The twisted cohomological equation over the geodesic flow

We study the twisted cohomoligical equation over the geodesic flow on $SL(2,\mathbb{R})/\Gamma$. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sobolev norms) between the twisted coboundary and the solution. We also give a tame splittings for non-homogeneous cohomological equations. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for partially hyperbolic actions in products of rank-one groups in future works.

1. Introduction 1.1. Motivation and results. The cohomological equations of the hococycle flow and the geodesic flow of the homogeneous spaces of SL(2, R) have been well understood, see [3] and [7]. In this paper, we extend the study to the twisted cohomological equation of the geodesic flow.
In fact, the study of the twisted cohomological equation provides a tool for obtaining local differentiable rigidity of algebraic actions by KAM type iteration scheme. The KAM scheme was used by Damjanovic and Katok to prove local rigidity for genuinely higher-rank partially hyperbolic actions on torus in [1]. Later, an improved version of the scheme was applied on homogeneous space of SL(2, R) × SL(2, R) to obtain weak local rigidity for certain parabolic algebraic actions [2]. To carry out the scheme, people need to solve the linearized equation: over the algebraic action α, where Λ is valued on the tangent space of the homogeneous space. The equation decomposes into the twisted cohomological equations of the form on the µ-eigenspace of Ad(α). Hence a complete and detailed description of twisted cohomological obstructions for the action α is necessary for the scheme.
In this paper, we give a complete solution to the twisted cohomological equation over the geodesic flow. We construct of the solution to the twisted coboundary equation, classify the the obstructions and obtain tame estimates of the solution. The results in the present paper will be used to prove local differentiable rigidity of the left translations of the two-dimensional subgroup where s, t ∈ R on SL(2, R) 3 /Γ, see [10].
1.2. History and method. Results concerning the cohomology of horocycle flow are due to Flaminio and Forni in [3]. They used Fourier analysis in each irreducible unitary representations of P SL(2, R) to obtain Sobolev estimates of the cohomological equation. These estimates satisfy a uniform upperbound condition, across the class of irreducible representations. Global estimates were then formed by glueing estimates together from each irreducible component. This scheme was further used in [9] to study the cohomological equation of the classical (discrete) horocycle map, and it was also used in [7] to study the cohomological equation of the classical geodesic flow.
In this paper, we follow the same general scheme as in [3] to study the twisted cohomological equation over the geodesic flow. In earlier papers, the obstructions to solving the equation can be constructed explicitly, which provides distributional solutions by Green's function. For the twisted equations, the obstructions are much more complex, which results in explicit construction is mostly likely impossible. This does seem to require some new techniques for handling it; the same is true in an attempt at obtaining Sobolev estimates of the solution.

Statement of results
2.1. Irreducible representations of SL(2, R). We choose as generators for sl(2, R) the elements The Casimir operator is then given by which generates the center of the enveloping algebra of sl(2, R). The Casimir operator acts as a constant u ∈ R on each irreducible unitary representation space and its value classifies them into three classes except the trivial representation. For Casimir parameter µ of SL(2, R), let ν = √ 1 − µ be a representation parameter. We denote by (π ν , H ν ) or (π µ , H µ ) the following models for the (1) principal series (ν ∈ iR); (2) complementary series (ν ∈ (−1, 1)\0); (3) the mock discrete series or the principal series (ν = 0); (4) discrete series representation spaces (ν ∈ Z\0). For the principal series, we also use the notation (π + ν , H + ν ) for the spherical model and (π − ν , H − ν ) for the non-spherical model. For the discrete series we also use (π + ν , H + ν ) to denote the upper half-plane model and (π − ν , H − ν ) to denote the lower half-plane model.
Any unitary representation (π, H) of SL(2, R) is decomposed into a direct integral (see [3] and [5]) with respect to a positive Stieltjes measure dS(µ) over the spectrum σ( ). The Casimir operator acts as the constant µ ∈ σ( ) on every Hilbert space H µ . Here ℓ(µ) is the (at most countable) multiplicity of the irreducible representation of SL(2, R) appearing in π. We say that π has a spectral gap (of u 0 ) if u 0 > 0 and S((0, u 0 ]) = 0 and π contains no non-trivial SL(2, R)fixed vectors.
In this paper, we only consider unitary representations of SL(2, R) with a spectral gap. That is, for complementary series, we assume there is 0 < u 0 < 1 such that ν ∈ (−u 0 , u 0 )\0. For the proofs involving the discrete series, we only consider the holomorphic case (ν ≥ 1) because there is a complex antilinear isomorphism between two series of the same Casimir parameter, but we list corresponding results for the anti-holomorphic case (ν ≤ −1).  (2) if g ∈ H s with s ≥ |m| 2 + 8, and D(g) = 0 for any (X − m)-invariant distribution D, then the equation has a solution f ∈ H s− |m| 2 −3 with estimates

