# Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

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## Abstract

In this paper we give several global characterisations of the Hörmander class \(\Psi ^m(G)\) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. 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, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.

## Keywords

Pseudo-differential operators compact Lie groups microlocal analysis elliptic operators global hypoellipticity Leibniz formula## Mathematics Subject Classification

Primary 35S05 Secondary 22E30## 1 Introduction

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 [12] and [10], the first two authors carried out a comprehensive non-commutative analysis of an analogue of the Kohn–Nirenberg quantization on \({{\mathbb R}^n}\), in terms of the representation theory of the group. This yields a full matrix-valued symbol defined on the product \(G\times {\widehat{G}}\), which can be viewed as a non-commutative version of the phase space. The calculus and other properties of this quantization resemble those well-known in the local theory on \({{\mathbb R}^n}\). However, due to the non-commutativity of the group in general, the full symbol is matrix-valued, with the dimension of the matrix symbol \(a(x,\xi )\) at \(x\in G\) and \([\xi ]\in {\widehat{G}}\) equal to the dimension of the representation \(\xi \). This construction is inspired by that of Taylor [20] but is carried out entirely in terms of the group \(G\) without referring to its Lie algebra and the Euclidean pseudo-differential classes there. It is parallel to the Kohn-Nirenberg quantization on \({{\mathbb R}^n}\) ([9]), see formula (2). In [2] (see also [21]), pseudo-differential operators have been characterised by their commutator properties with the vector fields. In [10], we gave a different version of this characterisation relying on the commutator properties in Sobolev spaces, which led in [12] (see also [10]) to characterisations of symbols of operators in \(\Psi ^m(G)\). 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 \({\widehat{G}}\). This will be given in Theorem 2.2.

In Theorem 2.6 we give another simple description of Hörmander’s classes on the group \({\mathrm{SU}(2)}\) and on the 3-sphere \({{\mathbb S}^3}\). This is possible due to the explicit analysis carried out in [12] and the explicit knowledge of the representation of \({\mathrm{SU}(2)}\). We note that this approach works globally on the whole sphere, e.g. compared to the analysis of Sherman [16] working only on the hemisphere (see also [17]).

We give two applications of the characterisation provided by Theorem 2.2. First, we characterise elliptic operators in \(\Psi ^m(G)\) by their matrix-valued symbols. Similar to the toroidal case (see [1]) this can be applied further to spectral problems. For example, this characterisation will be instrumental in establishing a version of the Gohberg lemma and estimates for the essential spectrum for operators on compact Lie groups, see [5].

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 [18]), 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 \(G\times {\widehat{G}}\).

Another application of Theorem 2.2 to obtain the sharp Gårding inequality on compact Lie groups appears in [13]. Moreover, in [15] it will play a crucial role in establishing \(L^p\)-multiplier theorems on general compact Lie groups, the result which was also announced in [14].

We will now introduce some notation. Let throughout this paper \(G\) be a compact Lie group of (real) dimension \(\dim G\) with the unit element \(e\), and denote by \(\widehat{G}\) 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 [10], but recall some basic facts for the convenience of the reader and in order to fix the notation. Each \([\xi ]\in \widehat{G}\) corresponds to a homomorphism \(\xi : G\rightarrow \mathrm U(d_\xi )\) with \(\xi (xy)=\xi (x)\xi (y)\) satisfying the irreducibility condition \(\mathbb C^{d_\xi } = {{\mathrm{span}}}\{ \xi (x) v : x\in G\}\) for any given \(v\in \mathbb C^{d_\xi }\setminus \{0\}\). The number \(d_\xi \) is referred to as the dimension of the representation \(\xi \).

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 \(\mathcal F : L^2(G) \rightarrow \ell ^2(\widehat{G})\).

### **Theorem 1.1**

The notion of Fourier series extends naturally to \(C^\infty (G)\) and to the space of distributions \(\mathcal D'(G)\) with convergence in the respective topologies.

