The Sobolev spaces that we have introduced are based on the Sobolev spaces corresponding to left-invariant vector fields and left-invariant positive Rockland operators. We could have considered the right Sobolev spaces L˜p s(G) defined via the Sobolev norms

$$f \mapsto \|| (\mathbf{I} + \tilde{\mathcal{R}})^{s/\nu} f \||\_{L^p}.$$

The relations between left and right vector fields in (1.11) easily implies that if f ∈ Lp(G) is such that Xαf ∈ Lp(G) then ˜f : x → f(x−1) is in Lp(G) and satisfies X˜ α ˜f ∈ Lp(G). By Lemma 4.4.19, we see that the map f → ˜f must map continuously Lp s → L˜p s for any p ∈ (1, ∞) and s a multiple of the homogeneous degrees of positive Rockland operators.

More generally, the spectral calculus, see (4.50), implies

$$(\mathbf{I} + \tilde{\mathcal{R}}\_2)^{s/\nu} f(x) = (\mathbf{I} + \mathcal{R}\_2)^{s/\nu} \tilde{f}(x^{-1}), \qquad f \in \mathcal{S}(G),$$

where, again, ˜f(x) = f(x−1), and thus for any p ∈ (1, ∞o),

$$\|(\mathbf{I} + \tilde{\mathcal{R}}\_p)^{s/\nu} f\|\_{L^p(G)} = \|(\mathbf{I} + \mathcal{R}\_p)^{s/\nu} \tilde{f}\|\_{L^p(G)}, \qquad f \in \mathcal{S}(G).$$

This easily implies that f → ˜f maps continuously Lp s → L˜p s for any p ∈ (1, ∞) and any real exponent s ∈ R. This is also an involution: ˜˜f = f. Hence the map

$$\begin{array}{ccc} \begin{array}{ccc} L\_s^p(G) & \longrightarrow & \tilde{L}\_s^p(G) \\ f & \longmapsto & \tilde{f} \end{array} \end{array}$$

is an isomorphism of vector spaces.

Even if the left and right Sobolev spaces are isomorphic, they are not equal in general. Note that in the commutative case of G = Rn, both left and right Sobolev spaces coincide. It is also the case on compact Lie groups, where the Sobolev spaces are associated with the Laplace-Beltrami operator (which is central) and coincide with localisation of the Euclidean Sobolev spaces [RT10a]. This is no longer the case in the nilpotent setting. Indeed, below we give an example of functions f (necessarily not symmetric, that is, ˜f = f), in some Lp s(G) but not in L˜p s(G).

Example 4.4.32. Let us consider the three dimensional Heisenberg group H1 and the canonical basis X, Y, T of its Lie algebra (see Example 1.6.4). Then X = ∂x − y 2 ∂t whereas X˜ = ∂x + y 2 ∂t thus X˜ − X = y∂t.

The Sobolev spaces are then associated with the natural sub-Laplacian X2 + Y 2, see Example 6.1.1. Hence it is covered by the work of Folland [Fol75] on Sobolev spaces associated with sub-Laplacian on stratified Lie groups and consequently, L2 1(G) is the space of functions f ∈ L2(H1) such that Xf and Y f are both in L2(H1) [Fol75, Corollary 4.13].

One can find a smooth function φ ∈ C∞(R) such that φ, φ ∈ L2(R) but R |z| 2|φ (z)| 2dz = ∞. For instance, we consider φ = φ1 ∗ ψ where ψ is a suitable smoothing function (i.e. ψ ∈ D(G) is valued in [0, 1] with a 'small' support around 0), and the graph of the function φ1 is given by isosceles triangles parametrised by - ∈ N, with vertex at points (-, β), and base on the horizontal axis and with length 2/α. We then choose α, β ∈ R with 2β ∈ (−3, −1) and 2α > 2β + 1. We also fix a smooth function χ : R → [0, 1] supported on [1/2, 2] with χ(1) = 1. We define f ∈ C∞(R3) via

$$f(x,y,t) = \phi\left(\frac{yx}{2} + t\right)\chi(x)\chi(t).$$

One checks easily that f, Xf and Y f are square integrable hence f ∈ L2 1(H1). However y∂tf is not square integrable. As X˜ − X = y∂tf, this shows that (−X + X˜)f /∈ L2(H1) and Xf ˜ can not be in L2 thus f is not in L˜2 1(H1).

# **4.5 Hulanicki's theorem**

We now turn our attention to Hulanicki's theorem which will be useful in the next chapter when we deal with pseudo-differential operators on graded Lie groups. An important consequence of Hulanicki's theorem is the fact that a Schwartz multiplier in (the L2-self-adjoint extension of) a positive Rockland operator has a Schwartz kernel. This section is devoted to the statement and the proof of Hulanicki's theorem and its consequence regarding Schwartz multiplier.

From now on, we will allow ourselves to keep the same notation R for a positive Rockland operator and its self-adjoint extension R2 on L2(G) when no confusion is possible. In particular, when we define functions of R2 (see Corollary 4.1.16), that is, a multiplier m(R2) defined using the spectral measure of R2 where m ∈ L∞(R+) is a function, we may often write

$$m(\mathcal{R}\_2) = m(\mathcal{R}),$$

in order to ease the notation. Furthermore, we denote the corresponding rightconvolution kernel of this operator by

$$m(\mathcal{R})\delta\_{o\cdot\cdot}$$

#### **4.5.1 Statement**

Hulanicki proved in [Hul84] that if multipliers m satisfy Marcinkiewicz properties, then the kernels of m(R) satisfy certain estimates:

**Theorem 4.5.1** (Hulanicki)**.**  Let |·| be a fixed homogeneous quasi-norm on G. For any M1 ∈ N, M2 ≥ 0 there exist C = CM1,M2 > 0 and k = kM1,M2 ∈ N0, k = k M1,M2 ∈ N0 such that for any m ∈ Ck[0, ∞), we have

$$\sum\_{\substack{\left[\alpha\right] \le M\_1}} \int\_G \left| X^{\alpha} m(\mathcal{R}) \delta\_o(x) \right| \left( 1 + |x| \right)^{M\_2} dx \le C \sup\_{\substack{\lambda > 0 \\ \ell = 0, \dots, k \\ \ell' = 0, \dots, k'}} \left( 1 + \lambda \right)^{\ell'} |\partial\_{\lambda}^{\ell} m(\lambda)|,$$

in the sense that if the right-hand side is finite then the left-hand side is also finite and the inequality holds.

The main consequence of Theorem 4.5.1 is the following:

**Corollary 4.5.2.**  If φ ∈ S(R) then the kernel φ(R)δo of φ(R) is Schwartz. Furthermore, the map associating a multiplier function with its kernel

$$\mathcal{S}(\mathbb{R}) \ni \phi \longmapsto \phi(\mathcal{R})\delta\_o \in \mathcal{S}(G),\tag{4.51}$$

is continuous between the Schwartz spaces.

The continuity of (4.51) means that for any continuous seminorm · on S(G) there exist C > 0 and N ∈ N such that for any m ∈ S(R) we have

$$\left\| m(\mathcal{R})\delta\_o \right\| \le C \sup\_{x \in \mathbb{R}, \ell \le N} \left| (1+|x|)^N \partial^\ell m(x) \right|.$$

Examples of such Schwartz seminorms are ·S(G),N , N ∈ N, defined in Section 3.1.9, and ·S,a,k,p, a > 0, k ∈ N0, p ∈ [1, ∞], defined in Proposition 4.4.27.

For completeness' sake, we include the proofs of Theorem 4.5.1 and Corollary 4.5.2 below. Before this, let us notice that Corollary 4.5.2 implies that the heat kernel of any Rockland operator is Schwartz. However, we will see that the proofs of Theorem 4.5.1 and Corollary 4.5.2 rely on the properties of the Bessel potentials which have been shown, in turn, using the properties of the heat kernel. Beside the properties of the Bessel potentials, the proof uses the functional calculus of R and the structure of G.

#### **4.5.2 Proof of Hulanicki's theorem**

This section is devoted to the proof of Theorem 4.5.1 and can be skipped at first reading.

We follow the essence of [Hul84], but we modify the original proof to take into account our presentation of the properties of Rockland operators as well as to bring some (small) simplifications. We also do not present some results obtained in [Hul84] on groups of polynomial growth. One of these simplifications is the fact that we fix a quasi-norm |·| which we assume to be a norm. Indeed, it is clear from the equivalence of quasi-norms (see Proposition 3.1.35) that it suffices to prove Hulanicki's theorem for one quasi-norm for it to hold for any quasi-norm. As a homogeneous norm exists by Theorem 3.1.39, we may assume that |·| is a norm without loss of generality. We could do without this but it simplifies the constants in the next pages.

#### **First step**

The first step in the proof can be summarised with the following lemma:

**Lemma 4.5.3.** Let m : [0, +∞) → C be a function and let o ∈ N. We define the function F : (−∞, 1) → C by

$$F(\xi) := \begin{cases} \ m \left( \xi^{-\frac{1}{T\_o}} - 1 \right) & \text{if } 0 < \xi < 1, \\ 0 & \text{if } \xi \le 0, \end{cases}$$

and we have

$$\forall \lambda \in [0, \infty) \qquad m(\lambda) = F\left((1 + \lambda)^{-\ell\_o}\right).$$

Furthermore, the following holds.

$$\begin{aligned} \text{Let } k \in \mathbb{N}. \text{ If } m \in C^k[0, +\infty) \text{ and } \\ \lim\_{\lambda \to +\infty} (1 + \lambda)^{1 + j + k\ell\_o} |m^{(j)}(\lambda)| = 0 \text{ for } j = 1, \dots, k', \end{aligned}$$

then the function F extends to a function in Ck(R)

3. Let k ∈ N and m ∈ Ck[0, ∞). We assume that the suprema

$$\sup\_{\lambda \ge 0} (1+\lambda)^{2+j+k\ell\_o} |m^{(j)}(\lambda)|, \qquad j = 0, \ldots, k.$$

are finite. Then we can construct an extension to R, still denoted by F, such that the function F ∈ Ck(R) is supported in [0, 2] and satisfies F-(0) = 0 and for every -∈ Z,

$$\left|\widehat{F}(\ell)\right| \le C(1+|\ell|)^{-k} \sup\_{\substack{\lambda \ge 0\\j=0,\dots,k}} (1+\lambda)^{1+j+k\ell\_o} |m^{(j)}(\lambda)|,$$

where C = Ck,o is a positive constant independent of m. Here F-(-), - ∈ Z, denotes the Fourier coefficients of F in the sense of

$$
\widehat{F}(\ell) := \int\_{-\pi}^{\pi} F(\xi) e^{-i\xi\ell} \frac{d\xi}{2\pi}.
$$

Proof. Part (1) is easy to prove. Part (2) in the case of k = 1 follows easily from the following observations.

• If ξ = (1 + λ)− 0 , λ > 0 then

$$\frac{F(\xi) - F(0)}{\xi} = (1 + \lambda)^{\ell\_o} m(\lambda).$$

• We can compute formally for ξ ∈ (0, 1):

$$F'(\xi) = \frac{-\frac{1}{\ell}\_o}{\xi^{\frac{1}{\ell}\_o+1}} m' \left(\xi^{-\frac{1}{\ell}\_o} - 1\right),$$

and in particular if ξ = (1 + λ)− 0 , λ > 0, then

$$F'(\xi) = -\frac{1}{\ell\_o} (1+\lambda)^{1+\ell\_o} m'(\lambda).$$

The general case of Part (2) follows from the following observation: F(k ) (ξ) is a linear combination over j = 1,...,k of

$$
\xi^{-\frac{1}{\tilde{\tau}\_o} - (k'-j) - j(\frac{1}{\tilde{\tau}\_o} + 1)} m^{(j)} \left( \xi^{-\frac{1}{\tilde{\tau}\_o}} - 1 \right) = \xi^{-\frac{1+j}{\tilde{\tau}\_o} - k'} m^{(j)} \left( \xi^{-\frac{1}{\tilde{\tau}\_o}} - 1 \right) .
$$

The details are left to the reader.

Let us prove Part (3). Let m ∈ Ck[0, ∞). Let Pk be the Taylor expansion of m at 0, that is, Pk is the polynomial of degree k such that we have for λ > 0 small,

$$m(\lambda) = P\_k(\lambda) + o(|\lambda|^k).$$

We fix an arbitrary smooth function χ supported in [0, 2] and satisfying χ ≡ 1 on [0, 1]. We construct an extension of F, still denoted F, by setting

$$F(\xi) := \begin{cases} 0 & \text{if } \xi \le 0, \\\ m\left(\xi^{-\frac{1}{\ell\_o}} - 1\right) & \text{if } 0 < \xi < 1, \\\ P\_k\left(\xi^{-\frac{1}{\ell\_o}} - 1\right) \chi(\xi) & \text{if } \xi \ge 1. \end{cases}$$

We assume that the suprema given in the statement of Part 3 are finite. Clearly F ∈ Ck(R) is supported in [0, 2]. The proof of Part 2 implies easily

$$\|F^{(k')}\|\_{\infty} \le C \sum\_{j=1}^{k'} \sup\_{\lambda \ge 0} (1+\lambda)^{1+j+\ell\_o k'} |m^{(j)}(\lambda)|,\tag{4.52}$$

where the constant C = Ck,o,χ > 0 is independent on m.

The Fourier coefficient of F at 0 is

$$\begin{split} \widehat{F}(0) &= \int\_{-\pi}^{\pi} F(\xi) \frac{d\xi}{2\pi} \\ &= \int\_{0}^{1} m(\xi^{-\frac{1}{\ell\_{o}}} - 1) \frac{d\xi}{2\pi} + \int\_{1}^{2} P\_{k} \left(\xi^{-\frac{1}{\ell\_{o}}} - 1\right) \chi(\xi) \frac{d\xi}{2\pi} \\ &= \int\_{0}^{\infty} m(\lambda) \frac{-\ell\_{o}}{2\pi} \frac{d\lambda}{(1 + \lambda)^{\ell\_{o} + 1}} + \int\_{1}^{2} P\_{k} \left(\xi^{-\frac{1}{\ell\_{o}}} - 1\right) \chi(\xi) \frac{d\xi}{2\pi} .\end{split}$$

We can always assume that the function χ was chosen so that

$$\int\_{1}^{2} P\_k\left(\xi^{-\frac{1}{\ell\_o}} - 1\right) \chi(\xi) \frac{d\xi}{2\pi} = \int\_{0}^{\infty} m(\lambda) \frac{\ell\_o}{2\pi} \frac{d\lambda}{(1+\lambda)^{\ell\_o+1}}.$$

Indeed, it suffices to replace χ by χ + cχ1 where χ1 ∈ D(R) is supported in (1, 2) and c a well chosen constant.

It is a simple exercise using integration by parts to show that the Fourier coefficients may be estimated by

$$\forall k'=0,\ldots,k \quad \exists C=C\_{k'} > 0 \quad \forall \ell \in \mathbb{Z} \qquad |\hat{F}(\ell)| \le C(1+|\ell|)^{-k'}||F^{(k')}||\_{\infty}.$$

This together with (4.52) concludes the proof of Part (3). -

#### **Second step**

The second step consists in noticing that, with the notation of Lemma 4.5.3, studying the multiplier m(R) and using the Fourier series of F leads to consider the operator ei(I+R)−o and, more precisely, the properties of its convolution kernel.

**Lemma 4.5.4.**  Let o ∈ N and Fo(ξ) := eiξ − 1, ξ ∈ R. Then, for any - ∈ Z, the convolution kernel of Fo(-(I + R)− o ) is an integrable function:

$$F\_o(\ell(\mathbf{I} + \mathcal{R})^{-\ell\_o})\delta\_o \in L^1(G).$$

Proof of Lemma 4.5.4. Since Fo(ξ) = ∞ j=1 (iξ)j j! , we have at least formally

$$\begin{aligned} \kappa\_{\ell} &:= \left\{ F\_o(\ell(\mathbf{I} + \mathcal{R})^{-\ell\_o}) \right\} \delta\_o \\ &= \sum\_{j=1}^{\infty} \frac{(i\ell)^j}{j!} (\mathbf{I} + \mathcal{R})^{-j\ell\_o} \delta\_0 = \sum\_{j=1}^{\infty} \frac{(i\ell)^j}{j!} \mathcal{B}\_{\nu j \ell\_o}, \end{aligned}$$

where Ba is the convolution kernel of the Bessel potentials, see Section 4.3.4, and ν is the degree of homogeneity of R. In fact, by Corollary 4.3.11, we know that

$$
\forall a \in \mathbb{C}\_{+} \quad \mathcal{B}\_{a} \in L^{1}(G) \quad \text{and} \quad \mathcal{B}\_{\nu j \ell\_{o}} = \mathcal{B}\_{\nu \ell\_{o}} \ast \dots \ast \mathcal{B}\_{\nu \ell\_{o}} := \mathcal{B}\_{\nu \ell\_{o}}^{\ast j}.
$$

Thus in the Banach algebra L1(G) endowed with the convolution product, the series

$$\sum\_{j=1}^{\infty} \frac{(i\ell)^j}{j!} \mathcal{B}\_{\nu\ell\_o}^{\*j} = \sum\_{j=1}^{\infty} \frac{(i\ell)^j}{j!} \mathcal{B}\_{\nu j \ell\_o},$$

is convergent in the L1-norm. It is then a routine exercise to justify that κ is in L1(G) and is also the convolution kernel of the multiplier Fo(-(I + R)− o ). -

Unfortunately the brute force estimate

$$\|\|F\_o(\ell(\mathbf{I}+\mathcal{R})^{-\ell\_o})\delta\_o\|\|\_{L^1(G)} \le \sum\_{j=1}^{\infty} \frac{|\ell|^j}{j!} \|\mathcal{B}\_{\nu\ell\_o}\|\_{L^1(G)}^j = \exp(\|\|\mathcal{B}\_{\nu\ell\_o}\|\_{L^1(G)}|\ell|) - 1,$$

is exponential in and would be of no use for us. However, we notice that we can already modify the proof above to show:

**Lemma 4.5.5.** We keep the notation of Lemma 4.5.4 and its proof. If |·| is a homogeneous quasi-norm on G, then for each - ∈ Z and ao > 0, the function Fo(-(I + R)− o )δo is integrable against a weight of the form (1 + |·|)ao . Moreover, if νo > Q/2, where Q is the homogeneous dimension of G, then the function Fo(-(I + R)− o )δo is in L2(G).

