$L^p$-bounds for pseudo-differential operators on compact Lie groups

Given a compact Lie group $G$, in this paper we establish $L^p$-bounds for pseudo-differential operators in $L^p(G)$. The criteria here are given in terms of the concept of matrix symbols defined on the non-commutative analogue of the phase space $G\times\widehat{G}$, where $\widehat{G}$ is the unitary dual of $G$. We obtain two different types of $L^p$ bounds: first for finite regularity symbols and second for smooth symbols. The conditions for smooth symbols are formulated using $\mathscr{S}_{\rho,\delta}^m(G)$ classes which are a suitable extension of the well known $(\rho,\delta)$ ones on the Euclidean space. The results herein extend classical $L^p$ bounds established by C. Fefferman on $\mathbb R^n$. While Fefferman's results have immediate consequences on general manifolds for $\rho>\max\{\delta,1-\delta\}$, our results do not require the condition $\rho>1-\delta$. Moreover, one of our results also does not require $\rho>\delta$. Examples are given for the case of SU(2)$\cong\mathbb S^3$ and vector fields/sub-Laplacian operators when operators in the classes $\mathscr{S}_{0,0}^m$ and $\mathscr{S}_{\frac12,0}^m$ naturally appear, and where conditions $\rho>\delta$ and $\rho>1-\delta$ fail, respectively.

In this work we study the L p boundedness of pseudo-differential operators on compact Lie groups. The investigation of the behaviour of pseudo-differential operators of Hörmander's class S m ρ,δ in L p is a fundamental problem in the theory of pseudodifferential operators. The fact that the class S 0 1,δ begets bounded operators on L p for every 1 < p < ∞ is well known (e.g. [Tay96,Ch. 13]). The boundedness on L p (R n ) for all 1 < p < ∞ fails for general operators with symbols in S 0 ρ,δ (R n ×R n ) with ρ < 1. Furthermore, when m > 0 is small, for operators with symbols in S −m ρ,δ (R n × R n ) with ρ < 1 one can only get L p (R n ) boundedness for finite intervals centered at p = 2, which is a consequence of C. Fefferman's estimates (cf. [Fef73]) and the work on multipliers of Hirschman (e.g. [Hir56]) and Wainger (cf. [Wai65]). The obstruction for the boundedness on L p (R n ) for all 1 < p < ∞ of operators in OpS 0 ρ,δ (R n × R n ) with ρ < 1 is explained in a more general setting by the works of Richard Beals [Bea79a] and [Bea79b]. The C. Fefferman's results were extended to symbols with finite regularity by Li and Wang in [LW84]. A version of L p Fefferman type bounds in the setting of S(m, g) classes has been established in [Del06].
The L p boundedness on compact groups for invariant operators (Fourier multipliers) with symbols of finite regularity has been studied in [RW13,RW15]. The case of the circle has been considered in [MW09].
In this paper we first establish L p bounds for finite regularity symbols by applying multiplier results of [RW13] and [RW15]. Secondly, we extend Fefferman's bounds to compact Lie groups obtaining some improvement with respect to the range of (ρ, δ) from the point of view of pseudo-differential operators on compact manifolds. Our analysis will be based on the global quantization developed in [RT10] and [RT13] as a noncommutative analogue of the Kohn-Nirenberg quantization of operators on R n . The classes S m ρ,δ (G) on a compact Lie group G extend the corresponding Hörmander ones when G is viewed as a manifold. The advantage here is that we will not impose the usual restriction 1 − δ ≤ ρ when dealing with those classes on manifolds. Thus, here we will allow ρ ≤ 1 2 and ρ = δ. In order to illustrate our main results we first recall the L p (R n ) bounds obtained by C. Fefferman ([Fef73]). In the following theorem, we denote by σ(x, D) the pseudodifferential operator with symbol σ(x, ξ), i.e.
(c) Let σ = σ(x, ξ) ∈ S −n(1−ρ)/2 ρ, δ The part (a) can be deduced from (c) by complex interpolation. The part (b) corresponds to the classical counter-example due to Hardy-Littlewood-Hirschman-Wainger (cf. [Zyg77], [Hir56], [Wai65]). The complex interpolation and the duality (H 1 ) ′ = BMO obtained by Stein and Fefferman in [FS72] reduce the proof of (c) to the estimation of L ∞ − BMO bounds. We note that the conditions on the symbol in Theorem A restrict the choice of the parameter ρ depending on the order of the symbol. Part (b) shows that part (a) is sharp with respect to the size of the interval around p = 2. Moreover the sharpness of the choice of the value ρ is explained by an estimate due to Hörmander [Hör67]. In this paper, one of our main results will give an analogue of Theorem A on compact Lie groups.