*matrix-valued full symbol*\(\sigma _A(x,\xi )\in \mathbb C^{d_\xi \times d_\xi }\) defined by

The first aim of this paper is to characterise Hörmander’s class of pseudo-differential operators \(\Psi ^m(G)\) by the behaviour of their full symbols, with applications to the ellipticity and global hypoellipticity of operators.

The plan of this paper 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 \(\widehat{G}\ni [\xi ] \mapsto \sigma (\xi ) \in \mathbb C^{d_\xi \times d_\xi }\) of varying dimension. We also give an application to pseudo-differential operators on the group \({\mathrm{SU}(2)}\). 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, Sect. 5 contains criteria for the global hypoellipticity of pseudo-differential operators and a number of examples.

## 2 Characterisations of Hörmander’s Class

*difference operator*of order \(k\) if it is given by

### **Definition 2.1**

*admissible*, if the corresponding functions \(q_1, \ldots , q_n\in C^\infty (G)\) satisfy \(\mathrm dq_j(e)\ne 0\), \(j=1,\ldots ,n\), and \({{\mathrm{rank}}}(\mathrm dq_1(e),\ldots ,\mathrm dq_n(e))=\dim G\). It follows, in particular, that \(e\) is an isolated common zero of the family \(\{q_j\}_{j=1}^n\). An admissible collection is called

*strongly admissible*if

### **Theorem 2.2**

- (A)
\(A\in \Psi ^m(G)\).

- (B)For every left-invariant differential operator \(P_x\in \mathrm {Diff}^k(G)\) of order \(k\) and every difference operator \(Q_\xi \in \mathrm {diff}^\ell (\widehat{G})\) of order \(\ell \) the symbol estimateis valid.$$\begin{aligned} \Vert Q_\xi P_x \sigma _A(x,\xi ) \Vert _{op} \le C_{Q_\xi P_x} \langle \xi \rangle ^{m-\ell } \end{aligned}$$
- (C)For an admissible selection \(\triangle _1,\ldots ,\triangle _n \in \mathrm {diff}^1(\widehat{G})\) we havefor all multi-indices \(\alpha ,\beta \). Moreover, \({{\mathrm{sing\,supp}}}R_A(x,\cdot ) \subseteq \{e\}\).$$\begin{aligned} \Vert \triangle _\xi ^\alpha \partial _x^{\beta } \sigma _A(x,\xi )\Vert _{op} \le C_{\alpha \beta } \langle \xi \rangle ^{m-|\alpha |} \end{aligned}$$
- (D)For a strongly admissible selection \(\triangle _1,\ldots ,\triangle _n \in \mathrm {diff}^1(\widehat{G})\) we havefor all multi-indices \(\alpha ,\beta \).

It follows from the proof that if conditions (C) or (D) are satisfied for one admissible (strongly admissible, resp.) selection of first order differences, they are automatically satisfied for any admissible (strongly admissible, resp.) selection of first order differences. The set of symbols \(\sigma _A\) satisfying either of equivalent conditions \((B)\)–\((D)\) will be denoted by \(\fancyscript{S}^{m}(G)\).

### **Corollary 2.3**

Let the symbol \(\sigma _A\) of a linear continuous operator from \(C^\infty (G)\) to \(\mathcal D'(G)\) satisfy either of the assumptions \((B)\), \((C)\), \((D)\) of Theorem 2.2. Then the symbols \(\sigma _{A_u}\) satisfy the same symbolic estimates for all \(u\in G\).

This statement follows immediately from the condition \((B)\) if we observe that the difference operators applied to the symbol \(\sigma _{A_u}\) just lead to a different set of difference operators in the symbolic inequalities. Alternatively, one can observe that the change of variables by \(\phi \) amounts to the change of the basis in the representation spaces of \({\widehat{G}}\), thus leaving the condition \((B)\) invariant again.

### *Example 2.4*

- (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 \(\xi \) involves only matrix entries from finitely many (neighbouring) representations. - (2)
*Taylor’s formula.*Any admissible selection of first order differences allows for a Taylor series expansion. - (3)
*Leibniz rule.*Leibniz rules have a particularly simple structure if one chooses difference operators of a certain form, see Proposition 4.3. Moreover, operators of this form yield a strongly admissible collection, see Lemma 4.4.

