Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} Fourier multipliers on compact Lie groups

In this paper we prove Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} Fourier multiplier theorems for invariant and also non-invariant operators on compact Lie groups in the spirit of the well-known Hörmander–Mikhlin theorem on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^n$$\end{document} and its variants on tori Tn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb T^n$$\end{document}. We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.

of view Theorem 2.1 provides a Mikhlin type multiplier theorem which reduces the assumptions on the symbol ensuring the L p -boundedness of the operator. In Theorem 3.5 we give a refinement of this describing precisely the difference operators that can be used for making assumptions on the symbol. For example, if G is semi-simple, only those associated to the root system suffice, which appears natural in the context.
We give several applications of the obtained result. Thus, in Corollary 5.1 we give a criterion for the L p -boundedness for a class of operators with symbols in the class S 0 ρ (G) of type ρ ∈ [0, 1]. Such operators appear e.g. with ρ = 1 2 as parametrices for the sub-Laplacian or for the "heat" operator, see Example 2.6, or with ρ = 0 for inverses of operators X + c, with X ∈ g and c ∈ C, see Corollary 2.7 on general G and Example 2.8 on SU(2) and S 3 . We note that although operators X + c are not locally hypoelliptic, we still get a-priori L p -estimates for them as a consequence of our result.
We illustrate Theorem 3.5 in Remark 2.9 in the special case of the tori T n . In different versions of multiplier theorems on T n , one usually imposes conditions on differences of order [ n 2 ] + 1 applied to the symbol. In Remark 2.9 we show that e.g. on T 2 or T 3 , it is enough to make an assumption on only one second order difference of a special form applied to the symbol.
In Theorem 5.2 we give an application to the L p -estimates for general operators from C ∞ (G) to D ′ (G), not necessarily invariant. This result is also a relaxation of the symbolic assumptions on the operator compared to those in the pseudo-differential classes.
The paper is organised as follows. In Section 2 we formulate the results with several application and give a number of examples. In Section 3 we introduce the necessary techniques and prove the results. In Section 4 we briefly discuss central multipliers and the meaning of the difference operator △ * in this case. Finally, in Section 5 we prove corollaries for operators with symbols in S 0 ρ (G) and for non-invariant operators.

Multiplier theorems on compact Lie groups
Let G be a compact Lie group with identity 1 and the unitary dual G. The following considerations are based on the group Fourier transform defined in terms of equivalence classes [ξ] of irreducible unitary representations ξ : Peter-Weyl theorem on G implies in particular that this pair of transforms is inverse to each other and that the Plancherel identity holds true for all φ ∈ L 2 (G). Here φ(ξ) 2 HS = trace( φ(ξ) φ(ξ) * ) denotes the Hilbert-Schmidt norm of matrices. The Fourier inversion statement (2.1) is valid for all φ ∈ D ′ (G) and the Fourier series converges in C ∞ (G) provided φ is smooth. It is further convenient to denote ξ = max{1, λ ξ }, where λ 2 ξ is the eigenvalue of the Casimir element (positive Laplace-Beltrami operator) acting on the matrix coefficients associated to the representation ξ. The Sobolev spaces can be characterised by Fourier coefficients as where ℓ 2 ( G) is defined as the space of matrix-valued sequences such that the sum on the right-hand side of (2.2) is finite.
