)|α(γn̄(γ^{−1}x̄))^{Re(λ)−ρ} \frac{dn̄(γ^{−1}x̄)}{dx̄}dx̄
= ∑_{γ∈Γ} ∫_{n̄(γV̄)} χ̃(x̄)|φ(γ)^{−1}f(x̄)|α(γ^{−1}x̄)^{−Re(λ)−ρ}dx̄
≤ ∑_{γ∈Γ} ∥φ(γ^{−1}∥ ∫N̄ χ̃(x̄)|f(x̄)|α(γ^{−1}x̄)^{−Re(λ)−ρ}dx̄ .

We now use

Lemma 4.8 There is c ∈ A_{+} such that c^{−1}a_{γ} ≤ α(γ^{−1}x̄) ≤ ca_{γ} uniformly for all x̄ ∈ supp(χf) and γ ∈ Γ.

Again assuming the lemma and using

∥φ(γ)^{−1}∥ ≤ C_{ϵ}a_{γ}^{δ_{φ}+ϵ}
∫N̄ χ̃(x̄)|f(x̄)|dx̄ < ∞

for arbitrary small 𝔞^{∗} ∋ ϵ > 0 we obtain

∫V̄ |f(x̄)|dx̄ ≤ C ∑_{γ∈Γ} a_{γ}^{−Re(λ)−ρ+δ_{φ}+ϵ} .

As above the sum on the right hand side is finite if Re(λ) > δ_{Γ} + δ_{φ} + ϵ.

Any element f ∈ B_{Γ}(σ_{λ}, φ) ( f ∈ S_{Γ}(σ_{λ}, φ) ) can be decomposed into a sum of functions which are either supported near the cusps of B_{Γ} or in the interior of B_{Γ}. Hence, we have shown the first part of the theorem asserting that f̂ is a regular distribution.

We will now show f̂ belongs to H^{p,r} ( H^{p,r^{0}} ). So far we have shown that f̂ ∈ H^{1,0}. In order to obtain the stronger estimate we modify the proof above. We give the details in the case that f ∈ B̃_{Γ}(σ_{λ}, φ).

Let P ∈ P and assume that supp(f) ⊂ B_{P}. Let X̄ ∈ U(𝔫̄) be homogeneous of degree −ν such that ν ≤ rα. We now must show that

∥f∥_{P,V̄ ,X̄,p}^{p} := ∫V̄ |f(x̄X̄)|^{p}dx̄ < ∞

Let Δ : U(𝔫̄) → U(𝔫̄) ⊗ U(𝔫̄) be the coproduct. We write Δ ⊗ id(Δ(X̄)) = ∑_{k} A_{k} ⊗ B_{k} ⊗ C_{k}. Then we compute

∫V̄ |f(x̄X̄)|^{p}dx̄
≤ ∑_{γ∈Γ^{P}} ∫V̄ |φ(γ)^{−1}f_{P} (γx̄X̄)|^{p}dx̄
= ∑_{γ∈Γ^{P}} ∫V̄ |φ(γ)^{−1} ∑_{k} α(γx̄A_{k})^{λ−ρ}σ(m(γx̄B_{k})^{−1})f_{P} (n̄(γx̄C_{k}))|^{p}dx̄
≤ ∑_{γ∈Γ^{P}} ∑_{k} ∫V̄ ∥φ(γ)^{−1}∥^{p}|f_{P} (n̄(γx̄C_{k}))|^{p}∥σ(m(γx̄B_{k})^{−1})∥^{p}|α(γx̄A_{k})^{λ−ρ}|^{p}dx̄
= ∑_{γ∈Γ^{P}} ∑_{k} ∫_{n̄(γV̄)} ∥φ(γ)^{−1}∥^{p}∥σ(m(γn̄(γ^{−1}x̄)B_{k})^{−1})∥^{p}|α(γn̄(γ^{−1}x̄)A_{k})^{λ−ρ}|^{p}
|f_{P} (n̄(γn̄(γ^{−1}x̄)C_{k}))|^{p}α(γ^{−1}x̄)^{−2ρ}dx̄ .