In Section 2 we recall basic elements of the theory of pseudo-differential operators on compact Lie groups. In Section 3 we establish our main results on L p boundedness, we consider two types of conditions, the first ones imposing finite regularity on the symbol and the second ones for C ∞ -smooth symbols.
To give a taste of our results we state two of our main theorems. Here, we rely on the global noncommutative analogue of the Kohn-Nirenberg quantization (3.7) on a compact Lie group G developed in [RT10,RT13] providing a one-to-one correspondence between matrix symbols σ on the noncommutative phase space G × G and the corresponding operators A ≡ σ(x, D) given by (1.2) We refer to Section 3 for the precise definitions of the appearing objects.
The following limited regularity result corresponds to Theorem 4.8. The notation D ξ will indicate a suitable difference operator with respect to the discrete unitary dual.
Theorem 1.1. Let G be a compact Lie group of dimension n, and let 0 ≤ δ, ρ ≤ 1. Denote by κ the smallest even integer larger than n 2 . Let 1 < p < ∞ and ℓ > n p with ℓ ∈ N. Let A : C ∞ (G) → D ′ (G) be a linear continuous operator such that its matrix symbol σ satisfies for all multi-indices α, β with |α| ≤ κ, |β| ≤ ℓ and for all x ∈ G and [ξ] ∈ G. Then the operator A is bounded from L p (G) to L p (G).
Here [ n p ] denotes the integer part of n p . For smooth symbols we will prove the following theorem which corresponds to Theorem 4.15. In particular, we will not impose the well known restrictions ρ > 1 2 or ρ ≥ 1 − δ when dealing with pseudo-differential operators on closed manifolds as an advantage of the global calculus employed here.
Theorem 1.2. Let G be a compact Lie group of dimension n. Let 0 ≤ δ < ρ < 1 and Let σ ∈ S −ν ρ,δ (G). Then σ(x, D) extends to a bounded operator from L p (G) to L p (G) Let us compare the statement of Theorem 1.2 on compact Lie groups with Theorem A on R n . First we can observe that Theorem A readily yields the corresponding L pboundedness result on a general compact manifold M for pseudo-differential operators with symbols in class S −ν ρ,δ . However, for these classes to be invariantly defined on M one needs the condition ρ ≥ 1 − δ (see e.g. [Shu01]). Together with condition ρ > δ this implies, in particular, that ρ > 1 2 . Therefore, Theorem A implies the statement of Theorem 1.2 under the additional assumption that ρ ≥ 1 − δ (and hence also ρ > 1 2 ), see also Remark 3.3 for the relation between operators in these symbol classes. Thus, the main point of Theorem 1.2 is to establish the L p -boundedness without this restriction. This is possible due to the global symbolic calculus available thanks to G being a group. We point out that the condition δ ≤ ρ has been also removed for the L 2 -boundedness on R n by J. Hounie [Hou86].

Motivation and applications
Let us give several examples of one type of applications and relevance of the obtained results. Let G = SU(2) ≃ S 3 be equipped with the usual matrix multiplication of SU(2), or with the quaternionic product on S 3 . Let X, Y, Z be three left-invariant vector fields, orthonormal with respect to the Killing form. Then we have the following properties, established in [RTW14]: (i) Let L sub = X 2 + Y 2 be the sub-Laplacian (hypoelliptic by Hörmander's sum of squares theorem). Then its parametrix L ♯ sub has symbol in the symbol class S −1 1 2 ,0 (G). (ii) Let H = X 2 + Y 2 − Z, it is also hypoelliptic by Hörmander's sum of squares theorem. Then its parametrix H ♯ has symbol in the symbol class S −1 1 2 ,0 (G). (iii) The operator Z + c is globally hypoelliptic if and only if ic ∈ 1 2 Z. In this case its inverse (Z + c) −1 exists and has symbol in the symbol class S 0 0,0 (G). We note that especially in the case (iii), the class S 0 0,0 (G) is not invariantly defined in local coordinates while our global definition makes sense. For examples (i) and (ii), the class S −1 1 2 ,0 (G) in local coordinates gives the Hörmander class S −1 1 2 , 1 2 (R 3 ). Consequently, Theorem A can not be applied since the condition ρ > δ is not satisfied in this case. Nevertheless, the results obtained in this paper apply, for example Theorem 1.1 works for the class S 0 0,0 (G), and both Theorem 1.1 and Theorem 1.2 for the class S −1 1 2 ,0 (G). Thus, this can be used to derive a-priori estimates in L p Sobolev spaces W p,s , for example , 1 < p < ∞, , 1 < p < ∞, , 1 < p < ∞.