For an arbitrary continuous linear operator A : C ∞ (G) → D ′ (G) we denote its Schwartz kernel as K A ∈ D ′ (G × G) and by a change of variables we associate the right-convolution kernel R A (g 1 , g 2 ) = K A (g 1 , g −1 1 g 2 ). Thus, at least formally, we write Following the analysis in [12] we denote the partial Fourier transform of the rightconvolution kernel with respect to the second variable as symbol of the operator, which is a distribution taking values in the set of moderate sequences of matrices Here we are concerned with left-invariant operators, which means that A • T g = T g • A for the left-translation T g : φ → φ(g −1 ·). This implies that the kernel K A satisfies the invariance K A (g 1 , g 2 ) = K A (g −1 g 1 , g −1 g 2 ) for all g ∈ G and hence R A is independent of the first argument. In consequence, also the symbol is independent of the first argument and we will write σ A (ξ) for it. In combination with Fourier inversion formula (2.1) this means that the operator A can be written as It follows 1 that is independent of g. We refer to operators of this form as non-commutative Fourier multipliers. The Plancherel identity (2.2) implies that the operator A is bounded on and · op is the operator norm on the unitary space C d ξ . We now define difference operators Q ∈ diff ℓ ( G) acting on sequences σ ∈ Σ( G) in terms of corresponding functions q ∈ C ∞ (G), which vanish to (at least) ℓth order in the identity element 1 ∈ G, and their interrelation with the group Fourier transform Note, that σ ∈ Σ( G) implies F −1 σ ∈ D ′ (G) and therefore the multiplication with a smooth function is well-defined. The main idea of introducing such operators is that applying differences to symbols of Calderon-Zygmund operators brings an improvement in the behaviour of Op(Qσ) since we multiply the integral kernel of Op(σ) by a function vanishing on its singular set. Different collections of difference operators have been explored in [14] in the pseudo-differential setting. Difference operators of particular interest arise from matrix-coefficients of representations. For a fixed irreducible representation ξ 0 we define the (matrix-valued) difference ξ 0 D = ( ξ 0 D ij ) i,j=1,...,d ξ 0 corresponding to the matrix elements of ξ 0 (g) − I, q ij (g) = ξ 0 (g) ij − δ ij , δ ij the Kronecker delta. If the representation is fixed, we omit the index ξ 0 . For a sequence of difference operators of this type, Among other things, it follows from [14] that an invariant operator A belongs to the usual Hörmander class of pseudo-differential operators Ψ 0 (G) defined by localisations if and only if its matrix symbol satisfies for all multi-indices α and for all [ξ] ∈ G. From this point of view the following condition (2.8) is a natural relaxation from the L p -boundedness of zero order pseudodifferential operators to a multiplier theorem.
Theorem 2.1. Denote by κ be the smallest even integer larger than 1 for all multi-indices α with |α| ≤ κ, and for all [ξ] ∈ G. Then the operator A is of weak type (1, 1) and L p -bounded for all 1 < p < ∞.
Remark 2.2. a) The assumptions given in the theorem can be relaxed. For the top order difference we need only one particular difference operator. Moreover, for the lower order difference operators we only need differences associated to the root system if G is semi-simple, and to an extended root system for a general compact Lie group. Such a refinement will be given in Theorem 3.5 once we introduced the necessary notation. b) Additional symmetry conditions for the operator imply simplifications. Later on we will show how the assumptions can be weakened for central multipliers. c) We have to round up the number of difference conditions to even integers. This seems to be for purely technical reasons, but was already observed similarly in [17] for central multipliers.
d) The conditions are needed for the week type (1, 1) property. Interpolation allows to reduce assumptions on the number of differences for L p -boundedness.
Before proceeding to the proof of the theorem, we will mention some applications. As first example let us consider the known case of the Riesz transform.
Example 2.3. Let us consider the partial Riesz transform R Z = (−∆) −1/2 • Z associated to a left-invariant vector-field Z ∈ g on a Lie group G. For simplicity we assume that Z is normalised with respect to the Killing form on g. The Riesz transform is a left-invariant operator acting on L 2 (G) with symbol σ R Z (ξ) = (λ ξ ) − 1 2 σ Z (ξ), σ Z (ξ) = (Zξ)(1) the symbol of the left-invariant vector field, and by definition of the Laplacian as sum of squares σ R Z (ξ) op ≤ 1. Note here, that λ ξ = 0 implies that ξ = 0 is the trivial representation and therefore also σ Z (ξ) = 0 as vector fields annihilate constants. It follows from Corollary 4.7 that this operator extends to a bounded operator on all L p (G), 1 < p < ∞ and is of weak type (1, 1), recovering the well-known result in [16].
