# \(L^p\) Fourier multipliers on compact Lie groups

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## Abstract

In this paper we prove \(L^p\) 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 \(\mathbb R^n\) and its variants on tori \(\mathbb T^n\). 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.

## Keywords

Multipliers Compact Lie groups Pseudo-differential operators## Mathematics Subject Classification

Primary 43A22 43A77 Secondary 43A15 22E30## 1 Introduction

In this paper we prove \(L^p\) multiplier theorems for invariant and then also for non-invariant operators on compact Lie groups. We are primarily interested in Fourier multipliers rather than in spectral multipliers.

The topic has been attracting intensive research for a long time. There is extensive literature providing criteria for central multipliers, see e.g. Weiss [23], Coifman and Weiss [5], Stein [22], Cowling [8], Alexopoulos [2], to mention only very few. There are also results for functions of the sub-Laplacian, for example on \({\mathrm{SU}(2)}\), see Cowling and Sikora [9].

The topic of the \(L^p\)-bounded multipliers has been extensively researched on symmetric spaces of noncompact type for multipliers corresponding to convolutions with distributions which are bi-invariant with respect to the subgroup, see e.g. Anker [3] and references therein. However, general results on compact Lie groups are surprisingly elusive. For the case of the group \({\mathrm{SU}(2)}\) a characterisation for operators leading to Calderon–Zygmund kernels in terms of certain symbols was given by Coifman and Weiss in [5] based on a criterion for Calderon–Zygmund operators from [4] (see also [6]). The proofs and formulations, however, rely on explicit formulae for representations and for the Clebsch–Gordan coefficients available on \({\mathrm{SU}(2)}\) and are not extendable to other groups. In general, in the case when we do not deal with functions of a fixed operator, it is even unclear in which terms to formulate criteria for the \(L^p\)-boundedness.

In this paper we prove a general result for arbitrary compact Lie groups \(G\). This becomes possible based on the tools initiated and developed by the first author and Turunen in [17] and [15], in particular the development of the matrix valued symbols and the corresponding quantization relating operators and their symbolic calculus with the representation theory of the group. In view of the results in [17, 18], pseudo-differential operators in Hörmander classes \(\Psi ^m(G)\) can be characterised in terms of decay conditions imposed on the matrix valued symbols using natural difference operators acting on the unitary dual \({\widehat{G}}\). From this point 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 \(\fancyscript{S}^0_\rho (G)\) of type \(\rho \in [0,1]\). Such operators appear e.g. with \(\rho =\frac{1}{2}\) as parametrices for the sub-Laplacian or for the “heat” operator, see Example 2.6, or with \(\rho =0\) for inverses of operators \(X+c\), with \(X\in {\mathfrak g}\) and \(c\in {\mathbb C}\), see Corollary 2.7 on general \(G\) and Example 2.8 on \({\mathrm{SU}(2)}\) and \({{\mathbb S}^3}\). 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 \({{\mathbb T}^n}\). In different versions of multiplier theorems on \({{\mathbb T}^n}\), one usually expects to impose conditions on differences of order \([\frac{n}{2}]+1\) applied to the symbol. In Remark 2.9 we show that e.g. on \({\mathbb T}^2\) or \({\mathbb T}^3\), it is enough to make an assumption on only one second order difference of a special form applied to the symbol. In particular, this improves by now classical theorems on \(L^p\)-multipliers requiring \(n\) differences, see e.g. Nikolskii [12, Sect. 1.5.3].

In Theorem 5.2 we give an application to the \(L^p\)-estimates for general operators from \(C^\infty (G)\) to \({\mathcal D}^{\prime }(G)\), not necessarily invariant. This result is also a relaxation of the symbolic assumptions on the operator compared to those in the pseudo-differential classes. In Theorem 5.2 we give a condition for symbols based on the \((1,0)\)-type behaviour. Since the number of imposed conditions is finite, it can be extended further to \((\rho ,\delta )\)-type conditions similarly to the case of multipliers in Sect. 5. In general, symbol classes of type \((\rho ,\delta )\) for matrix symbols on compact Lie groups were introduced in [18]. These symbols also satisfy a suitable version of the functional calculus, see the authors’ paper [20].

In [1], Fourier multiplier theorems have been recently obtained for operators to be bounded from \(L^p\) to \(L^q\) for \(1<p\le 2\le q<\infty \) in the setting of the compact Lie group SU(2). However, those results are different in nature as they explore only the decay rate of symbols rather than the much more subtle behaviour expressed in terms of difference operators in this paper.