The above examples show that similarly to the introduction of the classes S m ρ,δ by Hörmander in the analysis of hypoelliptic operators on R n , the classes S m ρ,δ (G) also appear in the analysis of (already) Fourier multipliers on Lie groups. Moreover, if a Lie group is acting on a homogeneous manifolds G/K, the Fourier analysis on G gives rise to the Fourier analysis on G/K in terms of class I representations of G.
We refer to [RTW14] and [RW15] for other examples of the appearance of the globally defined classes S −ν ρ,δ (G) in the context of compact Lie groups, also for noninvariant operators, but let us give one explicit example here.
Let f ∈ C ∞ (G) be a smooth function on G and let L be the Laplacian on G. Consider the (Schrödinger type) evolution problem Then, modulo lower order terms, the main term of its solution operator can be seen as a pseudo-differential operator with symbol where ξ stands for the eigenvalue of the elliptic operator (1 − L) 1/2 corresponding to the representation ξ. One can check, for example using the functional calculus from [RW14], that σ ∈ S 0 ρ,δ (G), with ρ = 1 − δ. In particular, we may have ρ ≤ δ, depending on the range of δ in (2.1).
This example can be extended further if we take L in (2.1) to be a sub-Laplacian, since the matrix symbol of the sub-Laplacian can be also effectively controlled, see e.g. [GR15], wth further dependence on indices, as e.g. already in the case of S 3 in (i) above.
There are other examples of problems that can be effectively treated by the global calculus rather than by localisations of the classical Hörmander calculus. For example, let L sub be a sub-Laplacian on G, i.e. a sum of squares of left-invariant vector fields satisfying Hörmander's commutator condition of order r. If we consider the Cauchy problem for the corresponding wave equation even with smooth function a > 0, it is weakly hyperbolic and its local analysis, while involving the microlocal structure of the sub-Laplacian, is rather complicated. However, the problem (2.2) can be effectively analysed using the global techniques of pseudo-differential operators on groups. Thus, such results have been obtained in [GR15] with sharp regularity estimates (depending on the Hörmander commutation order r) for the solutions of the Cauchy problem for (2.2) allowing a ≥ 0 to be also of Hölder regularity. The x-dependent pseudo-differential operators would appear if we allow a to also depend on x.
There is a variety of other problems in analysis that require the control of lower order terms of the operator which are not provided by the classical theory of pseudodifferential operators using localisations but can be controlled using the global theory of pseudo-differential operators on groups or on homogeneous spaces. For example, using such techniques, estimates for the essential spectrum of operators on compact homogeneous manifolds have been obtained in [DR16], spaces of Gevrey functions and ultradistributions have been described in [DR14] relating them with the representation theory of the group acting on the space, Besov and other function spaces have been related to the representation theory of groups in [NRT16]. It should be noted that many of the developed techniques work not only on groups but also on compact homogeneous manifolds G/K via class I representations of the compact Lie group G, thus covering the cases of real, complex or quaternionic spheres, projective spaces, and many other settings.
Furthermore, many techniques can be extended to non-compact situations, notably those of nilpotent Lie groups, see [FR16]. Nilpotent Lie groups, in turn, have a wide range of applications to various problems involving differential operators and equations on general manifolds due to the celebrated lifting techniques of Rothschild and Stein [RS76]. We refer to [FR16] for further explanations and examples of this (nilpotent) setting.

Preliminaries
In this section we recall some basic facts about the theory of pseudo-differential operators on compact Lie groups and we refer to [RT10] and [RT13] for a comprehensive account of such topics.
Given a compact Lie group G, we equip it with the normalised Haar measure µ ≡ dx on the Borel σ-algebra associated to the topology of the smooth manifold G. The Lie algebra of G will be denoted by g. We also denote by G the set of equivalence classes of continuous irreducible unitary representations of G and by Rep(G) the set of all such representations. Since G is compact, the set 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. By the Peter-Weyl theorem the collection is an orthonormal basis of L 2 (G). If f ∈ L 1 (G) we define its global Fourier transform at ξ by The Fourier inversion formula is a consequence of the Peter-Weyl theorem, so that Given a sequence of matrices a(ξ) ∈ C d ξ ×d ξ , we can define where the series can be interpreted in a distributional sense or absolutely depending on the growth of (the Hilbert-Schmidt norms of) a(ξ). For a further discussion we refer the reader to [RT10].