Statement of the results.
(3) if g ∈ H s with s ≥ |m| 2 + 8, and the equation (X + m)f = g has a solution f ∈ H Remark 2.2. In [8], Ramirez shows that for the regular representation of SL(2, R) the existence of an L 2 -solution of the cohomological equation Xf = g grantees the existence of an smooth solution if g is smooth. This is quite different from the twisted case, since the above theorem shows that for the twisted cohomological equation, an L 2 -solution always exists.
The next two theorems make a detailed study for the twisted equation in each non-trivial irreducible component of SL(2, R). Also, tame splittings are provided for non-homogeneous equations.
Moreover, we have Preliminaries on representation theory of SL(2, R) 3.1. Sobolev spaces. As in Flaminio-Forni [3], the Laplacian gives unitary representation spaces a natural Sobolev structure. Let π be a unitary representation of SL(2, R) on a Hilbert space H. The Sobolev space of order s > 0 is the Hilbert space H s ⊂ H that is the maximal domain determined by the inner product The subspace H ∞ coincides with the intersection of the spaces H s for all s ≥ 0. H −s , defined as the Hilbert space duals of the spaces H s , are subspaces of the space E(H) of distributions, defined as the dual space of H ∞ .
In addition to the decomposition (2.2), all the operators in the enveloping algebra are decomposable with respect to the direct integral decomposition (2.2). Hence there exists for all s ∈ R an induced direct decomposition of the Sobolev spaces: with respect to the measure dS(µ) (we refer to [11,Chapter 2.3] or [4] for more detailed account for the direct integral theory).
The existence of the direct integral decompositions (2.2), (3.1) allows us to reduce our analysis of the cohomological equation to irreducible unitary representations. This point of view is essential for our purposes.
3.2. Sobolev norms. There exists an orthogonal basis {u k } in H ν , basis of eigenvectors of the operator Θ = U − V and hence of the Laplacian operator ∆ = − 2Θ 2 , satisfying: and the norms of the u k are given recursively by From Section 3.1 the Sobolev norms of the vectors of the orthogonal basis {u k } are given by the identities Then the Sobolev norm of a vector f = k f k u k ∈ H s ν is: By the above lemma, u k 2 s ≈ (1 + |k|) 2s−Re(ν) . So it follows that,

The basic solutions of the twisted equation
In this section we study the twisted cohomological equation The action of X on the basis element {u k } is given by: For n ∈ N ν = n − 1 and k = n, the above equation must be read as (X + m)u n = mu n + nu n+2 .
Let f = k f k u k and g = k g k u k be the Fourier expansions of the distributions f , g with respect to the adapted basis of H ν . So the twisted equation (4.1) becomes for all k ∈ I ν ; for ν = n − 1 (discrete series) and k = n equation (4.2) should be read as g n = mf n − f n+2 .  Since X t is isomorphic to R we have a direct integral decomposition where u is a regular Borel measure and We see that f = R (m + χ ′ (0)) −1 g χ du(χ) is a formal solution of the equation (X + m)f = g.
On the other hand, if (X + m)f = 0 with f ∈ H, then we have (m + χ ′ (0))f χ = 0 for almost every χ ∈ " R with respect to u. This implies that f χ = 0 for almost every χ ∈ " R. Then we have f = 0. Hence we showed the uniqueness of the solution of the twisted equation. This completes the proof.

Explicit construction of basic solutions.
For any m ∈ R and n ∈ I ν \S ν , we want to find the vector such that there is a Θ-finite f {n} such that We write ; (4.5) and for any k ≥ 1 with n − 2k ≥ 0, we can obtain the sequence (b n,n−2k ) using a recursive rule along with the two initial elements b n,n−2 and b n,n−4 : if n < 0, k ≥ 1, and n + 2k ≤ 0.  It is clear that U n has the same form as in (4.3).
These f {n} , n ∈ Z are called basic solutions and will be used to construct explicit solution of the twisted equation (4.1) in next Section 5.2. We now make a slight digression to obtain upperbounds of |b n,n−2k | u n−2k u n , which will be used to estimate the Sobolev orders of the basic solutions, see Proposition 5.1. .
From (4.6) it suffices to consider the case of n ≥ 0 and n − 2k ≥ 0. The remaining part of this section will be dedicated to the proof of this proposition.
Then (4.10) follow immediately from the above inequality by letting b = |ν| and x = n − 2r + 1. Hence we finish the proof. Proof. We note that if n ≥ 0 and n − 2k ≥ 2 then In (1) we use the inequality log(1 + x) ≤ x, for any x ≥ 0 and integral inequalities.
The above inequality shows that if n − 2k ≥ 2 then By Lemma 3.1 we have u n−2k = u n . Then the above estimates and Lemma 4.5 imply the conclusion.
If ν > 2 − |m|, we have Here in (1) we use that if n − 2k − ν > 2|m|(|m| + 2). Hence we finish the proof. Proof. Similar to the proof of Corollary 4.6, by using log(1 + x) ≤ x and integral inequalities we have By Lemma 3.1 we have The above estimates and Lemma 4.7 imply the conclusion.
Then to prove that |b n,n−2k | u n−2k un ≤ c n−2k it suffices to show that 2|m| (4.14) For any b ∈ R and x ≥ max{2|m| 2 + 2, |b| + 6} and we have In (1) we use the inequality (3) we use the fact that if x > 2m 2 then in (4) we use that 1 √ Then (4.14) follows immediately from the above inequality by letting b = ν and x = n − 2k + 3. Hence we finish the proof. Proof. Similar to the proof of Corollary 4.6, by using log(1 + x) ≤ x and integral inequalities we have . This and Lemma 4.9 imply the conclusion.
It is clear that Proposition 4.4 is a direct consequence of Corollary 4.6, 4.8 and 4.10.

Sobolev estimates of the solution.
We are now ready to give the explicit solution of the twisted equation.
Hence we prove (5.3) for n > 0. The proof of the case of n < 0 is similar. Then we get (5.3). It is clear that (5.2) is a direct consequence of (5.3). Hence we finish the proof. 5.3. Proof of proposition 5.3. We need an additional step to get to the proof.
Since f − u k is also Θ-finite, the observation after Definition 4.2 implies that f = u k and n∈S δ ν D δ,m ν,n (g)u n = 0. Hence we finish the proof. Hence we finish the proof.
Following the same way as in (6.1), we have f s− |m| 2 −3 ≤ C m g s . Hence we finish the proof.