### *Example 2.5*

Following [10] and [12], we simplify the notation on \({\mathrm{SU}(2)}\) and \({{\mathbb S}^3}\) by writing \(\sigma _A(x,\ell )\) for \(\sigma _A(x,t^\ell )\), \(\ell \in \frac{1}{2}{\mathbb N}_0\), and we refer to these works for the explicit formulae for the difference operators \(\triangle _{+}, \triangle _{-}, \triangle _0\). We denote \(\triangle _\ell ^\alpha =\triangle _{+}^{\alpha _1} \triangle _{-}^{\alpha _2}\triangle _0^{\alpha _3}\). In [12] we proved that the operator and the entrywise supremum norms are uniformly equivalent for symbols of pseudo-differential operators from \(\Psi ^m({\mathrm{SU}(2)})\). As a corollary to Theorem 2.2, Corollary 2.3, and [10, Theorem 12.4.3] we get

### **Theorem 2.6**

We note that Theorem 2.6 holds in exactly the same way if we replace \({\mathrm{SU}(2)}\) by \({{\mathbb S}^3}\). Moreover, according to Theorem 1.1 the statement of Theorem 2.6 holds without the kernel condition \({{\mathrm{sing\,supp}}}R_A(x,\cdot ) \subset \{e\}\) provided that instead of the admissible family \(\triangle _+, \triangle _-, \triangle _0\) we take a strongly admissible family of difference operators in \(\mathrm {diff}^1(\widehat{{\mathrm{SU}(2)}})\), for example one given by four first order difference operators corresponding to functions \(q_-\), \(q_+\), \(t^{1/2}_{-1/2,-1/2}-1\) and \(t^{1/2}_{+1/2,+1/2}-1\).

## 3 Proof of Theorem 2.2

### 3.1 \((A)\implies (C)\)

By [10, Thm. 10.9.6] (and in the notation used there) we know that condition (A) is equivalent to \(\sigma _A\in \Sigma ^m(G) = \bigcap _k \Sigma _k^m(G)\), where \(\Sigma _0^m(G)\) already corresponds to assumption (C). There is nothing to prove.

### 3.2 \((C)\implies (D)\)

Evident.

### 3.3 \((D)\implies (B)\)

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 \(Q\in \mathrm {diff}^\ell (\widehat{G})\). Hence, we obtain

### **Lemma 3.1**

It remains to show that \(\fancyscript{S}^m\subset \fancyscript{R}^m_\infty \), which is done in the sequel.

### **Lemma 3.2**

Let \(\sigma _A\in \fancyscript{S}^{m}\). Then \(R_A(x,\cdot )\in H^{-m-c}(G)\) for all \(c>\frac{1}{2}\dim G\).

### *Proof*

### 3.4 \((B)\Longleftrightarrow (C)\Longleftrightarrow (D)\)

Evident from the above.

### 3.5 \((B) \implies (A)\)

- (1)
any symbol \(\sigma _A\) satisfying condition (B) gives rise to a bounded linear operator \(A: H^m(G)\rightarrow L^2(G)\);

- (2)
the symbol \(\sigma _{[X,A]}\) of the commutator of \(A\) and any left-invariant vector field \(X\) also satisfies (B).

### **Lemma 3.3**

### **Lemma 3.4**

Let \(A\) be a differential operator of order \(m\). Then \(\sigma _A\in \fancyscript{S}^m\).

### *Proof*

Lemma 3.4 is a consequence of the already proved implications \((A)\Longrightarrow (C)\Longrightarrow (D)\Longrightarrow (B)\). Explicit formulae for the difference operators applied to symbols of differential operators can be also found in [10, Prop. 10.7.4] and provide an elementary proof of this statement.

