Hormander class of pseudo-differential operators on compact Lie groups and global hypoellipticity

In this paper we give several global characterisations of the Hormander class of pseudo-differential operators on compact Lie groups. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.

1. Introduction 1.1. For a compact manifold M we denote by Ψ m (M) the set of Hörmander's pseudodifferential operators on M, i.e., the class of operators which in all local coordinate charts give operators in Ψ m (R n ). Operators in Ψ m (R n ) are characterised by the symbols satisfying |∂ α ξ ∂ β x a(x, ξ)| ≤ C(1 + |ξ|) m−|α| for all multi-indices α, β and all x, ξ ∈ R n . An operator in Ψ m (M) is called elliptic if all of its localisations are locally elliptic. The principal symbol of a pseudo-differential operator on M can be invariantly defined as function on the cotangent bundle T * M, but it is not possible to control lower order terms in the same way. If one fixes a connection, however, it is possible to make sense of a full symbol, see Widom [18].
However, on a Lie group G it is a very natural idea to use the Fourier analysis and the global Fourier transform on the group to analyse the behaviour of operators which are originally defined by their localisations. This allows one to make a full use of the representation theory and of many results available from the harmonic analysis on groups. In [9] and [7], the first two authors carried out a comprehensive non-commutative analysis of an analogue of the Kohn-Nirenberg quantisation on R n , in terms of the representation theory of the group. This yields a full matrix-valued symbol defined on the product G × G, which can be viewed as a non-commutative version of the phase space. The calculus and other properties of this quantisation resemble those well-known in the local theory on R n . However, due to the noncommutativity of the group in general, the full symbol is matrix-valued, with the dimension of the matrix symbol a(x, ξ) at x ∈ G and [ξ] ∈ G equal to the dimension of the representation ξ. This construction resembles that of Taylor [15] but is carried out entirely in terms of the group G without referring to its Lie algebra and the Euclidean pseudo-differential classes there. In [2] (see also [16]), pseudo-differential operators have been characterised by their commutator properties with the vector fields. In [7], we gave a different version of this characterisation relying on the commutator properties in Sobolev spaces, which led in [9] (see also [7]) to characterisations of symbols of operators in Ψ m (G). However, the results there still rely on commutator properties of operators and some of them are taken as assumptions. One aim of this paper is to eliminate such assumptions from the characterisation, and to give its different versions relying on different choices of difference operators on G. This will be given in Theorem 2.2.
In Theorem 2.6 we give another simple description of Hörmander's classes on the group SU(2) and on the 3-sphere S 3 . This is possible due to the explicit analysis carried out in [9] and the explicit knowledge of the representation of SU (2). We note that this approach works globally on the whole sphere, e.g. compared to the analysis of Sherman [11] working only on the hemisphere (see also [12]). Using the classification of representations in some cases (see, e.g., [10]) it is possible to draw conclusions of this type on other groups as well. However, this falls outside the scope of this paper.
We give two applications of the characterisation provided by Theorem 2.2. First, we characterise elliptic operators in Ψ m (G) by their matrix-valued symbols. Similar to the toroidal case (see [1]) this can be applied further to spectral problems. Second, we give a sufficient condition for the global hypoellipticity of pseudo-differential operators. Since the hypoellipticity depends on the lower order terms of an operator, a knowledge of the full symbol becomes crucial. Here, we note that while classes of hypoelliptic symbols can be invariantly defined on manifolds by localisation (see, e.g., Shubin [13]), the lower order terms of symbols can not. From this point of view conditions of Theorem 5.1 appear natural as they refer to the full symbol defined globally on G × G.
