Cohomological equation and cocycle rigidity of discrete parabolic actions

We study the cohomological equation for discrete horocycle maps on $SL(2, \mathbb{R})$ and $SL(2,\mathbb R)\times SL(2, \mathbb{R})$ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of $SL(2,\mathbb R)$. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $sl(2, \mathbb R)$, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for $SL(2, \mathbb R)$ in which all cases of irreducible, unitary representations of $SL(2, \mathbb R)$ can be studied simultaneously. Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.


Introduction
Cohomology arises in various problems in dynamical systems, such as those concerning the existence of invariant measures and mixing of suspension flows, and it is of central importance to rigidity and stability questions, see [3], [1], [2] and [10].
There is an abundance of rigidity results for higher-rank (partially) hyperbolic algebraic systems. Local rigidity and cocycle rigidity for higher-rank standard hyperbolic algebraic actions were proved in the 1990's by Katok and Spatzier [11], [12]. Then the results were extended to a large class of higher-rank partially hyperbolic actions, see for example [1], [10], [11], [9] and [21]. These results were proven using geometric arguments that rely on the fast separation of orbits.
Much less in known about parabolic systems. Parabolic systems are fundamentally different from partially hyperbolic ones in that nearby orbits separate from each other at a much slower rate, namely, at so-called polynomial rather than exponential speed. For this reason, the methods used to prove local or cocycle rigidity results for higher-rank partially hyperbolic systems do not work in the parabolic setting. All known examples of parabolic systems are homogeneous, such as horocycle flows and Heisenberg nilflows. So instead, a successful approach has been to make use of the underlying algebraic nature of the system and proceed via the theory of unitary representations.
Tame estimates for solutions to the cohomological equation of horocycle flows were obtained by Flaminio and Forni in [5]. Subsequently, several authors used these results to study the cohomological equation in some models of rank two continuous, parabolic actions on products of SL 2 with coefficients in R or C, see for example the works by Mieczkowski, Damjanovic-Katok and Ramirez in [15], [3] and [16], respectively. The second author of this paper studied rank two continuous parabolic actions on some general classes of simple, higher rank Lie groups in [20].
There have also been cohomology results in the nilmanifold setting, which require a diophantine condition. Flaminio and Forni proved tame estimates for the solution of the cohomoloigical equation of Heisenberg nilmanifolds in [6]. They later obtained non-tame estimates for solutions to the cohomological equation in higher step nilmanifolds in [7]. Cosentino and Flaminio extended [6] in a different direction, by increasing the dimension of the Heisenberg nilmanifold rather than the step. They proved a tame splitting for the isotropic subgroups of the higher dimensional Heisenberg group.
Unlike in the partially hyperbolic setting, cohomology results for parabolic maps tend to be significantly harder to obtain than the corresponding results for parabolic flows. For example, the space of obstructions is more complicated for maps than flows, as it tends to be infinite rather than finite dimensional in irreducible, unitary representations of the group. Consequently, results for maps are more recent. The first author proved non-tame estimates for solutions of the cohomological equation of horocycle maps in [17], which was improved to another non-tame estimate by the first author with Flaminio and Forni in [8]. In [4], Damjanovic and Tanis used such estimates to prove a non-tame splitting for Z 2 actions by horocycle maps on SL(2, R) × SL(2, R)/Γ, where Γ is an irreducible, cocompact lattice.
In this paper, we prove non-tame estimates for the solution to the cohomological equation of horocycle maps that are sharp up to a fixed finite loss of regularity, see Theorem 1.2, thus improving on results in [17] and [8]. Estimates in previous papers were not sharp because with the unitary models that were used, the discrete series was different and harder than the other cases. In this paper we were able to obtain sharp estimates by applying a suitable unitary transformation to the Fourier models of SL(2, R) used in [8], where all cases of irreducible representations (principal series, discrete series, etc. ) could be handled at once and in the same way. We believe this version of the Fourier model will have other applications as well.