For each [ξ] ∈ G, the matrix elements of ξ are the eigenfunctions for the Laplacian L G (or the Casimir element of the universal enveloping algebra), with the same eigenvalue which we denote by −λ 2 [ξ] , so that The weight for measuring the decay or growth of Fourier coefficients in this setting is 1 2 , the eigenvalues of the elliptic first-order pseudo-differential operator (I − L G ) 1 2 . The Parseval identity takes the form which gives the norm on ℓ 2 ( G).

For a linear continuous operator
i.e. by applying A to each component of the matrix ξ(x). Then one has ([RT10], [RT13]) the global quantization in the sense of distributions, and the sum is independent of the choice of a representation ξ from each equivalence class [ξ] ∈ G. If A is a linear continuous operator from C ∞ (G) to C ∞ (G), the series (3.7) is absolutely convergent and can be interpreted in the pointwise sense. The symbol σ can be interpreted as a matrix-valued function on G × G. We refer to [RT10,RT13] for the consistent development of this quantization and the corresponding symbolic calculus. If the operator A is left-invariant then its symbol σ does not depend on x. We often call such operators simply invariant.
The following inequality will be useful (e.g. [RT10, Theorem 12.6.1]): For A, B ∈ C n×n , we have AB HS ≤ A op B HS , (3.8) where · op denotes the operator norm of the matrix A.
Our criteria will be formulated in terms of norms of the matrix-valued symbols. In order to justify their appearance, we recall that if A ∈ Ψ m ρ,δ (G) on a compact Lie group G is a pseudo-differential operators in Hörmander's class Ψ m ρ,δ (G), i.e. if all of its localisations to R n are pseudo-differential operators with symbols in the class S m ρ,δ (R n ), then the matrix-symbol of A satisfies Here · op denotes the operator norm of the matrix multiplication by the matrix σ(x, ξ). For this fact, see e.g. [RT10, Lemma 10.9.1] or [RT13] in the (1, 0) case. For the complete characterisation of Hörmander classes Ψ m 1,0 (G) in terms of matrix-valued symbols see also [RTW14]. In particular, this motivates the usage of the operator norms of the matrix-valued symbols.
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., 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 by diff k ( G). For a given function q ∈ C ∞ (G) it will convenient to denote the associated difference operator, acting on Fourier coefficients, by Definition 3.1. A collection of k first order difference operators is called admissible, if the corresponding functions q 1 , . . . , q k ∈ C ∞ (G) satisfy ∇q j (e) = 0, j = 1, . . . , k, and rank(∇q 1 (e), . . . , ∇q k (e)) = dim G.
In particular, the group unit element e is an isolated common zero of the family For a given admissible selection of difference operators on a compact Lie group G we use the multi-index notation be a basis for the Lie algebra of G, and let ∂ j be the left-invariant first order differential operators corresponding to Y j . For α ∈ N n 0 , we denote ∂ α = ∂ α 1 1 . . . ∂ αn n . We will use the notation ∂ α x for ∂ α .
There is also a particular family of difference operators associated to representations that we will need and that we now describe following [RW13,RW15]. Such difference operators play an important role in the Mihlin multiplier theorem proved in the above papers.
For a fixed irreducible representation ξ 0 let us define the (matrix-valued) difference operator corresponding to the matrix elements of the matrix-valued function ξ 0 (x) − I. In other words, if we set with δ ij the Kronecker delta, and use the definition in (3.9), then If the representation is fixed, we may omit the index ξ 0 for brevity. For a sequence of difference operators of this type, Let us now fix a particular collection ∆ 0 of representations: Let ∆ 0 be the collection of the irreducible components of the adjoint representation, so that where ξ are irreducible representations and 1 is the trivial one-dimensional representation. In the case when the centre Z(G) of the group is nontrivial, we extend the collection ∆ 0 to some collection ∆ 0 by adding to ∆ 0 a family of irreducible representations such that their direct sum is nontrivial on Z(G), and such that the function (which vanishes only in x = e) would define the square of some distance function on G near the identity element. Such an extension is always possible, and we denote by ∆ 0 any such extension; in the case of the trivial centre we do not have to take an extension and we set ∆ 0 = ∆ 0 . We denote further by △ * the second order difference operator associated to ρ 2 (x), In the sequel, when we write D α , we can always assume that it is composed only of Such difference operators D α enjoy a number of additional algebraic properties compared to arbitrary difference operators, for example they satisfy the Leibniz formula, and lead to the distance function ρ(x) that gives rise to the Calderon-Zygmund theory in the spirit of Coifman and Weiss, see [RW15] for the details.