Remark 2.4. In [16, p. 58], E.M. Stein asked whether the Riesz transform R Z as well as the Riesz potentials (−∆) iγ (γ real) are pseudo-differential operators on G. This is in fact true on all closed Riemannian manifolds. Indeed, if p 0 denotes the projection to the zero eigenspace of −∆, then we have the identity (−∆) z = (−∆ + p 0 ) z − p 0 for all complex z. The operator (−∆ + p 0 ) z is pseudo-differential for Re z < −1 by [15] and p 0 is smoothing, implying that (−∆) z are pseudo-differential of order Re z/2. By calculus this extends to all z ∈ C. In particular, the L p boundedness in Example 2.3 also follows.
for some ρ ∈ [0, 1] and all α. Then A defines a bounded operator mapping Example 2.6. The previous example applies in particular to the parametrices constructed in [14]. Thus, if we consider the sub-Laplacian L s = D 2 1 + D 2 2 on S 3 it has a parametrix from opS −1 1 2 (S 3 ) and therefore L s u ∈ L p (S 3 ) implies regularity for u. More precisely, the sub-elliptic estimate holds true for all 1 < p < ∞. Similarly, the "heat" operator H = D 3 − D 2 1 − D 2 2 on S 3 has a parametrix from opS −1 Similar examples can be given for arbitrary compact Lie groups G. Operators in Example 2.6 are locally hypoelliptic, but the following corollary applies to operators which are only globally hypoelliptic.
Corollary 2.7. Let X be a left-invariant real vector field on G. Then there exists a discrete exceptional set C ⊂ iR, such that for any complex number c ∈ C the operator X + c is invertible with inverse in opS 0 0 (G). Consequently, the inequality holds true for all 1 < p < ∞ and all functions f from that Sobolev space.
We prove this corollary later, but now only give its refinement on SU(2).
Example 2.8. To fix the scaling on the Lie algebra su (2), let (φ, θ, ψ) be the (standard) Euler angles on SU(2) and let D 3 = ∂/∂ψ. Let X be a left-invariant vector field on SU(2) normalised so that X = D 3 with respect to the Killing norm. Then it was shown in [14] that iC = 1 2 Z, and X + c is invertible if and only if ic ∈ 1 2 Z. For such c, the inverse (X + c) −1 has symbol in S 0 0 (SU (2)). The same conclusions remain true if we replace SU(2) by S 3 . In particular, we get that holds true for all 1 < p < ∞ and all functions f from that Sobolev space. We note that this estimate is non-localisable since operators X + c are locally non-invertible and also not locally sub-elliptic (unless n = 1).
Remark 2.9. The Hörmander multiplier theorem [10], although formulated in R n , has a natural analogue on the torus T n . The refinement in Theorem 3.5 on the top order difference brings a refinement of the toroidal multiplier theorem, at least for some , where ξ ∈ Z n and e j is its jth unit basis vector in Z n .
A (translation) invariant operator A and its symbol σ A are related 2 by σ A (k) = e −2πix·k (Ae 2πix·k ) = (Ae 2πix·k )| x=0 and Aφ(x) = k∈Z n e 2πix·k σ A (k) φ(k). Thus, it follows from Theorem 3.5 that, for example on T 3 , a translation invariant operator A is weak (1,1) type and bounded on L p (T 3 ) for all 1 < p < ∞ provided that there is for all k ∈ Z 3 and all (three) unit vectors e j , j = 1, 2, 3. Here in (2.10) we do not make assumptions on all second order differences, but only on one of them.

Proofs
The proof of Theorem 2.1 is divided into several sections. First we introduce the tools we need to prove Calderon-Zygmund type estimates for convolution kernels. Later on we show how to reduce the above theorem to a statement of Coifman and de Guzman, see [3] and also [5]. Finally, we use properties of the root system with finite Leibniz rules for difference operators to prove the refinement of Theorem 2.1 given in Theorem 3.5.

3.1.
A suitable pseudo-distance on G. At first we construct a suitable pseudodistance on the group G in terms of a minimal set of representations. For simplicity we assume first that the group is semi-simple, i.e., that the centre Z(G) of the group G is trivial. Later on we will explain the main modifications for the general situation. We now define with n = dim G where Ad : G → U(g) ≃ U(dim G) denotes the adjoint representation of the Lie group G and Ad = (rank G)1⊕ ξ∈∆ 0 ξ its Peter-Weyl decomposition into irreducible components (which all appear with multiplicity one). Note, that ρ 2 (g) is nonnegative by definition and smooth. At first we claim that ρ defines a pseudo-distance d ρ (g, h) = ρ(g −1 h).