Proof of Lemma 4.5.4. One checks easily that if ω : G → [1, ∞) is a continuous function satisfying

$$
\exists C = C\_{\omega} \quad \forall x, y \in G \qquad \omega(xy) \le C\omega(x)\omega(y),
$$

the subspace L1(w) of L1(G) of functions f which are integrable against w, is a Banach algebra for the norm

$$\|f\|\_{L^1(\omega)} = \int\_G |f(x)|\omega(x)dx.$$

Examples of such ω's are precisely weights of the form (1 + |·|)a with |·| being a quasi-norm on G. By Lemma 4.3.15, for any a ∈ C+ and ao > 0,

$$\mathcal{B}\_a \in L^1((1+|\cdot|)^{a\_o}).$$

We keep the notation of the proof of Lemma 4.5.4 and proceed in the similar way but using L1((1 + |·|)ao ) instead of L1(G):

$$\|\kappa\_{\ell}\|\_{L^{1}((1+|\cdot|)^{a\_{0}})} \leq \sum\_{j=1}^{\infty} \frac{|\ell|^{j}}{j!} \|\mathcal{B}\_{\nu\ell\_{0}}\|\_{L^{1}((1+|\cdot|)^{a\_{0}})}^{j} = \exp(\|\mathcal{B}\_{\nu\ell\_{0}}\|\_{L^{1}((1+|\cdot|)^{a\_{0}})}|\ell|) - 1,$$

which is finite.

Let us now show that κ ∈ L2(G). We have

$$||\kappa\_{\ell}||\_2 \le \sum\_{j=1}^{\infty} \frac{|\ell|^j}{j!} ||\mathcal{B}^{\*j}\_{\nu \ell\_o}||\_2$$

and

$$\|\|\mathcal{B}\_{\nu\ell\_o}^{\*j}\|\|\_{2} \le \|\|\mathcal{B}\_{\nu\ell\_o}\|\|\_{1} \|\mathcal{B}\_{\nu\ell\_o}^{\*(j-1)}\|\|\_{2} \le \dots \le \|\|\mathcal{B}\_{\nu\ell\_o}\|\|\_{1}^{j-1} \|\|\mathcal{B}\_{\nu\ell\_o}\|\|\_{2} \cdot \|\|\mathcal{B}\_{\nu\ell\_o}\|\|\_{1}$$

We also know by Corollary 4.3.11 that Ba ∈ L2(G) whenever Re a > Q/2. Thus in this case,

$$||\kappa\_{\ell}||\_2 \le \sum\_{j=1}^{\infty} \frac{|\ell|^j}{j!} ||\mathcal{B}\_{\nu\ell\_o}||\_1^{j-1} ||\mathcal{B}\_{\nu\ell\_o}||\_2 < \infty,$$

finishing the proof. -

Using only the properties of Banach algebras, we obtain again that

$$\int\_{G} |F\_o(\ell(\mathbf{I} + \mathcal{R})^{-\ell\_o})\delta\_o|(1+|\cdot|)^{a\_o} \quad \text{and} \quad \int\_{G} |F\_o(\ell(\mathbf{I}+\mathcal{R})^{-\ell\_o})\delta\_o|^2,$$

explode exponentially with which is not good enough for our subsequent analysis. However, we are going to show that G |Fo(-(I+R)− o )δo|(1+|·|)ao actually grows polynomially in - (see Lemma 4.5.6) and this is the crucial technical point in the proof of Theorem 4.5.1.

#### **Main technical lemma**

**Lemma 4.5.6.** We keep the notation of Lemmata 4.5.4 and 4.5.5. We fix a homogeneous quasi-norm |·| on G and assume that νo > Q/2. Then for ao ≥ 0,

$$\int\_{G} (1+|x|)^{a\_o} |F\_o(\ell(\mathbf{I}+\mathcal{R})^{-\ell\_o})\delta\_o(x)| dx \le C|\ell|^{3(a\_o+\frac{Q}{2}+1)},\tag{4.53}$$

where C = Cao,o,R,G,|·| is a positive constant independent of -∈ Z.

In the proof of Lemma 4.5.6 we will need the following easy lemma:

**Lemma 4.5.7.** If μ1,...,μ2m are 2m measures in M(G), then for any continuous function φ,

$$\begin{aligned} &\int\_G \phi \, d\mu\_1 \ast \dots \ast \mu\_{2m} \\ &= \int\_{G^m} \int\_G \phi \, d\left\{ \mu\_2(y\_1^{-1} \cdot) \right\} \ast \dots \ast \left\{ \mu\_{2m}(y\_m^{-1} \cdot) \right\} d\mu\_1(y\_1) \dots d\mu\_{2m-1}(y\_m), \end{aligned}$$

whenever the right or the left hand side is finite.

Here we have denoted by μ(y ·) the y-left-translated measure of a positive or complex Borel measure μ, that is, the measure given by

$$\int\_{x \in G} \phi(x) d\mu\_1(yx) = \int\_G \phi(y^{-1}x) d\mu\_1(x), \qquad \phi \in C\_c(G).$$

Proof of Lemma 4.5.7. First let us observe that if μ1,...,μm are m measures in M(G), then, recursively, one can show readily that

$$\int\_{G} \phi \, d(\mu\_1 \* \dots \* \mu\_m) = \int\_{G^m} \phi(x\_1 \dots x\_m) \, d\mu\_1(x\_1) \dots d\mu\_m(x\_m) \,. \tag{4.54}$$

If μ1,...,μ2m are 2m measures in M(G), then applying (4.54) for these 2m measures yields

$$\begin{aligned} \int\_G \phi \, d(\mu\_1 \ast \dots \ast \mu\_{2m}) \\ = \int\_{G^{2m}} \phi(x\_1 x\_2 \dots x\_{2m-1} x\_{2m}) \\ \qquad d\mu\_1(x\_1) \, d\mu\_2(x\_2) \dots d\mu\_{2m-1}(x\_{2m-1}) \, d\mu\_{2m}(x\_{2m}) \\ = \int\_G \left( \int\_{G^m} \phi(x\_2' x\_4' \dots x\_{2m}') \right) \end{aligned}$$

$$\qquad d\mu\_2(x\_1^{-1} x\_2') d\mu\_4(x\_3^{-1} x\_4') \dots d\mu\_{2m}(x\_{2m-1}' x\_{2m}^{-1})$$

$$\qquad d\mu\_1(x\_1) \dots d\mu\_{2m-1}(x\_{2m-1}),$$

after the change of variables x 2 = x1x2,...,x 2m = x2m−1x2m. Using (4.54), we recognise an iterated convolution in the quantity in parenthesis. -

We now turn our attention to showing Lemma 4.5.6.

Proof of Lemma 4.5.6. By Theorem 3.1.39, we may assume that the homogeneous quasi-norm |·| is also a norm, that is, we assume that the triangle inequality holds with a constant = 1. Moreover, we notice that it suffices to prove (4.53) with -∈ N. Indeed, as

$$\kappa\_{\ell} := F\_o(\ell(\mathbf{I} + \mathcal{R})^{-\ell\_o})\delta\_0 = e^{i\ell(\mathbf{I} + \mathcal{R})^{-\ell\_o}}\delta\_0 - \delta\_0,$$

we have

$$
\bar{\kappa}\_{\ell} = e^{-i\ell(\mathbf{I} + \mathcal{R})^{-\ell\_o}} \delta\_0 - \delta\_0 = F\_o(-\ell(\mathbf{I} + \bar{\mathcal{R}})^{-\ell\_o})\delta\_0,
$$

and κ0 = 0. Since

$$F\_o(\ell \xi) = e^{i\ell \xi} - 1 = (e^{i\xi})^\ell - 1 = (F\_o(\xi) + 1)^\ell - 1,$$

we have in the unital Banach algebra Cδ0 ⊕ L1(G),

$$\kappa\_{\ell} = (\kappa\_1 + \delta\_o)^{\*\ell} - \delta\_0.$$

Let us fix -∈ N. We can decompose

$$
\kappa\_1 = f\_0 + f\_{\infty},
$$

where f0 and f∞ are the integrable functions defined via

$$f\_0(x) := \kappa\_1(x) \mathbf{1}\_{|x| \le \ell^2} \quad \text{and} \quad f\_\infty(x) := \kappa\_1(x) \mathbf{1}\_{|x| > \ell^2}. \tag{4.55}$$

We can also write

$$
\kappa\_1 + \delta\_0 = \mu\_0 + f\_{\infty} \qquad \text{where} \qquad \mu\_0 := f\_0 + \delta\_0.
$$

We now develop the non-commutative convolution product in Cδ0 ⊕ L1(G),

$$\begin{aligned} \left(\kappa\_1 + \delta\_o\right)^{\*\ell} &= \left(\mu\_0 + f\_\infty\right)^{\*\ell} = \left(\mu\_0 + f\_\infty\right) \* \dots \* \left(\mu\_0 + f\_\infty\right), \\ &= \sum\_{\alpha,\beta} \mu\_0^{\*\alpha\_1} \* f\_\infty^{\*\beta\_1} \* \dots \* \mu\_0^{\*\alpha\_\ell} \* f\_\infty^{\*\beta\_\ell}, \end{aligned}$$

where the summation is over all sequences α = (α1,...,α), β = (β1,...,β), of 0 and 1 such that α1 + ... + α + β1 + ... + β = -.

Let us fix two such sequences α and β. By Lemma 4.5.7 applied to

$$
\mu\_{2j-1} = \mu\_0^{\*\alpha\_j}, \quad \text{and} \quad \mu\_{2j} = f\_{\infty}^{\*\beta\_j},
$$

and φ = ωao where the positive function ω ∈ C(G) is defined by

$$
\omega(x) := 1 + |x|, \quad x \in G,
$$

we have for any ao > 0,

$$\int\_{G} \omega^{a\_{0}} d\mu\_{0}^{\*\alpha\_{1}} \* f\_{\infty}^{\*\beta\_{1}} \* \dots \* \mu\_{0}^{\*\alpha\_{\ell}} \* f\_{\infty}^{\*\beta\_{\ell}}$$

$$= \int\_{G^{\ell}} \int\_{G} \omega^{a\_{0}} \, d\left\{ \mu\_{0}^{\*\alpha\_{1}}(y\_{1}^{-\beta\_{1}} \cdot) \right\} \* \dots \* \left\{ \mu\_{0}^{\*\alpha\_{\ell}}(y\_{\ell}^{-\beta\_{\ell}} \cdot) \right\}$$

$$df\_{\infty}^{\*\beta\_{1}}(y\_{1}) \dots df\_{\infty}^{\*\beta\_{\ell}}(y\_{\ell}).$$

Let us also fix (y1,...,y) ∈ G. We notice that for any μ ∈ M(G) we have

$$(\delta\_0(y \cdot)) \* \mu = \mu(y \cdot),$$

since for any φ ∈ Cc(G),

$$\int\_G \phi \, d\left\{ (\delta\_0(y \cdot)) \* \mu \right\} = \int\_G \phi(xz) \, d\delta\_0(yx) \, d\mu(z) = \int\_G \phi(y^{-1}x) \, d\mu(x).$$

Therefore, if α1 = 0,

$$\begin{aligned} \left\{ \mu\_0^{\*\alpha\_1} (y\_1^{-\beta\_1} \cdot) \right\} \* \left\{ \mu\_0^{\*\alpha\_2} (y\_2^{-\beta\_2} \cdot) \right\} &= \left\{ \delta\_0 (y\_1^{-\beta\_1} \cdot) \right\} \* \left\{ \mu\_0^{\*\alpha\_2} (y\_2^{-\beta\_2} \cdot) \right\} \\ &= \left\{ \mu\_0^{\*\alpha\_2} (y\_2^{-\beta\_1} \cdot) \right\} (y\_1^{-\beta\_1} \cdot) \\ &= \left\{ \mu\_0^{\*\alpha\_2} (y\_2^{-\beta\_1} y\_1^{-\beta\_1} \cdot) \right\}, \end{aligned}$$

whereas if α1 = 1, we have

$$\begin{split} & \left\{ \mu\_{0}^{\*\alpha\_{1}}(y\_{1}^{-\beta\_{1}} \cdot \cdot) \right\} \* \left\{ \mu\_{0}^{\*\alpha\_{2}}(y\_{2}^{-\beta\_{2}} \cdot) \right\} \\ & \quad = \left\{ \delta\_{0}(y\_{1}^{-\beta\_{1}} \cdot) \right\} \* \left\{ \mu\_{0}^{\*\alpha\_{2}}(y\_{2}^{-\beta\_{2}} \cdot) \right\} + \left\{ f\_{0}(y\_{1}^{-\beta\_{1}} \cdot) \right\} \* \left\{ \mu\_{0}^{\*\alpha\_{2}}(y\_{2}^{-\beta\_{2}} \cdot) \right\} \\ & \quad = \left\{ \mu\_{0}^{\*\alpha\_{2}}(y\_{2}^{-\beta\_{1}}y\_{1}^{-\beta\_{1}} \cdot) \right\} + \left\{ f\_{0}(y\_{1}^{-\beta\_{1}} \cdot) \right\} \* \left\{ \mu\_{0}^{\*\alpha\_{2}}(y\_{2}^{-\beta\_{2}} \cdot) \right\}. \end{split}$$

Recursively, we find that

$$\begin{aligned} &\left\{\mu\_0^{\*\alpha\_1}(y\_1^{-\beta\_1}\cdot)\right\}\*\ldots\*\left\{\mu\_0^{\*\alpha\_\ell}(y\_\ell^{-\beta\_\ell}\cdot)\right\} \\ &=\sum\_{j:\alpha\_j=1}\left\{f\_0(y\_1^{-\beta\_1}\cdot\ldots y\_j^{-\beta\_j}\cdot)\right\}\*\left\{\mu\_0^{\*\alpha\_{j+1}}(y\_{j+1}^{-\beta\_{j+1}}\cdot)\right\}\*\ldots\*\left\{\mu\_0^{\*\alpha\_\ell}(y\_\ell^{-\beta\_\ell}\cdot)\right\} \end{aligned}$$

when α = 0. If α = 0, we compute directly:

$$\begin{aligned} \left\{ \mu\_0^{\*\alpha\_1} (y\_1^{-\beta\_1} \cdot) \right\} \* \dots \* \left\{ \mu\_0^{\*\alpha\_\ell} (y\_\ell^{-\beta\_\ell} \cdot) \right\} &= \left\{ \delta\_o (y\_1^{-\beta\_1} \cdot) \right\} \* \dots \* \left\{ \delta\_o (y\_\ell^{-\beta\_\ell} \cdot) \right\} \\ &= \quad \delta\_o \left( y\_\ell^{-\beta\_\ell} \cdot \dots y\_1^{-\beta\_1} \cdot \right), \end{aligned}$$

so that

$$\begin{split} \int\_{G} \omega^{a\_{o}} \, \, d \left\{ \mu\_{0}^{\ast \alpha\_{1}} (y\_{1}^{-\beta\_{1}} \cdot) \right\} &\ast \dots \ast \left\{ \mu\_{0}^{\ast \alpha\_{\ell}} (y\_{\ell}^{-\beta\_{\ell}} \cdot) \right\} \\ = \omega^{a\_{o}} (y\_{1}^{\beta\_{1}} \cdot \dots y\_{\ell}^{\beta\_{\ell}}) &= \left( 1 + |y\_{1}^{\beta\_{1}} \cdot \dots y\_{\ell}^{\beta\_{\ell}}| \right)^{a\_{o}} \\ \leq & \left( 1 + |y\_{1}^{\beta\_{1}}| + \dots + |y\_{\ell}^{\beta\_{\ell}}| \right)^{a\_{o}} \leq \left( 1 + \ell \max\_{j'=1,\dots,\ell} |y\_{j'}^{\beta\_{j'}}| \right)^{a\_{o}}, \end{split}$$

since the quasi-norm |·| is assumed to be a norm.

If αj = 0, we notice that the measure given by

$$\left\{ f\_0(y\_1^{-\beta\_1}\dots y\_j^{-\beta\_j}\cdot) \right\} \* \left\{ \mu\_0^{\*\alpha\_{j+1}}(y\_{j+1}^{-\beta\_{j+1}}\cdot) \right\} \* \dots \* \left\{ \mu\_0^{\*\alpha\_\ell}(y\_\ell^{-\beta\_\ell}\cdot) \right\},\tag{4.56}$$

is compactly supported. Indeed recall that f0 is supported in B¯2 where we denote by B¯R := {x ∈ G : |x| ≤ R} a closed ball of radius R about 0 for the chosen norm. Therefore

$$\text{supp}\,\mu\_0 \subseteq \{0\} \cup \text{supp}\,f\_0 \subseteq \bar{B}\_{\ell^2}.$$

The general properties

$$\begin{aligned} \{\forall \mu\_1, \mu\_2 \in M(G) \: \text{supp}(\mu\_1 \* \mu\_2) \quad \subset \quad \text{(supp}\mu\_1)(\text{supp}\mu\_2) \\ = \{x\_1 x\_2 \: : \ x\_1 \in \text{supp}\mu\_1, \ x\_2 \in \text{supp}\mu\_2\}, \end{aligned}$$

and

$$\forall \mu \in M(G), \ y \in G \qquad \text{supp}\left(\mu(\cdot y^{-1})\right) = \left(\text{supp}\mu\right)y = \{xy \: : \ x \in \text{supp}\mu\},$$

imply that the measure in (4.56) is supported in

$$
\bar{B}\_{\alpha\_1 \ell^2} y\_1^{\beta\_1} \dots \bar{B}\_{\alpha\_{j-1} \ell^2} y\_{j-1}^{\beta\_{j-1}} \ \bar{B}\_{\alpha\_\ell \ell^2} y\_\ell^{\beta\_\ell} \dots y\_j^{\beta\_j} \ .
$$

Using the properties of the norm |·|, it is easy to check that the above subset of G is included in the closed ball about 0 of radius

$$(\sum\_{j'=1}^{\ell} \alpha\_{j'})\ell^2 + (\sum\_{j'=1}^{\ell} \beta\_{j'}|y\_{j'}|) \le \ell^3 + \ell \max\_{j'=1,\dots,\ell} |y\_{j'}|,$$

since  j=1 αj and  j=1 βj are ≤ -.