## 4 Ellipticity and Leibniz Formula

As an application of Theorem 2.2 we will give a characterisation of the elliptic operators in \(\Psi ^m(G)\) in terms of their global symbols.

### **Theorem 4.1**

Thus, both statements are equivalent to the existence of \(B\in \Psi ^{-m}(G)\) such that \(I-BA\) and \(I-AB\) are smoothing. For the ellipticity condition (4) on the general matrix level it is not enough to assume that \(|\det \sigma _A(x,\xi )|^{1/d_\xi }\ge C \langle \xi \rangle ^{m}\) due to the in general growing dimension of the matrices. However, if we assume that the smallest singular value of the matrix \(\sigma _A(x,\xi )\) is greater or equal than \(C{\left\langle {\xi }\right\rangle }^{m}\) uniformly in \(x\) and (all but finitely many) \(\xi \), then condition (4) follows.

Let the collection \(q_1,\ldots ,q_n\) give an admissible collection of difference operators and let \(\partial _x^{(\gamma )}\) be the corresponding family of differential operators as in the Taylor expansion formula (3). As an immediate corollary of Theorem 4.1 and [10, Thm. 10.9.10] we get

### **Corollary 4.2**

*grand*\(k\)

*th order difference*\({\overrightarrow{{\mathbb D}}}^k\) by

### **Proposition 4.3**

### *Proof*

Difference operators \({\mathbb D}\) associated to the representations are particularly useful in view of the finite Leibniz formula in Proposition 4.3 and the fact that the collection of all differences \({\mathbb D}_{ij}\) over all \(\xi \) and \(i,j\) is strongly admissible:

### **Lemma 4.4**

The family of difference operators associated to the family of functions \(\{q_{ij}=\xi _{ij}-\delta _{ij}\}_{[\xi ]\in {\widehat{G}},\ 1\le i,j\le d_\xi }\) 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. \(\square \)

We note that on the group \({\mathrm{SU}(2)}\) or on \({{\mathbb S}^3}\), by this argument with \(N=2\), or by the discussion in Sect. 2 the four function \(q_{ij}\) corresponding to the representation \([t^\ell ]\in \widehat{{\mathrm{SU}(2)}}\) with \(\ell =\frac{1}{2}\) of dimension two, \(d_\ell =2\), 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 \(G\times {\widehat{G}}\). We formulate this for symbol classes with \((\rho ,\delta )\) behaviour as this will be also used in Sect. 5.

### **Lemma 4.5**

### *Proof*

### *Proof of Theorem 4.1*

## 5 Global Hypoellipticity

The knowledge of the full symbols allows us to establish an analogue of the well-known hypoellipticity result of Hörmander [7] on \({{\mathbb R}^n}\), requiring conditions on lower order terms of the symbol. The following theorem is a matrix-valued symbol criterion for (local) hypoellipticity.

### **Theorem 5.1**

### *Proof*

To obtain global hypoellipticity, it is sufficient to show that an operator \(A\in \Psi (G)\) has a parametrix \(B\) satisfying subelliptic estimates \(\Vert B f\Vert _{H^s} \lesssim \Vert f\Vert _{H^{s+m}}\) for some constant \(m\) independent of \(s\in {\mathbb R}\).

### 5.1 Examples

Difference operators.

\( \sigma _{\partial _0} \) | \( \sigma _{\partial _+} \) | \( \sigma _{\partial _-} \) | \( \sigma _{\mathcal L} \) | |
---|---|---|---|---|

\({\mathbb D}_{11} \) | \( \frac{1}{2} \sigma _I \) | \( 0 \) | \( 0 \) | \( -\sigma _{\partial _0}+\frac{1}{4} \sigma _I\) |

\({\mathbb D}_{12} \) | \(0\) | \(\sigma _I\) | \(0\) | \(-\sigma _{\partial _-}\) |

\({\mathbb D}_{21} \) | \(0\) | \(0\) | \(\sigma _I\) | \(-\sigma _{\partial _+}\) |

