Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey-Roumieu and Gevrey-Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace-Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their $\alpha$-duals in the sense of Koethe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.


Introduction
The spaces of Gevrey ultradifferentiable functions are well-known on R n and their characterisations exists on both the space-side and the Fourier transform side, leading to numerous applications in different areas. The aim of this paper is to obtain global characterisations of the spaces of Gevrey ultradifferentiable functions and of the spaces of ultradistributions using the eigenvalues of the Laplace-Beltrami operator L G (Casimir element) on the compact Lie group G. We treat both the cases of Gevrey-Roumieu and Gevrey-Beurling functions, and the corresponding spaces of ultradistributions, which are their topological duals with respect to their inductive and projective limit topologies, respectively.
If M is a compact homogeneous space, let G be its motion group and H a stationary subgroup at some point, so that M ≃ G/H. Our results on the motion group G will yield the corresponding characterisations for Gevrey functions and ultradistributions on the homogeneous space M. Typical examples are the real spheres S n = SO(n + 1)/SO(n), complex spheres (complex projective spaces) CP n = SU(n + 1)/SU(n), or quaternionic projective spaces HP n .
Working in local coordinates and treating G as a manifold the Gevrey(-Roumieu) class γ s (G), s ≥ 1, is the space of functions φ ∈ C ∞ (G) such that in every local coordinate chart its local representative, say ψ ∈ C ∞ (R n ), is such that there exist constants A > 0 and C > 0 such that for all multi-indices α, we have that holds for all x ∈ R n . By the chain rule one readily sees that this class is invariantly defined on (the analytic manifold) G for s ≥ 1. For s = 1 we obtain the class of analytic functions. This behaviour can be characterised on the Fourier side by being equivalent to the condition that there exist B > 0 and K > 0 such that | ψ(η)| ≤ Ke −B η 1/s holds for all η ∈ R n . We refer to [5] for the extensive analysis of these spaces and their duals in R n . However, such a local point of view does not tell us about the global properties of φ such as its relation to the geometric or spectral properties of the group G, and this is the aim of this paper. The characterisations that we give are global, i.e. they do not refer to the localisation of the spaces, but are expressed in terms of the behaviour of the global Fourier transform and the properties of the global Fourier coefficients.
Such global characterisations will be useful for applications. For example, the Cauchy problem for the wave equation is well-posed, in general, only in Gevrey spaces, if a(t) becomes zero at some points. However, in local coordinates (1.1) becomes a second order equation with spacedependent coefficients and lower order terms, the case when the well-posedness results are, in general 1 , not available even on R n . At the same time, in terms of the group Fourier transform the equation (1.1) is basically constant coefficients, and the global characterisation of Gevrey spaces together with an energy inequality for (1.1) yield the well-posedness result. We will address this and other applications elsewhere, but we note that in these problems both types of Gevrey spaces appear naturally, see e.g. [3] for the Gevrey-Roumieu ultradifferentiable and Gevrey-Beurling ultradistributional well-posedness of weakly hyperbolic partial differential equations in the Euclidean space. In Section 2 we will fix the notation and formulate our results. We will also recall known (easy) characterisations for other spaces, such as spaces of smooth functions, distributions, or Sobolev spaces over L 2 . The proof for the characterisation of Gevrey spaces will rely on the harmonic analysis on the group, the family of spaces ℓ p ( G) on the unitary dual introduced in [8], and to some extent on the analysis of globally defined matrix-valued symbols of pseudo-differential operators developed in [8,9]. The analysis of ultradistributions will rely on the theory of sequence spaces (echelon and co-echelon spaces), see e.g. Köthe [6], Ruckle [7]. Thus, we will first give characterisations of the so-called α-duals of the Gevrey spaces and then show that α-duals and topological duals coincide. We also prove that both Gevrey spaces are perfect spaces, i.e. the α-dual of its α-dual is the original space. This is done in Section 4, and the ultradistributions are treated in Section 5.