A principal application for our sharp estimates is toward the more general problem of obtaining sharp estimates (up to a fixed, finite loss of regularity) for solutions to the cohomological equation of discrete parabolic maps in regular representations of some classes of possibly higher-rank, simple Lie groups, see Theorem 2.1 of [18], where analogous results were already obtained for parabolic flows in [20]. A main reason that cohomological equations are difficult to study in the higher-rank setting is that the representation theory of higher rank simple Lie groups is very complicated. So the crucial point is to study the problem in various subgroups, such as SL(2, R), where the representation theory is much simpler than that of the entire group. Hence, our results on horocycle maps are used in [18] to prove analogous results for regular representations of (some classes of) higher-rank simple Lie groups.
Moreover, even though the solution is not tame, we prove that estimates are tame in a co-dimension one subspace of the Lie algebra, see Theorem 1.3, which is used in [18] to deduce tame cocycle rigidity for abelian discrete parabolic actions in the higher-rank setting. Specifically, tame estimates in a co-dimension one subspace allow us conclude that solutions to the cohomological equation of the discrete parabolic map in regular representations of (some classes of) higher-rank Lie groups are also tame in a co-dimension one subspace of the Lie algebra, and the tame cocycle rigidity result is deduced from this. An important application of tame cocycle rigidity is to establish local rigidity by the KAM iterative scheme, see [2], [3].
To complete the picture, in this paper we also prove tame cocycle rigidity for discrete actions by horocycle maps on SL(2, R) × SL(2, R), improving on the non-tame cocycle rigidity estimates obtained in [4]. This may also be viewed as a motivating example for the analogous higher-rank result appearing in [18].
Finally, to keep the paper self-contained, we give a simpler proof of the non-tame lower bound for solutions to the cohomological equation of horocycle maps than the one appearing in [18], which concerned general root unipotent maps in SL(n, R). To the best of our knowledge, this result, and the more general version in [18], are the first examples of non-tame solutions to a cohomological equation in the non-commutative, homogeneous setting.
1.1. Results. The Lie algebra of SL(2, R) is generated by the vector fields which satisfy the commutation relations Let (π, H) be a unitary representation of SL(2, R) and let be the Laplacian on H. The operator (I + △) is essentially self-adjoint on H, so for any s ≥ 0, the spectral theorem gives that the operator (I + △) s/2 is defined. Then let W s (H) ⊂ H be the maximal domain of (I + △) s/2 on H equipped with the inner product where f, g is the H inner product. For any s ≥ 0, define The above spaces have corresponding distributional dual spaces. For any s > 0, let W −s (H) be the distributional dual space to W s (H), and let W −∞ (H) = s>0 W −s (H) be the distributional dual space to W ∞ (H).
Next we present our results for regular representations of SL(2, R) and SL(2, R) × SL(2, R). We remark that they hold for general unitary representations (π, H) of SL(2, R) and SL(2, R) × SL(2, R) provided the relevant spectral gap property holds. Namely, the spectral gap property for SL(2, R) representations is described in Appendix A, and when H is an SL(2, R) × SL(2, R) representation, we require that the restriction of π to any SL(2, R) factor has a spectral gap.
1.1.1. One-parameter discrete actions. Let Γ ⊂ SL(2, R) be a lattice and M := SL(2, R)/Γ. Let L 2 (M ) be the complex-valued, square-integrable functions on M with respect to the SL(2, R)-invariant volume form. Let π be the regular representation of SL(2, R) on H = L 2 (M ), and for s ∈ R∪{±∞}, we use the notation W s (M ) := W s (L 2 (M )) . The vector field X generates the geodesic flow, and U and V generate the unstable and stable horocycle flows with respect to the geodesic flow, which all act by left multiplication on M . In this paper, we refer to the stable horocycle flow as the horocycle flow (h t ) t∈R . So for any t ∈ R, For a given L > 0 and coboundary g, we find Sobolev estimates for the solution f to the equation Invariant distributions for the horocycle map have already been classified in Theorem 1.1 of [17], and a more precise description of the regularity of these distributions in irreducible representations of SL(2, R) was given in Theorem 2.6 of [8]. Denote the spaces of invariant distributions for h L in W −s (M ) and W −∞ (M ), respectively, by These invariant distributions were first classified in Theorem 1.2 of [17], and sharper estimates of their regularity were provided in the following theorem.  The space I 0 (M ) is described in Theorem 1.1 of [5] as follows: It has infinite, countable dimension. It is a direct sum of the trivial representation I vol and irreducible, unitary representations I µ belonging to the principal series, the complementary series, the discrete series and the mock discrete series. Specifically, • The space I vol is spanned by the SL(2, R)-invariant volume; , and each subspace has dimension equal to the multiplicity of µ ∈ spec( ); • If µ ≥ 1 and H µ is a principal series representation, then I µ ⊂ W −s (M ) if and only if s > 1/2, and it has dimension equal to twice the multiplicity of µ ∈ spec( ); • If µ = −n 2 + 2n for n ∈ Z + , then Our main theorem on solutions of the cohomological equation of horocycle maps is the following improved bound. Our approach follows [8] and [18] in that we first prove corresponding results for the twisted equation for horocycle flows, and deduce the below result for horocycle maps from that. As mentioned above, we improved on previous estimates by finding a more suitable version of the Fourier model used in [18], where we obtain the bound for all representation cases (principal, complementary, mock discrete and discrete series) simultaneously. Let G V be the operator on the space of coboundaries for the horocycle flow given by which is uniquely defined (up to additive constants) on W s (M ) when s ≥ 0 and was studied in [5].