We compute

m(γn̄(γ^{−1}x̄)B_{k}) = m(γγ^{−1}x̄ñ(γ^{−1}x̄)^{−1}α(γ^{−1}x̄)^{−1}m(γ^{−1}x̄)^{−1}B_{k})
= m(γ^{−1}x̄)^{−1}m(B_{k}^{[ñ(γ^{−1}x̄)^{m(γ^{−1}x̄)}]^{−1}α(γ^{−1}x̄)^{−1}})
= α(γ^{−1}x̄)^{− deg(B_{k})}m(γ^{−1}x̄)^{−1}m(B_{k}^{[ñ(γ^{−1}x̄)^{m(γ^{−1}x̄)}]^{−1}})
α(γn̄(γ^{−1}x̄)A_{k}) = α(γγ^{−1}x̄ñ(γ^{−1}x̄)^{−1}α(γ^{−1}x̄)^{−1}m(γ^{−1}x̄)^{−1}A_{k})
= α(γ^{−1}x̄)^{− deg(A_{k})}α(A_{k}^{[ñ(γ^{−1}x̄)^{m(γ^{−1}x̄)}]^{−1}})
n̄(γn̄(γ^{−1}x̄)C_{k}) = n̄(γγ^{−1}x̄ñ(γ^{−1}x̄)^{−1}α(γ^{−1}x̄)^{−1}m(γ^{−1}x̄)^{−1}C_{k})
= α(γ^{−1}x̄)^{− deg(C_{k})}x̄n̄(C_{k}^{[m(γ^{−1}x̄)ñ(γ^{−1}x̄)]^{−1}}) .

We now employ the following lemma.

Lemma 4.9 There exists a compact subset U ⊂ N such that ñ(γ^{−1}x̄) ∈ U for all x̄ ∈ supp(f_{P}) and γ ∈ Γ^{P}.

Assuming the lemma for a moment we obtain uniformly for x̄ ∈ supp(f_{P}) and γ ∈ Γ^{P}

∥σ(m(γn̄(γ^{−1}x̄)B_{k})^{−1})∥ ≤ Cα(γ^{−1}x̄)^{− deg(B_{k})}
|α(γn̄(γ^{−1}x̄)A_{k})^{λ−ρ}| ≤ Cα(γ^{−1}x̄)^{− deg(A_{k})}α(γx̄)^{−Re(λ)+ρ} .

Furthermore, using Theorem 4.5 we have uniformly for γ ∈ Γ^{P}

∫N̄ |f_{P} (n̄(γn̄(γ^{−1}x̄)C_{k}))|^{p}dx̄

≤ C sup_{x̄∈supp(f_{P})} α(γ^{−1}x̄)^{−p deg(C_{k})}N̄∥f̂_{P}∥_{H^{p, \frac{deg(C_{k})}{α}}}^{p}
≤ C′ sup_{x̄∈supp(f_{P})} α(γ^{−1}x̄)^{−p deg(C_{k})} .

Using in addition Lemma 4.7 we find

∥f∥_{P,V̄ ,X,p} ≤ C_{ϵ} ( ∑_{γ∈Γ^{P}} ∑_{k} a_{γ}^{p(− deg(C_{k})−deg(B_{k})−deg(A_{k})−Re(λ)+ρ+δ_{φ}+ϵ)−2ρ})^{\frac{1}{p}}

for arbitrary small ϵ > 0. Since − deg(C_{k}) − deg(B_{k}) − deg(A_{k}) = −μ ≤ rα the sum over Γ^{P} on the right hand side is finite if

rα < Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} .

We conclude that f̂ ∈ H^{p,r} if f ∈ B̃_{Γ}(σ_{λ}, φ) is supported near the cusps.

In the case that f belongs to the Schwartz space the obvious modification of the argument above yields f̂ ∈ H^{p,r^{0}}.

Assume now that f ∈ C_{c}^{∞}(B_{Γ}, V (σ_{λ}, φ)). We choose the parabolic subgroup P ⊂ G and the maximal compact subgroup K such that (21) holds. We must show that

∥f̂∥_{P,V,X̄,p}^{p} := ∫V̄ |f(x̄X̄)|^{p}dx̄ < ∞ .

We compute

∫V̄ |f(x̄X̄)|^{p}dx̄
≤ ∑_{γ∈Γ} ∫V̄ |φ(γ)^{−1}(χf)(γx̄X̄)|^{p}dx̄
= ∑_{γ∈Γ} ∫V̄ | ∑_{k} φ(γ)^{−1}α(γx̄A_{k})^{λ−ρ}σ(m(γx̄B_{k})^{−1})(χ̃f)(n̄(γx̄C_{k}))|^{p}dx̄
≤ ∑_{γ∈Γ} ∑_{k} ∫V̄ ∥φ(γ)^{−1}∥^{p}|(χ̃f)(n̄(γx̄C_{k}))|^{p}∥σ(m(γx̄B_{k})^{−1})∥^{p}|α(γx̄A_{k})^{λ−ρ}|^{p}dx̄
= ∑_{γ∈Γ} ∑_{k} ∫_{n̄(γV̄)} ∥φ(γ)^{−1}∥^{p}∥σ(m(γn̄(γ^{−1}x̄)B_{k})^{−1})∥^{p}|α(γn̄(γ^{−1}x̄)A_{k})^{λ−ρ}|^{p}
|(χ̃f)(n̄(γn̄(γ^{−1}x̄)C_{k}))|^{p}α(γ^{−1}x̄)^{−2ρ}dx̄ .