We note that the case of the periodic Gevrey spaces, which can be viewed as spaces on the torus T n , has been characterised by the Fourier coefficients in [11]. However, that paper stopped short of characterising the topological duals (i.e. the corresponding ultradistributions), so already in this case our characterisation in Theorem 2.5 appears to be new.
In the estimates throughout the paper the constants will be denoted by letter C which may change value even in the same formula. If we want to emphasise the change of the constant, we may use letters like C ′ , A 1 , etc.

Results
We first fix the notation and recall known characterisations of several spaces. We refer to [8] for details on the following constructions.
Let G be a compact Lie group of dimension n. Let G denote the set of (equivalence classes of) continuous irreducible unitary representations of G. Since G is compact, G is discrete. For [ξ] ∈ G, by choosing a basis in the representation space of ξ, we can view ξ as a matrix-valued function ξ : G → C d ξ ×d ξ , where d ξ is the dimension of the representation space of ξ. For f ∈ L 1 (G) we define its global Fourier transform at ξ by where dx is the normalised Haar measure on G. The Peter-Weyl theorem implies the Fourier inversion formula For each [ξ] ∈ G, the matrix elements of ξ are the eigenfunctions for the Laplace-Beltrami operator L G with the same eigenvalue which we denote by −λ 2 [ξ] , so that −L G ξ ij (x) = λ 2 [ξ] ξ ij (x), for all 1 ≤ i, j ≤ d ξ . Different spaces on the Lie group G can be characterised in terms of comparing the Fourier coefficients of functions with powers of the eigenvalues of the Laplace-Beltrami operator. We denote ξ = (1 + λ 2 [ξ] ) 1/2 , the eigenvalues of the elliptic first-order pseudo-differential operator (I − L G ) 1/2 .
Then, it is easy to see that f ∈ C ∞ (G) if and only if for every M > 0 there exists C > 0 such that f (ξ) HS ≤ C ξ −M , and u ∈ D ′ (G) if and only if there exist For this and other occasions, we can write this as u(ξ) = u(ξ * ) in the matrix notation. The appearance of the Hilbert-Schmidt norm is natural in view of the Plancherel identity can be taken as the definition of the space ℓ 2 ( G). Here, of course, A HS = Tr(AA * ). It is convenient to use the sequence space In [8], the authors introduced a family of spaces ℓ p ( G), 1 ≤ p < ∞, by saying that σ ∈ Σ belongs to ℓ p ( G) if the norm There is also the space ℓ ∞ ( G) for which the norm is finite. These are interpolation spaces for which the Hausdorff-Young inequality holds, in particular, we have . We refer to [8,Chapter 10] for further details on these spaces. Usual Sobolev spaces on G as a manifold, defined by localisations, can be also characterised by the global condition For a multi-index α = (α 1 , . . . , α n ), we define |α| = |α 1 | + · · · + |α n | and α! = α 1 ! · · · α n !. We will adopt the convention that 0! = 1 and 0 0 = 1. Let X 1 , . . . , X n be a basis of the Lie algebra of G, normalised in some way, e.g. with respect to the Killing form. For a multi-index α = (α 1 , . . . , α n ), we define the leftinvariant differential operator of order |α|, ∂ α := Y 1 · · · Y |α| , with Y j ∈ {X 1 , · · · , X n }, 1 ≤ j ≤ |α|, and j:Y j =X k 1 = α k for every 1 ≤ k ≤ n. It means that ∂ α is a composition of left-invariant derivatives with respect to vectors X 1 , · · · , X n , such that each X k enters ∂ α exactly α k times. There is a small abuse of notation here since we do not specify in the notation ∂ α the order of vectors X 1 , · · · , X n entering in ∂ α , but this will not be important for the arguments in the paper. The reason we define ∂ α in this way is to take care of the non-commutativity of left-invariant differential operators corresponding to the vector fields X k .