L p -boundedness
In this section we establish the main results on the boundedness of operators on L p (G). We observe that from Theorem A by Fefferman and the equivalence of classes described in Remark 3.3, one can extend the Fefferman bounds to compact Lie groups for symbols in S m ρ,δ (G) as an immediate consequence, but assuming that δ < ρ and 1 − δ ≤ ρ. In particular, this type of argument leads to the restriction ρ > 1 2 . So the case of interest to us will be the one allowing ρ ≤ 1 2 and ρ = δ. Moreover, we will also obtain some L p bounds for symbols with finite regularity, which can not be deduced from the aforementioned equivalence of classes for smooth symbols. In the latter case we even allow δ > ρ.
4.1. Symbols of finite regularity. In order to deduce some consequences the following lemma proved in [DR14] will be of importance to us.
Lemma 4.1. Let G be a compact Lie group. Then we have In the next lemma we deduce a sufficient condition for the L ∞ (G)-boundedness. Then A extends to a bounded operator from L ∞ (G) to L ∞ (G), and Proof. We first observe that completing the proof.
The next statement gives a sufficient condition for an operator to be Hilbert-Schmidt on L 2 (G) and bounded on L p (G) for 2 ≤ p ≤ ∞. It also shows an example of when the condition (4.1) is satisfied, see also Corollary 4.5. It will also imply further results on the L p -boundedness. This is a particular feature of the compact situation.
Proposition 4.3. Let G be a compact Lie group. Let σ be the symbol of a linear continuous operator A : Then (4.1) holds and A extends to a Hilbert-Schmidt bounded operator from L 2 (G) to L 2 (G), and to a bounded operator from L p (G) to L p (G) for all 2 ≤ p ≤ ∞.
The Hilbert-Schmidt norm of A is given by Proof. We observe that the Cauchy-Schwarz inequality and the Parseval identity (3.5) imply Hence by Lemma 4.2 the operator A is bounded from L ∞ (G) to L ∞ (G). Now, from the proof of Lemma 4.2 we see that the kernel K A of A is given by Hence A is a Hilbert-Schmidt operator on L 2 (G), (4.3) holds and in particular A is bounded on L 2 (G). By interpolating between p = 2 and p = +∞, we conclude the proof. (ii) It is not hard to see that an analogous condition to (4.2) does not hold in the case of non-compact groups. Indeed, for G = R n consider a symbol σ of the form but σ / ∈ L 2 (R n × R n ) and σ does not beget a Hilbert-Schmidt operator on L 2 (R n ).
As a consequence we obtain the next corollary with a condition in terms of the size of the symbol measured with the operator norm.
Corollary 4.5. Let G be a compact Lie group and let m be a real number such that m > dim G 2 . Let σ be the symbol of a linear continuous operator A : Then A extends to a Hilbert-Schmidt bounded operator from L 2 (G) to L 2 (G), and to a bounded operator from L p (G) to L p (G) for all 2 ≤ p ≤ ∞.
Proof. By applying (3.8) to the decomposition σ(x, ξ) = σ(x, ξ)I d ξ , where I d ξ is the identity matrix in C d ξ ×d ξ , Lemma 4.1 and the assumption on the symbol, we obtain for every x ∈ G: Then, sup Remark 4.6. (i) The condition (4.5) on the order of the symbol, m > dim M 2 , is well known (cf. [Shu01]) in the context of compact manifolds M as a sharp order to ensure that a pseudo-differential operator is Hilbert-Schmidt. Here, in contrast, we do not assume any smoothness on the symbol nor do we require it to satisfy inequalities for the derivatives.
(ii) We observe that in the case of the torus T n , if the symbol σ only depends on ξ and for k > n 2 , k ∈ N, satisfies the inequalities |∆ α ξ σ(ξ)| ≤ C ξ −|α| , for all ξ ∈ Z n , (4.6) and all multi-indices α such that |α| ≤ k, then the operator A is bounded on L p (T n ) for all 1 < p < ∞. Here ∆ α ξ = ∆ α 1 ξ 1 · · · ∆ αn ξn are the usual partial difference operators on the lattice Z n . This classical result has been extended to non-invariant operators on the torus, replacing (4.6) by and all multi-indices α, β such that |α| ≤ k, |β| ≤ k.
(iii) Moreover, recently in [RW15, Theorem 2.1] a version of the condition (4.6) has been obtained for compact Lie groups. Let κ be the smallest even integer larger than dim G 2 . Let A : C ∞ (G) → D ′ (G) be a left-invariant linear continuous operator. Among other things it was shown in [RW15] that if its matrix symbol σ satisfies 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 < ∞.