Lemma 4.10 There exists a compact subset U ⊂ N such that ñ(γ^{−1}x̄) ∈ U for all x̄ ∈ supp(χf) and γ ∈ Γ.

Using this lemma we conclude that for arbitrary small ϵ > 0

∥f∥_{P,V̄ ,X̄,p}^{p} ≤ C_{ϵ} ∑_{γ∈Γ} ∑_{k} a_{γ}^{p(− deg(C_{k})−deg(B_{k})−deg(A_{k})−Re(λ)+ρ+δ_{φ}+ϵ)−2ρ}

for arbitrary small ϵ > 0. Note that deg(C_{k}) + deg(B_{k}) + deg(A_{k}) = deg(X). We conclude that f̂ ∈ H^{p,r^{0}}.

Again, since any element f ∈ B_{Γ}(σ_{λ}, φ) ( f ∈ S_{Γ}(σ_{λ}, φ) ) can be decomposed into a sum of functions which are either supported near the cusps of B_{Γ} or in the interior of B_{Γ} we have shown that f̂ ∈ H^{p,r} ( f̂ ∈ H^{p,r^{0}} ), if r ∈ ℕ_{0} ( r^{0} ∈ ℕ_{0} ) satisfy the assumptions of the theorem.

Proof. [Lemma 4.7] For g ∈ G let g = κ(g)a(g)n(g) ∈ KAN be the Iwasawa decomposition. Furthermore, if g ∈ G \ wP, then we have the Bruhat decomposition g = n̄(g)m(g)α(g)ñ(g) ∈ N̄MAN. In particular,

κ(g)a(g)n(g) = n̄(g)m(g)α(g)ñ(g)
= κ(n̄(g))a(n̄(g))n(n̄(g))m(g)α(g)ñ(g)
= κ(n̄(g))m(g)a(n̄(g))α(g)n(n̄(g))^{α(g)^{−1}m(g)^{−1}} ñ(g) ,

so that a(g) = a(n̄(g))α(g). Since a(n̄) ≥ 1 we conclude that α(g) ≤ a(g).

Note that there is at most one γ_{0} ∈ Γ^{P} such that wP ∈ γ^{−1}supp(f_{P}). If this element exists, then we define Γ_{0}^{P} := Γ^{P} \ {γ_{0}}. If not, then we set Γ_{0}^{P} := Γ^{P}.

There is a compact subset V ⊂ N̄ such that n̄(γ^{−1}x̄) ∈ V for all x̄ ∈ supp(f_{P}) and γ ∈ Γ_{0}^{P}. In particular, we have a(n̄(γ^{−1}x̄)) ≤ c_{1} for some c_{1} ∈ A. We conclude that

c_{1}^{−1}a(γ^{−1}x̄) ≤ α(n̄(γ^{−1}x̄)) ≤ a(γ^{−1}x̄) .

Furthermore note that a(γ^{−1}x̄) = a(γ^{−1}κ(x̄))a(x̄), and 1 ≤ a(x̄) ≤ c_{2} for all x̄ ∈ supp(f_{P}). Thus

c_{1}^{−1}a(γ^{−1}κ(x̄)) ≤ α(n̄(γ^{−1}x̄)) ≤ c_{2}a(γ^{−1}κ(x̄)) .  (22)

We now want to apply Lemma [5], Lemma 2.3 asserting that there is c_{3} ∈ A_{+} such that for all γ ∈ Γ^{P} and x̄ ∈ supp(f_{P})

c_{3}^{−1}a_{γ} ≤ a(γ^{−1}κ(x̄)) ≤ a_{γ} .  (23)

In order to verify the assumption of this Lemma we must show that W A_{+}K ∩Γ^{P} is finite, where W is some neighbourhood of κ(supp(f_{P}))M. Note that suppf_{P} ⊂ Γ_{P} e_{P} (B_{P}). We choose W such that κ(supp(f_{P}))M ⊂ W ⊂ κ(Γ_{P} e_{P} (B_{P}))M.