\({\mathbb D}_{22} \) | \( -\frac{1}{2} \sigma _I \) | \( 0 \) | \( 0 \) | \( \sigma _{\partial _0}+\frac{1}{4} \sigma _I \) |

Invariant vector fields corresponding to the basis Pauli matrices in \(\mathfrak {su}(2)\) are expressible as \(\mathrm D_1 = -\frac{\mathrm i}{2}(\partial _-+\partial _+) \), \(\mathrm D_2=\frac{1}{2} (\partial _--\partial _+)\) and \(\mathrm D_3= -\mathrm i \partial _0\). We refer to [10, 12] for the details on these operators and for the explicit formulae for their symbols. We also note that the analysis of invariant operators by Coifman and Weiss in [4] can be interpreted in terms of these difference operators as well.

### *Example 5.2*

The statement is sharp: the spectrum of \(\mathrm D_3\) 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 \(-\ell \le \mathrm i c\le \ell \) and \(\mathrm i c+\ell \in \mathbb Z\)). In particular, eigenfunctions can be irregular, e.g., distributions which do not belong to certain negative order Sobolev spaces.

### *Example 5.3*

By conjugation formulae in [10, 12] this example can be extended to give examples of more general left-invariant operators. Namely, let \(X\in \mathfrak g\) be normalised, \(\Vert X\Vert =\Vert \mathrm D_3\Vert \) with respect to the Killing norm, and let \(\partial _X\) be the derivative with respect to the vector field \(X\). Then it was shown that the vector field \(X\) can be conjugated to \(\mathrm D_3\) so that the operator \(\partial _X+c\) is globally hypoelliptic if and only if \(\mathrm i c \not \in \frac{1}{2}\mathbb Z\).

We note that operators from Example 5.3 are not covered by Hörmander’s sum of the squares theorem [8]. Although the following example of the sub-Laplacian is covered by Hörmander’s theorem, Theorem 5.1 provides its inverse as a pseudo-differential operator with the global matrix-valued symbol in the class \(\fancyscript{S}^{-1}_{\frac{1}{2},0}(G)\).

### *Example 5.4*

*Sub-Laplacian.*The sub-Laplacian on \(G\) is defined as \(\mathcal L_s = \mathrm D_1^2 + \mathrm D_2^2\in \Psi ^2(G)\) and has the symbol

### *Example 5.5*

*Heat operator.*We consider the analogue of the heat operator on \({\mathrm{SU}(2)}\), namely the operator \(H=\mathrm D_3-\mathrm D_1^2- \mathrm D_2^2\). Writing \(H=\mathrm D_3-{\mathcal L}_s\), we see that its symbol is

### *Example 5.6*

*Schrödinger operator.*We consider the analogue of the Schrödinger operator on \(G\), namely the operators \(S_\pm =\pm \mathrm i \mathrm D_3-\mathrm D_1^2- \mathrm D_2^2\). We will treat both \(\pm \) cases simultaneously by keeping the sign \(\pm \) or \(\mp \). Writing \(S_\pm =\pm \mathrm i \mathrm D_3-{\mathcal L}_s\), we see that their symbols are

### *Example 5.7*

*D’Alembertian.*We consider \(W=\mathrm D_3^2-\mathrm D_1^2-\mathrm D_2^2\). Writing \(W=\mathrm D_3^2-\mathcal L_s=2\mathrm D_3^2-{\mathcal L}\) with the Laplace operator \({\mathcal L}= \mathrm D_1^2+\mathrm D_2^2+\mathrm D_3^2\), we see that the symbol of \(W\in \Psi ^2(G)\) is given by

Since the only non-zero triangular number which is also a cube is \(1\) (see, e.g., [22]) we see that the operator \(2\mathrm i \mathrm D_3^3-{\mathcal L}\) is globally hypoelliptic.