Note that if f is a measurable function supported in B¯R then

$$\begin{aligned} |\int\_G \omega^{a\_o}(x) f(x) dx| &\le \int\_{|x| \le R} (1+|x|)^{a\_o} |f(x)| dx \\ &\le \quad (1+R)^{a\_o} \int\_{|x| \le R} f(x) dx \\ &\le \quad (1+R)^{a\_o} |\bar{B}\_R|^{1/2} \|f\|\_2. \end{aligned}$$

We apply this to the function f in (4.56) and R = -3 + maxj=1,..., |yj |. This leads us to look at the L2-norm of this function which can be written as

$$f = T\_{\mu\_0^{\*\alpha\_\ell}(y\_\ell^{-\beta\_\ell} \cdot)} \dots T\_{\mu\_0^{\*\alpha\_{j+1}}(y\_{j+1}^{-\beta\_{j+1}} \cdot)} \left( f\_0(y\_1^{-\beta\_1} \dots y\_j^{-\beta\_j} \cdot) \right) \dots$$

Here we have used our usual notation for the convolution operator Tκ : φ → φ ∗ κ with right-convolution (distributional) kernel κ. Since such convolution operators are left-invariant, we have

$$\left\| T\_{\mu\_0^{\*\alpha\_{j'}}(y\_{j'}^{-\beta\_{j'}} \cdot \cdot)} \right\| \mathcal{e}(L^2(G)) = \left\| T\_{\mu\_0^{\*\alpha\_{j'}}} \right\| \mathcal{e}(L^2(G)) = \left\| T\_{\mu\_0} \right\|\_{\mathcal{L}^\theta(L^2(G))}^{\alpha\_{j'}}.$$

Again by left-invariance,

$$\left\| f\_0(y\_1^{-\beta\_1}\dots y\_j^{-\beta\_j} \cdot) \right\|\_{L^2(G)} = \|f\_0\|\_{L^2(G)} \le \|\kappa\_1\|\_{L^2(G)},$$

which is finite by Lemma 4.5.5 since we assume νo > Q/2. Therefore,

$$\|f\|\_{2} \le \|\kappa\_1\|\_{L^2(G)} \|T\_{\mu\_0}\|\_{\mathcal{C}(L^2(G))}^{\sum\_{j$$

We have obtained that for any (y1,...,y) ∈ G, α, β with α = 0,

$$\begin{split} \left| \int\_{G} \omega^{a\_{o}} d \left\{ f\_{0} (y\_{1}^{-\beta\_{1}} \cdot \ldots {}\_{j}^{-\beta\_{j}} \cdot ) \right\} \* \left\{ \mu\_{0}^{\*\alpha\_{j+1}} (y\_{j+1}^{-\beta\_{j+1}} \cdot) \right\} \* \ldots \* \left\{ \mu\_{0}^{\*\alpha\_{\ell}} (y\_{\ell}^{-\beta\_{\ell}} \cdot) \right\} \right| \\ \leq \sum\_{j:\alpha\_{j}=1} \left( 1 + \ell^{3} + \ell \max\_{j'=1,\ldots,\ell} |y\_{j'}| \right)^{a\_{o}} \left| \bar{B}\_{\ell^{3} + \ell \max\_{j'=1,\ldots,\ell} |y\_{j'}|} \right|^{1/2} \\ \left\| T\_{\mu\_{0}} \right\|\_{\mathcal{L}^{\ell}(L^{2}(G))}^{\sum\_{j$$

with C = κ1L2(G)|B¯1| 1/2. We now integrate against df ∗β1 ∞ (y1). . . df ∗β ∞ (y) to obtain

$$\begin{split} & \left| \int\_{G} \omega^{a\_{o}} d\mu\_{0}^{\*\alpha\_{1}} \ast f\_{\infty}^{\*\beta\_{1}} \ast \dots \ast \mu\_{0}^{\*\alpha\_{\ell}} \ast f\_{\infty}^{\*\beta\_{\ell}} \right| \\ & \leq \int\_{G^{\ell}} \left| \int\_{G} \omega^{a\_{o}} \, d\left\{ \mu\_{0}^{\*\alpha\_{1}} (y\_{1}^{-\beta\_{1}} \cdot) \right\} \ast \dots \ast \left\{ \mu\_{0}^{\*\alpha\_{\ell}} (y\_{\ell}^{-\beta\_{\ell}} \cdot) \right\} \right| \\ & \qquad \qquad \qquad d|f\_{\infty}^{\*\beta\_{1}}| (y\_{1}) \dots d|f\_{\infty}^{\*\beta\_{\ell}}| (y\_{\ell}) \\ & \leq C \| T\_{\mu\_{0}} \|\_{\mathcal{L}(L^{2}(G))}^{\sum\_{j$$

Let us estimate this last integral:

$$\begin{aligned} &\int\_{G^\ell} \ell \left(1 + \ell^3 + \ell \max\_{j'=1,\dots,\ell} |y\_{j'}|\right)^{a\_o + Q/2} d|f^{\*\beta\_1}\_{\infty}|(y\_1)\dots d|f^{\*\beta\_\ell}\_{\infty}|(y\_\ell) \\ &\leq \sum\_{q=0}^\infty \left(1 + \ell^3 + \ell(q+1)\right)^{a\_o + Q/2} \int\_{q \leq \max\_{j'} |y\_{j'}| < q+1} d|f^{\*\beta\_1}\_{\infty}|(y\_1)\dots d|f^{\*\beta\_\ell}\_{\infty}|(y\_\ell). \end{aligned}$$

For each of these integrals, we see that either

$$\int\_{q \le \max\_{j'} |y\_{j'}| < q+1} d|f^{\*\beta\_1}\_{\infty}|(y\_1)\dots d|f^{\*\beta\_\ell}\_{\infty}|(y\_\ell) = \begin{cases} \ 0 & \text{if } \beta = 0, \ q \ge 1 \text{ or } \beta \ne 0, \ q < \ell^2, \\\ 1 & \text{if } \beta = 0, \ q = 0, \end{cases}$$

or if β = 0 and q ≥ -2,

$$\begin{aligned} &\int\_{q\le\max\_{j'}\left|y\_{j'}\right|$$

for any a > 0 that will be suitably chosen. Indeed, by Lemma 4.5.5, this last integral is finite. Hence if β = 0, then

$$\begin{aligned} &\int\_{G^\ell} \ell \left(1 + \ell^3 + \ell \max\_{j'=1,\dots,\ell} |y\_{j'}| \right)^{a\_o + Q/2} d|f\_{\infty}^{\*\beta\_1}|(y\_1)\dots d|f\_{\infty}^{\*\beta\_\ell}|(y\_\ell) \\ &\leq \|f\_\infty\|\_1^{|\beta|-1} \left(\int\_G |\kappa\_1|\omega^a \right) \sum\_{q=\ell^2}^\infty \ell \left(1 + \ell^3 + \ell(q+1) \right)^{a\_o + Q/2} (1+q)^{-a}. \end{aligned}$$

We choose a := 2(ao + Q/2 + 2) so that the sum

$$\begin{aligned} \ell \sum\_{q=\ell^2}^{\infty} \left( 1 + \ell^3 + \ell(q+1) \right)^{a\_o + Q/2} (1+q)^{-a} &\leq \ell (1+2\ell)^{a\_o + Q/2} \sum\_{q=\ell^2}^{\infty} (q+1)^{a\_o + Q/2 - a\_o} \\ &\leq \ell (1+2\ell)^{a\_o + Q/2} \int\_{\ell^2}^{\infty} x^{a\_o + Q/2 - a} dx \leq \ell (1+2\ell)^{a\_o + Q/2} \frac{\ell^{2(a\_o + Q/2 - a + 1)}}{a\_o + Q/2 - a + 1}, \end{aligned}$$

is finite and bounded independently of -. We set

$$C\_a := \max\left(1, \left(\int\_G |\kappa\_1| \omega^a \right) \max\_{\ell \in \mathbb{N}} \sum\_{q=\ell^2}^{\infty} \left(1 + \ell^3 + \ell(q+1) \right)^{a\_o + Q/2} (1+q)^{-a} \right).$$

We have obtained in the case α and β both non-zero:

$$\begin{aligned} & \left| \int\_G \omega^{a\_0} d\mu\_0^{\*\alpha\_1} \ast f\_{\infty}^{\*\beta\_1} \ast \dots \ast \mu\_0^{\*\alpha\_\ell} \ast f\_{\infty}^{\*\beta\_\ell} \right| \\ & \leq CC\_a \sum\_{j:\alpha\_j=1} \left|| T\_{\mu\_0} \right||\_{\mathcal{L}^\ell(L^2(G))}^{\sum\_{j$$

whereas in the case α = 0 and β = 0,

$$\begin{aligned} \left| \int\_G \omega^{a\_o} d\mu\_0^{\*\alpha\_1} \ast f\_{\infty}^{\*\beta\_1} \ast \dots \ast \mu\_0^{\*\alpha\_\ell} \ast f\_{\infty}^{\*\beta\_\ell} \right| &\leq C \sum\_{j:\alpha\_j=1} \|T\_{\mu\_0}\|\_{\mathcal{L}^p(L^2(G))}^{\sum\_{j$$

and in the case β = 0 and α = 0,

$$\begin{split} & \left| \int\_{G} \omega^{a\_{o}} d\mu\_{0}^{\*\alpha\_{1}} \ast f\_{\infty}^{\*\beta\_{1}} \ast \dots \ast \mu\_{0}^{\*\alpha\_{\ell}} \ast f\_{\infty}^{\*\beta\_{\ell}} \right| \\ & \leq \int\_{G^{\ell}} \left( 1 + \ell \max\_{j'=1,\dots,\ell} |y\_{j'}^{\beta\_{j'}}| \right)^{a\_{o}} d|f\_{\infty}^{\*\beta\_{1}}|(y\_{1}) \dots d|f\_{\infty}^{\*\beta\_{\ell}}|(y\_{\ell}) \\ & \leq C \sum\_{q=\ell^{2}}^{\infty} \left( 1 + \ell q \right)^{a\_{o}} \left\| f\_{\infty} \right\|\_{1}^{|\beta|-1} \left( \int\_{G} |\kappa\_{1}| \omega^{a} \right) \left( 1 + q \right)^{-a} \\ & \leq CC\_{a} \|f\_{\infty}\|\_{1}^{|\beta|-1} . \end{split}$$

We can now sum over α and β to obtain

$$\begin{split} \int\_{G} |\kappa\_{\ell}| \omega^{a\_{o}} &\leq \sum\_{\beta \neq 0} CC\_{a} \|f\_{\infty}\|\_{1}^{|\beta|-1} \\ &+ \sum\_{\alpha \neq 0} C(1+\ell^{3})^{a\_{o}+Q/2} \ell \max(1, \|T\_{\mu\_{0}}\|\_{\mathcal{L}(L^{2}(G))})^{|\alpha|} \\ &+ \sum\_{\begin{subarray}{c} \alpha \neq 0 \\ \beta \neq 0 \end{subarray}} CC\_{a} \ell \max(1, \|T\_{\mu\_{0}}\|\_{\mathcal{L}(L^{2}(G))})^{|\alpha|} \|f\_{\infty}\|\_{1}^{|\beta|-1}. \end{split}$$

We need to estimate the operator norm of

$$T\_{\mu\_0} = \mathbf{I} + T\_{\kappa\_1} - T\_{f\_{\infty}}.$$

By functional calculus, the operator

$$\mathcal{I} + T\_{\kappa\_1} = \mathcal{I} + F\_o((\mathcal{I} + \mathcal{R})^{-\ell\_o}) = \exp(i(\mathcal{I} + \mathcal{R})^{-\ell\_o}),$$

has norm

$$\begin{aligned} \|\|\mathbf{I} + T\_{\kappa\_1}\|\|\_{\mathcal{L}^\ell(L^2(G))} &= \|\|\exp(i(\mathbf{I} + \mathcal{R})^{-\ell\_o})\|\|\_{\mathcal{L}^\ell(L^2(G))} \\ &\le \sup\_{\lambda \ge 0} |\exp(i(\mathbf{1} + \lambda)^{-\ell\_o})| \le 1. \end{aligned}$$

For Tf∞, we have

$$\|T\_{f\_{\infty}}\|\_{\mathcal{Q}}(L^2(G)) \le \|f\_{\infty}\|\_{L^1(G)}.$$

Hence

$$\|T\_{\mu\_o}\|\_{\mathcal{L}'(L^2(G))} \le \|\mathcal{I} + T\_{\kappa\_1}\|\_{\mathcal{L}'(L^2(G))} + \|T\_{f\_\infty}\|\_{\mathcal{L}'(L^2(G))} \le 1 + \|f\_\infty\|\_{L^1(G)},$$

and

$$\|f\_{\infty}\|\_1 + \max(1, \|T\_{\mu\_0}\|\_{\mathcal{L}^\varrho(L^2(G))}) \le 1 + 2\|f\_{\infty}\|\_1.$$

Let us also estimate

$$\begin{aligned} \|f\_{\infty}\|\_{L^{1}(G)} &= \int\_{|x| \ge \ell^{2}} |\kappa\_{1}(x)| \omega(x) \frac{1}{1+|x|} dx \\ &\le \quad \frac{1}{1+\ell^{2}} \int\_{G} |\kappa\_{1}| \omega. \end{aligned} \tag{4.57}$$

By Lemma 4.5.5,

$$c' := \int\_G |\kappa\_1|\omega$$

is finite. Thus we have obtained so far that

$$\begin{split} \int\_{G} |\kappa\_{\ell}| \omega^{a\_{o}} &\leq \quad CC\_{a} \ell \sum\_{\beta \neq 0, \alpha} (1+2\|f\_{\infty}\|\_{1})^{|\alpha|} \|f\_{\infty}\|\_{1}^{|\beta|-1} \\ &+ CC\_{a} (1+\ell^{3})^{a\_{o} + \frac{Q}{2} + 1} \sum\_{\alpha \neq 0} (1+2\|f\_{\infty}\|\_{1})^{|\alpha|} \\ &\leq \quad CC\_{a} \ell \sum\_{\beta \neq 0, \alpha} \left(1 + \frac{2c'}{1+\ell^{2}}\right)^{|\alpha|} \left(\frac{c'}{1+\ell^{2}}\right)^{|\beta|-1} \\ &+ CC\_{a} (1+\ell^{3})^{a\_{o} + \frac{Q}{2} + 1} \sum\_{\alpha \neq 0} \left(1 + \frac{2c'}{1+\ell^{2}}\right)^{|\alpha|} \\ &\leq \quad CC\_{a} (1+\ell^{3})^{a\_{o} + \frac{Q}{2} + 1} \sum\_{\beta, \alpha} \left(1 + \frac{2c'}{1+\ell^{2}}\right)^{|\alpha|} \left(\frac{c'}{1+\ell^{2}}\right)^{|\beta|} \\ &= CC\_{a} (1+\ell^{3})^{a\_{o} + \frac{Q}{2} + 1} \left(1 + \frac{3c'}{1+\ell^{2}}\right)^{\ell} .\end{split}$$

Since

$$\max\_{\ell \in \mathbb{N}} \left( 1 + \frac{3c'}{1 + \ell^2} \right)^{\ell} < \infty,$$

we have proved what we wanted, that is,

$$\int\_G |\kappa\_\ell| \omega^{a\_o} \le C'(1 + \ell^3)^{a\_o + \frac{Q}{2} + 1},$$

completing the proof of Lemma 4.5.6. -

The proof of Lemma 4.5.6 can be modified to obtain a similar property for Xαo Fo(-(I + R)− o )δo:

**Lemma 4.5.8.** We keep the notation of Lemmata 4.5.4, 4.5.5 and 4.5.6. Then for any αo ∈ Nn 0 with νo > [αo] + Q/2 and ao ≥ 0, we have

$$\int\_{G} (1+|x|)^{a\_o} |X^{\alpha\_o} F\_o(\ell(\mathbf{I}+\mathcal{R})^{-\ell\_o}) \delta\_o(x)| dx \le C|\ell|^{3(a\_o+\frac{Q}{2}+1)},\tag{4.58}$$

where C = Cαo,ao,o,R,G,|·| is a positive constant independent of -∈ Z.

The proof of Lemma 4.5.8 follows the one of Lemma 4.5.6 but with the derivatives Xαo now applied to the last term of the convolution products. These convolution products have to be understood as convolutions between compactly supported distributions and integrable functions since we replace the definition of f0 and f∞ given in (4.55) by

$$f\_0 := \kappa\_1 \chi(\ell^{-1}|\cdot|') \quad \text{and} \quad f\_{\infty} := \kappa\_1 (1 - \chi)(\ell^{-1}|\cdot|'),$$

where

We also need to replace ω =1+ |·|ao with

$$
\omega(x) = \begin{cases} 1 & \text{if } |x| \le 1, \\\ |x| & \text{if } |x| \ge 1, \end{cases}
$$

in order to make sense of the distribution Xαo δ0 with support at 0 being applied to the function ω which is smooth around 0. The conditions on the parameters given in Lemma 4.5.8 and the following lemma ensure that the new quantities G |Xαo κ| 2 and G |Xαo κ|ωa appearing in the proof are finite.

**Lemma 4.5.9.** Let R be a Rockland operator of homogeneous degree ν on a graded Lie group G and let Ba, a ∈ C+, be the kernels of its Bessel potentials.