(iv) Further, a condition of type (ρ, 0) has been also obtained in [RW15, Corollary 5.1]. Let ρ ∈ [0, 1] and let κ be as above. If A : C ∞ (G) → D ′ (G) is left-invariant and its matrix symbol σ satisfies for all multi-indices α with |α| ≤ κ and for all [ξ] ∈ G, then the operator A is bounded from the Sobolev space W p,r (G) to L p (G) for 1 < p < ∞ and Here the Sobolev space W p,r (G) consists of all the distributions f such that (I − L G ) r 2 f ∈ L p (G). (v) Lemma 4.2, Proposition 4.3 and Corollary 4.5 admit suitable extensions to general compact topological groups and nilpotent groups, see e.g. [FR14].
for all multi-indices α with |α| ≤ κ and for all [ξ] ∈ G, 1 < p < ∞, then the operator A is bounded from L p (G) to L p (G).
We note that if the condition (4.10) holds for ρ = 1, then r = 0, and A is bounded on L p (G) for every 1 < p < ∞. Hence, Theorem 4.7 absorbs the condition (4.10). We now derive a (ρ, δ)-type condition following the main idea in the proof of Theorem 5.2 in [RW15] and using Theorem 4.7.
Theorem 4.8. Let 0 ≤ δ, ρ ≤ 1 and dim G = n. Denote by κ the smallest even integer larger than n 2 . Let 1 < p < ∞ and let ℓ > n p be an integer. If A : C ∞ (G) → D ′ (G) is a linear continuous operator such that its matrix symbol σ satisfies for all x, with for all multi-indices α, β with |α| ≤ κ, |β| ≤ ℓ and for all [ξ] ∈ G, then the operator A is bounded from L p (G) to L p (G).
Proof. We first write denotes the right-convolution kernel of A. Let

Now we see that
By applying the Sobolev embedding theorem, we obtain

Now, by Fubini Theorem we have
Thus, the operator A will be bounded on L p (G) provided that the left-invariant operators f → f * ∂ γ y r A (y) are uniformly bounded on L p (G) with respect to y ∈ G, |γ| ≤ ℓ. We shall now estimate for each y ∈ G, the conditions under which such operators are bounded on L p (G) according to Theorem 4.7. The symbol of the left-invariant operator D y,γ : f → f * ∂ α y r A (y) is given by σ Dy,γ (ξ) = ∂ γ y σ(y, ξ). From (4.11) we have for all multi-indices α, β with |α| ≤ κ, |γ| ≤ ℓ and for all [ξ] ∈ G.
Hence D y,γ is bounded on L p (G) provided that and |γ| ≤ ℓ. But this follows from the condition since ℓ ≥ [ n p ] + 1 and (4.12) holds for |γ| ≤ [ n p ] + 1. Then, the operators D y,γ are uniformly bounded on L p (G) with respect to y ∈ G, |γ| ≤ ℓ, which concludes the proof. (ii) In Theorem 4.8 the usual condition δ ≤ ρ is not imposed, and in particular δ > ρ is allowed. (iii) For relatively large values of p, for instance if p > n, the Theorem 4.8 only requires smoothness of the first order with respect to x for the symbol σ, i.e. reduces to conditions on first order derivatives only. (iv) By (ii) and (iii), in the extreme situation ρ = 0, δ = 1, and for p > n, the condition (4.11) with m 0 = κ(1 − ρ) 1 p − 1 2 + δ([ n p ] + 1) takes the form for all multi-indices α, β with |α| ≤ κ, |β| ≤ 1 and for all [ξ] ∈ G. We note that in such situation, the required regularity (= 1) is independent of the dimension dim G = n, which is in contrast to the situation in the Euclidean setting. In particular, that is the case of the finite regularity improved version of Fefferman's bounds obtained by Li and Wang in [LW84].
For values near p = 2 the situation is opposite and the condition in Theorem 4.8 does not improve C. Fefferman's type conditions that can be obtained for C ∞smooth symbols. We will obtain sharper conditions on the symbol but requiring C ∞smoothness. In particular the usual restriction ρ ≥ 1 2 for (ρ, δ) classes on manifolds will not be imposed here as an advantageous consequence of the global symbolic calculus on compact Lie groups at our disposal.