We argue by contradiction. So assume that W A_{+}K ∩ Γ^{P} is infinite. Then we can find a sequence γ_{i} ∈ W A_{+}K ∩ Γ^{P} such that γ_{i} → Λ_{Γ^{P}}, where Λ_{Γ^{P}} denotes the set of accumulation points in ∂X of Γ^{P} O. Let D(O, Γ_{P}) be the Dirichlet domain of Γ_{P} with respect to the origin. By construction of Γ^{P} we have Λ_{Γ^{P}} ⊂ \overline{D(O, Γ_{P})} \ Γ_{P} e_{P} (Ȳ_{P}). Since W ⊂ κ(Γ_{P} e_{P} (B_{P}))M we have γ_{i} ∉ W A_{+}K for i >> 0. This is the contradiction.

Combining (22) and (23) we obtain the assertion of Lemma 4.7. □

Proof. [Lemma 4.8] Note that there is at most one γ_{0} ∈ Γ such that wP ∈ γ^{−1}supp(χf). If this element exists then we define Γ_{0} := Γ \ {γ_{0}}. If not, then we set Γ_{0} := Γ.

There is a compact subset V ⊂ N̄ such that n̄(γ^{−1}x̄) ∈ V for all x̄ ∈ supp(χf) and γ ∈ Γ_{0}. We argue as in the proof of Lemma 4.7 that there is c_{1} ∈ A_{+} such that

c_{1}^{−1}a(γ^{−1}κ(x̄)) ≤ α(n̄(γ^{−1}x̄)) ≤ c_{1}a(γ^{−1}κ(x̄)) .  (24)

We again want to apply Lemma [5], Lemma 2.3, which gives c_{2} ∈ A_{+} such that for all γ ∈ Γ and x̄ ∈ supp(χf)

c_{2}^{−1}a_{γ} ≤ a(γ^{−1}κ(x̄)) ≤ a_{γ} .  (25)

In order to verify the assumption of this Lemma we must show that W A_{+}K ∩ Γ is finite, where W is some neighbourhood of κ(supp(χf))M. Note that supp(χf) ⊂ Ω_{Γ}. We choose a compact W such that κ(supp(χf))M ⊂ W ⊂ κ(Ω_{Γ})M.

We again argue by contradiction. So assume that W A_{+}K ∩ Γ is infinite. Then we can find a sequence γ_{i} ∈ W A_{+}K ∩ Γ such that γ_{i} → Λ_{Γ}. Since W ⊂ κ(Ω_{Γ})M we have γ_{i} ∉ W A_{+}K for i >> 0. This is the contradiction.

Combining (24) and (25) we obtain the assertion of Lemma 4.8. □

Proof. [Lemma 4.9] We have

(γ^{−1}x̄)^{−1}wP = ñ(γ^{−1}x̄)^{−1}α(γ^{−1}x̄)^{−1}m(γ^{−1}x̄)^{−1}n̄(γ^{−1}x̄)^{−1}wP
= ñ(γ^{−1}x̄)^{−1}wP

Assume that there are sequences x̄_{i} ∈ supp(f_{P}) and γ_{i} ∈ Γ^{P} such that ñ(γ_{i}^{−1}x̄_{i}) → ∞. Then lim_{i→∞} x̄_{i}^{−1}γ_{i}wP = ∞_{P}. Taking a subsequence we can assume that x̄_{i} converges to x̄_{0} ∈ supp(f_{P}). Then lim_{i→∞} γ_{i}wP = x̄_{0}∞_{P} ∈ supp(f_{P}). This is impossible since Λ_{Γ^{P}} ∩supp(f_{P}) = ∅.

□

Proof. [Lemma 4.10] The proof is similar to that of Lemma 4.9. Assume that there are sequences x̄_{i} ∈ supp(χf) and γ_{i} ∈ Γ such that ñ(γ_{i}^{−1}x̄_{i}) → ∞. Then lim_{i→∞} x̄_{i}^{−1}γ_{i}wP = ∞_{P}. Taking a subsequence we can assume that x̄_{i} converges to x̄_{0} ∈ supp(χf). Then lim_{i→∞} γ_{i}wP = x̄_{0}∞_{P} ∈ supp(f_{P}). This is impossible since Λ_{Γ} ∩ supp(χf) = ∅. □

We now use the freedom to choose the exponent p in the assertion of Theorem 4.6 in order to extend the result to fractional Sobolev order.