The paper is organised as follows. In Sect. 2 we formulate the results with several application and give a number of examples. In Sect. 3 we introduce the necessary techniques and prove the results. In Sect. 4 we briefly discuss central multipliers and the meaning of the difference operator Open image in new window in this case. Finally, in Sect. 5 we prove corollaries for operators with symbols in \(\fancyscript{S}^0_\rho (G)\) and for non-invariant operators.

Some of the results of this paper have been announced in [19] without proof.

## 2 Multiplier theorems on compact Lie groups

^{1}that

### **Theorem 2.1**

### *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 [23] for central multipliers.

- (d)
The conditions are needed for the weak 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*

### *Remark 2.4*

### *Example 2.5*

### *Example 2.6*

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**

We prove this corollary later, but now only give its refinement on \({\mathrm{SU}(2)}\).

### *Example 2.8*

### *Remark 2.9*

^{2}by

## 3 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 [4] and also [6]. 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 \(G\)

### **Lemma 3.1**

- (1)
\(\rho ^2(g)\ge 0\) and \(\rho ^2(g) = 0\) if and only if \(g=1\) is the identity in \(G\);

- (2)
\(\rho ^2\) vanishes to second order in \(g=1\);

- (3)
\(\rho ^2\) is a class function, in particular it satisfies \(\rho ^2(g^{-1}) ={\rho ^2(g)}\) and \(\rho ^2(gh^{-1}) = \rho ^2(h^{-1}g)\);

- (4)
\(|\rho (gh^{-1})- \rho (g)| \le C \rho (h)\) for some constant \(C>0\) and all \(g,h\in G\);

- (5)
\(\rho (gh^{-1}) \le C (\rho (g)+\rho (h))\) for some constant \(C>0\) and all \(g,h\in G\).

### *Proof*

- (1)
At first we note that for any (not necessarily irreducible) unitary representation \(\xi \) trivially \(|{{\mathrm{trace}}}\xi (g)|\le d_\xi \) and therefore \(\mathfrak {R}(d_\xi - {{\mathrm{trace}}}\xi (g)) \ge 0\). Furthermore, \({{\mathrm{trace}}}\xi (g)=d_\xi \) is equivalent to \(\xi (g)=\mathrm I\). Therefore, \(\rho (g)=0\) implies that \(\mathrm {Ad}(g)=\mathrm I\) and therefore \(g\in Z(G)\), i.e., \(g=1\).

- (2)Differentiating the identity \(\xi (g) \xi (g)^* = \mathrm I\) twice at the identity element and denoting \(\xi ^*(g)=\xi (g)^*\) implies the equationsthe first implying that \((\mathfrak {R}{{\mathrm{trace}}}\xi )^{\prime }(1)=0\), while the second one gives for each \(v\in \mathfrak g=\mathrm T_1G\) the quadratic form$$\begin{aligned}&\xi ^{\prime }(1) + {\xi ^*}^{\prime }(1) = 0,\\&\xi ''(1) + 2 \xi ^{\prime }(1)\otimes {\xi ^*}^{\prime }(1) + {\xi ^*}''(1) = 0, \end{aligned}$$Summing this over \(\xi \in \Delta _0\) implies$$\begin{aligned} (v , (\mathfrak {R}{{\mathrm{trace}}}\xi )''(1) v) = - \Vert \xi ^{\prime }(1) v\Vert _{{\mathtt {HS}}}^2. \end{aligned}$$and, therefore, if \(v\in \mathfrak g\) is such that the left-hand side vanishes, then \(v\in \cap _{\xi }\ker \xi ^{\prime }(1)\). By \(Z(G)=\{1\}\) and the definition of \(\rho ^2(g)\) this implies \(v=0\).$$\begin{aligned} (v, \mathrm {Hess}\, {\rho ^2}(1) v) = - \sum _{\xi \in \Delta _0} \Vert \xi ^{\prime }(1) v\Vert _{{\mathtt {HS}}}^2, \end{aligned}$$
- (3)
Obvious by construction.

- (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 \(G\).

- (5)
follows directly by (4).

### *Remark 3.2*

If the centre of the group is non-trivial, we have to make a slight change to the definition of \(\rho ^2(g)\). We have to include \(2\dim Z(G)\) additional representations to the set \(\Delta _0\) defined by the choice of an isomorphism \(Z(G)\simeq \mathbb T^\ell =\mathbb R^\ell /{\mathbb Z^\ell }\). For each coordinate \(\theta _j\) we include both \(\theta \mapsto \mathrm e^{\pm 2\pi \mathrm i \theta _j}\), 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 \(\Delta _0\) and \(\rho (g)\) are defined in this way. In general, for the statements below to be true, any extension of \(\Delta _{0}\) will work as long as the function \(\rho ^{2}(g)\) in (3.1) is the square of a distance function on \(G\) in a neighbourhood of the neutral element.