Moreover, the estimate is tame on a co-dimension one subspace of sl(2, R). Theorem 1.3. Using the assumptions in the above theorem, we have Remark 1.4. Theorem 1.2 of [5] shows that for any s ≥ 1 and for any ǫ > 0, Then the above discussion and estimates (5) implies that It has already been shown from the lower bound in Theorem 2.2 of [18] that the above non-tame estimate is sharp with respect to a loss of regularity of 1/2 derivatives. However that proof was written for representations of SL(n, R), for any n ≥ 2, and is overly complicated for the special case of SL(2, R). Hence, we provide a simpler and more transparent version of that proof in Section 2, which is specific to SL(2, R). Theorem 1.5. [special case of Theorem 2.2 of [18]] For any s ≥ 0, for any σ ∈ [0, s + 1/2) and for any L > 0, the following holds. For every constant C > 0 there is a function g ∈ W ∞ (M ) with a solution f ∈ W ∞ (M ) to the equation be an irreducible lattice, and let M = SL(2, R) × SL(2, R)/Γ. Also let L 2 (M ) and W s (M ) be defined analogously as in the above SL(2, R) case. We consider the following unipotent maps acting by left multiplication on M : The next theorem shows that in contrast to the one-parameter setting, tame estimates hold for solutions to the cohomological equation of two-parameter actions.
Theorem 1.6. Let s ≥ 0 and suppose f, g ∈ W s+3 (M ) and satisfies the cocycle equation 1.2. Direct decompositions of Sobolev space. For any Lie group G of type I, such as SL(2, R) and SL(2, R) × SL(2, R), with a unitary representation ρ, there is a decomposition of ρ into a direct integral of irreducible unitary representations for some measure space (Z, µ) (we refer to [22,Chapter 2.3] or [13] for more detailed account of direct integral theory). All the operators in the enveloping algebra are decomposable with respect to the direct integral decomposition (10). Hence there exists for all s ∈ R an induced direct decomposition of the Sobolev spaces: with respect to the measure dµ(z).
The existence of the direct integral decompositions (10), (11) allows us to reduce our analysis of the cohomological equation to irreducible unitary representations. This point of view is essential for our purposes. See for example, Theorem 1.1 of [4].
2. Proof of Theorems 1.2 and 1.5 We begin with Theorem 1.2, which has already been proven for the principal, complementary and mock discrete series in Theorems 3.4 and 3.19 of [8]. Next we discuss the model that we use to prove Theorem 1.2, where we can handle all representation cases together, including the discrete series. The estimates follow the approach in the proof of Theorem 3.4 of [8].
2.1. A Fourier model for SL(2, R). The Hilbert space models for the principal, complementary, discrete and mock discrete series are discussed in Appendix A. We use the representation parameter for convenience, and we denote the real part of ν by Reν.