(4) We observe that both the left and the right hand side vanish exactly in h = 1 to first order. The existence of the constant C follows therefore just by compactness of G.
Remark 3.2. If the centre of the group is non-trivial, we have to make a slight change to the definition of ρ 2 (g). We have to include 2 dim Z(G) additional representations to the set ∆ 0 defined by the choice of an isomorphism Z(G) ≃ T ℓ = R ℓ /Z ℓ . For each coordinate θ j we include both θ → e ±2πiθ j , suitably extended to the maximal torus and then to G. The statement of Lemma 3.1 remains true for both modifications. In the following we assume that ∆ 0 and ρ(g) are defined in this way.
(1) We can find a chart in the neighbourhood of the identity element such that ρ(g) = |x| and dg = ν(x)dx for some smooth density ν with ν(0) = 0. Then direct calculation yields for small r (2) follows from (1) by interpolation with the normalisation condition used.
(3) this follows from ϕ r being a class function.
Furthermore, the first expression is non-zero for small r only if either of the terms is non-zero, which gives ρ(g) r 1/n or ρ(gh −1 ) r 1/n . This corresponds for small r to two balls of radius r 1/n , i.e., volume r. Integration over g ∈ G implies the desired statement.
As ψ r and ρ n satisfy all assumptions of [3], we have the following criterion.
for some constants ǫ > 0 and C > 0 uniform in r. Then A is of weak type (1, 1) and bounded on all L p (G) for 1 < p < ∞.
Later on we will need some more properties of the functions ψ r . We collect them as follows Lemma 3.4. Let q ∈ C ∞ (G) be a smooth function vanishing to order t in 1. Then for all s ∈ [0, 1 + n 2 ]. Proof. Note that the statement is purely local in a neighbourhood of 1. For sufficiently small r we find local co-ordinates near 1 supporting ψ r (g) and satisfying ρ(g) = |x|. We write q as Taylor polynomial q N (g) of degree t + N plus remainder R N (g) = O(ρ t+N +1 (g)) and decompose q(g)ψ r (g) accordingly. First, we observe where ξ is (abusing notation) the Fourier co-variable to x. The integral is split into r 1/n |ξ| 1 and r 1/n |ξ| 1. Second, we consider the remainder and show that it is smaller. Indeed, and choosing N > s − 1 the desired smallness follows. Assumptions we had to make were −2s+2+n ≥ 0, i.e., s ≤ 1+ n 2 and n−2s−M < 0, i.e., M > n − 2s. Furthermore, we need s ≥ 0. The lemma is proven.
Denote by △ * the difference operator associated to the symbol ρ 2 (g) defined in (3.1), By Lemma 3.1 (2), this is the second order difference operator, △ * ∈ diff 2 ( G), and in view of (3.1) it can be decomposed as Therefore, after summation of the Leibniz rules (3.5) we observe that Iterating this, we observe that In the sum the orders of difference operators always add up to 4. Similar we obtain for higher orders m, for some difference operators Q ℓ,j ∈ diff ℓ ( G) andQ ℓ,j ∈ diff 2m−ℓ ( G).

3.4.
Proof of Theorem 2.1. We note that Theorem 2.1 follows from its refined version which we give as Theorem 3.5 below. Let ∆ 0 be an extended root system as in Remark 3.2, and we define the family of first order difference operators associated to ∆ 0 by where δ ij is the Kronecker delta. We write D k for the family of operators of the form D α = D α 1 1 · · · D α l l , where D 1 , . . . , D l ∈ D 1 , and for multi-indices α = (α 1 , . . . , α l ) of any length but such that |α| ≤ k. We note that for even κ, in view of (3.8), the difference operator △ * κ/2 is a linear combination of operators in D κ . In general, clearly D k ⊂ diff k ( G).
Theorem 3.5. Denote by κ be the smallest even integer larger than 1 2 for all operators D α ∈ D κ−1 , and for all [ξ] ∈ G. Then the operator A is of weak type (1, 1) and L p -bounded for all 1 < p < ∞.