Corollary 4.11 1. Let f ∈ B_{Γ}(σ_{λ}, φ) and assume that max_{[P]_{Γ}∈P} (δ_{φ_{|Γ_{P}}} − ρ^{Γ_{P}} ) < 0 and Re(λ) > δ_{Γ} + δ_{φ}. In addition we assume that for p ∈ (1, ∞) we have that

ϵ := [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} ] − [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}]

satisfies |\frac{ϵ}{α}| > \frac{ρ+δ_{Γ}}{ρ−δ_{Γ}} , \frac{ρ−δ^{Γ}}{p} > α, \frac{ρ−δ^{Γ}}{q} > α, and there is an integer k ∈ ℕ such that

0 < [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} ] < kα ≤ [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}] if ϵ < 0 (26)
0 < [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}] < kα ≤ [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} ] if ϵ > 0 .(27)

Then f̂ ∈ H^{p,r} provided r ∈ ℝ satisfies

rα < min (Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} , −2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}) .

2. Let f ∈ S_{Γ}(σ_{λ}, φ) and assume that Re(λ) > δ_{Γ}+δ_{φ}. In addition we assume that for p ∈ (1, ∞) we have \frac{ρ−δ_{Γ}}{p} > α, \frac{ρ−δ^{Γ}}{q} > α, and Re(λ) − ρ − δ_{φ} + \frac{ρ−δ_{Γ}}{p} > 0. Then f̂ ∈ H^{p,r^{0}} provided r^{0} ∈ ℝ satisfies

r^{0}α < Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} .

Proof. We begin with the first assertion in case ϵ > 0. Let k ∈ ℕ be the minimal number such that (27) holds true. For sufficiently small δ > 0 we define 1 < p_{0} < p < p_{1} < ∞ such that

(k − 1)α + δ = [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p_{1}} ]
kα + δ = [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p_{0}} ] .

If δ is sufficiently small, then we have

(k − 1)α < min [[Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{1}} ] , [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p_{1}} ]]
kα < min [[Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{0}} ] , [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p_{0}} ]] .

We now apply Theorem 4.6 in order to conclude that f̂ ∈ H^{p_{0},k} ∩ H^{p_{1},k−1}. We define θ ∈ (0, 1) such that \frac{1}{p} = \frac{1}{p_{1}} + θ(\frac{1}{p_{0}} − \frac{1}{p_{1}} ). Then interpolation gives f̂ ∈ H^{p,s} for s := k − 1 + θ. Note that s depends on δ. If we choose δ > 0 sufficiently small, then s > r, and the assertion of the corollary follows.

Let now ϵ < 0. Let k ∈ ℕ be the minimal number such that (26) holds true. For sufficiently small δ > 0 we define 1 < p_{0} < p < p_{1} < ∞ such that

(k − 1)α + δ = [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{1}} ]
kα + δ = [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{0}} ] .

If δ is sufficiently small, then we have

(k − 1)α < min [[Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{1}} ] , [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p_{1}} ]]
kα < min [[Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{0}} ] , [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p_{0}} ]] .

We now apply Theorem 4.6 in order to conclude that f̂ ∈ H^{p_{0},k} ∩ H^{p_{1},k−1}. We define θ ∈ (0, 1) such that \frac{1}{p} = \frac{1}{p_{1}} + θ(\frac{1}{p_{0}} − \frac{1}{p_{1}} ). Then interpolation gives f̂ ∈ H^{p,s} for s := k − 1 + θ. If we choose δ > 0 sufficiently small, then s > r, and the assertion of the corollary follows again.

Now we consider the case that f ∈ S_{Γ}(σ_{λ}, φ). For sufficiently small δ > 0 we define 1 < p_{0} < p < p_{1} < ∞ such that

(k − 1)α + δ = [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{1}} ]
kα + δ = [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p_{0}} ] ,

where k ∈ ℕ is uniquely determined. We now apply Theorem 4.6 in order to conclude that f̂ ∈ H^{p_{0},k} ∩ H^{p_{1},k−1}. We define θ ∈ (0, 1) such that \frac{1}{p} = \frac{1}{p_{1}} + θ(\frac{1}{p_{0}} − \frac{1}{p_{1}} ). Then interpolation gives f̂ ∈ H^{p,s} for s := k − 1 + θ. If we choose δ > 0 sufficiently small, tean s > r, and the assertion of the corollary follows in this case, too. □

We now deduce the following consequence of Corollary 4.11.

Corollary 4.12 Let p ∈ (1, ∞) and λ ∈ 𝔞_{ℂ}^{∗} satisfy the assumption of Corollary 4.11, 1. Furthermore, let (f_{μ})_{μ}, f_{μ} ∈ B_{Γ}(σ_{μ}, φ), be a germ at λ of a meromorphic family and f̂_{μ} = ∑_{l}(μ − λ)^{l}f̂_{l} be the Laurent expansion at λ of (f̂_{μ})_{μ}. Then f̂_{l} ∈ H^{p,<r}, where

rα = min (Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} , −2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}) .