The Fourier transform for functions in the line model H µ of the principal or complementary series is defined bŷ Following [8], for functions in the upper half-plane model of the discrete series or mock discrete series, also denoted by H µ , the Fourier transform is given byf Cauchy's theorem implies thatf is supported on R + when f ∈ H ∞ µ (see Lemma 3.15 of [8]). By direct computation from the vector field formulas given in Appendix A, we haveV = − √ −1ξ and In all cases (i.e. principal, complementary, discrete and mock discrete series), Lemma 3.1 and Lemma 3.16 of [7] give a constant C Reν > 0 such that In the discrete series, the number Reν = ν may be large. The measure then ξ −Reν dξ created difficulties in [8] and [17] that led to weak estimates with respect to loss of regularity for the solution of the cohomological equation in the discrete series. By an appropriate change of variable, the above norm is evaluated by an integral with the measure ξ −1 dξ for all irreducible representation cases, so we can handle all cases simultaneously and obtain sharp non-tame estimates.
To this end, fix any irreducible, unitary representation of the regular representation. For any J ⊆ R, define be the unitary transformation given by (14) Af One can verify that these formulas give the usual commutation relations (16) [ In light of (2) and (13), for any s > 0, we have

2.2.
Proof of Theorem 2.1. As in [8], we derive estimates for the solution to the cohomological equation of horocycle maps from estimates of the solution f to the twisted equation where λ ∈ R * . The distributions that are invariant under the operator (V + √ −1λ) have already been studied in [8]. For any irreducible, unitary representation H µ of SL(2, R), define Lemmas 3.3 and 3.14 of [8] show that for any ǫ > 0, We derive sharp bounds in Theorem 1.2 from the following sharp abounds on solutions to the twisted equation (18). The strategy of deriving bounds for maps for from those of the twisted equation was already used in [8].
Theorem 2.1. For any s ≥ 0, for any ǫ, there is a constant C s,ǫ > 0 such that the following holds. For any irreducible, unitary representation H µ of SL(2, R). For any λ ∈ R * and for any g ∈ Ann λ,2s+1 µ , there is a solution solution f ∈ H s µ to the twisted equation (18) such that As above, the estimate is tame when the norm involves only the X and V derivatives, First let s ∈ N. By the triangle inequality, the commutation relations (1) and formula (17), there is a constant C s > 0 such that Because A is unitary, estimates of solutions to the twisted equation (18) are equivalently obtained in L 2 ν (R) by solving (V + √ −1λ)Af = Ag using the vector fields (15). It follows that Next, as in [8], set (Af ) λ (ξ) = (Af )(λξ) for any function f , which means To simplify notation until the end of the argument, for any function f , set (20) is reduced to solving Notice that for any σ ∈ N, we have Hence, we can obtain an estimate of (19) by an estimate involving only the U and X derivatives. In light of (21), let The next two propositions estimate each term of the above inequality. The below proposition does not use a distributional assumption on g.
Proof. We start with a technical lemma.
Proof. By the product rule for ordinary differentiation, we get Hence, for any α ∈ N, there are coefficients (a Next, a short computation gives we get by induction that By the above equality and (23), we get constants (c We now finish the proof of Proposition 2.2. In the above lemma, let g = g 1 and g 2 = (ξ − 1) −1 . An induction argument shows there are coefficients (c Then from the definition of X , and because we get a constant C l,m,n > 0 such that for any ξ ∈ R \ I, By the triangle inequality and Lemma 2.3, it follows that there is a constant C α,β > 0 such that The proof of Proposition 2.2 is now complete.
In the next proposition, we use the distributional assumption to estimate f over the interval I. Specifically, it is shown in [8] that the space I 1 µ is one-dimensional and spanned by the functional α,β > 0 such that for any g ∈ Ann 1,α+2β+1 Proof. First let ξ t = t(ξ − 1) + 1 .
Then for any α, β ∈ N and for any g ∈ Ann 1,α+2β+1 µ , we have Proof. Hence, as in formula (48) of [8], we have Hence, for any α ∈ N, we have Next, we have Using ξt = ξ t − (1 − t), we get . Then by the above two equalities, we have Observe Now we compute the formula for U β f (ξ). Observe that for any m ∈ Z and for any ξ ∈ I, we have Then for W 0 , W 1 as in the statement of the lemma, by an induction argument using the above commutation relations and by (32), we get where c (s k ) ∈ C is a constant depending on the sequence (s k ) ∈ L (β) l . The lemma follows from the above equality and formula (29).