1. If |·| is a homogeneous quasi-norm on G, b ≥ 0 and α ∈ Nn 0 with Re a + b > [α], then

$$\int\_G |x|^b |X^\alpha \mathcal{B}\_a(x)| dx < \infty.$$

2. If α ∈ Nn 0 with Re a > [α] + Q/2, then

$$\int\_G |X^\alpha \mathcal{B}\_a(x)|^2 dx < \infty.$$

Proof of Lemma 4.5.9. For the first part, we generalise the first part of Lemma 4.3.15, that is,

$$\int\_G |x|^b |X^\alpha \mathcal{B}\_a(x)| dx \le \frac{1}{|\Gamma(\frac{a}{\nu})|} \int\_0^\infty t^{\frac{\operatorname{Re} a}{\nu} - 1} e^{-t} \int\_G |x|^b |X^\alpha h\_t(x)| dx \, dt,$$

and using the homogeneity of the heat kernel (see (4.17)) and the change of variables y = t − 1 ν x, we get

$$\int\_{G} |x|^b |X^\alpha h\_t(x)| dx = \int\_{G} |t^{\frac{1}{\nu}} y|^b t^{-\frac{[\alpha]}{\nu}} |X^\alpha h\_1(y)| dy = c\_{b,\alpha} t^{\frac{b-[\alpha]}{\nu}},$$

where cb,α = |y| bXαh1(y)L1(dy) is a finite constant since h1 ∈ S(G). Thus,

$$\int\_{G} |x|^b |X^{\alpha} \mathcal{B}\_a(x)| dx \le \frac{c\_{b,\alpha}}{|\Gamma(\frac{a}{\nu})|} \int\_0^\infty t^{\frac{\operatorname{Ra} a}{\nu} - 1 + \frac{b - \lceil \alpha \rceil}{\nu}} e^{-t} dt$$

is finite whenever Re a + b − [α] > 0.

#### 4.5. Hulanicki's theorem 267

For the second part of Lemma 4.5.9, we can not adapt the proof of the square integrability of Ba since we can not relate Xαh¯(x) with (Xαh)(x−1). We proceed differently. Let φ ∈ D(G). From the properties of the heat kernel and the definition of Ba we have

$$\mathcal{B}\_a(x^{-1}) = \bar{\mathcal{B}}\_a(x) = \overline{\mathcal{B}\_{\bar{a}}(x)}$$

and so

$$\begin{aligned} \left| \int\_G (X^\alpha \mathcal{B}\_a) \phi \right| &= \left| \int\_G \mathcal{B}\_a X^\alpha \phi \right| \\ &= \left| (X^\alpha \phi) \* \mathcal{B}\_{\bar{a}}(0) \right| \\ &\leq \left\| (X^\alpha \phi) \* \mathcal{B}\_{\bar{a}} \right\|\_{\infty} \\ &\leq \left\| C \right\| \left( \mathbf{I} + \bar{\mathcal{R}}\_2 \right)^{\frac{\mathsf{s}}{\mathsf{b}}} (X^\alpha \phi) \* \bar{\mathcal{B}}\_{\bar{a}} \right\|\_{2} \end{aligned}$$

by the Sobolev embeddings, see Theorem 4.4.25 for s > Q/2. But, since Ba¯ is the right-convolution kernel of the Bessel potential (I + R¯2) −a ν , we have

$$\begin{aligned} \left\| \left( \mathbf{I} + \bar{\mathcal{R}}\_2 \right)^{\frac{s}{\nu}} (X^{\alpha} \phi) \* \bar{\mathcal{B}}\_{\bar{a}} \right\|\_{2} &= \left\| \left( \mathbf{I} + \bar{\mathcal{R}}\_2 \right)^{\frac{s - \operatorname{Ra} a}{\nu}} (X^{\alpha} \phi) \right\|\_{2} \\ &\leq \left\| \left( \mathbf{I} + \bar{\mathcal{R}}\_2 \right)^{\frac{s - \operatorname{Ra} a + [\alpha]}{\nu}} \phi \right\|\_{2}, \end{aligned}$$

by Theorem 4.4.16. Hence we have obtained

$$\left| \int\_{G} X^{\alpha} \mathcal{B}\_{a} \phi \right| \leq CC' \left\| \left( \mathbf{I} + \bar{\mathcal{R}}\_{2} \right)^{\frac{s - \operatorname{Re} a + \lceil a \rceil}{\nu}} \phi \right\|\_{2} \leq C\_{1} \|\phi\|\_{2},$$

if s − Re a + [α] ≤ 0, from the Sobolev inclusions, see Theorem 4.4.3 (4). Hence, under this condition, XαBa ∈ L2(G). -

Proceeding as in the proof of Lemma 4.5.5, Lemma 4.5.9 yields

The details of the proof of Lemma 4.5.8 are left to the interested reader.

#### **Last step**

We can now conclude the proof of Theorem 4.5.1.

End of the proof of Theorem 4.5.1. We fix ao ∈ N and αo ∈ Nn 0 . We consider k ∈ N and o to be chosen in terms of ao and αo. Let m ∈ Ck[0, ∞) satisfying the

,

hypotheses of Lemma 4.5.3 Part 3 for o and k, and let F be the corresponding function in Ck(R). Since supp F ⊂ [0, 2], we may develop F in the Fourier series:

$$F(\xi) = \sum\_{\ell \in \mathbb{Z}} \widehat{F}(\ell) e^{i\xi \ell} = \sum\_{\ell \in \mathbb{Z} \backslash \{0\}} \widehat{F}(\ell) e^{i\xi \ell}, \quad \xi \in (-\pi, \pi),$$

as F-(0) = 0. Since

$$0 = F(0) = \sum\_{\ell \in \mathbb{Z}\backslash\{0\}} \widehat{F}(\ell),$$

we also have for any ξ ∈ [0, 2] that

$$F(\xi) = \sum\_{\ell \in \mathbb{Z}\backslash\{0\}} \widehat{F}(\ell)(e^{i\xi\ell} - 1).$$

This yields

$$m(\lambda) = \sum\_{\ell \in \mathbb{Z}\backslash\{0\}} \widehat{F}(\ell)(e^{i\ell(1+\lambda)^{-\ell\_o}} - 1), \quad \lambda \ge 0,$$

and at least formally

$$m(\mathcal{R})\delta\_o = \sum\_{\ell \in \mathbb{Z} \backslash \{0\}} \widehat{F}(\ell)\kappa\_\ell,\tag{4.59}$$

where κ := ei(I+R)−o δ0 − δ0 as before. By Lemma 4.5.8, if νo > [αo] + Q/2, then

$$\begin{split} &\sum\_{\ell\in\mathbb{Z}\backslash\{0\}}|\hat{F}(\ell)|\int\_{G}(1+|x|)^{a\_{o}}|X^{a\_{o}}\kappa\_{\ell}(x)|dx \\ &\leq C\sum\_{\ell\in\mathbb{Z}\backslash\{0\}}|\hat{F}(\ell)||\ell|^{3(a\_{o}+\frac{Q}{2}+1)} \\ &\leq CC\_{k}\sup\_{\begin{subarray}{c}\lambda\in\mathbb{R}\\j=0,\ldots,k\end{subarray}}(1+|\lambda|)^{1+j+k\ell\_{o}}|m^{(j)}(\lambda)|\sum\_{\ell\in\mathbb{Z}\backslash\{0\}}(1+|\ell|)^{-k}|\ell|^{3(a\_{o}+\frac{Q}{2}+1)},\end{split}$$

see Lemma 4.5.3 for the estimates of |F-(-)|. This last sum converges provided that we have chosen k > 3(ao + Q 2 + 1) + 2. We assume that we have chosen such o and k. One can now show easily that m(R)δo ∈ L1(G) and that (4.59) holds in L1(G). Furthermore, Xαom(R)δo ∈ L1(G) and

$$\begin{aligned} \int\_G (1+|x|)^{a\_o} |X^{\alpha\_o} m(\mathcal{R}) \delta\_0(x)| dx &\leq \sum\_{\ell \in \mathbb{Z} \backslash \{0\}} |\widehat{F}(\ell)| \int\_G (1+|x|)^{a\_o} |X^{\alpha\_o} \kappa\_\ell(x)| \\ &\leq \|C\_k' \sup\_{\begin{subarray}{c} \lambda \geq 0 \\ j=0,\ldots,k \end{subarray}} (1+|\lambda|)^{1+j+k\ell\_o} |m^{(j)}(\lambda)|. \end{aligned}$$

This concludes the proof of Theorem 4.5.1. -

#### **4.5.3 Proof of Corollary 4.5.2**

Let us show Corollary 4.5.2.

Proof of Corollary 4.5.2. Applying Theorem 4.5.1 to the restriction of m ∈ S(R) to [0, ∞), we have for any α and a > 0 that

$$\|m(\mathcal{R})\delta\_0\|\_{\mathcal{S},a,[\alpha],1} \le C \sup\_{\substack{\lambda \ge 0\\j=0,\dots,k}} (1+|\lambda|)^{1+j+k\ell\_o} |m^{(j)}(\lambda)|,$$

with k := 3(a + Q 2 + 1) + 3 and o the smallest integer such that νo > [α] + Q/2. Clearly the right-hand side of this inequality is less than a S(R)-seminorm of m, up to a constant depending on this seminorm. By Proposition 4.4.27, this concludes the proof of Corollary 4.5.2. -

We may simplify the proof of Corollary 4.5.2 by modifying the proof of Theorem 4.5.1 and choosing F independently of k for m ∈ S(R). Indeed, it suffices to set

$$F(\xi) := \begin{cases} \ m \left( \xi^{-\frac{1}{\xi}\_o} - 1 \right) \chi\_1(\xi) & \text{if } \xi > 0, \\ 0 & \text{if } \xi \le 0, \end{cases}$$

where χ1 ∈ D(R) is supported in [−1, 2] and satisfies χ1 ≡ 1 on [0, 1] together with F-(0) = 0.

Remark 4.5.10. Behind this technical point lays the fact that the spectrum of R2 is contained in [0, ∞) thus we can modify any function m as we see fit on (−∞, 0) without changing the operator m(R).

We had already used this idea in the proof of Theorem 4.5.1 indirectly, since different extensions of the function F will lead to the same formula in (4.59).

Note that we may also modify the function m at 0, see Remark 4.2.8 (3), but we did not use this point in the proofs of Theorem 4.5.1 or Corollary 4.5.2.

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# **Chapter 5**

# **Quantization on graded Lie groups**

In this chapter we develop the theory of pseudo-differential operators on graded Lie groups. Our approach relies on using positive Rockland operators, their fractional powers and their associated Sobolev spaces studied in Chapter 4. As we have pointed out in the introduction, the graded Lie groups then become the natural setting for such analysis in the context of general nilpotent Lie groups.

The introduced symbol classes Sm ρ,δ and the corresponding operator classes

$$
\Psi^{m}\_{\rho,\delta} = \operatorname{Op} S^{m}\_{\rho,\delta},
$$

for (ρ, δ) with 1 ≥ ρ ≥ δ ≥ 0 and δ = 1, have an operator calculus, in the sense that the set m∈R Ψm ρ,δ forms an algebra of operators, stable under taking the adjoint, and acting on the Sobolev spaces in such a way that the loss of derivatives is controlled by the order of the operator. Moreover, the operators that are elliptic or hypoelliptic within these classes allow for a parametrix construction whose symbol can be obtained from the symbol of the original operator.

During the construction of the pseudo-differential calculus ∪m∈RΨm ρ,δ on graded Lie groups in this chapter, there are several difficulties one has to overcome and which do not appear in the case of compact Lie groups as described in Chapter 2. The immediate one is the need to find a natural framework for discussing the symbols to which we will be associating the operators (quantization) and we will do so in Section 5.1. In Section 5.2 we define symbol classes leading to algebras of symbols and operators and discuss their properties. The symbol classes that we introduce are based on a positive Rockland operator on the group and contain all the left-invariant differential operators. As with Sobolev spaces, the symbol classes can be shown to be actually independent of the choice of a positive Rockland operator used in their definition. In Section 5.3 we show that the multipliers of Rockland operators are in the introduced symbol classes. We investigate the behaviour of the kernels of operators corresponding to these symbols in Section 5.4, both at 0 and at infinity and show, in particular, that they are Calder´on-Zygmund (in the sense of Coifman and Weiss, see Sections 3.2.3 and A.4). The symbolic calculus is established in Section 5.5. In Section 5.7 we show that the operators satisfy an analogue of the Calder´on-Vaillancourt theorem. The construction of parametrices for elliptic and hypoelliptic operators in the calculus is carried out in Section 5.8.

#### **Conventions**

Throughout Chapter 5, G is always a graded Lie group, endowed with a family of dilations with integer weights. Its homogeneous dimension is denoted by Q. Also throughout, R will be a homogeneous positive Rockland operator of homogeneous degree ν. If G is a stratified Lie group, we can choose R = −L with L a sub-Laplacian, or another homogeneous positive Rockland operator. Since it is a left-invariant differential operator, we denote by π(R) the operator described in Definition 1.7.4. Both R and π(R) and their properties have been extensively discussed in Chapter 4, especially Section 4.1.

Finally, when we write

sup π∈G-

we always understand it as the essential supremum with respect to the Plancherel measure on G-.

# **5.1 Symbols and quantization**

The global quantization naturally occurs on any unimodular Lie (or locally compact) group of type 1 thanks to the Plancherel formula, see Subsection 1.8.2 for the Plancherel formula. The quantization was first noticed by Michael Taylor in [Tay86, Section I.3]. The case of locally compact type 1 groups was studied recently in [MR15]. The case of the compact Lie groups was described in Section 2.2.1. Here we describe the particular case of graded nilpotent Lie groups, with an emphasis on the technical meaning of the objects involved. A very brief outline of the constructions of this chapter appeared in [FR14a].

Formally, for a family of operators σ(x, π) on Hπ parametrised by x ∈ G and π ∈ G-, we associate the operator T = Op(σ) given by

$$T\phi(x) := \int\_{\widehat{G}} \text{Tr}\left(\pi(x)\sigma(x,\pi)\widehat{\phi}(\pi)\right)d\mu(\pi). \tag{5.1}$$

Again formally, the Fourier inversion formula implies that if σ(x, π) does not depend on x and is the group Fourier transform of some function κ, i.e. if σ(x, π) = κ-(π), then Op(σ) is the convolution operator with right-convolution kernel κ, i.e. Op(σ)φ = φ∗κ. We would like this to be true not only for (say) integrable functions κ but also for quite a large class of distributions, in order

$$\text{to quantize } X^{\alpha} = \text{Op}(\sigma) \text{ by } \sigma(x,\pi) = \pi(X)^{\alpha},$$

with π(X) as in Definition 1.7.4.

The first problem is to make sense of the objects above. The dependence of σ on x is not problematic for the interpretation in the formula (5.1), but we have identified a unitary irreducible representation π with its equivalence class and the families of operators may be measurable in π ∈ Rep G but not defined for all π ∈ G-. More worryingly, we would like to consider collections of operators which are unbounded, for instance such as π(X)α, π ∈ G-. For these reasons, it may be difficult to give a meaning to the formula (5.1) in general.

Thus, our first task is to define a large class of collections of operators σ(x, π), x ∈ G, π ∈ G-, for which we can make sense of the quantization procedure. We will use the realisations

$$
\mathcal{K}(G), \ L^{\infty}(\hat{G}), \quad \text{and} \quad \mathcal{Q}\_L(L^2(G)).
$$

of the von Neumann algebra of the group G described in Section 1.8.2. We will also use their generalisations

$$\mathcal{K}\_{a,b}(G), \ L\_{a,b}^{\infty}(\hat{G}), \quad \text{and} \quad \mathcal{E}\_L^{\rho}(L\_a^2(G), L\_b^2(G)).$$

which we define in Section 5.1.2. In order to do so we use a special feature of our setting, namely the existence of positive Rockland operators and the corresponding L2-Sobolev spaces.

#### **5.1.1 Fourier transform on Sobolev spaces**

In Section 4.3, we have discussed in detail the fractional powers of a positive Rockland operator R and of the operator I +R. In the sequel, we will also need to understand powers of the operators π1(I + R), π1 ∈ Rep G. We now address this, and use it to extend the group Fourier transform to the Sobolev spaces L2 a(G).

From now on we will keep the same notation for the operators R and π1(R) (where π1 ∈ Rep (G)) and their respective self-adjoint extensions, see Proposition 4.1.15. We note that by Proposition 4.2.6 the operator π1(R) is also positive. We can consider the powers of I + R and π1(I + R)= I+ π1(R) as defined by the functional calculus

$$(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}} = \int\_0^\infty (1 + \lambda)^{\frac{a}{\nu}} dE(\lambda), \quad \pi\_1 (\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}} = \int\_0^\infty (1 + \lambda)^{\frac{a}{\nu}} dE\_{\pi\_1}(\lambda),$$

where E and Eπ1 are the spectral measures of R and π1(R), respectively, and ν is the homogeneous degree of R, see Corollary 4.1.16.

Remark 5.1.1. If a/ν is a positive integer, there is no conflict of notation between

Indeed, if a = ν, the two coincide. If a = ν, - ∈ N, then the operator π1(I + R) a ν defined by functional calculus coincides with the --th power of π1(I+R). The case a = 0 is trivial.

We can describe more concretely the operators π1(I + R) a ν , π1 ∈ Rep G.

**Lemma 5.1.2.** Let R be a positive Rockland operator of homogeneous degree ν. As in Corollary 4.3.11, we denote by Ba the right-convolution kernels of its Bessel potentials (I + R)− a ν , Re a > 0.

If a ∈ C with Re a < 0, then B−a is an integrable function and

$$
\forall \pi\_1 \in \text{Rep} \, G \qquad \pi\_1(\mathbf{I} + \mathcal{R})^{\triangleq} = \widehat{\mathcal{B}}\_{-a}(\pi\_1).
$$

For any a ∈ C and any π1 ∈ Rep G, the operator π1(I + R) a ν maps H∞ π1 onto H∞ π1 bijectively. Furthermore, the inverse of π1(I + R) a ν is π1(I + R)− a ν as operators acting on H∞ π1 .

Proof. Let a ∈ C, Re a < 0. Then the Bessel potential (I + R) a ν coincides with the bounded operator with right-convolution kernel B−a ∈ L1(G), see Corollary 4.3.11. Therefore, (I + R) a ν ∈ LL(L2(G)) and

$$\mathcal{F}\_G\{(\mathcal{I}+\mathcal{R})^{\frac{a}{\nu}}f\} = \mathcal{F}\_G\{f\*\mathcal{B}\_{-a}\} = \widehat{\mathcal{B}}\_{-a}\widehat{f}, \quad f \in L^2(G).$$

Now we apply Corollary 4.1.16 with the bounded multiplier given by φ(λ) = (1 + λ) a ν , λ ≥ 0. By Equality (4.5) in Corollary 4.1.16, we obtain

$$\mathcal{F}\_G\{(\mathbf{I}+\mathcal{R})^{\frac{a}{\nu}}f\} = \pi(\mathbf{I}+\mathcal{R})^{\frac{a}{\nu}}\widehat{f}, \quad f \in L^2(G).$$

The injectivity of the group Fourier transform on K(G) yields that B-−a(π) = π(I + R) a ν for any π ∈ G-, and the first part of the statement is proved.