Remark 4.10. Recently, the Mihlin multiplier theorem obtained by the second author and Wirth in [RW13,RW15] has been reobtained by Fischer in [Fis16] using different collections of difference operators. If the integer part of n/2 is odd, the orders of required difference operators coincide, while if it is even, the order is improved by one. However, at the same time, the collections of difference operators one has to work with are different: the difference operators in [Fis16] come from fundamental representations of the group, while our collection D α comes from the finite decomposition of the adjoint representation into irreducible components. However, since they are related, it is probable that the evenness of κ can be removed for our collection of difference operators D α .
In any case, if in (4.11) one replaces the collection D α of difference operators by the collection of difference operators associated to fundamental representations of the group, a simple modification of the proof yields the statement of Theorem 4.8 with κ being the smallest integer larger than n 2 without requiring its evenness, giving an improvement of the order by one for half of the dimensions. 4.2. C ∞ -smooth symbols. We turn now to a different perspective by looking for conditions for C ∞ -smooth symbols. We will employ the geodesic distance on the group G and it will be denoted by d. For the distances from the unit element e we will write |y| = d(y, e). The corresponding BMO space with respect to this distance will be denoted by BMO(G). The following lemma will be useful to obtain L ∞ − BMO(G) bounds by applying partitions of unity.
Lemma 4.11. Let G be a compact Lie group of dimension n and let 0 < a < 1. Let for some R > 0. Then σ(x, D) extends to a bounded operator from L ∞ (G) to L ∞ (G), and for l ≥ n 2 we have with C independent of σ, f and R.
Proof. Let σ ∈ S −m ρ,δ (G) be supported in for some fixed R > 0. In order to prove Lemma (4.11) we will apply Lemma 4.2. We split G into the form By applying Cauchy-Schwarz inequality, Parseval identity and the inequality (3.8) to the decomposition with C independent of R and σ. For the inequality (4.17) we have applied the estimate (4.14). We now consider the integral |y|≥b |F −1 G σ(x, ·)(y)|dy. To analyse it we take the difference operator ∆ q associated to q that vanishes at e of order l and e is its isolated zero, i.e., there exist constants C 1 , C 2 > 0 such that C 1 |y| l ≤ |q(y)| ≤ C 2 |y| l .
We first note that |q(y)| ≤ C|y| l , for small |y|, e.g. |q(y)| ≤ Cd l , for |y| ≤ d for some suitable d. We have, using the boundedness of r, with C independent of R and σ. For the estimation of the integral {|y|≥b} |y| −2l dy we note that the essential case is b small, and so the bound can be reduced to a local estimation.
We will now establish a L ∞ (G) − BMO(G) estimate which will have as a consequence the main results for smooth symbols. The space BMO(G) correspond to the system of balls B(x, r) determined by the geodesic distance d.
Theorem 4.12. Let G be a compact Lie group of dimension n and let 0 < ρ < 1.
Proof. Here and everywhere we will write a := 1 − ρ. Let us fix f ∈ L ∞ (G) and B = B(x 0 , r) ⊂ G. We will show that there exist an integer k and a constant C > 0 independents of f and B such that (4.20) where we have written g = σ(x, D)f and g B = 1 µ (B(x 0 ,r)) B
We now apply Lemma 4.11 to the pieces ρ j obtaining

An application of the Mean Value Theorem gives us
which gives (4.20) for σ 0 . We now consider the term σ 1 , we recall that we fixed a ball B(x 0 , r) ⊂ G. We now also fix a cut-off function φ over G, with 0 ≤ φ ≤ 10, φ ≥ 1 on B(x 0 , r) and such that its Fourier transform φ verifies supp ( φ) 1−a,δ (G), we note that σ 1 (x, D)•L is a pseudo-differential operator in OpS 0 1−a,δ . We also have (4.28) By the L 2 boundedness for operators of order 0 (cf. [Fis15, Proposition 8.1]), applied to the operator σ 1 (x, D) • L, we deduce the existence of a constant C and an integer l 0 such that On the other hand, by (4.28) we have . We also observe that φ(ξ) 2 HS ≤ C ′ d ξ . Indeed, by the definition of the Fourier transform on compact groups, since ξ(x) is unitary and 0 ≤ φ ≤ 10 we have where we have used the identities ξ * (x) 2 HS = Tr(ξ * (x)ξ(x)) = Tr(I d ξ ) = d ξ . Since L is a positive operator, we obtain For the estimation of φ 2 H − na 2 we have used the following inequalities: For the last inequality we have applied the estimate (4.14). Thus (4.30) By the Cauchy-Schwarz inequality we get This proves the desired estimated for I.