### 3.2 A special family of mollifiers

### **Lemma 3.3**

- (1)
\(\sup _g|\varphi _r(g)| \sim c_r \sim r^{-1}\) as \(r\rightarrow 0\).

- (2)
\(\Vert \varphi _r\Vert _2 \sim r^{-1/2}\) as \(r\rightarrow 0\).

- (3)
\(\varphi _r*\varphi _s=\varphi _s*\varphi _r\).

- (4)
\( \int _{\rho (g)\ge t^{1/n}} \varphi _r(g)\mathrm dg \le C_N \big (\frac{r}{t} \big )^N\) for all \(N\ge 0\).

- (5)
\(\int _G|\varphi _r(gh^{-1}) -\varphi _r(g)| \mathrm dg \le C^{\prime }\frac{\rho (h)}{r^{1/n}}\).

### *Proof*

- (1)We can find a chart in the neighbourhood of the identity element such that \(\rho (g) = |x|\) and \(\mathrm dg = \nu (x)\mathrm dx\) for some smooth density \(\nu \) with \(\nu (0)\ne 0\). Then direct calculation yields for small \(r\)$$\begin{aligned} c_r^{-1}&= \int _G\tilde{\varphi }(r^{-1/n} |x| )\nu (x)\mathrm dx = \int _0^1 \tilde{\varphi }(r^{-1/n} s) s^{n-1} \int _{\mathbb S^{n-1}} \nu (s\theta ) \mathrm d\theta \mathrm ds\\&\lesssim \int _0^1 \tilde{\varphi }(r^{-1/n} s) s^{n-1} \mathrm ds \sim r. \end{aligned}$$
- (2)
follows from (1) by interpolation with the normalisation condition used.

- (3)
this follows from \(\varphi _r\) being a class function.

- (4)Again direct computation of the left-hand side yields for sufficiently small \(r\)for a function \(F\in C_0^\infty (\mathbb R_+)\), which implies in particular the desired estimate.$$\begin{aligned}&c_r \int _{s\ge t^{1/n}} \tilde{\varphi }(r^{-1/n} s) s^{n-1} \int _{\mathbb S^{n-1}} \nu (s\theta )\mathrm d\theta \mathrm ds\\&\qquad \lesssim c_r \int _{s\ge t^{1/n}} \tilde{\varphi }(r^{-1/n} s) s^{n-1} \mathrm ds \sim F(\textstyle \frac{t}{r}) \end{aligned}$$
- (5)Using that \(\tilde{\varphi }\in C^\infty _0(\mathbb R)\) the mean value theorem implies in combination with Lemma 3.1(4)Furthermore, the first expression is non-zero for small \(r\) only if either of the terms is non-zero, which gives \(\rho (g)\lesssim r^{1/n}\) or \(\rho (gh^{-1})\lesssim r^{1/n}\). This corresponds for small \(r\) to two balls of radius \(r^{1/n}\), i.e., volume \(r\). Integration over \(g\in G\) implies the desired statement.$$\begin{aligned} |\varphi _r(gh^{-1})-\varphi _r(g)|&= c_r |\tilde{\varphi }\left( r^{-1/n}\rho (gh^{-1})\right) -\tilde{\varphi }\left( r^{-1/n}\rho (g)\right) |\\&\lesssim c_r r^{-1/n} |\rho \left( gh^{-1}\right) -\rho (g)| \lesssim c_r r^{-1/n} \rho (h). \end{aligned}$$

As \(\psi _r\) and \(\rho ^n\) satisfy all assumptions of [4], we have the following criterion.

### **Criterion**

Later on we will need some more properties of the functions \(\psi _r\). We collect them as follows

### **Lemma 3.4**

### *Proof*

Assumptions we had to make were \(-2s+2+n \ge 0\), i.e., \(s\le 1 +\frac{n}{2}\) and \(n-2s-M<0\), i.e., \(M>n-2s\). Furthermore, we need \(s\ge 0\). The lemma is proven.\(\square \)

### 3.3 Difference operators and Leibniz rules

### 3.4 Proof of Theorem 2.1

### **Theorem 3.5**

### *Remark 3.6*

Note, that the number of difference conditions is \(\varkappa =2m\), where \(\frac{n}{2} < \varkappa \le 2+\frac{n}{2}\), as we have to assure that \(\epsilon >0\) and that Lemma 3.4 is applicable.

## 4 Applications to central multipliers

We turn to some applications of Theorem 2.1. First we collect some statements about central sequences \(\sigma \in \Sigma (\widehat{G}), \sigma (\xi ) = \sigma _\xi \mathrm I\). Particular examples of interest are defined in terms of \(d_\xi \) or \(\lambda _\xi \) 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 \(\sigma _\xi \) is defined on the full weight lattice \(\Lambda \subset \mathfrak t^*\) for the Cartan subalgebra \(\mathfrak t\), and treat \(\widehat{G}\) as subset of \(\Lambda \), representations identified with their dominant highest weights. We refer to e.g. [10] 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.