Let a ∈ C. We apply Corollary 4.1.16 with the multiplier given by φ(λ) = (1+λ) a ν , λ ≥ 0. Although this multiplier is unbounded, simple modifications of the proof show that Equality (4.5) in Corollary 4.1.16 still holds for f in the domain of the operator. Recall that the domain of (I + R) a ν contains S(G) by Corollary 4.3.16 and moreover (I + R) a ν S(G) = S(G).  Recall that π1(f)v ∈ H∞ π1 when f ∈ S(G) by Proposition 1.7.6 (iv), hence here π1{(I+R) a ν f}v ∈ H∞ π as well. By Lemma 1.8.19, π1(I+R) a ν maps H∞ π1 to H∞ π1 . The spectral calculus implies that as operators acting on H∞ π1 , we have

$$
\pi\_1(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}} \pi\_1(\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}} = \mathbf{I}\_{\mathcal{H}^{\infty}\_{\pi\_1}} \quad \text{and} \quad \pi\_1(\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}} \pi\_1(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}} = \mathbf{I}\_{\mathcal{H}^{\infty}\_{\pi\_1}}.
$$

Consequently, the inverse of π1(I + R) a ν is π1(I + R)− a ν as operators defined on H∞ π1 and π1(I + R) a ν H∞ π1 = H∞ π1 . -

Lemma 5.1.2 and Remark 4.1.17 now imply easily

**Corollary 5.1.3.** Let R be a positive Rockland operator of homogeneous degree ν. For any a ∈ C, {π(I + R) a ν : H∞ π → H∞ π , π ∈ G-} is a measurable G--field of operators acting on smooth vectors (in the sense of Definition 1.8.14).

Lemma 5.1.2 together with the Plancherel formula (see Section 1.8.2) and Corollary 4.3.11 also imply

**Corollary 5.1.4.** Let R be a positive Rockland operator of homogeneous degree ν. For any a ∈ R, we have

$$a > Q/2 \qquad \implies \quad \{\pi(\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}}, \pi \in \widehat{G}\} \in L^2(\widehat{G}),$$

and also, for a > Q/2,

$$\|\|\pi(\mathcal{I}+\mathcal{R})^{-\frac{a}{\nu}}\|\|\_{L^2(\widehat{G})} = \|\|\widehat{\mathcal{B}}\_a(\pi)\|\|\_{L^2(\widehat{G})} = \|\|\mathcal{B}\_a\|\|\_{L^2(G)} < \infty.$$

Note that an analogue of Corollary 5.1.4 for compact Lie groups may be obtained by noticing that (2.15) yields

$$m > n/2 \qquad \implies \qquad \sum\_{\pi \in \widehat{G}} d\_{\pi} ||\pi(\mathbf{I} - \mathcal{L}\_{G})^{-\frac{m}{2}}||\_{\mathbf{HS}}^{2} = \sum\_{\pi \in \widehat{G}} d\_{\pi}^{2} \langle \pi \rangle^{-2m} < \infty.$$

The following statement describes an important property of the field {π(I + R) a ν , π ∈ G-}, in relation with the right Sobolev spaces (see Section 4.4.8 for right Sobolev spaces):

**Proposition 5.1.5.** Let R be a positive Rockland operator on G of homogeneous degree ν. Let also a ∈ R.

If f ∈ L˜2 a(G), then (I + R˜) a ν f ∈ L2(G) and there exists a field of operators {σπ : H∞ π → Hπ , π ∈ G-} such that

$$\{\sigma\_{\pi}\pi(\mathbf{I}+\mathcal{R})^{\frac{a}{\nu}}:\mathcal{H}\_{\pi}^{\infty}\to\mathcal{H}\_{\pi},\ \pi\in\hat{G}\}\in L^{2}(\hat{G}),\tag{5.2}$$

and for almost all π ∈ G-,

$$\mathcal{F}\_G\{(\mathbf{I}+\tilde{\mathcal{R}})^{\frac{a}{\nu}}f\}(\pi)=\sigma\_\pi\pi(\mathbf{I}+\mathcal{R})^{\frac{a}{\nu}}.\tag{5.3}$$

Conversely, if {σπ : H∞ π → Hπ , π ∈ G-} satisfies (5.2) then there exists a unique function f ∈ L˜2 a(G) satisfying (5.3).

In Proposition 5.1.5, σππ(I + R) a ν is not obtained as the composition of (possibly) unbounded operators as in Definition A.3.2. Instead, for σππ(I + R) a ν , it is viewed as the composition of a field of operators defined on smooth vectors with a field of operators acting on smooth vectors, see Section 1.8.3.

In Proposition 5.1.5, we use the right Sobolev spaces associated with the positive Rockland operator R. These spaces are in fact independent of the choice of a positive Rockland operator used in their definition, see Sections 4.4.5 and 4.4.8. Consequently, if (5.2) holds for one positive Rockland operator then (5.2) and (5.3) hold for any positive Rockland operator and the Sobolev norm of f ∈ L2(G), using one particular positive Rockland operator R, is equal to the L2(G-)-norm of (5.2).

Proof of Proposition 5.1.5. If f ∈ L˜2 a(G), then by Theorem 4.4.3 (3) (see also Section 4.4.8), we have that fa := (I +R˜) a ν f is in L2(G) and its Fourier transform is a field of bounded operators (in fact in the Hilbert-Schmidt class). By Lemma 5.1.2, π(I + R)− a ν maps H∞ π onto itself. Hence we can define

$$
\sigma\_{\pi} := \pi(f\_a)\pi(\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}},
$$

as an operator defined on H∞ π . One readily checks that the operators σπ, π ∈ G-, satisfy (5.2) and (5.3).

For the converse, if {σπ : H∞ π → Hπ : π ∈ G-} satisfies (5.2) then we define the function

$$L^2(G) \ni f\_a := \mathcal{F}\_G^{-1}\{\sigma\_\pi \pi (\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}}\},$$

which is square integrable by the Plancherel theorem (see Theorem 1.8.11), and the function

$$f := (\mathbf{I} + \tilde{\mathcal{R}})^{-\frac{a}{\nu}} f\_a,$$

which will be in L˜2 a(G) by Theorem 4.4.3 (3). One readily checks that the function f satisfies the properties described in the statement. -

We now aim at stating and proving a property similar to Proposition 5.1.5 for the left Sobolev spaces. It will use the composition of a field with π(I+R) a ν on the left and this is problematic when we consider any general field σ = {σπ : H∞ π → Hπ} without utilising the composition of unbounded operators as in Definition A.3.2. To overcome this problem, we introduce the following notion:

**Definition 5.1.6.** Let π1 ∈ Rep G and a ∈ R. We denote by Ha π1 the Hilbert space obtained by completion of H∞ π1 for the norm

$$\|\cdot\|\|\_{\mathcal{H}^{a}\_{\pi\_1}} : v \longmapsto \|\pi\_1(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}}v\|\|\_{\mathcal{H}\_{\pi\_1}} := \|v\|\|\_{\mathcal{H}^{a}\_{\pi\_1}},$$

where R is a positive Rockland operator on G of homogeneous degree ν.

We may call them the Hπ1 -Sobolev spaces. Note that in the case of the Schr¨odinger representation for the Heisenberg group, they coincide with Shubin-Sobolev spaces, see Section 6.4.3. More generally, if we realise an element π ∈ G- as a representation π1 acting on some L2(Rm) via the orbit methods, see Section 1.8.1, then we view the corresponding Sobolev spaces as tempered distributions: Ha π1 ⊂ S (Rm).

The following lemma is a routine exercise.

# **Lemma 5.1.7.** Let π1 ∈ Rep G and a ∈ R.

$$a < b \implies \mathcal{H}\_{\pi\_1}^b \subset \mathcal{H}\_{\pi\_1}^a.$$

For any a, b ∈ R, the operator π1(I + R) a ν maps Hb π1 to Hb−a π1 injectively and continuously. In this way, Ha π1 and H−a π1 are in duality via

$$\langle u, v \rangle\_{\mathcal{H}^{a}\_{\pi\_1} \times \mathcal{H}^{-a}\_{\pi\_1}} := (\pi\_1(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}} u, \pi\_1(\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}} \bar{v})\_{\mathcal{H}\_{\pi\_1}}.$$

This duality extends the Hπ1 duality in the sense that

$$\forall u \in \mathcal{H}\_{\pi\_1}^a \cap \mathcal{H}\_{\pi\_1}, \ v \in \mathcal{H}\_{\pi\_1}^{-a} \cap \mathcal{H}\_{\pi\_1} \qquad \langle u, v \rangle\_{\mathcal{H}\_{\pi\_1}^a \times \mathcal{H}\_{\pi\_1}^{-a}} = (u, \bar{v})\_{\mathcal{H}\_{\pi\_1}}.$$

4. If π2 is another strongly continuous representation such that π1 ∼T π2, that is, T is a unitary operator satisfying T π1 = πT2, then T maps H∞ π1 to H∞ π2 bijectively by Lemma 1.8.12 and extends uniquely to an isometric operator Ha π1 → Ha π2 .

Lemma 5.1.7, especially Part 4, shows that G--fields with domain or range on these Sobolev spaces make sense:

**Definition 5.1.8.** Let a ∈ R. A G--field of operators σ = {σπ : H∞ π → Hπ, π ∈ G-} defined on smooth vectors is defined on the Sobolev spaces Ha π when for each π1 ∈ Rep G, the operator σπ1 is bounded on Ha π1 in the sense that

$$\exists C \quad \forall v \in \mathcal{H}\_{\pi\_1}^{\infty} \quad \|\sigma\_{\pi\_1}v\|\_{\mathcal{H}\_{\pi\_1}} \leq C\|v\|\_{\mathcal{H}\_{\pi\_1}^a}.$$

Thus, by density of H∞ π1 in Ha π1 , σπ1 extends uniquely to a bounded operator defined on Ha π1 for which we keep the same notation σπ1 : Ha π1 → Hπ1 .

Example 5.1.9. For any positive Rockland operator of degree ν, the field {π(I + R) a ν , π ∈ G-}, is defined on the Sobolev spaces Ha π. This is an easy consequence of Lemma 5.1.7, especially Part 3.

We will allow ourselves the shorthand notation

$$\sigma = \{ \sigma\_{\pi} : \mathcal{H}\_{\pi}^{a} \to \mathcal{H}\_{\pi}, \pi \in \hat{G} \}, \rho$$

to indicate that the G--field of operators is defined on the Sobolev spaces Ha π.

Instead of Definition 5.1.8, we could also have defined G--fields of operators defined on Ha π-Sobolev spaces in a way similar to Definition 1.8.13 (where G- fields of operators defined on smooth vectors were defined). Naturally, these two viewpoints are equivalent since H∞ π1 is dense in Ha π1 .

However, in order to define G--fields of operators with range in the Ha π-Sobolev spaces, we have to adopt the latter viewpoint in the sense that we modify Definitions 1.8.13 and 1.8.14 (in this way, we make no further assumptions on the fields or on the Sobolev spaces):

**Definition 5.1.10.** Let a ∈ R.

• A G--field of operators defined on smooth vectors with range in the Sobolev spaces Ha π is a family of classes of operators {σπ, π ∈ G-} where

$$\sigma\_{\pi} := \{ \sigma\_{\pi\_1} : \mathcal{H}\_{\pi\_1}^{\infty} \to \mathcal{H}\_{\pi\_1}^{a}, \pi\_1 \in \pi \}$$

for each π ∈ G viewed as a subset of Rep G, satisfying for any two elements σπ1 and σπ2 in σπ:

$$
\pi\_1 \sim\_T \pi\_2 \implies \sigma\_{\pi\_2} T = T \sigma\_{\pi\_1} \text{ on } \mathcal{H}^{\infty}\_{\pi}.
$$

(Here we have kept the same notation for the intertwining operator T and its unique extension between Sobolev spaces Ha π1 → Ha π2 , see Lemma 5.1.7 Part 4.)

• It is measurable when for one (and then any) choice of realisation π1 ∈ π and any vector vπ1 ∈ Ha π1 , as π runs over G-, the resulting field {σπvπ, π ∈ G-} is μ-measurable whenever G vπ2 Ha π dμ(π) < ∞. (Here we assume that all the Ha π-norms are realised via a fixed positive Rockland operator.)

Unless otherwise stated, a G--field of operators defined on smooth vectors with range in the Sobolev spaces Ha π is always assumed measurable. We will allow ourselves the shorthand notation

$$\sigma = \{ \sigma\_{\pi} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}^{a}, \pi \in \hat{G} \}$$

to indicate that the G--field of operators has range in the Sobolev space Ha π.

Naturally, if a G--field of operators is defined on smooth vectors σ = {σπ : H∞ π → Hπ, π ∈ G-} with the usual range Hπ = H0 π, then it has range in the Sobolev spaces Ha π when for each π1 ∈ Rep G and any v ∈ H∞ π1 , we have σπ1 v ∈ Ha π1 .

Moreover, the following property of composition is easy to check: if σ1 has range in Ha π and σ2 is defined on Ha π,

$$\text{i.i.e.} \quad \sigma\_1 = \{\sigma\_{1,\pi} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}^{a}, \pi \in \widehat{G}\} \quad \text{and} \quad \sigma\_2 = \{\sigma\_{2,\pi} : \mathcal{H}\_{\pi}^{a} \to \mathcal{H}\_{\pi}, \pi \in \widehat{G}\},$$

then the following field

$$\sigma\_2 \sigma\_1 := \{ \sigma\_{2,\pi} \sigma\_{1,\pi} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}, \pi \in \dot{G} \}$$

makes sense as a G--field of operators defined on smooth vectors. This coincides or extends the definition of composition of fields (the first one acting on smooth vectors) given in Section 1.8.3.

We can apply this property of composition to σ = {σπ : H∞ π → Ha π, π ∈ G-} and {π(I + R) a ν , π ∈ G-}, see Example 5.1.9 for the latter, to obtain the G--field defined on smooth vectors by

$$\pi(\mathbf{I}+\mathcal{R})^{\triangleq}\sigma = \{\pi(\mathbf{I}+\mathcal{R})^{\triangleq}\sigma\_{\pi} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}, \pi \in \widehat{G}\}.\tag{5.4}$$

We can now state the proposition which will enable us to define the group Fourier transform of a function in a left or right Sobolev space.

### **Proposition 5.1.11.** Let a ∈ R.

(L) If f ∈ L2 a(G), then (I + R) a ν f ∈ L2(G) and there exists a field of operators {σπ : H∞ π → Ha π , π ∈ G-} such that

$$\{\pi(\mathbf{I}+\mathcal{R})^{\triangleq}\boldsymbol{\sigma}\_{\pi}:\mathcal{H}\_{\pi}^{\infty}\to\mathcal{H}\_{\pi},\ \pi\in\hat{G}\}\in L^{2}(\hat{G}),\tag{5.5}$$

$$\mathcal{F}\_G\{(\mathbf{I}+\mathcal{R})^\# f\}(\pi) = \pi(\mathbf{I}+\mathcal{R})^\# \sigma\_\pi,\quad \text{for almost all } \pi \in \widehat{G},\tag{5.6}$$

where R is a positive Rockland operator on G of homogeneous degree ν.

Conversely, if {σπ : H∞ π → Ha π , π ∈ G-} satisfies (5.5) for one positive Rockland operator R, then there exists a unique function f ∈ L2 a(G) satisfying (5.6).

(R) If f ∈ L˜2 a(G), then the (unique) field σ obtained in Proposition 5.1.5 can be extended uniquely into a field {σπ : Ha π → Hπ , π ∈ G-} defined on Ha π.

Properties (L) and (R) are independent of the choice of R.

In Proposition 5.1.11, π(I + R) a ν σπ is not obtained as the composition of (possibly) unbounded operators as in Definition A.3.2 but is understood via (5.4).

In Proposition 5.1.11, we use the left and right Sobolev spaces associated with the positive Rockland operator R. These spaces are in fact independent of the choice of a positive Rockland operator used in their definition, see Sections 4.4.5 and 4.4.8. Consequently, if (5.5) hold for one positive Rockland operator then (5.5) and (5.6) hold for any positive Rockland operator and the Sobolev norm of f ∈ L2(G), using one particular positive Rockland operator R, is equal to the L2(G-)-norm of (5.5).

Proof of Proposition 5.1.11. Property (L). If f ∈ L2 a(G), then by Theorem 4.4.3 (3), we have that fa := (I + R) a ν f is in L2(G) and its Fourier transform is a field of bounded operators (in fact in the Hilbert-Schmidt class). By (5.4) we can define σ = {σπ : H∞ π → Ha π} via σπ := π(I + R)− a ν π(fa). One readily checks that the field σ satisfies (5.2) and (5.3).

For the converse, if {σπ : H∞ π → Ha π : π ∈ G-} satisfies (5.2) then we define the function

$$L^2(G) \ni f\_a := \mathcal{F}\_G^{-1}\{\pi(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}}\sigma\_\pi\},$$

which is square integrable by the Plancherel theorem (see Theorem 1.8.11), and the function

$$f := (\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}} f\_a,$$

which will be in L2 a(G) by Theorem 4.4.3 (3). One readily checks that the function f satisfies the properties described in the statement. This shows the property (L).

Property (R) follows easily from (5.2). -

From the proof above, one can check easily that if f ∈ L2 a(G) or L˜2 a(G) is also in any of the spaces where the group Fourier transform has already been defined, namely, L2(G) or K(G), then σ = {σπ : H∞ π → Hπ, π ∈ G-} will coincide with the group Fourier transform of f. Hence we can extend the definition of the group Fourier transform to Sobolev spaces:

**Definition 5.1.12.** Let a ∈ R. The group Fourier transform of f ∈ L2 a(G) or f ∈ L˜2 a(G) is the field σ of operators defined on smooth vectors given in Proposition 5.1.11.