We will distinguish between two families of Sobolev spaces over L 2 . The first one is defined by The second one is defined for k ∈ N 0 ≡ N ∪ {0} by Obviously, H k ≃ W k,2 for any k ∈ N 0 but for us the relation between norms will be of importance, especially as k will tend to infinity.
Let 0 < s < ∞. We first fix the notation for the Gevrey spaces and then formulate the results. In the definitions below we allow any s > 0, and the characterisation of α-duals in the sequel will still hold. However, when dealing with ultradistributions we will be restricting to s ≥ 1.
Definition 2.1. Gevrey-Roumieu(R) class γ s (G) is the space of functions φ ∈ C ∞ (G) for which there exist constants A > 0 and C > 0 such that for all multi-indices α, we have Functions φ ∈ γ s (G) are called ultradifferentiable functions of Gevrey-Roumieu class of order s.
For s = 1 we obtain the space of analytic functions, and for s > 1 the space of Gevrey-Roumieu functions on G considered as a manifold, by saying that the function is in the Gevrey-Roumieu class locally in every coordinate chart. The same is true for the other Gevrey space: Functions φ ∈ γ (s) (G) are called ultradifferentiable functions of Gevrey-Beurling class of order s.
if and only if there exist B > 0 and K > 0 such that Expressions appearing in the definitions can be taken as seminorms, and the spaces are equipped with the inductive and projective topologies, respectively 2 . We now turn to ultradistributions.
. These are well-defined since G is compact and hence ξ(x) are actually analytic.
The proof of Theorem 2.5 follows from the characterisation of α-duals of 3 the Gevrey spaces in Theorem 4.2 and the equivalence of the topological duals and αduals in Theorem 5.2.
The result on groups implies the corresponding characterisation on compact homogeneous spaces M. First we fix the notation. Let G be a compact motion group of M and let H be the stationary subgroup of some point. Alternatively, we can start with a compact Lie group G with a closed subgroup H. The homogeneous space M = G/H is an analytic manifold in a canonical way (see, for example, [2] or [10] as textbooks on this subject). We normalise measures so that the measure on H is a probability one. Typical examples are the spheres S n = SO(n + 1)/SO(n) or complex spheres (complex projective spaces) PC n = SU(n + 1)/SU(n).
We denote by G 0 the subset of G of representations that are class I with respect to the subgroup H. This means that [ξ] ∈ G 0 if ξ has at least one non-zero invariant vector a with respect to H, i.e. that ξ(h)a = a for all h ∈ H. Let H ξ denote the representation space of ξ(x) : H ξ → H ξ and let B ξ be the space of these invariant vectors. Let k ξ = dim B ξ . We fix an orthonormal basis of H ξ so that its first k ξ vectors are the basis of B ξ . The matrix elements ξ ij (x), 1 ≤ j ≤ k ξ , are invariant under the right shifts by H. We refer to [12] for the details of these constructions.
We can identify Gevrey functions on M = G/H with Gevrey functions on G which are constant on left cosets with respect to H. Here we will restrict to s ≥ 1 to see the equivalence of spaces using their localisation. This identification gives rise to the corresponding identification of ultradistributions. Thus, for a function f ∈ γ s (M) we can recover it by the Fourier series of its canonical lifting f (g) := f (gH) to G, f ∈ γ s (G), and the Fourier coefficients satisfy f (ξ) = 0 for all representations with With this, we can write the Fourier series of f (or of f , but as we said, from now on we will identify these and denote both by f ) in terms of the spherical functions ξ ij of the representations ξ, [ξ] ∈ G 0 , with respect to the subgroup H. Namely, the Fourier series (2.1) becomes In view of this, we will say that the collection of Fourier coefficients if and only if its Fourier coefficients are of class I with respect to H and, moreover, for every B > 0 there exists K B > 0 such that It would be possible to extend Theorem 2.6 to the range 0 < s < ∞ by adopting Definition 2.1 starting with a frame of vector fields on M, but instead of obtaining the result immediately from Theorem 2.3 we would have to go again through arguments similar to those used to prove Theorem 2.3. Since we are interested in characterising the standard invariantly defined Gevrey spaces we decided not to lengthen the proof in this way. On the other hand, it is also possible to prove the characterisations on homogeneous spaces G/H first and then obtain those on the group G by taking H to be trivial. However, some steps would become more technical since we would have to deal with frames of vector fields instead of the basis of left-invariant vector fields on G, and elements of the symbolic calculus used in the proof would become more complicated.