For the proof it is enough to check (3.3) with appropriate ǫ. If we choose ǫ such that n(1 + ǫ) = 4m, m ∈ N, the condition is equivalent to Applying the Leibniz rule (3.10) to the left-hand side implies for fixed [ξ] ∈ G that we have for certain differences Q ℓ,j ∈ diff ℓ ( G) of order ℓ arising from Leibniz rule and corresponding differencesQ ℓ,j ∈ diff 2m−ℓ ( G). Summing d ξ times these inequalities over [ξ] ∈ G and using the assumptions of Theorem 3.5, we can apply Lemma 3.4 in the form Under the assumption that 2m ≤ 2 + n 2 this implies the desired estimate (3.12).
Remark 3.6. Note, that the number of difference conditions is κ = 2m, where n 2 < κ ≤ 2 + n 2 , as we have to assure that ǫ > 0 and that Lemma 3.4 is applicable.

Applications to central multipliers
We turn to some applications of Theorem 2.1. First we collect some statements about central sequences σ ∈ Σ( G), σ(ξ) = σ ξ I. Particular examples of interest are defined in terms of d ξ or λ ξ or appear in connection with invariant multipliers on homogeneous spaces with respect to massive subgroups. For the sake of simplicity we assume in the sequel that σ ξ is defined on the full weight lattice and invariant under the Weyl group W (and not on the positive cone of maximal weights). We refer to [9] for Weyl group, Weyl dimension and Weyl character formula. We will use a notion of difference operators on the weight lattice; difference operators of higher order are understood as iterates of first order forward differences on this lattice.
for any difference operator △ k of order k acting on the weight lattice.
Proof. We recall first, that the dimension d ξ can be expressed in terms of the heighest weights (for simplicity also denoted by the variable ξ ∈ Λ ⊂ t * , t = T 1 T for T ⊂ G a maximal torus of G) by Weyl's dimension formula 3 The sum goes over the positive roots α ∈ ∆ + 0 , which form a subset of the set ∆ 0 used before. Weyl's dimension formula directly implies (4.1) from ξ ∼ 1 + ||ξ||.
In order to prove (4.2) we consider first an arbitrary difference of first order of the form △ τ d ξ = d ξ+τ − d ξ for a suitable lattice vector τ ∈ Λ. Then, an elementary calculation shows that and, therefore, we see that the right-hand side indeed behaves like ξ −1 for all d ξ = 0. The full statement follows in analogy.
For the following we assume that σ ∈ Σ( G) is central. This corresponds to a distribution F −1 σ ∈ D ′ (G) invariant under the adjoint action of the group. The following lemma explains the action of the difference operator △ * on σ. We always understand σ ξ as scalar sequence on the lattice of heighest weights extended to the full lattice by the action of the Weyl group.
Lemma 4.2. There exists a second order difference operator △ 2 acting on the lattice of heighest weights such that Proof. It suffices to prove the formula for elementary sequences σ ξ which are 1/d ξ * for some ξ = ξ * and 0 otherwise. Then △ * σ is the Fourier transform of ρ 2 (g)χ ξ * (g), which in turn can be calculated based on a multiplication formula for characters evalutated in points exp x ∈ T ⊂ G following directly from Weyl character formula, the Weyl denominator. Crucial point is that the expression in brackets on the third line is invariant under changes of ω ′ . Hence, The difference operator △ 2 annihilates linear functions on the lattice and is therefore of second order.
Example 4.3. On the group S 3 ≃ SU(2) we obtain for △ * ∼ d 1/2 − trace t 1/2 (in the notation of [12]) that central sequences σ ℓ satisfy (4.4) with △ 2 σ ℓ = 2σ ℓ − σ ℓ − − σ ℓ + , which is (up to sign) the usual second order difference on 1 2 Z. W the Weyl group and again ρ the Weyl vector, in place of our distance function ρ 2 (g) = dim G−trace Ad(g). This function seems to simplify the treatment of central multipliers (as the associated difference operator δ acts in a much simpler way on central sequences), but it does not allow the use of a finite Leibniz rule which is important for our proof in the non-central case. It is remarkable that δd ξ = 0.

4.2.