The global hypoellipticity of vector fields and second order differential operators on the torus has been extensively studied in the literature, see e.g. [3], [4], and references therein. In this paper we study the global hypoellipticity problem for pseudo-differential operators on general compact Lie groups, and discuss a number of explicit examples on the group S 3 ∼ = SU (2). We say that an operator A is globally hypoelliptic on G if Au = f and f ∈ C ∞ (G) imply u ∈ C ∞ (G). Thus, we give examples of first and second order differential operators which are not locally hypoelliptic and are not covered by Hörmander's sum of the squares theorem but which can be seen to be globally hypoelliptic by the techniques described in this paper. For example, if X is a left-invariant vector field on S 3 and ∂ X is the derivative with respect to X, then the operator ∂ X + c is globally hypoelliptic if and only if ic ∈ 1 2 Z. We note that this operator is not locally hypoelliptic, but the knowledge of the global symbol allows us to draw conclusions about its global properties. It is interesting to observe that the number theory plays a role in the global properties, in some way similar to the appearance of the Liouville numbers in the global properties of vector fields on the torus. For example, the failure of the D'Alembertian D 2 3 − D 2 1 − D 2 2 to be globally hypoelliptic can be related to properties of the so-called triangular numbers, and an explicit counterexample to the global hypoellipticity can be consequently constructed based on these numbers. We also look at the cases of the sub-Laplacian and other operators, which are covered by Hörmander's theorem, but for which we construct an explicit inverse which turns out to be a pseudo-differential operator with the global matrix-valued symbol of type ( 1 2 , 0). Such examples can also be extended to variable coefficient versions.
Another application of Theorem 2.2 to obtain the sharp Gårding inequality on Lie groups will appear elsewhere.
1.2. We will now introduce some notation. Let throughout this paper G be a compact Lie group of (real) dimension n = dim G with the unit element e, and denote by G the set of all equivalence classes of continuous irreducible unitary representations of G. For necessary details on Lie groups and their representations we refer to [7], but recall some basic facts for the convenience of the reader and in order to fix the notation. Each [ξ] ∈ G corresponds to a homomorphism ξ : G → U(d ξ ) with ξ(xy) = ξ(x)ξ(y) satisfying the irreducibility condition C d ξ = span{ξ(x)v : x ∈ G} for any given v ∈ C d ξ \ {0}. The number d ξ is referred to as the dimension of the representation ξ.
We always understand the group G as a manifold endowed with the (normalised) bi-invariant Riemannian structure. Of major interest for us is the following version of the Peter-Weyl theorem defining the group Fourier transform F : L 2 (G) → ℓ 2 ( G).
Theorem 1.1. The space L 2 (G) decomposes as the orthogonal direct sum of biinvariant subspaces parameterised by G, the decomposition given by the Fourier series Furthermore, the following Parseval identity is valid, The notion of Fourier series extends naturally to C ∞ (G) and to the space of distributions D ′ (G) with convergence in the respective topologies. Any operator A on G mapping C ∞ (G) to D ′ (G) gives rise to a matrix-valued full symbol holds as D ′ -convergent series. For such operators we will also write A = Op(σ A ). For a rather comprehensive treatment of this quantisation we refer to [7] and [9]. We denote the right-convolution kernel of A by R A , so that where dy is the normalised Haar measure on G. The symbol σ A and the rightconvolution kernel R A are related by σ A (x, ξ) = G R A (x, y) ξ * (y) dy. The first aim of this paper is to characterise Hörmander's class of pseudo-differential operators Ψ m (G) by the behaviour of their full symbols, with applications to the ellipticity and global hypoellipticity of operators.
The plan of this note is as follows. In the next section we give several equivalent symbolic characterisations of pseudo-differential operators by their symbols. These characterisations are based on difference operators acting on sequences of matrices G ∋ [ξ] → σ(ξ) ∈ C d ξ ×d ξ of varying dimension. We also give an application to pseudo-differential operators on the group SU (2). Section 3 contains the proof and some useful lemmata on difference operators. Section 4 contains the criteria for the ellipticity of operators in terms of their full matrix-valued symbols as well as a finite version of the Leibniz formula for difference operators associated to the group representations. Finally, Section 5 contains criteria for the global hypoellipticity of pseudo-differential operators and a number of examples.

Characterisations of Hörmander's class
The following statement is based on difference operators. They are defined as follows. We say that Q ξ is a difference operator of order k if it is given by for a function q = q Q ∈ C ∞ (G) vanishing of order k at the identity e ∈ G, i.e., (P x q Q )(e) = 0 for all left-invariant differential operators P x ∈ Diff k−1 (G) of order k − 1. We denote the set of all difference operators of order k as diff k ( G). Definition 2.1. A collection of m ≥ n first order difference operators △ 1 , . . . , △ m ∈ diff 1 ( G) is called admissible, if the corresponding functions q 1 , . . . , q m ∈ C ∞ (G) satisfy dq j (e) = 0, j = 1, . . . , m, and rank(dq 1 (e), . . . , dq m (e)) = n. It follows, in particular, that e is an isolated common zero of the family {q j } m j=1 . An admissible collection is called strongly admissible if j {x ∈ G : q j (x) = 0} = {e}.