We also have the ultradistributional result following from Theorem 2.5.
and only if its Fourier coefficients are of class I with respect to H and, moreover, for every B > 0 there exists K B > 0 such that Finally, we remark that in the harmonic analysis on compact Lie groups sometimes another version of ℓ p ( G) spaces appears using Schatten p-norms. However, in the context of Gevrey spaces and ultradistributions eventual results hold for all such norms. Indeed, given our results with the Hilbert-Schmidt norm, by an argument similar to that of Lemma 3.2 below, we can put any Schatten norm · Sp , 1 ≤ p ≤ ∞, instead of the Hilbert-Schmidt norm · HS in any of our characterisations and they still continue to hold.

Gevrey classes on compact Lie groups
We will need two relations between dimensions of representations and the eigenvalues of the Laplace-Beltrami operator. On one hand, it follows from the Weyl character formula that , with the latter 4 also following directly from the Weyl asymptotic formula for the eigenvalue counting function for L G , see e.g. [8,Prop. 10.3.19]. This implies, in particular, that for any 0 ≤ p < ∞ and any s > 0 and B > 0 we have On the other hand, the following convergence for the series will be useful for us: Proof. We notice that for the δ-distribution at the unit element of the group, δ(ξ) = I d ξ is the identity matrix of size d ξ × d ξ . Hence, in view of (2.4) and (2.5), we can write By using the localisation of H −t (G) this is finite if and only if t > n/2.
We denote by G * the set of representations from G excluding the trivial representation. For [ξ] ∈ G, we denote |ξ| := λ ξ ≥ 0, the eigenvalue of the operator (−L G ) 1/2 corresponding to the representation ξ. For [ξ] ∈ G * we have |ξ| > 0 (see e.g. [4]), and for [ξ] ∈ G\ G * we have |ξ| = 0. From the definition, we have |ξ| ≤ ξ . On the other hand, let λ 2 1 > 0 be the smallest positive eigenvalue of −L G . Then, for [ξ] ∈ G * we have λ ξ ≥ λ 1 , implying We will need the following simple lemma which we prove for completeness. Let a ∈ C d×d be a matrix, and for 1 ≤ p < ∞ we denote by ℓ p (C) the space of such matrices with the norm , and for p = ∞, a ℓ ∞ (C) = sup 1≤i,j≤d |a ij |. We note that a ℓ 2 (C) = a HS . We adopt the usual convention c ∞ = 0 for any c ∈ R.
Lemma 3.2. Let 1 ≤ p < q ≤ ∞ and let a ∈ C d×d . Then we have Proof. For q < ∞, we apply Hölder's inequality with r = q p and r ′ = q q−p to get implying (3.4) for this range. Conversely, we have proving the other part of (3.4) for this range. For q = ∞, we have a ℓ p (C) ≤ d i,j=1 a p ℓ ∞ (C) 1/p ≤ a ℓ ∞ (C) d 2/p . Conversely, we have trivially a ℓ ∞ (C) ≤ a ℓ p (C) , completing the proof.
We observe that the Gevrey spaces can be described in terms of L 2 -norms, and this will be useful to us in the sequel.