For the estimation of II, we begin by writing [φ, σ 1 (x, D)] f (x) in the form where θ(x, ξ) is a suitable symbol. To calculate θ(x, ξ) we write B := σ 1 (x, D), consider the convolution kernel k x of B and observe that Hence, θ(x, ξ) is given by On the other hand, by using Taylor expansions on compact Lie groups (see e.g. [RT10] or [RTW14]), we can write (4.32) For the last inequality we have used the following estimate: By (4.32) and from the inequalities for the support of θ j one has According to Lemma 4.11 applied to the symbols θ j and l > n 2 we have Since φ ≥ 1 on B(x 0 , δ), by using (4.31) and (4.33) into (4.26) we have 1 which concludes the proof.
We now establish a theorem for symbols in S m ρ,δ (G). We recall that the L 2 boundedness holds for operators with symbols in S 0 ρ,δ (G) (cf. [Fis15, Proposition 8.1]). As a consequence of real interpolation between L 2 boundedness and the previous L ∞ − BMO boundedness we have: Theorem 4.13. Let G be a compact Lie group of dimension n and let 0 < ρ < 1.
Proof. We write A = σ(x, D). The symbol σ satisfies the condition of Theorem 4.12. Hence A is bounded from L ∞ (G) to BMO(G). Moreover A is bounded from L 2 (G) to L 2 (G). This implies the boundedness of A from L p (G) to L p (G) for 2 ≤ p ≤ ∞. On the other hand, since σ A * ∈ S 0 ρ,δ (G) then A * : L p (G) → L p (G) is bounded for 2 ≤ p ≤ ∞ and hence by duality we get that A : L p (G) → L p (G) is bounded also for 1 ≤ p ≤ 2.
Remark 4.14. The index n(1−ρ) 2 in Theorem 4.13 can not be improved, i.e., if one takes instead an index ν 0 < n(1−ρ) 2 , one only gets L p boundedness for some finite interval around p = 2 and not for any p outside that interval. This situation will be explained in more detail by the next theorem and Remark 4.16.
Hence γ z k,S 0 1−a,δ ≤ C k σ k,S −ν 1−a,δ , (4.34) with C k > 0 independent of z. More precisely, C k only depends on finitely many semi-norms of σ. By the L 2 boundedness for S 0 ρ,δ classes there exist a constant C > 0 and an integer N such that T z f L 2 (G) ≤ C γ z N,S 0 1−a,δ f L 2 (G) . (4.35) From (4.34) and (4.35) we obtain for a suitable constant C 1 > 0. It is clear that the family {T z } 0≤Re z≤1 is analytic in the strip and continuous in S. Thus, the family {T z } 0≤Re z≤1 defines an analytic family of operators uniformly bounded on L(L 2 (G), L 2 (G)). In order to apply the complex interpolation we observe that where C 1 is independent of f .
An application of Theorem 4.12 to the operator T is gives The complex interpolation for an analytic family of operators gives us where p = 2 t , 0 < t ≤ 1. The corresponding symbol of the operator T t is given by γ t (x, ξ) = e −t 2 σ(x, ξ) ξ ν+ na 2 (t−1) . Since 0 ≤ ν ≤ na 2 , there exists t, 0 ≤ t ≤ 1 such that ν = na 2 (1 − t).
Hence the operator A = σ(x, D) is bounded from L p (G) into L p (G) for p = 2 t and σ(x, D)f L p (G) ≤ C p σ k,S −ν 1−a,δ f L p (G) .
We note that By interpolation between p = 2 and p = 2 t we obtain the L p (G) boundedness for p verifying We can now apply a duality argument for the case 1 ≤ p ≤ 2. Since the symbol of the operator A * = σ(x, D) * also belongs to S −ν 1−a,δ we have A * : L p (G) → L p (G), for Remark 4.16. The index n(1−ρ) 2 in Theorem 4.15 is sharp. Indeed, for G = T 1 , if ν 0 < 1−ρ 2 one only gets boundedness on finite intervals around p = 2. This is a consequence of the well-known classical multiplier theory on the torus (cf. [Hir56]) and Wainger (cf. [Wai65]). Indeed, let G = T 1 and 0 < ρ < 1, 0 < ν 0 < 1−ρ 2 and consider σ(ξ) = e i ξ a ξ ν 0 , for ξ ∈ Z. Then σ ∈ S −ν 0 ρ,0 (T 1 ) and the corresponding operator σ(D) is bounded on L p (T 1 ) for the interval 1 p − 1 2 < ν 0 a and is not bounded for p outside that interval. The L p boundedness inside the interval with centre at p = 2 can also be obtained from the general result in Theorem 4.15.