### 4.1 Some auxiliary statements on central sequences

First, we consider the sequence \(d_\xi \) of dimensions of representations. We extend the sequence \(d_\xi \) to the full weight lattice by Weyl’s dimension formula (after fixing the set \(\Delta _0^+\) of positive roots).

### **Lemma 4.1**

### *Proof*

### **Lemma 4.2**

### *Proof*

### **Lemma 4.3**

### **Lemma 4.4**

### *Proof*

### *Example 4.5*

On the group \(\mathbb S^3\simeq \mathrm {SU}(2)\) we obtain for Open image in new window (in the notation of [15]) that central sequences \(\sigma ^\ell \) satisfy (4.10) with \(\triangle _2\sigma ^\ell =2 \sigma ^\ell - \sigma ^{\ell -1} - \sigma ^{\ell +1}\), which is (up to sign) the usual second order difference on \(\frac{1}{2}\mathbb Z\).

### *Remark 4.6*

The statement of Lemma 4.4 extends to arbitrary compact groups. The additional representations used to define \(\rho ^2(g)\) give more summands adding up to another second order difference operator on the lattice.

### *Remark 4.7*

### 4.2 Functions of the Laplacian

### **Lemma 4.8**

### *Remark 4.9*

It is enough to assume the asymptotic expansion up to fixed finite order \(\varkappa \) as in Theorem 2.1.

### *Proof*

We identify \(\mathfrak t^*\) with \(\mathbb R^t, t={{\mathrm{rank}}}G\), which is the space \(V\) in definition (4.11).

In a first step let \(f_k(\eta )\) be smooth and homogoneous of degree \(-k\) on \(|\eta |\ge 1\). Then \(f_k\in S^{-k}(\mathbb R^t)\) and by the arguments of [15, Theorem 4.5.3] we immediately get that the restriction of \(f\) to the lattice belongs to the symbol class \(\fancyscript{S}^{-k}_1({\mathcal T})\).

Furthermore, lattice differences preserve \(\mathcal O\big ((1+|\eta |)^{-N}\big )\) for any \(N\). Therefore, choosing \(N\) in dependence on the order of the difference we immediately see that the restriction of \(f\) to the lattice belongs to \(\fancyscript{S}^{0}_1({\mathcal T})\).

### **Corollary 4.10**

Assume \(f:\mathbb R_+\rightarrow \infty \) has an asymptotic expansion up to order \(\varkappa \) into homogeneous components at \(\infty \). Then \(f(-\Delta )\) is bounded on \(L^p(G)\) for \(1<p<\infty \).

### *Proof*

### *Remark 4.11*

Coifman and G. Weiss showed in [7] that central multipliers correspond to \(L^p(G)\)-bounded operators if \(\fancyscript{D}(d_\xi \sigma _\xi )\) is an \(L^p(\mathcal T)\)-bounded multiplier on the corresponding lattice, where \(\fancyscript{D}\) is the product of elementary (backward) differences \(\triangle _{-\alpha }\) corresponding to the positive roots \(\alpha \in \Delta _0^+\).

## 5 Applications to non-central operators

In this section we give applications to invariant and non-invariant operators. Difference operators \({\mathbb D}^\alpha \) 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.

### **Corollary 5.1**

### *Proof*

Similar to Remark 5.3, Corollary 5.1 remains true if in (5.1) we take only the single difference Open image in new window of order \(\varkappa \) and only those differences that are associated to the extended root system \(\Delta _0\) for \(|\alpha |\le \varkappa -1\), if we apply Theorem 3.5 instead of Theorem 2.1 in the proof.

We also note that the variable coefficient version \(\fancyscript{S}^m_{\rho ,\delta }(G)\) of these classes \(\fancyscript{S}^m_{\rho }(G)\), especially the class \(\fancyscript{S}^m_{1,\frac{1}{2}}(G)\), played an important role in the proof of the sharp Gårding inequality on compact Lie groups in [16].

### 5.2 Proof of Corollary 2.7

^{3}that the bases of the representation spaces are chosen such that \(\sigma _X(\xi )\) is diagonal for all \([\xi ]\in \widehat{G}\). Let further \([\eta ]\in \widehat{G}\) be a fixed representation with associated differences \(\mathbb D_{ij}={}_{\eta }\mathbb D_{ij}\). Then for some \(\tau _{ij}\) we have

### 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 pseudo-differential operators.

### **Theorem 5.2**

### *Remark 5.3*

### *Proof*

## Footnotes

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