We also have φ ∈ γ (s) (G) if and only if for every A > 0 there exists C A > 0 such that for all multi-indices α we have Proof. We prove the Gevrey-Roumieu case (R) as the Gevrey-Beurling case (B) is similar. For φ ∈ γ s (G), (3.5) follows in view of the continuous embedding L ∞ (G) ⊂ L 2 (G) with f L 2 ≤ f L ∞ since the measure is normalised. Now suppose that for φ ∈ C ∞ (G) we have (3.5). In view of (2.3), and using Lemma 3.1 with an integer k > n/2, we obtain 5 with constant C k depending only on G. Consequently we also have Using the inequalities in view of (3.6) and (3.5) we get k and A 2 independent of α, implying that φ ∈ γ s (G) and completing the proof.
The following proposition prepares the possibility to passing to the conditions formulated on the Fourier transform side. Proposition 3.4. We have φ ∈ γ s (G) if and only if there exist constants A > 0 and C > 0 such that holds for all k ∈ N 0 . Also, φ ∈ γ (s) (G) if and only if for every A > 0 there exists C A > 0 such that for all k ∈ N 0 we have 5 Note that this can be adopted to give a simple proof of the Sobolev embedding theorem.
Proof. We prove the Gevrey-Roumieu case (3.8) and indicate small additions to the argument for γ (s) (G). Thus, let φ ∈ γ s (G). Recall that by the definition there exist some A > 0, C > 0 such that for all multi-indices α we have We will use the fact that for the compact Lie group G the Laplace-Beltrami operator L G is given by L G = X 2 1 + X 2 2 + ... + X 2 n , where X i , i = 1, 2, . . . , n, is a set of left-invariant vector fields corresponding to a normalised basis of the Lie algebra of G. Then by the multinomial theorem 6 and using (3.7), with Y j ∈ {X 1 , . . . , X n }, 1 ≤ j ≤ |α|, we can estimate with A 1 = 2nA, implying (3.8). For the Gevrey-Beurling case γ (s) (G), we observe that we can obtain any A 1 > 0 in (3.9) by using A = A 1 2n in the Gevrey estimates for φ ∈ γ (s) (G).
We can now pass to the Fourier transform side. Proof. We will treat the case γ s since γ (s) is analogous. Using the fact that the Fourier transform is a bounded linear operator from L 1 (G) to l ∞ ( G), see (2.3), and using Proposition 3.4, we obtain We can now prove Theorem 2.3.
So for a given r > 0 there exists some x 0 = x 0 (r) > 0 such that We will be interested in large r, in fact we will later set r = |ξ| A , so we can assume that r is large. Consequently, in (3.18) and later, we can assume that x 0 is sufficiently large. Thus, we can take an even (sufficiently large) integer m 0 such that m 0 ≤ x 0 < m 0 + 2. Using the trivial inequalities It follows from this, (3.18) and (3.19), that Now we will use the following simple inequality, t N N ! ≤ e t for t > 0. Setting later m = 2k and a = B 2 , we estimate Using this inequality and (3.22) we obtain Consequently, arguing as in case (R) we get (3.21), i.e.
The same argument as in the case (R) now completes the proof.
"If" part. For a given A > 0 define B > 0 by solving A = 2 s B s and take C A big enough as in the case of (R), so that we get Therefore, φ ∈ γ (s) (G) by Proposition 3.4.
First we analyse α-duals of Gevrey spaces regarded as sequence spaces through their Fourier coefficients.
We can embed γ s (G) or γ (s) (G) in the sequence space Σ using the Fourier coefficients and Theorem 2.3. We denote the α-dual of such the sequence space γ s (G) (or γ (s) (G)) as with a similar definition for γ (s) (G). The proof of this lemma in (R) and (B) cases will be different. For (R) we can show this directly, and for (B) we employ the theory of echelon spaces by Köthe [6].