This leads us to define the following spaces of fields of operators:

**Definition 5.1.13.** (L) Let L2 a(G-) denote the space of fields of operators σ with range in Ha π and satisfying (5.5), that is,

$$\begin{aligned} \sigma &= \{ \sigma\_{\pi} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}^{a}, \,\pi \in \hat{G} \}, \\ \{ \pi (\mathbf{I} + \mathcal{R})^{\#} \sigma\_{\pi} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}, \,\pi \in \hat{G} \} &\in L^{2}(\hat{G}), \end{aligned}$$

for one (and then any) positive Rockland operator of homogeneous degree ν. We also set

$$\|\|\sigma\|\|\_{L^{2}\_{a}(\widehat{G})} := \|\pi(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}}\sigma\_{\pi}\|\_{L^{2}(\widehat{G})}.\tag{5.7}$$

#### 5.1. Symbols and quantization 281

(R) Let L˜2 a(G-) denote the space of fields of operators σ defined on Ha π and satisfying (5.2), that is,

$$\begin{aligned} \sigma &= \{ \sigma\_{\pi} : \mathcal{H}\_{\pi}^{a} \to \mathcal{H}\_{\pi} \text{ }, \pi \in \overset{\circ}{G} \}, \\ \{ \sigma\_{\pi} \pi (\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi} \text{ }, \pi \in \overset{\circ}{G} \} &\in L^{2}(\widehat{G}), \end{aligned}$$

for one (and then any) positive Rockland operator of homogeneous degree ν. We also set

$$\|\sigma\|\_{\hat{L}^2\_a(\hat{G})} := \|\sigma\_\pi \pi(\mathbf{I} + \mathcal{R})^\frac{a}{\nu}\|\_{L^2(\hat{G})}.$$

It is a routine exercise, using Proposition 5.1.11 and the properties of the Sobolev spaces (see Section 4.4), to show that

**Proposition 5.1.14.** Let a ∈ R. If R is a positive Rockland operator of homogeneous degree ν, the map ·L2 a(G-) given by (5.7) is a norm on the vector space L2 a(G-). Endowed with this norm, L2 a(G-) is a Banach space which is independent of R. Two norms corresponding to any two choices of Rockland operators via (5.7) are equivalent.

The Fourier transform FG is an isomorphism between Banach spaces acting from L2 a(G) onto L2 a(G-). It coincides with the usual Fourier transform on L2(G) for a = 0.

Let σ = {σπ, π ∈ G-} be in L2 a(G-). Then ασπ, π ∈ G-

$$is\ in\ L^2\_{a-[\alpha]}(\widehat{G})\ for\ any\ \alpha\in\mathbb{N}^n\_0,\ and.$$

$$\{\pi(\mathbf{I} + \mathcal{R})^{s/\nu}\sigma\_{\pi}, \pi \in \widehat{G}\}$$

}

is in L2 a−s(G-) for any s ∈ R. Furthermore, if f = F−1 G σ ∈ L2 a(G) then

{π(X)

$$\mathcal{F}\_G(X^\alpha f)(\pi) = \pi(X)^\alpha \widehat{f}(\pi) \quad \text{and} \quad \mathcal{F}\_G((\mathcal{I} + \mathcal{R})^{s/\nu} f)(\pi) = \pi(\mathcal{I} + \mathcal{R})^{s/\nu} \widehat{f}(\pi).$$

We have similar results for the right Sobolev spaces. Furthermore the adjoint map σ → σ∗ maps L2 a(G-) → L˜2 a(G-) and L˜2 a(G-) → L2 a(G-) isomorphically as Banach spaces.

Recall that the tempered distributions Xαf and (I + R)s/νf used in the statement just above are respectively defined via

$$
\langle X^{\alpha}f, \phi \rangle = \langle f, \{X^{\alpha}\}^{t} \phi \rangle, \quad \phi \in \mathcal{S}(G), \tag{5.8}
$$

and

$$<\langle (\mathbf{I} + \mathcal{R})^{s/\nu} f, \phi \rangle = \langle f, (\mathbf{I} + \bar{\mathcal{R}})^{s/\nu} \phi \rangle, \quad \phi \in \mathcal{S}(G). \tag{5.9}$$

For (5.9), see Definition 4.3.17. For (5.8), this is the composition of the formula obtained for one vector field (with polynomial coefficients) by integration by parts. See also (1.10) for the definition of {Xα}t .

In Corollary 1.8.3, we stated the inversion formula valid for any Schwartz function on any connected simply connected Lie group. Here we weaken the hypothesis using the Sobolev spaces in the context of a graded Lie group G:

**Proposition 5.1.15** (Fourier inversion formula)**.** Let f be in the left Sobolev space L2 s(G) or in the right Sobolev space L˜2 s(G) with s > Q/2. Then for almost every π ∈ Rep G, the operator f -(π) is trace class with

$$\int\_{\widehat{G}} \text{Tr}|\widehat{f}(\pi)|d\mu(\pi) < \infty. \tag{5.10}$$

Furthermore, f is continuous on G, and for every x ∈ G we have

$$f(x) = \int\_{\hat{G}} \text{Tr}\left(\pi(x)\hat{f}(\pi)\right)d\mu(\pi) = \int\_{\hat{G}} \text{Tr}\left(\hat{f}(\pi)\pi(x)\right)d\mu(\pi). \tag{5.11}$$

In the statement above, as s > Q/2 > 0, the field f is in L2(G-), it is then a field of bounded operators (even in Hilbert-Schmidt classes) and so can be composed on the left and the right with π(x). The (possibly infinite) traces

$$\text{Tr}\left|\pi\_1(x)\widehat{f}(\pi\_1)\right|, \quad \text{Tr}\left|\widehat{f}(\pi\_1)\pi\_1(x)\right| \quad \text{and} \quad \text{Tr}\left|\widehat{f}(\pi\_1)\right|.$$

are equal for π1 ∈ Rep G as π1 is unitary. They are constant on the class of π1 ∈ Rep G in G and are, therefore, treated as depending on π ∈ G-. They are finite for μ-almost all π ∈ Gin view of (5.10).

Note that (5.10) implies not only that the two expressions

$$\int\_{\widehat{G}} \text{Tr}\left(\pi(x)\widehat{f}(\pi)\right)d\mu(\pi) \quad \text{and} \quad \int\_{\widehat{G}} \text{Tr}\left(\widehat{f}(\pi)\pi(x)\right)d\mu(\pi)$$

make sense but that they are also equal by the properties of the trace since π(x) is bounded.

Proof of Proposition 5.1.15. Let R be a positive Rockland operator of homogeneous degree ν. Let f ∈ L2 s(G) with s > Q/2. We set

$$f\_s := (\mathcal{I} + \mathcal{R})^{\frac{s}{\nu}} f \in L^2(G).$$

The properties of the trace imply

$$|\mathrm{Tr}|\widehat{f}(\pi)| = \mathrm{Tr}\left|\pi(\mathbf{I} + \mathcal{R})^{-\frac{\mu}{\nu}}\widehat{f}\_s(\pi)\right| \le ||\pi(\mathbf{I} + \mathcal{R})^{-\frac{\mu}{\nu}}||\_{\mathrm{HS}}||\widehat{f}\_s(\pi)||\_{\mathrm{HS}}.$$

Integrating against the Plancherel measure, we obtain by the Cauchy-Schwartz inequality

$$\int\_{\widehat{G}} \text{Tr} |\widehat{f}(\pi)| d\mu(\pi) \le ||\pi(\mathbf{I} + \mathcal{R})^{-\frac{\kappa}{\nu}}||\_{L^2(\widehat{G})} ||\widehat{f}\_s||\_{L^2(\widehat{G})}.$$

By Corollary 5.1.4, Cs := π(I + R)− s ν L2(G-) is a positive finite constant. Since f s(π)L2(G-) is equal to fL2 s(G) which is finite, we have obtained (5.10).

Let φ ∈ S(G). By the Plancherel formula, especially (1.30), we have

$$\begin{split} (f, \phi)\_{L^{2}(G)} &= \quad (f\_{s}, (\mathbf{I} + \mathcal{R})^{-\frac{s}{\nu}} \phi)\_{L^{2}(G)} \\ &= \quad \int\_{\hat{G}} \text{Tr}\left(\mathcal{F}\_{G} \{f\_{s}\}(\pi) \left(\mathcal{F}\_{G} \{(\mathbf{I} + \mathcal{R})^{-\frac{s}{\nu}} \phi\right\}(\pi)\right)^{\*}\right) d\mu(\pi) \\ &= \quad \int\_{\hat{G}} \text{Tr}\left(\pi (\mathbf{I} + \mathcal{R})^{\frac{s}{\nu}} \hat{f}(\pi) \left\{\hat{\phi}(\pi)^{\*} \pi (\mathbf{I} + \mathcal{R})^{-\frac{s}{\nu}}\right\} d\mu(\pi) \\ &= \quad \int\_{\hat{G}} \text{Tr}\left(\hat{f}(\pi) \left\{\hat{\phi}(\pi)^{\*}\right\} d\mu(\pi). \end{split}$$

Note that the two functions fs and (I + R) s ν φ are both square integrable so all the traces above are finite.

We now fix a non-negative function χ ∈ D(G) with compact support containing 0 and satisfying G χ = 1. We apply what precedes to φ := χ given by

$$\chi\_{\epsilon}(y) := \epsilon^{-Q} \chi(\epsilon^{-1} y), \quad \epsilon > 0, \ y \in G,$$

and obtain

$$(f, \chi\_{\epsilon})\_{L^{2}(G)} = \int\_{\widehat{G}} \text{Tr}\left(\widehat{f}(\pi) \, \widehat{\chi}\_{\epsilon}(\pi)^{\*}\right) d\mu(\pi). \tag{5.12}$$

Let us show that the right hand-side of (5.12) converges to

$$\int\_{\widehat{G}} \text{Tr}\left(\widehat{f}(\pi) \left. \widehat{\chi}\_{\epsilon}(\pi)^{\*}\right| d\mu(\pi) \longrightarrow\_{\epsilon \to 0} \int\_{\widehat{G}} \text{Tr}\left(\widehat{f}(\pi)\right) d\mu(\pi). \tag{5.13}$$

Note that the right hand-side of (5.13) is finite by (5.10).

The integrand on the left-hand side is bounded by

$$\left| \text{Tr} \left( \widehat{f}(\pi) \begin{array}{c} \widehat{\chi}\_{\epsilon}(\pi)^{\*} \end{array} \right) \right| \leq \left\| \widehat{\chi}\_{\epsilon}(\pi) \right\| \left| \mathcal{A}^{\epsilon}(\mathcal{H}\_{\pi}) \text{Tr} \right| \widehat{f}(\pi) |, \epsilon$$

and

$$\|\widehat{\chi}\_{\epsilon}(\pi)\|\_{\mathcal{L}^{\ell}(\mathcal{H}\_{\pi})} \le \|\chi\_{\epsilon}\|\_{L^{1}(G)} = \|\chi\|\_{L^{1}(G)}.$$

Hence

$$\left| \text{Tr} \left( \widehat{f}(\pi) \begin{array}{c} \widehat{\chi}\_{\epsilon}(\pi)^{\*} \end{array} \right) \right| \leq \||\chi||\_{L^{1}(G)} \text{Tr} |\widehat{f}(\pi)|,$$

and the right-hand side is μ-integrable by (5.10).

Let us show the convergence for every π ∈ G-

$$\text{Tr}\left(\hat{f}(\pi)\,\hat{\chi}\_{\epsilon}(\pi)^{\*}\right) \longrightarrow\_{\epsilon \to 0} \text{Tr}\left(\hat{f}(\pi)\right). \tag{5.14}$$

In order to do this, we want to estimate the difference

$$\begin{split} \left| \text{Tr} \left( \widehat{f}(\pi) \left. \widehat{\chi}\_{\epsilon}(\pi)^{\*} \right. \right) - \text{Tr} \left( \widehat{f}(\pi) \right) \right| &=& \left| \text{Tr} \left( \widehat{f}(\pi) \left. \left( \widehat{\chi}\_{\epsilon}(\pi)^{\*} - \text{I} \right) \right) \right| \\ &\leq& \left\| \left. \widehat{\chi}\_{\epsilon}(\pi)^{\*} - \text{I} \right| \right\|\_{L^{\infty}(\widehat{G})} \text{Tr} \left| \widehat{f}(\pi) \right| . \end{split} $$

Since

$$
\widehat{\chi}\_{\epsilon}(\pi)^{\*} = \int\_{G} \chi\_{\epsilon}(y)\pi(y) dy = \int\_{G} \epsilon^{-Q} \chi(\epsilon^{-1}y)\pi(y) dy = \int\_{G} \chi(z)\pi(\epsilon z) dz,
$$

and as G χ = 1, we have

$$\begin{split} \|\left|\widehat{\chi}\_{\epsilon}(\pi)^{\*}-\mathrm{I}\right|\left|\mathcal{L}(\mathsf{\boldsymbol{\mu}}\_{\pi})\right| &=& \|\int\_{G} \chi(z) \left(\pi(\epsilon z)-\mathrm{I}\right) dz\|\left|\mathcal{L}(\mathsf{\boldsymbol{\mu}}\_{\pi})\right| \\ &\leq& \int\_{G} |\chi(z)| \, \left\|\pi(\epsilon z)-\mathrm{I}\right\|\_{\mathcal{L}^{\ell}(\mathsf{\boldsymbol{\mu}}\_{\pi})} dz \\ &\leq& \sup\_{z\in\mathsf{supp}\chi} \left\|\pi(\epsilon z)-\mathrm{I}\right\|\_{\mathcal{L}^{\ell}(\mathsf{\boldsymbol{\mu}}\_{\pi})} \int\_{G} |\chi(z)|dz. \end{split}$$

As π is strongly continuous and suppχ compact, we know that

$$\sup\_{z \in \text{supp}\chi} ||\pi(\epsilon z) - \mathbf{I}||\mathcal{A}(\mathcal{H}\_{\pi}) \longrightarrow\_{\epsilon \to 0} 0.$$

This implies the convergence in (5.14) for each π ∈ G-.

We can now apply Lebesgue's dominated convergence theorem to obtain the convergence in (5.13).

By the Sobolev embeddings (see Theorem 4.4.25), f is continuous on G and it is a simple exercise to show that the left hand-side of (5.12) converges to

$$(f, \chi\_{\epsilon})\_{L^{2}(G)} \longrightarrow\_{\epsilon \to 0} f(0).$$

Hence we have obtained the inversion formula given in (5.11) at x = 0. Replacing f by its left translation f(x ·) which is still in L2 s(G) with the same Sobolev norm, it is then easy to obtain (5.11) for every x ∈ G.

For the case of f ∈ L˜2 s(G) with s > Q/2, we set fs := (I + R˜) s ν f ∈ L2(G) and we obtain similar properties as above, ending by using right translations to obtain (5.11). -

#### **5.1.2 The spaces** Ka,b(G)**,** LL(L2 a(G), L2 b (G))**, and** L∞ a,b(G-)

In this section we describe the spaces Ka,b(G), LL(L2 a(G), L2 b (G)) and L∞ a,b(G-), extending the notion of the group von Neumann algebras discussed in Section 1.8.2, to the setting of Sobolev spaces.

**Definition 5.1.16** (Spaces LL(L2 a(G), L2 b (G)) and Ka,b(G))**.** Let a, b ∈ R. We denote by

$$\mathcal{QC}\_L(L\_a^2(G), L\_b^2(G))$$

the subspace of operators T ∈ L (L2 a(G), L2 b (G)) which are left-invariant.

We denote by

$$\mathcal{K}\_{a,b}(G)$$

the subspace of tempered distributions f ∈ S (G) such that the operator S(G) φ → φ ∗ f extends to a bounded operator from L2 a(G) to L2 b (G).

If a positive Rockland operator R of homogeneous degree ν is fixed, then the Ka,b(G)-norm is defined for any f ∈ Ka,b(G), as the operator norm of φ → φ ∗ f viewed as an operator from L2 a(G) to L2 b (G), i.e.

$$\|f\|\kappa\_{a,b} := \|\phi \mapsto \phi \ast f\|\_{\mathcal{A}'(L^2\_a(G), L^2\_b(G))}.\tag{5.15}$$

Here we have considered the Sobolev norms φ → (I + R) c ν φ2 for c = a, b for L2 a(G) and L2 b (G), respectively.

The vector space LL(L2 a, L2 b ) is a Banach subspace of L (L2 a, L2 b ). Since the Sobolev spaces L2 a(G) are independent of the choice of a positive Rockland operator R (see Section 4.4.5), so are LL(L2 a(G), L2 b (G)) and also Ka,b(G). However, the norms on these spaces do depend on a choice of a positive Rockland operator R.

We may often write Ka,b instead of Ka,b(G) to ease the notation when no confusion is possible.

We have the immediate properties:

**Proposition 5.1.17.** 1. If a = b = 0 then

$$
\mathcal{K}\_{0,0} = \mathcal{K} \quad \text{and} \quad \mathcal{QC}\_L(L\_a^2, L\_b^2) = \mathcal{QC}\_L(L^2).
$$

The norms ·K0,0 and ·K (defined in (5.15) and in (1.37) respectively) coincide. For any f ∈ K we have

$$\|f^\*\|\_{\mathcal{K}} = \|f\|\_{\mathcal{K}}\quad where\quad f^\*(x) = \bar{f}(x^{-1}),$$

and

$$\forall r > 0 \qquad \|f \circ D\_r\|\_{\mathcal{K}} = r^{-Q} \|f\|\_{\mathcal{K}}.$$

$$
\mathcal{K}\_{a,b}(G) \subset \mathcal{S}'(G).
$$

Moreover if Tf denotes the convolution operator φ → φ ∗ f for f ∈ S (G), then the following are equivalent:

$$\begin{split} f \in \mathcal{K}\_{a,b} &\iff T\_f \in \mathcal{QC}\_L(L^2\_a(G), L^2\_b(G)) \\ &\iff (\mathbf{I} + \mathcal{R})^{\frac{b}{\nu}} T\_f (\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}} \in \mathcal{QC}\_L(L^2(G)) \\ &\iff (\mathbf{I} + \mathcal{R})^{\frac{b}{\nu}} (\mathbf{I} + \tilde{\mathcal{R}})^{-\frac{a}{\nu}} f \in \mathcal{K}(G), \end{split}$$

where R is any positive Rockland operator of homogeneous degree ν.