By the above lemma and Minkowski's inequality, and because the interval I is bounded away from zero, we get a constant C β > 0 such that Because d dξ = −1 2ξ X , it follows that Notice that for any ξ ∈ I and any t ∈ [0, 1], there is a constant C > 0, depending only on I, such that Then using the commutation relations from (33) again and expanding the expression, we get a constant C α,β > 0 such that Proof of Theorem 2.1. By Propositions 2.2 and 2.4, we get a constant C α,β > 0 such that The above gives the estimate in Theorem 2.1 for functions (Af ) λ , where (Af ) λ (ξ) = (Af )(λξ). For the general result, we argue as in formula (58) of [8]. Observe that on smooth functions f , we have Hence, Note that V commutes with (V + √ −1λ) (see also equation (22)). Replacing functions f with V γ f in (37), we get a constant C α > 0 such that Recall that we simplified notation by writing f = Af , and we have an expression (13) for converting Sobolev norms in · to Sobolev norms in L ν 2 (R). Then using the commutation relations given in (1), it follows that for any s ∈ N, there is a constant C s > 0 such that By interpolation, the above estimate holds for any s ≥ 0, which is the second estimate in Theorem 2.1. Next, we consider the full Sobolev norm. As above, for any s ∈ N, there is a constant C s > 0 such that The case for s ≥ 0 now follows by interpolation.  [5]. In particular, they defined a Green operator G V µ for the horocycle flow on the space of coboundaries in H µ for the horocycle flow, which is uniquely defined up to additive constants on smooth functions by the cohomological equation Recalling that A is unitary, we simplify notation as in the previous section and write g = Ag.
In the model L 2 ν (R), the cohomological equation for the horocycle map h L is We recall from Theorem 1.1 that the space of invariant distributions for the map that are not invariant for the flow is I L,twist (M ) ⊂ W −(1/2+ǫ) (M ) for any ǫ > 0. Restricting to the the line or upper half-plane model of SL(2, R), an explicit spanning set was first described in [17]. In the Fourier model L 2 ν (R) and in [8], this spanning set is given by where m ∈ Z if H µ is in the principal or complementary series, and m ∈ N + otherwise. Below we use that each distribution D m is of the same form as the distribution (27) used to study the twisted equation (18).
Proof of Theorem 1.2. First let α, β ∈ N and g ∈ Ann α+2β+1+ǫ L , where L > 0. It will be convenient to estimate We first focus on f restricted to the interval I L given by Because g is also a coboundary for the horocycle flow, from (39) we can write π], and ξL ∈ [−π, π] whenever ξ ∈ I L . Then by Lemma 2.3, for any α, β ∈ N, we get coefficients (b Recall that X = −2ξ d dξ , so there is a constant C α,β > 0 such that

So the triangle inequality gives
When ξ ∈ R \ I L , we only consider the case ξ ∈ R + , as the case ξ ∈ R − is analogous. Write .
By the triangle inequality, We estimate each term. For the first one, notice For the remaining terms, Lemma 2.3 gives coefficients (b where φ a,(ǫ) l,m,n (ξ) := (ξ Notice that for any ξ ≥ 0 and any ǫ > 0, Now to estimate the uniform norm of φ A short computation further shows that because (l, m, n) = (0, 0, 0), the index k in the above sum satisfies k ≥ 1. Now for any k ∈ N \ {0}, By the above formula, and by formula (47), we get a constant C α,β > 0 such that By formulas (45), (46) and the above estimate, we get By (41), for any a ∈ N \ {0}, we have that g ∈ Ann 2πa/L ,α+2β+1+ǫ µ . Hence, we use the estimate (36) for the twisted equation, and get that for any j, k ∈ N, Combining the above estimate with (50), we get Then the triangle inequality, the above estimate and (43) imply there is a constant C α,β > 0 such that We simplified notation by writing f = Af . From the definition of the unitary map A : H µ → L 2 ν (R) given in formula 14, we have that where F is the Fourier transform. Then Because the analogous estimate to (51) also holds on R − , we conclude using the above equality that Notice that V commutes with the translation operator h L , so f •h L −f = g implies (V γ f ) • h L − V γ f = V γ g, for any γ ∈ N. Then by the commutation relations (1) and the above estimate, for any s ∈ N, there is a constant C s > 0 such that Theorem 1.2 now follows for s ≥ 0 by interpolation .