For a given admissible selection of difference operators on a compact Lie group G we use multi-index notation △ α ξ = △ α 1 1 · · · △ αm m and q α (x) = q 1 (x) α 1 · · · q m (x) αm . Furthermore, there exist corresponding differential operators ∂ (α) x ∈ Diff |α| (G) such that Taylor's formula (2) f holds true for any smooth function f ∈ C ∞ (G) and with h(x) the geodesic distance from x to the identity element e. An explicit construction of operators ∂ (α) x in terms of q α (x) can be found in [7,Section 10.6]. In addition to these differential operators ∂ (α) x ∈ Diff |α| (G) we introduce operators ∂ α x as follows. Let ∂ x j ∈ Diff 1 (G), j = 1, . . . , n, be a collection of left invariant first order differential operators corresponding to some linearly independent family of the left-invariant vector fields on G. We denote ∂ α x = ∂ α 1 x 1 · · · ∂ αn xn . We note that in most estimates we can freely replace operators ∂ (α) x by ∂ α x and the other way around since they can be clearly expressed in terms of each other.
After fixing the notion of difference operators, we have to specify orders. Each of the bi-invariant subspaces H ξ of Theorem 1.1 is an eigenspace of the Laplacian L on G with the corresponding eigenvalue −λ 2 ξ . Based on these eigenvalues we define ξ = (1 + λ 2 ξ ) 1/2 . In particular we recover the familiar characterisation We are now in a position to formulate the main result of this paper.
Then the following statements are equivalent: The set of symbols σ A satisfying either of conditions (B)-(D) will be denoted by S m (G).
Among other things, this theorem removes the assumption of the conjugation invariance from the list of conditions used in [7].
On the other hand, as a corollary of Theorem 2.2 this can be expressed in terms of difference operators, showing that the difference conditions are conjugation invariant. In fact, to draw such a conclusion, we even do not need to know that conditions (B)-(D) characterise the class Ψ m (G): This statement follows immediately from the condition (B) if we observe that the difference operators applied to the symbol σ Au just lead to a different set of difference operators in the symbolic inequalities. Alternatively, one can observe that the change of variables by φ amounts to the change of the basis in the representation spaces of G, thus leaving the condition (B) invariant again.
Example 2.4. On the torus T n = R n /Z n , the family of functions q j (x) = e 2πix j − 1, j = 1, . . . , n, gives rise to a strongly admissible collection of difference operators with ξ ∈ T n ≃ Z n , where e j is the j th unit vector in Z n . As an immediate consequence of Theorem 2.2 we recover the fact that the class Ψ m (T n ) can be characterised by the difference conditions on their toroidal symbols: for all x ∈ T n and ξ ∈ Z n . We refer to [8] for details of the corresponding toroidal quantisation of operators on T n . See also [1].
In Theorem 2.2 we characterise Hörmander operators entirely by symbol assumptions based on a set of admissible difference operators. There does not seem to be a canonic choice for difference operators, for different applications different selections of them seem to be most appropriate. We will comment on some of them.