4. For any c1, c2 ≥ 0 we have the inclusions

$$\mathcal{QC}\_L(L\_a^2, L\_b^2) \subset \mathcal{QC}\_L(L\_{a+c\_1}^2, L\_{b-c\_2}^2)$$

and

$$
\mathcal{K}\_{a,b} \subset \mathcal{K}\_{a+c\_1, b-c\_2 \cdots}
$$

5. If f ∈ Ka,b then Xαf ∈ Ka,b−[α] for any α ∈ Nn 0 and (I + R)s/νf ∈ Ka,b−s for any s ∈ R. Furthermore, Xα and (I + R)s/ν are bounded on Ka,b:

$$\|X^{\alpha}f\|\_{\mathcal{K}\_{a,b-\left[\alpha\right]}} \leq C\_{a,b,\left[\alpha\right]}\|f\|\_{\mathcal{K}\_{a,b}}$$

and

$$\| (\mathcal{I} + \mathcal{R})^{s/\nu} f \|\_{\mathcal{K}\_{a,b-s}} \le C\_{a,b,s}' \| f \|\_{\mathcal{K}\_{a,b}}$$

for some positive finite constants Ca,b,[α] and C a,b,s independent of f.

If −a and b are in νN0, a norm equivalent to the Ka,b-norm is

$$f \longmapsto \sum\_{\left[\alpha\right] \leq -a,\left[\beta\right] \leq b} \|\tilde{X}^{\alpha} X^{\beta} f\|\_{\mathcal{K}},$$

and if a ∈ [a, 0] and b ∈ [0, b] then

$$||f||\_{\mathcal{K}\_{a',b'}} \le C\_{a,b,a',b,\mathcal{R}} \sum\_{\left[\alpha\right] \le -a,\left[\beta\right] \le b} ||\tilde{X}^{\alpha} X^{\beta} f||\_{\mathcal{K}}.$$

The definitions of the tempered distributions Xαf and (I + R)s/νf were recalled in (5.8) and (5.9) respectively. For the proper definition of the operators (I + R) b ν , (I + R˜)− a ν , see Definitions 4.3.17 and 4.4.31.

Proof of Proposition 5.1.17. Part (1) follows from the properties of the von Neumann-algebras K(G) and LL(L2(G)) as well as from the following two easy observations:

$$\forall \psi \in L^2(G) \qquad \|\psi \circ D\_r\|\_2 = r^{-\frac{Q}{2}} \|\psi\|\_2,$$

and for any f ∈ K, φ ∈ S(G) and r > 0,

$$
\phi\*(f\circ D\_r)(x) = r^{-Q}\left(\left(\phi\circ D\_{\frac{1}{r}}\right)\*f\right)(rx).
$$

Part (2) is easy to check. Part (3) follows from the Schwartz kernel theorem, see Corollary 3.2.1. Parts (4) and (5), follow easily from the properties of the Sobolev spaces and Part (3). -

We now show that we can make sense of convolution of distributions in some Ka,b(G)-spaces. The following lemma is almost immediate to check.

**Lemma 5.1.18.** Let f ∈ Ka,b(G) and g ∈ Kb,c(G) for a, b, c ∈ R, and let Tf : φ → φ ∗ f and Tg : φ → φ ∗ g be the associated operators. Then the operator TgTf is continuous from L2 a(G) to L2 c (G) and its right-convolution kernel (as a continuous linear operator from S(G) to S (G)) is denoted by h ∈ Ka,c(G).

If (fn) and (gn) are sequences of Schwartz functions converging to f in Ka,b(G) and g in Kb,c(G), respectively, then h is the limit of fn ∗ gn in Ka,c(G).

Consequently, with the notation of the lemma above, h coincides with the convolution of f with g whenever the convolution of f with g makes any technical sense, for instance, if the tempered distributions f and g (which are already assumed to be in Ka,b(G) and Kb,c(G) respectively) satisfy

Hence we may extend the notation and define:

**Definition 5.1.19.** If f ∈ Ka,b(G) and g ∈ Kb,c(G) for a, b, c ∈ R, and Tf : φ → φ∗f, Tg : φ → φ ∗ g are the associated operators, we denote by f ∗ g the distribution in Ka,c(G) which is the right convolution kernel of TgTf .

We obtain easily the following properties:

**Corollary 5.1.20.** Let f ∈ Ka,b(G) and g ∈ Kb,c(G) for a, b, c ∈ R. Then we have the following property of associativity for any φ ∈ S(G)

$$
\phi \* (f \* g) = (\phi \* f) \* g,
$$

and more generally for any h ∈ Kc,d(G) (where d ∈ R)

$$f \ast (g \ast h) = (f \ast g) \ast h,$$

as convolutions of an element of Ka,b(G) with an element of Kb,d(G) for the lefthand side, and of an element of Ka,c(G) with an element of Kc,d(G) for the righthand side.

The rest of this section is devoted to the definition of the group Fourier transform of a distribution in Ka,b(G). We start by defining what will turn out to be the image of the group Fourier transform on Ka,b(G). We recall that L∞(G-) is the space of measurable fields of operators on G which are uniformly bounded, see Definition 1.8.8.

**Definition 5.1.21.** Let a, b ∈ R. We denote by L∞ a,b(G-) the space of fields of operators σ = {σπ : H∞ π → Hb π , π ∈ G-} satisfying

$$\exists C > 0 \quad \forall \phi \in \mathcal{S}(G) \qquad \|\sigma \hat{\phi}\|\_{L^2\_b(\widehat{G})} \le C \|\phi\|\_{L^2\_a(G)}.\tag{5.16}$$

Here we assume that a positive Rockland operator has been fixed to define the norms on L2 b (G-) and L2 a(G).

For such a field σ, σL∞a,b(G-) denotes the infimum of the constant C > 0 satisfying (5.16).

We may sometimes abuse the notation and write σπL∞a,b(G-) when no confusion is possible.

Note that as φ ∈ S(G), its group Fourier transform acts on smooth vectors, see Example 1.8.18. Hence the composition σφ above makes sense, see Section 1.8.3.

Naturally, the space L∞ a,b(G-) introduced in Definition 5.1.21 is independent of the choice of a Rockland operator used to define the norms on L2 b (G-) and L2 a(G):

**Lemma 5.1.22.** If {σπ : H∞ π → Hb π, π ∈ G-} satisfies the condition in Definition 5.1.21 for one positive Rockland operator, then it satisfies the same property for any positive Rockland operator. Moreover, if R1 and R2 are two positive Rockland operators, and if σL∞a,b,R1 (G-) and σL∞a,b,R2 (G-) denote the corresponding infima, then there exists C > 0 independent of σ such that

$$C^{-1} \|\sigma\|\_{L^{\infty}\_{a,b,\mathcal{R}\_2}(\hat{G})} \le \|\sigma\|\_{L^{\infty}\_{a,b,\mathcal{R}\_1}(\hat{G})} \le C \|\sigma\|\_{L^{\infty}\_{a,b,\mathcal{R}\_2}(\hat{G})}.$$

Proof. This follows easily from the independence of the Sobolev spaces on G and G of the positive Rockland operators, see Section 4.4.5 and Proposition 5.1.14. -

If the field acts on smooth vectors, we can simplify Definition 5.1.21:

**Lemma 5.1.23.** Let σ = {σπ : H∞ π → H∞ π , π ∈ G-} be a field acting on smooth vectors. Then σ ∈ L∞ a,b(G-) if and only if

$$\{\pi(\mathbf{I}+\mathcal{R})^{\frac{\mathbf{h}}{\nu}}\boldsymbol{\sigma}\_{\pi}\,\pi(\mathbf{I}+\mathcal{R})^{-\frac{\mathbf{a}}{\nu}}:\mathcal{H}\_{\pi}^{\infty}\to\mathcal{H}\_{\pi}^{\infty},\pi\in\hat{G}\}\in L^{\infty}(\hat{G}),\tag{5.17}$$

where R is a positive Rockland operator of degree ν, and in this case,

$$\|\sigma\|\_{L\_{a,b}^{\infty}(\widehat{G})} = \|\pi(\mathbf{I} + \mathcal{R})^{\frac{b}{\nu}}\sigma\_{\pi}\pi(\mathbf{I} + \mathcal{R})^{-\frac{a}{\nu}}\|\_{L^{\infty}(\widehat{G})}.$$

Proof. This follows easily from the density of S(G) in L2 b (G). -

Note that the composition in (5.17) makes sense as all the fields involved act on smooth vectors. In Corollary 5.1.30, we will see a sufficient condition (which will be useful later) for a field to be acting on smooth vectors.

We can now characterise the elements of Ka,b(G) in terms of L∞ a,b(G-):

## **Proposition 5.1.24.** Let a, b ∈ R.

(i) If σ ∈ L∞ a,b(G-), then the operator Tσ : S(G) → S (G) defined via

$$\widehat{T\_{\sigma}\phi}(\pi) := \sigma\_{\pi}\widehat{\phi}(\pi), \quad \phi \in \mathcal{S}(G), \ \pi \in \widehat{G}, \tag{5.18}$$

extends uniquely to an operator in L (L2 a, L2 b ). Moreover,

$$\|T\_{\sigma}\|\_{\mathcal{L}^{\ell}(L\_a^2, L\_b^2)} = \|\sigma\|\_{L\_{a,b}^{\infty}(\widehat{G})},\tag{5.19}$$

where the Sobolev norms are defined using a chosen positive Rockland operator R with homogeneous degree ν. The right convolution kernel f ∈ S (G) of Tσ is in Ka,b(G).

(ii) Conversely, if f ∈ Ka,b(G) then there exists a unique σ ∈ L∞ a,b(G-) such that

$$
\widehat{\phi \ast f}(\pi) = \sigma\_{\pi} \widehat{\phi}(\pi), \quad \phi \in \mathcal{S}(G), \ \pi \in \widehat{G}. \tag{5.20}
$$

Furthermore, if f is also in any of the spaces where the group Fourier transform has already been defined, namely any Sobolev space L2 a(G) or K(G), then σ = {σπ, π ∈ G-} will coincide with the group Fourier transform of f.

Proof. The properties of Tσ in Part (i) follow from the Plancherel theorem (Theorem 1.8.11) and the density of S(G) in L2(G). The right convolution kernel f ∈ S (G) of Tσ is in Ka,b(G) by Proposition 5.1.17.

Conversely, let f ∈ Ka,b(G). By assumption the operator Tf : S(G) φ → φ ∗ f admits a bounded extension from L2 a(G) to L2 b (G). Thus the operator (I + R) b ν Tf (I + R)− a ν is bounded on L2(G) and we denote by fa,b ∈ K(G) its right convolution kernel. For any φ ∈ S(G), we have φa := (I + R) a ν φ ∈ S(G) by Corollary 4.3.16 thus φa ∗ fa,b ∈ L2(G) and we have

$$T\_f \phi \in L\_b^2(G) \quad \text{with} \quad T\_f \phi = (\mathbf{I} + \mathcal{R})^{-\frac{b}{\nu}} (\phi\_a \* f\_{a,b}).$$

Consequently FG(Tfφ) ∈ L2 b (G-) and

$$\mathcal{F}\_G(T\_f\phi) = \pi(\mathbf{I} + \mathcal{R})^{-\frac{b}{\nu}}\widehat{f}\_{a,b}\widehat{\phi}\_a = \pi(\mathbf{I} + \mathcal{R})^{-\frac{b}{\nu}}\widehat{f}\_{a,b}\pi(\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}}\widehat{\phi}\_{-a}$$

One checks easily that {σπ : H∞ π → Hb π, π ∈ G-} defined via

$$\sigma\_{\pi} := \pi (\mathbf{I} + \mathcal{R})^{-\frac{b}{\nu}} \widehat{f}\_{a,b}(\pi) \,\pi (\mathbf{I} + \mathcal{R})^{\frac{a}{\nu}}$$

is in L∞ a,b(G-) and satisfies (5.20). The rest of the proof of Part (ii) follows easily from the computations above and the uniqueness of the group Fourier transforms already defined. -

Thanks to Proposition 5.1.24, we can extend the definition of the group Fourier transform to Ka,b(G):

**Definition 5.1.25** (The group Fourier transform on Ka,b(G))**.** The group Fourier transform of f ∈ Ka,b(G) is the field of operators {σπ : H∞ π → Hb π, π ∈ G-} in L∞ a,b(G-) associated to f by Proposition 5.1.24, and we write

$$\widehat{f}(\pi) := \pi(f) := \sigma\_{\pi}, \quad \pi \in \widehat{G}.$$

As the next example implies, any left-invariant vector field is in some Ka,b(G) and their Fourier transform can be defined via Definition 5.1.25. As is shown in the proof below, this coincides with the infinitesimal representation of the corresponding element of U(g) defined in Section 1.7.

Example 5.1.26. Let α ∈ Nn 0 . The operator Xα is in L (L2 [α] (G), L2(G)) and more generally in L (L2 [α]+s(G), L2 s(G)) for any s ∈ R. Its right convolution kernel is the distribution Xαδ0 defined via (see (5.8))

$$
\langle X^{\alpha} \delta\_0, \phi \rangle = \langle \delta\_0, \{X^{\alpha}\}^t \phi \rangle = \{X^{\alpha}\}^t \phi(0),
$$

which is in K[α],0, and more generally in Ks+[α],s for any s ∈ R. Its group Fourier transform is

$$\mathcal{F}\_G(X^\alpha \delta\_0)(\pi) = \pi(X^\alpha) = \pi(X)^\alpha$$

and coincides with the infinitesimal representation on U(g). It is in L∞ s+[α],s(G-) for any s ∈ R.

Proof. By Theorem 4.4.16, Xα maps L2 [α] (G) continuously to L2(G) and, more generally, L2 s+[α] (G) continuously to L2 s(G).

By Proposition 1.7.6, we have for any φ ∈ S(G)

$$\mathcal{F}\_G(X^\alpha \phi)(\pi) = \pi(X^\alpha)\widehat{\phi}(\pi) = \pi(X)^\alpha \widehat{\phi}(\pi).$$

This shows that FG(Xαδ0) coincides with {π(Xα), π ∈ G-}. -

As our next example shows, when multipliers in a positive Rockland operator are in LL(L2 s(G), L2 s−b(G)), the group Fourier transform of their right convolution kernels can also be given via the functional calculus of the Rockland operators:

Example 5.1.27. Let R be a positive Rockland operator of homogeneous degree ν. Let m be a measurable function on [0, ∞) satisfying

$$\exists C > 0 \quad \forall \lambda \ge 0 \quad |m(\lambda)| \le C(1+\lambda)^{\frac{h}{\nu}}.$$

Then the operator m(R) defined by the functional calculus of R extends uniquely to an operator in LL(L2 s+b(G), L2 s(G)) for any s ∈ R. Its right convolution kernel m(R)δ0 is in Ks+b,s for any s ∈ R. Its group Fourier transform is

$$\mathcal{F}\_G(m(\mathcal{R})\delta\_0)(\pi) = m(\pi(\mathcal{R})) $$

defined by the functional calculus of π(R). It is in L∞ s+b,s(G-) for any s ∈ R. For a fixed s ∈ R, we have

$$\begin{aligned} \|m(\mathcal{R})\|\_{\mathcal{L}\_L(L\_{s+b}^2(G), L\_s^2(G))} &= \|m(\mathcal{R})\delta\_0\|\_{\mathcal{K}\_{s+b,s}} = \|m(\pi(\mathcal{R}))\|\_{L\_{s+b,s}^\infty(\widehat{G})} \\ &\leq \sup\_{\lambda>0} (1+\lambda)^{-\frac{b}{\nu}} |m(\lambda)|, \end{aligned}$$

if we realise the Sobolev norms with R.

We refer to Section 4.1.3 and Corollary 4.1.16 for the properties of the functional calculus of R2 and π(R).

Proof. The function m1 given by

$$m\_1(\lambda) := m(\lambda)(1+\lambda)^{-\frac{b}{\nu}}, \quad \lambda \ge 0,$$

is measurable and bounded on [0, ∞). The operator m1(R) defined by the functional calculus of R is therefore bounded on L2(G) with

$$\|m\_1(\mathcal{R})\|\_{\mathcal{C}(L^2(G))} \le \sup\_{\lambda \ge 0} |m\_1(\lambda)|.$$

Again from the properties of the functional calculus of R, we also have

$$m(\mathcal{R}) \supset m\_1(\mathcal{R})(\mathbf{I} + \mathcal{R})^{\frac{b}{\nu}},$$

in the sense of operators. Since Dom(I + R)b/ν ⊃ S(G) (see Corollary 4.3.16), this shows that the domain of m(R) contains S(G) and that

$$m\_1(\mathcal{R}) = m(\mathcal{R})(\mathrm{I} + \mathcal{R})^{-\frac{b}{\nu}} \quad \text{on } \mathcal{S}(G).$$

The properties of the functional calculus of R yield for any s ∈ R,

$$\begin{aligned} \|m\_1(\mathcal{R})\|\_{\mathcal{L}^\ell(L^2(G))} &= \|m\_1(\mathcal{R})\|\_{\mathcal{L}^\ell(L^2\_s(G))} \\ &= \|m(\mathcal{R})(\mathcal{I}+\mathcal{R})^{-\frac{b}{\nu}}\|\_{\mathcal{L}^\ell(L^2\_s(G))} \\ &= \|m(\mathcal{R})\|\_{\mathcal{L}^\ell(L^2\_{s+b}(G), L^2\_s(G))}. \end{aligned}$$

By Corollary 4.1.16, the kernel of m1(R) is the tempered distribution m1(R)δ0 with Fourier transform {m1(π(R)), π ∈ G-}. Adapting the proof of Corollary 4.1.16, we see that

$$m\_1(\pi(\mathcal{R})) = m(\pi(\mathcal{R})) (\mathbf{I} + \pi(\mathcal{R}))^{-\frac{b}{\nu}} \quad \text{on } \mathcal{H}^{\infty}\_{\pi}, \quad \pi \in \widehat{G}.$$

It is now straightforward to check that the kernel of the operator m(R) is in Ks+b,s and its Fourier transform is {m(π(R)), π ∈ G-}. -

Naturally, any Schwartz function is in any Ka,b and one can readily estimate the associated norm:

Example 5.1.28. If φ ∈ S(G), then for any a, b ∈ R, the operator Tφ : ψ → ψ ∗ φ is in L (L2 a(G), L2 b (G)), φ ∈ Ka,b and φ- ∈ L∞ a,b. If we fix a positive Rockland operator R of homogeneous degree ν, then we have

$$\|T\_{\boldsymbol{\Phi}}\|\_{\mathcal{H}(L^{2}\_{a}(G),L^{2}\_{b}(G))} = \|\boldsymbol{\phi}\|\_{\mathcal{K}\_{a,b}} = \|\widehat{\boldsymbol{\phi}}\|\_{L^{\infty}\_{a,b}} \le \|(\mathbf{I}+\mathcal{R})^{\frac{b}{\nu}}(\mathbf{I}+\tilde{\mathcal{R}})^{-\frac{a}{\nu}}\boldsymbol{\phi}\|\_{L^{1}(G)} < \infty,$$

where the norms on L (L2 a(G), L2 b (G)), Ka,b and L∞ a,b are defined with R.