Proof of Theorem 1.3. This is immediate from (52).

2.4.
Proof of Theorem 1.5. We use the Fourier transform of the line model of the principal series. Recall the formulas for vector fieldsÛ ,V and X are given in formula (12). We first consider Sobolev estimates for the solution to the equation (18), so which reduces to studying , as in (21). Let |ν| ≥ 4, and let be defined for any ξ ∈ R by (55)ĝ(ξ) :=q(ξ)(ξ ν+1 − 1) .
Notice that g ∈ H ∞ µ andĝ(1) = 0. Then by Theorem 3.4 of [8], the equation (18) has a solution f ∈ H ∞ µ . The following property will be used several times: for all m ∈ Z, Lemma 2.6. Let s ≥ 0. Then there is a constant C s > 0 such that Proof. First let s ∈ 2N. By the commutation relations (1), and by the triangle inequality, we have a constant C s > 0 such that From formula (39) and Lemma 3.7 of [8], we have Leibniz-type formulas for the operatorsX andÛ . Specifically, there are universal coefficients (a ijkm ) such that for any pair of functions g 1 , g 2 , we have Set By (56), and becauseX is only a first order differential operator, it follows that there is a constant C s,q > 0 such that X mÛ nĝ 0 ≤ C s,q (1 + |ν|) m+n . Becauseĝ is supported on a bounded interval, it follows from the formula forÛ and the above estimate that (57) ≤ C s,q (1 + |ν|) s .
Because q is fixed, we have now proven the lemma in the case that s ∈ 2N. The lemma for s ≥ 0 follows by interpolation. Now we focus on a lower bound for the Sobolev norm of the solution f , given by (54). Observe that for any s ≥ 0, Proof. Clearly, We prove that there is a constant C s > 0 such that We begin by considering integer powers ofV . Let β ∈ N \ {0}. Becausê g(1) = 0, the fundamental theorem of calculus shows that for all ξ ∈ R, . Then for any t ∈ [0, 1], set Formula (51) of [8] gives, by a short computation, that Next, we now expand and rewrite the expression [Û + (t − 1) d 2 dξ 2 ] β . Let W 0 := ∂ 2 ∂ξ 2 and W 1 :=Û . For each 0 ≤ m ≤ β − 1 and 1 ≤ n ≤ β, let S m,n be the set of all sequences of length m + n consisting of m 0's and n 1's. Then where, A m,n := Hence, Then in the above expression, we show that the first term is relatively large, and the sum is relatively small. Proposition 2.7 will follow by the triangle inequality.
Lemma 2.8. For any β ∈ N, there is a constant C β > 0 such that for any ξ ∈ I ν , and for any |ν| ≥ 4, Proof. Notice that for any k ∈ N \ {0}, So by (56) and by the conditions on the sum (61) there is a constant C β > 0 such that for any ξ ∈ I ν , Hence, there is a constant C β > 0 such that for all ξ ∈ I ν and t ∈ [0, 1], It remains to prove the following lower bound.
Proof of Theorem 1.5. Using the Fourier transform in the line model, the cohomological equation (7) for unipotent maps has the form .
Further define H on R by and notice that where becauseq is supported on [ 3 4 , 4 3 ], we get thatf andĝ are supported on 2π L [ 3 4 , 4 3 ] . Then for any s ∈ N, the commutation relations give a constant C s > 0 such that The Leibniz-type formula forV (see (58)) gives universal coefficients (b Define H L by H(ξ) = H L (Lξ). Then because H L is smooth and independent of ν and L, for each m + i ≤ β, we have a constant C β > 0 such that Therefore, Moreover, a calculation shows that for any constant c and for each j ∈ N, Then by Lemma 2.6, there is a constant C β > 0 such that Finally, because H ·ĝ twist is compactly supported and the derivativesÛ and X contribute at most one power of ν, we get a constant C β > 0 such that It follows by interpolation that for any s ≥ 0, there is a constant C s > 0 such that g s ≤ C (2) s (1 + L 2s )|ν| s . On the other hand, by (71) and by Proposition 2.7, for any s ≥ 0, there are constants c (1) s > 0 such that for any |ν| ≥ c (1) s , So let σ ∈ [0, s + 1/2) and C > 0. Then for any |ν| large enough that s+σ,n (1 + L 2(s+σ) )|ν| s+σ , we get s+σ,n (1 + L 2(s+σ) )|ν| s+σ > C g s+σ .