Functions of the Laplacian. We say a bounded function f defined on a normed linear space V has an asymptotic expansion at ∞, if there exist functions f k (η), homogeneous of order k for large η, such that holds true for certain constants C N . We fix a maximal torus T of G.
Lemma 4.5. Assume f : t * → C is bounded and has an asymptotic expansion into smooth components at ∞ and denote f (ξ) its restriction to the weight lattice Λ ⊂ t * . Then the central sequence f (ξ)I defines an L p -bounded multiplier on G for all 1 < p < ∞.
Remark 4.6. It is enough to assume the asymptotic expansion up to fixed finite order κ as in Theorem 2.1.
Proof. We identify t * with R t , t = rank G, which is the space V in definition (4.5).
In a first step let f k (η) be smooth and homogoneous of degree −k on |η| ≥ 1. Then f k ∈ S −k (R t ) and by the arguments of [12,Theorem 4.5.3] we immediately get that the restriction of f to the lattice belongs to the symbol class S −k 1 (T ). Furthermore, lattice differences preserve O (1 + |η|) −N for any N. Therefore, choosing N in depenendence on the order of the difference we immediately see that the restriction of f to the lattice belongs to S 0 1 (T ). In order to obtain the L p -boundedness we follow the proof of Theorem 2.1. Note that ψ r is defined in terms of the pseudo-distance ρ and therefore central. Hence ψ r (ξ) is a central sequence (also denoted by ψ r (ξ) for the moment) and thus by Lemma 4.2 in combination with Lemma 4.1 we obtain the desired bounds for the HS-norm of and corresponding higher differences with respect to △ * . Proof. This follows from the fact that λ 2 ξ = ||ξ +ρ|| 2 −||ρ|| 2 has the desired asymptotic expansion in ξ. This implies that f (λ 2 ξ ) also has an asymptotic expansion, see Remark 4.6, and one-dimensionality allows one to chose the components of the expansion as smooth functions.

Applications to non-central operators
In this section we give applications to invariant and non-invariant operators. Difference operators D α in this section correspond to those in Theorem 2.1 for simplicity of the formulations. However, in Remark 5.3 we explain that those associated to the extended root system analogously to those in Theorem 3.5 will suffice. 5.1. Mapping properties of operators of order zero. As a second main example we consider operators associated to symbols S 0 ρ (G) of type ρ ∈ [0, 1], i.e. matrix symbols for which holds for all α and all [ξ] ∈ G, and ask for mapping properties of such operators within Sobolev spaces over L p (G). Such symbol classes appear naturally as parametrices for non-elliptic operators, see Example 2.6 and Corollary 2.7. We now give a refined version of a multiplier theorem for such operators: Corollary 5.1. Let ρ ∈ [0, 1] and let κ be the smallest even integer larger than 1 2 dim G. Assume that A is a left-invariant operator on G with matrix symbol σ A satisfying The proof follows by interpolation from two end point statements, the trivial one for p = 2 and the fact that ξ −κ(1−ρ) σ A (ξ) defines an operator of weak type (1, 1) on L 1 (G). The latter follows from Theorem 2.1 in combination with Leibniz rule (3.6) for difference operators, which can be estimated by ξ −|α| whenever |α| ≤ κ.
Similar to Remark 5.3, Corollary 5.1 remains true if in (5.1) we take only the single difference △ * of order κ and only those differences that are associated to the extended root system ∆ 0 for |α| ≤ κ − 1, if we apply Theorem 3.5 instead of Theorem 2.1 in the proof.
We also note that the variable coefficient version S m ρ,δ (G) of these classes S m ρ (G), especially the class S m 1, 1 2 (G), played an important role in the proof of the sharp Gårding inequality on compact Lie groups in [13].

5.2.