### *Example 5.8*

## References

- 1.Agranovich, M.S.: Elliptic pseudodifferential operators on a closed curve. (Russian). Trudy Mosk. Mat. Obshch.
**47**, 22–67 (1984)zbMATHMathSciNetGoogle Scholar - 2.Beals, R.: Characterization of pseudo-differential operators and applications. Duke Math. J.
**44**, 45–57 (1977)CrossRefzbMATHMathSciNetGoogle Scholar - 3.Bergamasco, A., Zani, S., Luís, S.: Prescribing analytic singularities for solutions of a class of vector fields on the torus. Trans. Am. Math. Soc.
**357**, 4159–4174 (2005)CrossRefzbMATHGoogle Scholar - 4.Coifman, R., Weiss, G.: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes. Lecture Notes in Mathematics. Springer, Berlin (1971)Google Scholar
- 5.Dasgupta, A., Ruzhansky, M.: Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups.
*to appear in, J. Anal. Math*. arXiv:1306.0041 - 6.Himonas, A., Petronilho, G.: Global hypoellipticity and simultaneous approximability. J. Funct. Anal.
**170**, 356–365 (2000)CrossRefzbMATHMathSciNetGoogle Scholar - 7.Hörmander, L.: Pseudo-differential operators and hypoelliptic equations. In: Proceeding of symposia pure mathematics, pp. 138–183. Amer. Math. Soc., 1967Google Scholar
- 8.Hörmander, L.: Hypoelliptic second order differential equations. Acta Math.
**119**, 147–171 (1967)CrossRefzbMATHMathSciNetGoogle Scholar - 9.Kohn, J.J., Nirenber, L.: An algebra of pseudo-differential operators. Comm. Pure Appl. Math.
**18**, 269–305 (1965)CrossRefzbMATHMathSciNetGoogle Scholar - 10.Ruzhansky, M., Turunen, V.: Pseudo-Differential Operators and Symmetries. Birkhäuser, Basel (2010)CrossRefzbMATHGoogle Scholar
- 11.Ruzhansky, M., Turunen, V.: Quantization of pseudo-differential operators on the torus. J. Fourier Anal. Appl.
**16**, 943–982 (2010)CrossRefzbMATHMathSciNetGoogle Scholar - 12.Ruzhansky, M., Turunen, V.: Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere. Int. Math. Res. Notices IMRN
**11**, 2439–2496 (2013). doi: 10.1093/imrn/rns122 MathSciNetGoogle Scholar - 13.Ruzhansky, M., Turunen, V.: Sharp Gårding inequality on compact Lie groups. J. Funct. Anal.
**260**, 2881–2901 (2011)CrossRefzbMATHMathSciNetGoogle Scholar - 14.Ruzhansky, M., Wirth, J.: On multipliers on compact Lie groups. Funct. Anal. Appl.
**47**, 72–75 (2013)CrossRefzbMATHMathSciNetGoogle Scholar - 15.Ruzhansky, M., Wirth, J.: \(L^p\) Fourier multipliers on compact Lie groups. arXiv:1102.3988v1.
- 16.Sherman, T.: Fourier analysis on the sphere. Trans. Am. Math. Soc.
**209**, 1–31 (1975)CrossRefzbMATHMathSciNetGoogle Scholar - 17.Sherman, T.: The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one. Acta Math.
**164**, 73–144 (1990)CrossRefzbMATHMathSciNetGoogle Scholar - 18.Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
- 19.Sloane, N.J.A. (ed.): The On-Line Encyclopedia of Integer Sequences. Published electronically at www.research.att.com/~njas/sequences/. Sequence A001108
- 20.Taylor, M.E.: Noncommutative microlocal analysis. I. Mem. Amer. Math. Soc.
**52**(313) (1984)Google Scholar - 21.Taylor, M.E.: Beals-Cordes -type characterizations of pseudodifferential operators. Proc. Am. Math. Soc.
**125**, 1711–1716 (1997)CrossRefzbMATHGoogle Scholar - 22.Weisstein, E.W.: Cubic Triangular Number, from MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/CubicTriangularNumber.html
- 23.Widom, H.: A complete symbolic calculus for pseudodifferential operators. Bull. Sci. Math.
**104**, 19–63 (1980)zbMATHMathSciNetGoogle Scholar