(1) Simplicity. Difference operators have a simple structure, if the corresponding functions are just matrix coefficients of irreducible representations. Then application of difference operators at fixed ξ involves only matrix entries from finitely many (neighbouring) representations. Example 2.5. We will conclude this section with an application for the particular group SU (2). Certain explicit calculations on SU (2) have been done in [9] and the background on the necessary representation theory of SU(2) can be also found in [7]. Representations on SU(2) are parametrised by half-integers ℓ ∈ 1 2 N (so-called quantum numbers) and are of dimension d ℓ = 2ℓ+1. Thus, irreducible representations of SU(2) are given by matrices t ℓ (x) ∈ C (2ℓ+1)×(2ℓ+1) , ℓ ∈ 1 2 N, after some choice of the basis in the representation spaces. One admissible selection of difference operators corresponds to functions q + , q 0 , q 0 ∈ C ∞ (SU(2)) defined by Our analysis on SU(2) is in fact equivalent to the corresponding analysis on the 3sphere S 3 ∼ = SU(2), the isomorphism given by the identification of SU(2) with S 3 ⊂ H in the quaternion space H, with the quaternionic product on S 3 corresponding to the matrix multiplication on SU (2). Writing explicitly an isomorphism Φ : S 3 → SU (2) as the identity matrix e ∈ SU(2) corresponds to the vector 1 in the basis decomposition We note that the family △ + := △ q + , △ − := △ q − and △ 0 := △ q 0 is not strongly admissible because in addition to e ∈ SU(2) (or to 1 ∈ H) they have another common zero at −e (or at −1).
Following [7] and [9], we simplify the notation on SU(2) and S 3 by writing σ A (x, ℓ) for σ A (x, t ℓ ), ℓ ∈ 1 2 N, and we refer to these works for the explicit formulae for the difference operators In [9] we proved that the operator and the Hilbert-Schmidt norms are uniformly equivalent for symbols of pseudo-differential operators from Ψ m (SU (2) (2)) to D ′ (SU (2)).
Moreover, in these cases we also have the rapid off-diagonal decay property of symbols, namely, we have (2), for every N ≥ 0, all ℓ ∈ 1 2 N, every multi-indices α, β ∈ N 3 0 , and for all matrix column/row numbers i, j.
We note that Theorem 2.6 holds in exactly the same way if we replace SU(2) by S 3 . Moreover, according to Theorem 1.1 the statement of Theorem 2.6 holds without the kernel condition sing supp R A (x, ·) ⊂ {e} provided that instead of the admissible family △ + , △ − , △ 0 we take a strongly admissible family of difference operators in diff 1 ( SU(2)).

Proof of Theorem 2.2
3.1. (A) =⇒ (C). By [7, Thm. 10.9.6] (and in the notation used there) we know that condition (A) is equivalent to σ A ∈ Σ m (G) = k Σ m k (G), where Σ m 0 (G) already corresponds to assumption (C). There is nothing to prove.

(C) =⇒ (D). Evident.
Also the following partial converse is true. If (D) is satisfied, then the symbol estimates imply for the corresponding right-convolution kernel R A (x, ·) of A that sing supp R A (x, ·) ⊆ n j=1 {y ∈ G : q j (y) = 0} = {e}.

(D) =⇒ (B)
. For a given strongly admissible selection of first order differences, we can apply Taylor's formula to the function q Q corresponding to the difference operator Q ∈ diff ℓ ( G). Hence, we obtain Lemma 3.1. Let Q ∈ diff ℓ ( G) be an arbitrary difference operator of order ℓ. Then for suitable constants c α ∈ C and a difference operator Q N ∈ diff N ( G).
Consequently, (D) together with (B) for differences of order larger than N implies (B). More precisely, with the notation σ(x, ξ) C ∞ for 'any' C ∞ -seminorm of the form sup x P x σ(x, ·) op and Assumption (D) corresponds to the first intersection S m = ℓ S m ℓ . It remains to show that S m ⊂ R m ∞ , which is done in the sequel. Let Q ξ ∈ diff ∞ ( G), i.e. q Q ∈ C ∞ (G) is vanishing to infinite order at e. Let further p(y) = n j=1 q 2 j (y). Then p(y) = 0 for y = e and p vanishes of the second order at y = e. Hence, q Q (y)/|p(y)| N ∈ C ∞ (G) for arbitrary N and, therefore, can be estimated in terms of symbol bounds for σ A ∈ S m . Indeed, by Lemma 3.2 it follows that |p(y)  The nuclearity of C ∞ (G) allows one to split the first statement into the corresponding results for purely x-and purely ξ-dependent symbols; both situations are evident. As for the second statement, we use the symbolic calculus established in [7,Thm. 10.7.9] for classes of symbols including, in particular, the class S m , which writes the symbol of the composition as the asymptotic sum The first term σ X σ A − σ A σ X is on the level of right-convolution kernels R A (x, ·) given by where X R is the right-invariant vector field tangent to X at e (and the index y describes the action as a differential operator with respect to the y-variable). Apparently, X y − X R y is a smooth vector field on G and can thus be written as n j=1 q j (y)∂ R j with q j (e) = 0 and ∂ R j suitable right-invariant derivatives. Hence, on the symbolic level σ X σ A − σ A σ X = n j=1 △ j (σ A σ ∂ j ) with △ j ∈ diff 1 ( G), defined by △ j f (ξ) = q j f (ξ). Now the statement follows from Lemma 3.4 by the aid of the asymptotic Leibniz rule [7, Thm. 10.7.12]: Lemma 3.3. Let △ 1 , . . . , △ n ∈ diff 1 ( G) be a set of admissible differences. For any for any (fixed) set of admissible differences.