Proof of Corollary 2.7. Let X be left-invariant vector field on the group G with σ X (ξ) = (Xξ)(1) as its symbol. We assume 4 that the bases of the representation spaces are chosen such that σ X (ξ) is diagonal for all [ξ] ∈ G. Let further [η] ∈ G be a fixed representation with associated differences D ij = η D ij . Then for some τ ij we have D ij σ X = (Xη ij )(1)I d ξ ×d ξ = τ ij I d ξ ×d ξ as can be seen immediately on the Fourier side and follows from F δ 1 = I d ξ ×d ξ . By our choice of representation spaces, τ ij = 0 for i = j and j τ jj = 0. The latter one is just another formulation of the fact that the character χ η (x) = trace η(x) vanishes to second order in the identity element 1. Now σ X+c (ξ) = σ X (ξ) + cI is invertible for all ξ, whenever c ∈ spec(−X) ⊂ iR. For such c the Leibniz rule (3.5) for D ij implies so that D ij σ −1 X+c = 0 for i = j and c + τ jj ∈ spec(−X), and Using this recursion formula we see that σ −1 X+c ∈ S 0 0 (G) provided all appearing matrix inverses exist, which means c ∈ spec(−X) − iN[τ 11 , . . . , τ ll ], where the latter stands for the set of all linear combinations of τ 11 , . . . , τ ll with integer coefficients. Outside this exceptional set of parameters by Corollary 5.1 we conclude the L p -estimate for all 1 < p < ∞.

5.3.
Non-invariant pseudo-differential operators. The result for multipliers implies the L p -boundedness for non-invariant operators if we assume sufficient regularity of the symbol. Again, such a result is an extension of the L p -boundedness of pseudodifferential operators.
Let A : C ∞ (G) → D ′ (G) be a linear continuous operator (not necessarily invariant). Following [12], we define its matrix symbol σ A : G× G → [ξ]∈ G C d ξ ×d ξ so that for each (x, [ξ]) ∈ G × G, the matrix σ A (x, ξ) ∈ C d ξ ×d ξ is given by σ A (x, ξ) = ξ(x) * (Aξ)(x). In particular, for the left-invariant operators we have (2.5). Consequently, it was shown in [12] that such symbols are well-defined on G × G and that the operator A can be quantised as
We also have the relation (2.3) in this setting.
Let ∂ x j , 1 ≤ j ≤ 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 ∂ β x = ∂ β 1 x 1 · · · ∂ βn xn . In [12], and completed in [14], it was shown that the Hörmander class Ψ m (G) of pseudo-differential operators on G defined by localisations can be characterised in terms of the matrix symbols. In particular, we have A ∈ Ψ m (G) if and only if its matrix symbol σ A satisfies ∂ β x D α σ A (x, ξ) op ≤ C α,β ξ m−|α| for all multi-indices α, β, for all x ∈ G and [ξ] ∈ G. For the L p -boundedness it is sufficient to impose such conditions up to finite orders as follows, extending Theorem 2.1 to the non-invariant case: Theorem 5.2. Denote by κ be the smallest even integer larger than n 2 , n the dimension of the group G. Let 1 < p < ∞ and let l > n p be an integer. Let A : C ∞ (G) → D ′ (G) be a linear continuous operator such that its matrix symbol σ A satisfies (5.2) ∂ β x D α σ A (x, ξ) op ≤ C α,β ξ −|α| for all multi-indices α, β with |α| ≤ κ and |β| ≤ l, for all x ∈ G and [ξ] ∈ G. Then the operator A is bounded on L p (G).
Remark 5.3. The modifications similar to other formulations of multiplier theorems regarding the choice of difference operators remain true in a straightforward way. For example, it is enough to impose difference conditions D α in (5.2) only with respect to the (extended) root system. Thus, in analogy with Theorem 3.5, the conclusion of Theorem 5.2 remains true if we impose ∂ β x △ * κ/2 σ A (x, ξ) op ≤ C ξ −κ as well as (5.2) only for D α ∈ D κ−1 , for all |β| ≤ l. Similarly, Corollary 5.1 can be extended to the general (non-invariant) case.
Proof. Let Af (x) = (f * r A (x))(x), where r A (x)(y) = R A (x, y) denotes the rightconvolution kernel of A. Let A y f (x) = (f * r A (y))(x), so that A x f (x) = Af (x). Then By an application of the Sobolev embedding theorem we get sup y∈G |A y f (x)| p ≤ C |α|≤l G |∂ α y A y f (x)| p dy.
Therefore, using the Fubini theorem to change the order of integration, we obtain , where the last inequality holds due to Theorem 2.1.