With Definition 5.1.25, we can reformulate Proposition 5.1.24 and parts of Proposition 5.1.17 and Corollary 5.1.20 as the following proposition.

**Proposition 5.1.29.** 1. Let a, b ∈ R. The Fourier transform FG maps Ka,b(G) onto L∞ a,b(G-). Furthermore, FG : Ka,b(G) → L∞ a,b(G-) is an isomorphism between Banach spaces. In particular, for f ∈ Ka,b(G),

$$\|f\|\_{\mathcal{K}\_{a,b}} = \|\dot{f}\|\_{L^{\infty}\_{a,b}(\widehat{G})}.$$

It coincides with the Fourier transform on K(G) for a = b = 0.

2. If σ1 ∈ L∞ a1,b1 (G-) and σ2 ∈ L∞ a2,b2 (G-) with b2 = a1, then their product σ1σ2 makes sense as the element of L∞ a2,b1 (G-) given by the Fourier transform of (F−1 G σ2) ∗ (F−1 G σ1).

In other words, if f1 ∈ Ka1,b1 (G-) and f2 ∈ Ka2,b2 (G-) with b2 = a1, then the Fourier transform of f2 ∗ f1 ∈ Ka2,b1 (G-) is

$$
\mathcal{F}\_G(f\_2 \* f\_1) = \mathcal{F}\_G(f\_1)\mathcal{F}\_G(f\_2).
$$

3. Let σ = {σπ : H∞ π → Hπ, π ∈ G-} ∈ L∞ a,b(G-). Then we have for any α ∈ Nn 0 ,

$$\{\pi(X)^{\alpha}\sigma\_{\pi}:\mathcal{H}\_{\pi}^{\infty}\to\mathcal{H}\_{\pi},\pi\in\hat{G}\}\in L\_{a,b-[\alpha]}^{\infty}(\hat{G}),\tag{5.21}$$

and for any s ∈ R,

$$\{\pi(\mathbf{I}+\mathcal{R})^{s/\nu}\sigma\_{\pi}:\mathcal{H}\_{\pi}^{\infty}\to\mathcal{H}\_{\pi},\pi\in\widehat{G}\}\in L\_{a,b-s}^{2}(\widehat{G}).\tag{5.22}$$

Furthermore, if f = F−1 G σ ∈ Ka,b(G) then

$$\mathcal{F}\_G(X^\alpha f)(\pi) = \pi(X)^\alpha \widehat{f}(\pi) \quad \text{and} \quad \mathcal{F}\_G((\mathbf{I} + \mathcal{R})^{s/\nu} f)(\pi) = \pi(\mathbf{I} + \mathcal{R})^{s/\nu} \widehat{f}(\pi).$$

The fields of operators in (5.21) and (5.22) are understood as compositions of fields of operators in L∞ a2,b2 and L∞ a1,b1 with b2 = a1, see Part 2 and Examples 5.1.26 and 5.1.27.

With the help of Proposition 5.1.29, we can now give a usefull sufficient condition for a field to act on smooth vectors and reformulate Corollary 4.4.10 into

**Corollary 5.1.30.** Let a, b ∈ R and let {γ, - ∈ Z} be a sequence of real numbers which tends to ±∞ as - → ±∞. Let σ ∈ L∞ a+γ,b+γ (G-) for every - ∈ Z. Then σ is a field of operators acting on smooth vectors:

$$\sigma = \{ \sigma\_{\pi} : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}^{\infty}, \pi \in \hat{G} \}.$$

Furthermore σ ∈ L∞ a+γ,b+γ(G-) for every γ ∈ R and for any c ≥ 0, we have

$$\sup\_{|\gamma| \le c} \left\| \sigma \right\|\_{L^{\infty}\_{a+\gamma,b+\gamma}(\widehat{G})} \le C\_c \max\left( \left\| \sigma \right\|\_{L^{\infty}\_{a+\gamma,b+\gamma\_\ell}(\widehat{G})}, \left\| \sigma \right\|\_{L^{\infty}\_{a+\gamma-\ell,b+\gamma\_{-\ell}}(\widehat{G})} \right),$$

where -∈ N0 is the smallest integer such that γ ≥ c and −γ− ≥ c.

Proof. By Proposition 5.1.29, π(X)ασ ∈ L∞ a+γ,b+γ−[α] for every α ∈ Nn 0 and every - ∈ Z. Thus choosing γ ≥ [α] − b, we have π(X)ασφ- ∈ L2(G-) for every φ ∈ S(G). Realising π ∈ G as a representation of G and fixing v ∈ H∞ π , this implies that the mapping x → π(x)σπφ-(π)v is smooth. Hence σπφ-(π)v is smooth and σφ acts on smooth vectors. As this holds for every φ ∈ D(G), so does σ by Lemma 1.8.19. We conclude with Corollary 4.4.10. -

We end this section with one more technical property:

**Lemma 5.1.31.** Let σ ∈ L∞ a,b(G-) where a, b ∈ R. Let φ ∈ S(G). Then we have σφ- ∈ L˜2 s(G-) for any s ∈ R and

$$\int\_{\widehat{G}} \text{Tr}\left| \sigma\_{\pi} \widehat{\phi}(\pi) \right| d\mu(\pi) < \infty. \tag{5.23}$$

Setting f := F−1 G σ ∈ Ka,b, the function φ∗ f is smooth and we have for any x ∈ G the equality

$$\phi \ast f(x) = \int\_{\widehat{G}} \text{Tr}\left(\pi(x)\sigma\_{\pi}\widehat{\phi}(\pi)\right)d\mu(\pi).$$

Remark 5.1.32. The composition σφ makes sense since σ is defined on smooth vectors and φ acts on smooth vectors. The composition π(x)σππ(φ) makes sense since π(x) is bounded and σφ is bounded (even in Hilbert Schmidt classes) since it is stated first that σφ- ∈ L˜2 s(G-) for any s, hence in particular in L2(G-).

Proof. Let Tσ be the operator with right convolution kernel f := F−1 G σ. Then Tσ ∈ L (L2 a(G), L2 b (G)) and T ∗ σ Tσ extends to an operator in L (L2 a(G)). For any φ ∈ S(G), the definition of the adjoint and the duality between Sobolev spaces yield

$$\begin{aligned} \|T\_{\sigma}\phi\|\_{L^{2}(G)}^{2} &= \left\langle T\_{\sigma}^{\*}T\_{\sigma}\phi, \bar{\phi} \right\rangle\_{L^{2}\_{a}(G)\times L^{2}\_{-a}(G)} \\ &\leq \quad \|T\_{\sigma}^{\*}T\_{\sigma}\|\_{\mathcal{L}^{\ell}(L^{2}\_{a}(G))} \|\phi\|\_{L^{2}\_{a}(G)} \|\phi\|\_{L^{2}\_{-a}(G)}. \end{aligned}$$

This last expression is finite since T ∗ σ Tσ ∈ L (L2 a(G)) and S(G) ⊂ L2 s (G) for any s ∈ R. Thus Tσφ ∈ L2(G) and its Fourier transform is σφ- ∈ L2(G-). For any s ∈ R, we may replace φ with φs = (I + R)s/νφ ∈ S(G) and σφs ∈ L2(G-) yields σφ- ∈ L2 s(G-).

Applying Proposition 5.1.15 to σφ- ∈ L˜2 s(G-) for some s > Q/2, we obtain (5.23). Note that f := F−1 G σ is a tempered distribution so φ ∗ f is smooth (see Lemma 3.1.55). The group Fourier transform of φ ∗ f is σφ by Proposition 5.1.29 Part 2 and Example 5.1.28. We now conclude with the inversion formula given in Proposition 5.1.15. -

#### **5.1.3 Symbols and associated kernels**

In this section we aim at establishing a one-to-one correspondence between a collection σ of operators parametrised by G × G and a function κ; this function will turn out to be the kernel of the operator naturally associated to σ. For the abstract setting behind measurable fields of operators and some of their properties we refer to Section B.1.6, especially to Proposition B.1.17, as well as Section 1.8.3.

**Definition 5.1.33** (Symbols)**.** A symbol is a field of operators {σ(x, π) : H∞ π → Hπ, π ∈ G-} depending on x ∈ G, satisfying for each x ∈ G

$$\exists a, b \in \mathbb{R} \qquad \sigma(x, \cdot) := \{\sigma(x, \pi) : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}, \pi \in \hat{G}\} \in L\_{a,b}^{\infty}(\hat{G}).$$

Here we use the usual identifications of a strongly continuous irreducible unitary representation from Rep G with its equivalence class in G-, and of a field of operators acting on the smooth vectors parametrised by G with its equivalence class with respect to the Plancherel measure μ.

We will usually assume that the symbols are uniformly regular in x:

**Definition 5.1.34** (Continuous and smooth symbols)**.**

• A symbol {σ(x, π) : H∞ π → Hπ, π ∈ G-} is said to be continuous in x ∈ G whenever there exists a, b ∈ R such that

$$\forall x \in G \quad \sigma(x, \cdot) := \{ \sigma(x, \pi) : \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}, \pi \in \hat{G} \} \in L\_{a,b}^{\infty}(\hat{G}),$$

and the map x → σ(x, ·) is continuous from G ∼ Rn to the Banach space L∞ a,b(G-).

• A symbol σ = {σ(x, π) : H∞ π → Hπ, π ∈ G-} is said to be smooth in x ∈ G whenever it is a field of operators depending smoothly in x ∈ G (see Remark 1.8.16) and, for every β ∈ Nn 0 , the field {∂β x σ(x, π) : H∞ π → Hπ, π ∈ G-} is continuous.

**Important note:** In the sequel, whenever we talk about symbols (on graded Lie groups), we always mean the symbols which are smooth in x ∈ G in the sense of Definition 5.1.34 unless stated otherwise.

For a symbol as in Definition 5.1.34, we will usually write

$$\sigma = \{ \sigma(x,\pi), (x,\pi) \in G \times \check{G} \},$$

but we may sometimes abuse the notation and refer to the symbol simply as σ(x, π).

**Lemma 5.1.35.** If σ = {σ(x, π),(x, π) ∈ G × G-} is a symbol, then

$$\kappa\_x := \mathcal{F}\_G^{-1}\{\sigma(x,\cdot)\}$$

is a tempered distribution and the map

$$G \ni x \longmapsto \kappa\_x \in \mathcal{S}'(G)$$

is smooth.

In other words,

$$
\kappa \in C^{\infty}(G, \mathcal{S}'(G)).
$$

Here C∞(G, S (G)) denotes the set of smooth functions from G to S (G).

Proof. As σ is a smooth symbol, for every β ∈ Nn 0 , there exists aβ, bβ ∈ R such that G x → ∂β x σ(x, ·) ∈ L∞ aβ,bβ (G-) is continuous. By Proposition 5.1.29, composing this with F−1 G implies that G x → ∂β x κx ∈ Kaβ,bβ is continuous. Since the inclusion Kaβ,bβ ⊂ S (G) is continuous, this implies that each map G x → ∂β x κx ∈ S (G) is continuous. Hence G x → κx ∈ S (G) is smooth. - **Definition 5.1.36** (Associated kernels)**.** If σ is a symbol, then the tempered distribution

$$\kappa\_x := \mathcal{F}\_G^{-1}\{\sigma(x,\cdot)\} \in \mathcal{S}'(G),$$

is called its associated kernel, sometimes its right convolution kernel, or just a kernel. We may also call the smooth map G x → κx ∈ S (G) or the map (x, y) → κx(y) = κ(x, y) the kernel associated with σ.

The smoothness of the map x → σ(x, ·) implies easily:

**Lemma 5.1.37.** If σ = {σ(x, π)} is a symbol with kernel κx then for any β ∈ Nn 0 ,

$$X^{\beta}\sigma := \{X\_x^{\beta}\sigma(x,\pi)\}, \quad \tilde{X}^{\beta}\sigma := \{\tilde{X}\_x^{\beta}\sigma(x,\pi)\}, \text{ and } \partial\_x^{\beta}\sigma := \{\partial\_x^{\beta}\sigma(x,\pi)\},$$

are symbols with respective kernels

$$X\_x^{\beta} \kappa\_x, \quad \tilde{X}\_x^{\beta} \kappa\_x, \quad and \quad \partial\_x^{\beta} \kappa\_x.$$

Examples of symbols are the symbols in the classes Sm ρ,δ(G) defined later on. Here are more specific examples of symbols which do not depend on x ∈ G.

Example 5.1.38. If f ∈ Ka,b(G), then f -= {f -(π) : H∞ π → Hπ, π ∈ G-} is a symbol with kernel f.

The following are particular instances of this case:

$$\exists C > 0 \quad \forall \lambda \ge 0 \quad |m(\lambda)| \le C(1+\lambda)^{b/\nu},$$

then {m(π(R)) : H∞ π → Hπ, π ∈ G-} is a symbol with kernel m(R)δ0, see Example 5.1.27. By Corollary 5.1.30, this symbol also acts on smooth vectors

$$\{m(\pi(\mathcal{R})): \mathcal{H}\_{\pi}^{\infty} \to \mathcal{H}\_{\pi}^{\infty}, \pi \in \dot{G}\}.$$

#### **5.1.4 Quantization formula**

With the notion of symbol explained in Section 5.1.3, our quantization makes sense:

**Theorem 5.1.39** (Quantization)**.** The quantization defined by formula (5.1) makes sense for any symbol σ = {σ(x, π)}. More precisely, for any φ ∈ S(G) and x ∈ G, we have

$$\operatorname{Op}(\sigma)\phi(x) = \int\_{\widehat{G}} \operatorname{Tr}\left(\pi(x)\sigma(x,\pi)\widehat{\phi}(\pi)\right)d\mu(\pi) = \phi\*\kappa\_x(x),\tag{5.24}$$

where κx denotes the kernel of σ. The integral over G in (5.24) is well-defined and absolutely convergent. We also have Op(σ)φ ∈ C∞(G). Furthermore, the quantization mapping σ → Op(σ) is one-to-one and linear.

Proof. Lemma 5.1.31 (see also Remark 5.1.32) implies that the integral in (5.24) is well defined, absolutely convergent and is equal to φ ∗ κx(x).

By Lemma 3.1.55, for each x ∈ G, the function φ ∗ κx is smooth. By Lemma 5.1.35, x → κx ∈ S (G) is smooth. Hence by composition, x → φ∗κx(x) is smooth.

The quantization is clearly linear. Since the kernel is in one-to-one linear correspondence with the operator, and by Lemma 5.1.35 also with the symbol, the quantization σ → Op(σ) is one-to-one. -

**Definition 5.1.40** (Notation)**.** If an operator T is given by the formula (5.24) with symbol σ(x, π), so that T = Op(σ), we will also write

$$
\sigma = \sigma\_T \quad \text{or} \quad \sigma(x,\pi) = \sigma\_T(x,\pi) \quad \text{or even} \quad \sigma = \text{Op}^{-1}(T).
$$

This notation is justified since the quantization given by (5.24) is one-to-one by Theorem 5.1.39.

The operators associated with the symbols given in Example 5.1.38 are the ones alluded to in the introduction of this Section:

Continued Example 5.1.38: If f ∈ Ka,b(G), then Op(f -) is the convolution operator φ → φ ∗ f with the right convolution kernel f.

The following are particular instances of this case:

• Op(I) = I and, more generally, for any α ∈ Nn 0 , Op(π(X)α) = Xα.

These relations can also be expressed as

$$
\sigma\_\mathcal{I}(x,\pi) = \mathcal{I}\_{\mathcal{H}\_\pi} \text{ and } \sigma\_{X^\alpha}(x,\pi) = \pi(X)^\alpha.
$$

• If R is a positive Rockland operator of homogeneous degree ν and if m is a measurable function on [0, ∞) satisfying

> ∃C > 0 ∀λ ≥ 0 |m(λ)| ≤ C(1 + λ) b/ν,

then Op(m(π(R))) = m(R).

In these examples, the symbols are independent of x. However it is easy to produce x-dependent symbols out of them using the following two observations.

$$\sum\_{\{\alpha\} \leq M} c\_{\alpha}(x)X^{\alpha} \quad \text{with smooth coefficients } c\_{\alpha} \tag{5.25}$$

may be quantized, in the sense that [α]≤M cα(x)π(X)α is a (smooth) symbol and we have

$$\sum\_{\substack{[\alpha]\le M}} c\_{\alpha}(x)X^{\alpha} = \operatorname{Op}\left(\sum\_{[\alpha]\le M} c\_{\alpha}(x)\pi(X)^{\alpha}\right).$$

The differential calculus is, by definition, the space of differential operators of the form

$$\sum\_{|\alpha| \le d} b\_{\alpha}(x) \partial\_x^{\alpha} \quad \text{with smooth coefficients } b\_{\alpha},$$

or, equivalently, of the form (5.25), see (3.1.5). Hence, we have obtained that the differential calculus may be quantized. This could be viewed as 'the minimum requirement' for a notion of symbol and quantization on a manifold.

2. In order to achieve this, we had to consider and use fields of operators defined on smooth vectors in our definition of symbol. Indeed, for instance, the symbol associated to a left-invariant vector field X is {π(X)} while π(X) are defined on H∞ π but is not bounded on Hπ.