Ellipticity and Leibniz formula
As an application of Theorem 2.2 we will give a characterisation of the elliptic operators in Ψ m (G) in terms of their global symbols.
Theorem 4.1. An operator A ∈ Ψ m (G) is elliptic if and only if its matrix valued symbol σ A (x, ξ) is invertible for all but finitely many [ξ] ∈ G, and for all such ξ satisfies Thus, both statements are equivalent to the existence of B ∈ Ψ −m (G) such that I − BA and I − AB are smoothing smoothing. For the ellipticity condition (3) on the general matrix level it is not enough to assume that | det σ A (x, ξ)| 1/d ξ ≥ C ξ m due to the in general growing dimension of the matrices. However, if we assume that the smallest singular value of the matrix σ A (x, ξ) is greater or equal than C ξ m uniformly in x and (all but finitely many) ξ, then condition (3) follows.
Let the collection q 1 , . . . , q n give an admissible collection of difference operators and let ∂ (γ) x be the corresponding family of differential operators as in the Taylor expansion formula (2). As an immediate corollary of Theorem 4.1 and [7, Thm. 10.9.10] we get for large ξ, and the symbols σ B k are defined recursively by First, we give some preliminary results. In Lemma 3.3 we gave an asymptotic Leibniz formula but here we will present its finite version. Given a continuous unitary matrix representation ξ = ξ ij 1≤i,j≤ℓ : G → C ℓ×ℓ , ℓ = d ξ , let q(x) = ξ(x) − I (i.e. q ij = ξ ij − δ ij with Kronecker's deltas δ ij ), and define D ij f (ξ) := q ij f (ξ).
In the previous notation, we can write D ij = ∆ q ij . However, here, to emphasize that q ij 's correspond to the same representation, and to distinguish with difference operators with a single subindex, we will use the notation D instead. For a multi-index γ ∈ N ℓ 2 0 , we will write |γ| = ℓ i,j=1 |γ ij |, and for higher order difference operators we write D γ = D γ 11 11 D γ 12 12 · · · D γ ℓ,ℓ−1 ℓ,ℓ−1 D γ ℓℓ ℓℓ . For simplicity, we will abbreviate writing a = a(ξ) and b = b(ξ). Let k ∈ N 0 and let α, β ∈ N k be such that 1 ≤ α j , β j ≤ ℓ for all j ∈ {1, · · · , k}. Let us define a grand where we may note that the first order differences D α j β j commute with each other. We may compute with "matrices" − → D k a = − → D k αβ a α,β∈{1,··· ,ℓ} k using the natural operations, e.g.
If now f, g ∈ C ∞ (G), we consequently get which on the Fourier transform side gives (5). For k ≥ 2, let us argue inductively. Taking differences is linear, so the main point is to analyse the application of a first order difference D α 1 β 1 to the term where α = (α 1 , α 2 ) and β = (β 1 , β 2 ). Proceeding by induction, we notice that applying a first order difference to the term ( − → D 1≤i,j≤d ξ is strongly admissible. Moreover, this family has a finite subfamily associated to finitely many representations which is still strongly admissible. Proof. We observe that there exists a homomorphic embedding of G into U(N) for a large enough N and this embedding itself is a representation of G of dimension N. Decomposing this representation into irreducible components gives a finite collection of representations. The common zero set of the corresponding family {q ij } is e which means that it is strongly admissible.