Proof. Choose ϵ > 0 sufficiently small such that F_{μ} := (μ − λ)^{−ord_{μ=λ}f_{μ}}f_{μ} is defined and holomorphic on {|μ − λ| ≤ ϵ}, and r − 2ϵ > 0. The proofs of Theorem 4.6 and Corollary 4.11 show that ∥F_{μ}∥_{H^{p,r−2ϵ}} is locally bounded. Since F̂_{μ} = ext^{Γ}F_{μ} is a germ of a holomorphic family of distributions we conclude that F_{μ} is weakly continuous in H^{p,r−2ϵ}. Since the embedding H^{p,r−2ϵ} ↪ H^{p,r−3ϵ} is compact the family F_{μ} is holomorphic with values in H^{p,r−3ϵ}. The coefficients f̂_{l} can be expressed as

f̂_{l} = \frac{1}{2πi} ∫_{|μ−λ|=ϵ}(μ − λ)^{ord_{μ=λ}f_{μ}−l−1}F_{μ} ,

and therefore f̂_{l} ∈ H^{p,r−3ϵ}. Since ϵ > 0 can be chosen arbitrarily small the assertion follows. □

4.4 Proof of Theorem 4.3

We begin with the first assertion of the theorem. For a generic set of p ∈ (1, ∞) we must show that if ϕ ∈ Fam_{Γ}^{st}(Λ_{Γ}, σ_{λ}, φ), then there exists twisting and embedding data such that i_{∗}ϕ ∈ H^{p,<r_{p,λ}(Γ)′}(∂X, V (σ′_{λ′}, φ′)).

From now on p is called generic if p is irrational and

ϵ := [Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} ] − [−2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}] ≠ 0 .

It is easy to see that the set of generic p is a dense subset of (1, ∞) since either ρ − δ_{Γ} ≠ 2ρ or Γ is elementary hyperbolic and has no cusps.

Since ϕ is stably deformable we can choose twisting and embedding data such that

(i_{(1)})_{∗}ϕ ∈ ^{Γ}C^{−∞}(∂X^{(1)}, V^{(1)}(σ_{λ^{(1)}}^{(1)}, φ^{(1)}))

is deformable. By twisting further if necessary we can assume that σ^{(1)} is the trivial representation 1 ∈ M̂^{(1)} (see [5], (33)). Choosing further embedding data we can in addition assume that

(i_{(2)})_{∗}(i_{(1)})_{∗}ϕ ∈ ^{Γ}C^{−∞}(∂X^{(2)}, V^{(2)}(1_{λ^{(2)}}, φ^{(1)})) ,

such that

Re(λ^{(2)}) < 0  (28)
max_{[P]_{Γ}∈P^{(2)}} (δ_{φ_{|Γ_{P}}^{(1)}} − (ρ^{(2)})^{Γ_{P}} ) < 0  (29)
r_{p,λ^{(2)}}^{(2)}(Γ) < 0  (30)
\frac{ρ^{(2)} + δ_{Γ}^{(2)}}{ρ^{(2)} − δ_{Γ}^{(2)}} < |\frac{ϵ}{α}|  (31)
(ρ^{(2)} − δ_{Γ}^{(2)}) min (\frac{1}{p} , \frac{1}{q}) > α ,  (32)

and

min (−Re(λ^{(2)}) − ρ^{(2)} − δ_{φ^{(2)}} + \frac{ρ^{(2)} − δ_{Γ}^{(2)}}{p} , −2 max_{[P]_{Γ}∈P^{(2)}} [δ_{φ_{|Γ_{P}}^{(2)}} + ρ_{Γ_{P}}^{(2)}] + 2\frac{ρ^{(2)}}{p} ) > 0 . (33)

Let d denote the codimension of the embedding. Then λ^{(2)} = λ^{(1)} − \frac{1}{2}dα, ρ^{(2)} = ρ^{(1)} + \frac{1}{2}dα, (ρ^{(2)})^{Γ_{P}} = (ρ^{(1)})^{Γ_{P}} + \frac{1}{2}dα, and δ_{Γ}^{(2)} = δ_{Γ}^{(1)} − \frac{1}{2}dα. So we can get (28) for sufficiently large d.

Moreover, we have the relations

max_{[P]_{Γ}∈P^{(2)}} (δ_{φ_{|Γ_{P}}^{(1)}} − (ρ^{(2)})^{Γ_{P}} ) = max_{[P]_{Γ}∈P^{(1)}} (δ_{φ_{|Γ_{P}}^{(1)}} − (ρ^{(1)})^{Γ_{P}} ) − \frac{1}{2}dα .