Proof of Theorems 1.6
Recall the definition of the translation operators h   [4] shows that there is a solution p ∈ W ∞ (H) such that with estimates p s ≤ C s,L 2 f 3s+6 , s ≥ 0. In [4] H = L 2 0 (G/Γ) and π is the regular representation, where Γ is an irreducible lattice in G and L 2 0 (G/Γ) is the space of square integrable functions on G/Γ with zero average. Note that the result can be extended to unitary representation of G such that such that the restriction of π to any SL(2, R) factor has a spectral gap. We have the decomposition: where H µ and H θ range over all non-trivial irreducible representations of SL(2, R). The solution p was constructed, as well as its estimates were obtained in [4] in each H µ ⊗ H θ . Then discussion in Section 1.2 shows that p is a bona fide solution.
Hence, it suffices to obtain the tame estimates of p with respect to f and g. We use X 1 , V 1 and U 1 to denote the basis Lie algebra for the first copy of SL(2, R) and X 2 , V 2 and U 2 for the second copy. For Z ∈ {X 2 , V 2 , U 2 }, we note that Since the restriction of π on the first copy of SL(2, R) is still a unitary representation with spectral gap, by using (6), it follows that Similarly, for Y ∈ {X 1 , V 1 , U 1 }, we have Then (9) follows directly from (73), (74) and the following elliptic regularity theorem (see [19, Chapter I, Corollary 6.5 and 6.6]): Theorem 3.1. Let π be a unitary representation of a Lie group G with Lie algebra g on a Hilbert space H. Fix a basis {Y j } for g and set L 2m = Y 2m j , m ∈ N.
where C m is a constant only dependent on m and {Y j }.
Appendix A. Unitary representations of SL(2, R) Section 1.2 already discussed the direct integral decomposition for unitary representations of general type I Lie groups, but we can say more in the special case of SL(2, R).
Recall that sl(2, R) is generated by the vector fields The Casimir operator is given by and generates the center of the enveloping algebra of sl(2, R). Any unitary representation (π, H) of SL(2, R) is decomposed into a direct integral (see [5] and [14]) 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 representation space K µ , which does not need to be irreducible. In fact, K µ is in general the direct sum of an (at most countable) number of unitarily equivalent representation spaces H µ equal to the spectral multiplicity of µ ∈ σ( ). We say that π has a spectral gap if there is some u 0 > 0 such that s((0, u 0 ]) = 0. It is clear that if π has a spectral gap then π contains no non-trivial SL(2, R)-fixed vectors.
The representation spaces H µ have unitarily equivalent models, which we also write as H µ , and each is one of the following four classes: • If µ ∈ (0, 1), then H µ is in the complementary series.
• If µ > 1, then H µ is in the principal series.
• If µ = 1, then H µ is in the mock discrete series or the principal series. • Otherwise if µ ≤ 0, then H µ is in the discrete series. It will be convenient to describe these representations via models that use a representation parameter ν satisfying where it is sufficient to take ν = √ 1 − µ. Below are standard models from the literature. The line model of the principal or complementary series consists of functions on R and has the following norm. For µ ≥ 1, f Hµ := f L 2 (R) , and when µ > 1,

The group action on H µ is given by
where x ∈ R and (76) A = a b c d ∈ SL(2, R) .

This yields the vector fields
The holomorphic discrete or mock discrete series consists of holomorphic functions on the upper half-plane, H. Its counterpart, the anti-holomorphic discrete or mock discrete series, does not need to be considered because of a complex anti-linear isomorphism between the two spaces. The holomorphic discrete or mock discrete series norm is given by f Hµ = H |f (x + iy)| 2 y ν−1 dx dy The group action is analogous to the one for the line model. For z ∈ H and A given by (76), we have which yields the vector fields