We note that on the group SU(2) or on S 3 , by this argument, or by the discussion in Section 2 the four function q ij corresponding to the representation [t ℓ ] ∈ SU(2) with ℓ = 1 2 of dimension two, d ℓ = 2, already give a strongly admissible collection of four difference operators. We can now apply Proposition 4.3 to the question of inverting the symbols on G × G. We formulate this for symbol classes with (ρ, δ) behaviour as this will be also used in Section 5.
Lemma 4.5. Let m ≥ m 0 , 1 ≥ ρ > δ ≥ 0 and let us fix difference operators {D ij } 1≤i,j≤d ξ 0 corresponding to some representation ξ 0 ∈ G. Let the matrix symbol a = a(x, ξ) satisfy D γ ∂ β x a(x, ξ) op ≤ C βγ ξ m−ρ|γ|+δ|β| for all multi-indices β, γ. Assume also that a(x, ξ) is invertible for all x ∈ G and [ξ] ∈ G, and satisfies for all x ∈ G and [ξ] ∈ G, and if m 0 = m in addition that for all x ∈ G and [ξ] ∈ G. Then the matrix symbol a −1 defined by a −1 (x, ξ) = a(x, ξ) −1 satisfies Proof. Let us denote b(x, ξ) := a(x, ξ) −1 and estimate ∂ β x b first. Suppose we have proved that (11) ∂ β x b(x, ξ) op ≤ C β ξ −m 0 +δ|β| whenever |β| ≤ k. We proceed by induction. Let us study the order k + 1 cases ∂ x j ∂ β x b with |β| = k. Since a(x, ξ)b(x, ξ) = I, by the usual Leibniz formula we get From (7), (8), (9) and (11) we obtain the desired estimate The argument is more complicated than that for the ∂ xderivatives because the Leibniz formula in Proposition 4.3 has more terms. Indeed, the Leibniz formula in Proposition 4.3 applied to a(x, ξ)b(x, ξ) = I gives Writing these equations for all 1 ≤ i, j ≤ d ξ 0 gives a linear system on {D ij b} ij with coefficients of the form I + bD kl a for suitable sets of indices k, l. Since bD kl a op ≤ C ξ −ρ , we can solve it for {D ij b} ij for large ξ . The inverse of this system is bounded, and Suppose now we have proved that We proceed by induction. Let us study the order k + 1 cases where by Proposition 4.3 the right hand side is a sum of terms of the form (D ε a) (D η b) , where |ε|, |η| ≤ k + 1 ≤ |ε| + |η|. Especially, we look at the terms D ε a D η b with |η| = k + 1, since the other terms can be estimated by ξ −ρ(k+1) due to (12) and (7). Writing the linear system of equations on D ij D η ′ b produced by the Leibniz formula for all i, j and all |η ′ | = k, we see that the matrix coefficients in front of matrices D ij D η ′ b are sums of the terms of the form D ε a. The main term in each coefficient is a(x, ξ) corresponding to ε = 0, while the other terms corresponding to ε = 0 can be estimated by ξ m−ρ in view of (12) and (7). It follows that we can solve D ij D η ′ b from this system and the solution matrix can be estimated by ξ −m 0 in view of (8) since its main term is a(x, ξ). Therefore, all terms of the type D ij D η ′ can be estimated by which is what was required to prove. By combining these two arguments we obtain estimate (10).
Proof of Theorem 4.1. Assume first that A ∈ Ψ m (G) is elliptic. Then it has a parametrix B ∈ Ψ −m (G) such that AB − I, BA − I ∈ Ψ −∞ (G). By the composition formulae for the matrix valued symbols in [7, Thm. 10.7.9] and Theorem 2.2 it follows that Op(σ A σ B ) − I ∈ Ψ −1 (G). Consequently, the product σ A (x, ξ)σ B (x, ξ) is an invertible matrix for all sufficiently large ξ and (σ A σ B ) −1 op ≤ C. It follows then also that σ A (x, ξ) is an invertible matrix for all sufficiently large ξ . From this and the equality σ −1 We can disregard representations ξ with bounded ξ because they correspond to a smoothing operator. From Theorem 2.2 applied with the difference operators from Lemma 4.4, we get that Op(σ −1 Lemma 4.5 with m 0 = m, ρ = 1 and δ = 0. Since Op(σ −1 A ) ∈ Ψ −m (G), Theorem 10.9.10 in [7] implies that A has a parametrix (given by formula (4)). This implies that A is elliptic. The proof is complete.