Thus we can obtain (29) for sufficiently large d. Since r_{p,λ^{(2)}}^{(2)}(Γ) = r_{p,λ^{(1)}}^{(1)}(Γ) − \frac{d}{q} we can also get (30). In order to obtain (33) note that embedding in codimension d increases the left hand side by dα. For (31) and 32 note that ϵ does not change under embedding and twisting, but \frac{ρ^{(2)}+δ_{Γ}^{(2)}}{ρ^{(2)}−δ_{Γ}^{(2)}} → 0 and _{\frac{α}{ρ^{(2)}−δ_{Γ}^{(2)}}} → 0 as d → ∞. If ϵ > 0, then we consider the non-empty interval

I^{(1)} := ([−2 max_{[P]_{Γ}∈P^{(1)}} [δ_{φ_{|Γ_{P}}^{(1)}} + ρ_{Γ_{P}}^{(1)}] + 2\frac{ρ^{(1)}}{p} ] , [Re(λ^{(1)}) − ρ^{(1)} − δ_{φ}^{(1)} + \frac{ρ^{(1)} − δ_{Γ}^{(1)}}{p} ]) .

If ϵ < 0, then we set

I^{(1)} := ([Re(λ^{(1)}) − ρ^{(1)} − δ_{φ}^{(1)} + \frac{ρ^{(1)} − δ_{Γ}^{(1)}}{p} ] , [−2 max_{[P]_{Γ}∈P^{(1)}} [δ_{φ_{|Γ_{P}}^{(1)}} + ρ_{Γ_{P}}^{(1)}] + 2\frac{ρ^{(1)}}{p} ]) .

Then I^{(2)} = I^{(1)} + \frac{dα}{p}. Since p is irrational there are arbitrary large d such that I^{(2)} contains an integer. We now fix d such that the inequalities above are satisfied and I^{(2)} contains an integer k ∈ ℕ.

In order to simplify the notation we now replace ϕ by (i_{(2)})_{∗}(i_{(1)})_{∗}ϕ, and we omit the superscripts ()^{(2)} everywhere. Thus we can assume that

ϕ ∈ Fam_{Γ}(Λ_{Γ}, 1_{λ}, φ)
Re(λ) < 0
max_{[P]_{Γ}∈P} (δ_{φ_{|Γ_{P}}} − ρ^{Γ_{P}} ) < 0
r_{p,λ}(Γ) < 0
min (−Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} , −2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}) > 0
\frac{ρ + δ_{Γ}}{ρ − δ_{Γ}} < |\frac{ϵ}{α}|
(ρ − δ_{Γ}) min (\frac{1}{p} , \frac{1}{q}) > α,

and I contains an integer k.

We must show that ϕ ∈ H^{p,<r_{p,λ}(Γ)}. Since ϕ is deformable and strongly supported on the limit set we can find a germ at λ of a holomorphic family (ϕ_{μ})_{μ}, ϕ_{μ} ∈ ^{Γ}C^{−∞}(∂X, V (1_{μ}, φ)), such that ϕ_{λ} = ϕ and (res^{Γ}(ϕ_{μ}))_{|μ=λ} = 0. By Lemma 2.12 we have that ϕ ∈ Ext_{Γ}^{sing}(1_{λ}, φ). Since ext^{Γ} has at most finite-dimensional singularities and B_{Γ}(1_{λ}, φ) ⊂ D_{Γ}(1_{λ}, φ) is dense we can find a germ of a holomorphic family (ψ_{μ})_{μ}, ψ_{μ} ∈ B_{Γ}(1_{μ}, φ), such that ψ_{λ} = 0 and (ext^{Γ}ψ_{μ})_{|μ=λ} = ϕ. We now define the germ at −λ of a meromorphic family (f_{μ})_{μ}, f_{μ} ∈ B_{Γ}(1_{μ}, φ), by f_{−μ} := Ŝ_{1_{μ},φ}^{w,Γ}ψ_{μ}. Here Ŝ_{1_{μ},φ}^{w,Γ} : D_{Γ}(1_{μ}, φ) → D_{Γ}(1_{−μ}, φ) (note that 1^{w} = 1) is the scattering matrix defined in 2.14, and we employ Lemma 2.15 in order to see that f_{μ} ∈ B_{Γ}(1_{μ}, φ). Let p_{1}(λ) be the Plancherel density. It is a meromorphic function on 𝔞_{ℂ}^{∗}. We then have the following identities:

ext_{−μ}^{Γ} ∘ Ŝ_{1_{μ},φ}^{w,Γ} = Ĵ_{1_{μ},φ}^{w} ∘ ext_{μ}^{Γ}
p_{1}(−μ)Ĵ_{1_{−μ},φ}^{w^{−1}} ∘ Ĵ_{1_{μ},φ}^{w} = id .