(B) For any B > 0 we consider the so-called echelon space, Now, by diagonal transform we have E B ∼ = l 1 and hence E B ∼ = l ∞ , and it is easy to check that E B is given by By Theorem 2.3 we know that φ ∈ γ (s) (G) if and only if φ(ξ) Using Köthe's theory relating echelon and co-echelon spaces [6,Ch. 30.8], we have, But this means that for some B > 0 we have Finally, we observe that this is equivalent to (4.1) if we use Lemma 3.2 and (3.2).
We now give the characterisation for α-duals. Proof. We prove the case (R) only since the proof of (B) is similar. First we deal with "If" part. Let v ∈ Σ be such that (4.2) holds for every B > 0. Let ϕ ∈ γ s (G). Then by Theorem 2.3 there exist some constants A > 0 and C > 0 such that φ(ξ) HS ≤ Ce −A ξ 1/s . Taking B = A/2 in (4.2) we get that Let v ∈ [γ s (G)] ∧ and let B > 0. Then by Lemma 4.1 we have that This implies that the exists a constant K B > 0 such that e −B ξ 1/s ||v(ξ)|| HS ≤ K B , yielding (4.2).
We now want to show that the Gevrey spaces are perfect in the sense of Köthe. We define the α−dual of [γ s (G)] ∧ as and similarly for [γ (s) (G)] ∧ . First, we prove the following lemma. Proof. We first show the Beurling case as it is more straightforward.
"If" part. Here we are given w ∈ Σ such that for every B > 0 the series (4.3) converges. Let us take any v ∈ [γ (s) (G)] ∧ . By Theorem 4.2 there exist B > 0 and by the assumption (4.3), which shows that w ∈ γ (s) (G) ∧ .
(R) For B > 0 we consider the echelon space By diagonal transform we have D B ∼ = l ∞ , and since l ∞ is a perfect sequence space, we have D B ∼ = l 1 , and it is given by By Theorem 4.2 we know that γ s (G) ∧ = B>0 D B , and hence γ s (G) This means that w ∈ γ s (G) ∧ if and only if there exists B > 0 such that we have Consequently, by Lemma 3.2 we get completing the proof of the "only if" part. Conversely, given (4.3) for some 2B > 0, we have Now we can show that the Gevrey spaces are perfect spaces (sometimes called Köthe spaces). Proof. We will show this for γ s (G) since the proof for γ (s) (G) is analogous. From the definition of [ γ s (G)] ∧ we have γ s (G) ⊆ [ γ s (G)] ∧ . We will prove the other direction, The series makes sense due to Lemma 4.3, and we have φ(ξ) HS = w ξ HS . Now since s ||w ξ || HS < ∞, which implies that for some C > 0 we have e B ξ 1/s ||w ξ || HS < C ⇒ || φ(ξ)|| HS ≤ Ce −B ξ 1/s . By Theorem 2.3 this implies φ ∈ γ s (G). Hence γ s (G) = [ γ s (G)] ∧ , i.e. γ s (G) is a perfect space.
We can take the Laplace-Beltrami operator in Definition 5.1 because of the equivalence of norms given by Proposition 3.4.
We have the following theorem showing that topological and α-duals of Gevrey spaces coincide.
(B) This case is similar but we give the proof for completeness. "If" part. Let v ∈ γ (s) (G) ∧ and for any φ ∈ γ (s) (G) define v(φ) by (5.1). By a similar argument to the case (R), it is a well-defined linear functional on γ (s) (G). To check the continuity, suppose φ j → φ in γ (s) (G), that is, for every A > 0 we have for a sequence C j → 0 as j → ∞, for every A > 0. From the proof of Theorem 2.3 it follows that for every B > 0 we have φ j (ξ) − φ(ξ) HS ≤ K j e −B ξ 1/s , where K j → 0 as j → ∞. Hence we can estimate as j → ∞ since K j → 0 as j → ∞, and where we now take B > 0 to be such that [ξ]∈ G d ξ e −B ξ 1/s v ξ HS < ∞ by Lemma 4 .1 and (3.2). Therefore, we have v ∈ γ ′ (s) (G).