Global hypoellipticity
We now turn to the analysis of hypoelliptic operators. We recall that an operator A is globally hypoelliptic on G if Au = f and f ∈ C ∞ (G) imply u ∈ C ∞ (G). We use the notation S m ρ,δ (G) for the class of symbols for which the corresponding (ρ, δ)versions of symbol estimates are satisfied, namely, σ A ∈ S m ρ,δ (G) if for a strongly admissible selection △ 1 , . . . , △ n ∈ diff 1 ( G) we have is valid for all 1 ≥ ρ > δ ≥ 0. By Op(S m ρ,δ (G)) we denote the class of all operators A of the form (1) with symbols σ A ∈ S m ρ,δ (G). Such operators can be readily seen to be continuous on C ∞ (G). In the previous notation we have S m (G) = S m 1,0 (G), and Ψ m (G) = Op(S m 1,0 (G)) by Theorem 2.2. 5.1. The knowledge of the full symbols allows us to establish an analogue of the well-known hypoellipticity result of Hörmander [5] on R n , requiring conditions on lower order terms of the symbol. The following theorem is a matrix-valued symbol criterion for (local) hypoellipticity.
Theorem 5.1. Let m ≥ m 0 and 1 ≥ ρ > δ ≥ 0. Let A ∈ Op(S m ρ,δ (G)) be a pseudo-differential operator with the matrix-valued symbol σ A (x, ξ) ∈ S m ρ,δ (G) which is invertible for all but finitely many [ξ] ∈ G, and for all such ξ satisfies Assume also that for a strongly admissible collection of difference operators we have (14) σ Proof. First of all, we can assume at any stage of the proof that (13) and (14) hold for all [ξ] ∈ G since we can always modify symbols for small ξ which amounts to adding a smoothing operator. Then we observe that if we apply Theorem 2.2 with difference operators D from Lemma 4.4, it follows from Lemma 4.5 that σ −1 A ∈ S −m 0 ρ,δ (G). Let us show next that σ −1 . Differentiating the equality . From this and (14) it follows that ∂ For differences, the Leibniz formula in Proposition 4.3 applied to (15) gives Writing these equations for all i, j gives a linear system on (14) and Theorem 2.2, we can solve it for matrices j. An induction argument similar to the one in the proof of Lemma 4.5 shows that σ −1 One can readily check that B is a parametrix of A (this is a special case of [7, Exercise 10.9.12]). We claim that σ B N ∈ S −m 0 −N ρ,δ (G). We already know that Thus, B ∈ Op(S −m 0 ρ,δ (G)) is a parametrix for A. Finally, the pseudo-locality of the operator A (which can be proved directly from the symbolic calculus) implies that sing supp Au ⊂ sing supp u. Writing u = B(Au)− Ru with R ∈ Ψ −∞ (G), we get that sing supp u ⊂ sing supp(Au) sing supp(Ru), so that sing supp Au = sing supp u because Ru ∈ C ∞ (G).
To obtain global hypoellipiticity, it is sufficient to show that an operator A ∈ Ψ(G) has a parametrix B satisfying subelliptic estimates Bf H s f H s+m for some constant m independent of s ∈ R.

5.2.
Examples. We will conclude with a collection of examples on the group SU (2). These examples can be also viewed as examples on the 3-sphere S 3 , see Example 2.5.
The statement is sharp: the spectrum of D 3 consists of all imaginary half-integers and all eigenspaces are infinite-dimensional (stemming from the fact that each such imaginary half-integer hits infinitely many representations for which −ℓ ≤ ic ≤ ℓ and ic + ℓ ∈ Z). In particular, eigenfunctions can be irregular, e.g., distributions which do not belong to certain negative order Sobolev spaces.
Since the only non-zero triangular number which is also a cube is 1 (see, e.g., [17]) we see that the operator 2iD 3 3 − L is globally hypoelliptic.