Thus we can write

ϕ = (\frac{1}{p_{1}(−μ)}Ĵ_{1_{−μ},φ}^{w^{−1}}(ext^{Γ}f_{−μ}))_{|μ=λ} .

Let f̂_{μ} := ∑_{k}(\frac{μ+λ}{α})^{k}f̂_{k} be the Laurent expansion of the germ (ext^{Γ}f_{μ})_{μ} at −λ. Then by Corollary 4.12 we have f̂_{k} ∈ H^{p,<r′}, where r′ is determined by

r′α := min (−Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} , −2 max_{[P]_{Γ}∈P} [δ_{φ_{|Γ_{P}}} + ρ_{Γ_{P}} ] + 2\frac{ρ}{p}) .

Let

\frac{1}{p_{1}(μ)}Ĵ_{1_{μ},φ}^{w^{−1}} = ∑_{l}(\frac{μ + λ}{α})^{l}B_{l}

be the Laurent expansion of (\frac{1}{p_{1}(μ)} Ĵ_{1_{μ},φ}^{w^{−1}})_{μ} at −λ. Since Re(λ) < 0 we can apply Cor. 3.8 in order to get B_{l} : H^{p,<r′} → H^{p,<r′+2 \frac{Re(λ)}{α}}. Since r′ + 2\frac{Re(λ)}{α} = r_{p,λ}(Γ) and ϕ = ∑_{l+k=0} B_{l}(f̂_{k}) we conclude that ϕ ∈ H^{p,<r_{p,λ}(Γ)}.

Now we show the second assertion of Theorem 4.3. Let ϕ ∈ Cusp_{Γ}(σ_{λ}, φ) and λ ∉ I_{𝔞}. Again, after twisting and embedding we can assume that all cusps of Γ have smaller rank,

ϕ ∈ Cusp_{Γ}(1_{λ}, φ) ∩ Fam_{Γ}(Λ_{Γ}, 1_{λ}, φ)
Re(λ) < 0
r_{p,λ}^{0}(Γ) < 0
−Re(λ) − ρ − δ_{φ} + \frac{ρ − δ_{Γ}}{p} > 0
(ρ − δ_{Γ}) min (\frac{1}{p} , \frac{1}{q}) > α.

Let (f_{μ})_{μ}, f_{μ} ∈ B_{Γ}(1_{μ}, φ), be constructed as above. Because we assume λ ∉ I_{𝔞} this family is regular at μ = −λ.

We show that f_{−λ} ∈ S_{Γ}(1_{−λ}, φ). Let [P]_{Γ} ∈ P. We choose a P -invariant the cut-off function χ_{P} in order to define the map T_{P} (see Subsection 2.1). Then we can define the constant term

(T_{P} f_{−λ})_{P} := ∫_{Γ_{P} \P_{Γ}} π^{1_{−λ},φ}(x)T_{P} f_{−λ} dx .

Taking the constant term commutes with multiplication by χ_{P}, restriction and the intertwining operators. Therefore we have

(T_{P} f_{−λ})_{P} = (χ_{P} res^{Γ_{P}} Ĵ_{1_{−λ},φ}^{w}ϕ_{λ})_{P}
= χ_{P} res^{Γ_{P}} Ĵ_{1_{−λ},φ}^{w} (ϕ_{λ})_{P} .

Since ϕ is a cusp form we conclude that (T_{P} f_{−λ})_{P} = 0. Since AS(T_{P} f_{−λ}) = AS (T_{P} f_{−λ})_{P} = 0 for all P ∈ P we see that f_{−λ} ∈ S_{Γ}(1_{−λ}, φ).

Using

ϕ = (\frac{1}{p_{1}(−λ)}Ĵ_{1_{−λ},φ}^{w^{−1}}(ext^{Γ}f_{−λ}^{mod}))

and Corollary 4.11, 2.) we obtain ext^{Γ}f_{−λ} = f̂_{−λ} ∈ H^{p,<r′} for r′α := −Re(λ) − ρ − δ_{φ} + \frac{ρ−δ_{Γ}}{p}. It follows from Corollary 3.8 and r_{p,λ}^{0}(Γ) = r′ + 2\frac{Re(λ)}{α} that ϕ ∈ H^{p,<r_{p,λ}^{0}(Γ)}. □

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