Functional Calculus on Non-Homogeneous Operators on Nilpotent Groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $\mathcal{L}$ on a connected and simply connected nilpotent Lie group $G$ with the aid of the corresponding \emph{Rockland} operator $\mathcal{L}_0$ on the `local' contraction $G_0$ of $G$, as well as of the corresponding Rockland operator $\mathcal{L}_\infty$ on the `global' contraction $G_\infty$ of $G$. We provide asymptotic estimates of the Riesz potentials associated with $\mathcal{L}$ at $0$ and at $\infty$, as well as of the kernels associated with functions of $\mathcal{L}$ satisfying Mihlin conditions of every order. We also prove some Mihlin-H\"ormander multiplier theorems for $\mathcal{L}$ which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the `Plancherel measure' associated with $\mathcal{L}$ from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.


Introduction
This paper deals with functional calculus on non-homogeneous left-invariant hypoelliptic self-adjoint differential operators on nilpotent Lie groups.
The approach introduced in [28] indicates that it is possible to transfer information on operators that are functions of a (positive) Rockland operator L on a connected and simply connected graded group G, or on its convolution kernel, to analogous information relative to the projection of L on a general connected and simply connected, but not necessarily homogeneous, quotient group.
Let G = G/I be the quotient group, where we assume that I is not dilation invariant to avoid trivialities. The one-parameter family of isomorphic quotient groups G s /I s , where I s is I dilated by s ∈ R + , admits two limits G 0 = G/I 0 and G ∞ = G/I ∞ (no longer isomorphic to G), where I 0 and I ∞ are dilation invariant, so that G 0 and G ∞ admit induced gradations from G.
Correspondingly, the operator L induces a family (L s ) s∈[0,+∞] , of projected operators on the different quotients. The limit operators L 0 , L ∞ are Rockland, while the other L s lack homogeneity, remaining however hypoelliptic. More precisely, they are weighted subcoercive, according to the definition introduced in [9]. 1 The starting point in the analysis of [28] is a weighted generating family X 1 , . . . , X n of the Lie algebra g of G. The (Lie algebra of the) group G is then interpreted as the quotient of the free nilpotent Lie algebra F of sufficiently high step with generators X 1 , . . . , X n ; the Lie algebra F is then endowed with the (unique) gradation obtained assigning to each X j a degree equal to the weight of X j . Thus, in the above notation, F is the Lie algebra of G and the quotient map is uniquely determined by the condition that each X j is mapped onto X j . A non-commutative homogeneous polynomial P in n indeterminates (endowed with the same weights of X 1 , . . . , X n ) is then considered under the assumption that the operator L = P( X 1 , . . . X n ) is hypoelliptic (hence Rockland). In particular, also the operator L = P(X 1 , . . . , X n ) is hypoelliptic; examples of such operators are the sums of even powers of generating vector fields.
It was proved in [28] that there is a fundamental solution K of L satisfying the asymptotic relations 2 where K 0 and K ∞ are fundamental solutions of L 0 on G 0 and of L ∞ on G ∞ , respectively, while P is a suitable polynomial on G 0 .
The results of the present paper can be divided into four parts. The first part concerns the heat kernels associated to the operator L, i.e., the kernels of the operators e −tL . In Section 2 we recall the basic constructions of [28] and then we introduce a (somewhat redundant) family of left-invariant vector fields X s,j on each group G s , s ∈ [0, ∞], which behaves nicely under dilation (which can no longer be defined as automorphisms of the group G s , but rather as isomorphisms between different G s ). We then introduce two moduli | · | s and | · | s, * on each G s : the former behaves nicely under dilation and equals a homogeneous norm on G 0 near the identity e and a homogeneous norm on G ∞ near ∞, under suitable identifications; the latter, inspired by [18,20], is a compromise between the modulus | · | s and the Riemannian distance from e associated with the vectors X s,j . The importance of | · | s, * lies in the fact that it grows much faster than | · | s at ∞, in general, so that it leads to better multiplier theorems. In Section 3 we then make use of the vector fields X s,j and the moduli | · | s, * to prove uniform 'Gaussian' estimates for the kernels h s,t of the e −tLs (Theorem 3.1); we also consider estimates of the derivatives in s of the h s,t , appropriately defined.
In the second part (Section 4) we extend the asymptotic estimates in [28] to general complex powers of L (Theorem 4.4), defined by analytic continuation in the same fashion of the Euclidean case. Even though it would be possible to use the same techniques employed in [28], we shall rely as much as possible on the estimates on h s,t provided in Theorem 3.1; in this way, we are able to describe more precisely also the higher order terms of the obtained developments, in some specific situations (Theorem 4.7).
In the third part (Section 5) we give asymptotic estimates to kernels of more general multiplier operators (Theorem 5.12) and prove some multiplier theorems of Mihlin-Hörmander type (Theorems 5. 15 and 5.17). For what concerns the asymptotic estimates, here we consider more general functions of the operator L -namely, functions satisfying Mihlin conditions of every order up to the multiplication by a fractional power. Even though these functions include the complex powers of L, Theorem 3.1 is not completely contained in Theorem 5.12, since several terms of the developments obtained in the latter are only defined up to polynomials. We then pass to some multiplier theorems, which are generalization of some of the results presented in [20] to the non-homogeneous case. While Theorem 5.15 is stated in full generality and gives non-homogeneous Mihlin-Hörmander conditions on the multipliers in the fashion of [1,32], Theorem 5.17 makes use, in a quite more specific situation, of the techniques introduced in [14,15] and then systematically developed in [18,20] to lower the regularity threshold up to half the topological dimension of G (instead of half the growth of the volume of G as in Theorem 5.15). Optimality is achieved when G is a product of Métivier and abelian groups, and L is (any) hypoelliptic sub-Laplacian thereon.
The fourth part (Section 6) deals with the spectral Plancherel measure β L and its comparison with β L0 and β L∞ (Theorem 6.4), when L is 'quasi-homogeneous', following [32]. Here we both extend the results of [32] to sums of even powers of generating homogeneous vector fields (instead of quasi-homogeneous sub-Laplacians), and we also observe that the estimates on the density of β L with respect to the Lebesgue measure on R + automatically improve to asymptotic expansions at 0 and at ∞.

General Setting
In this section we shall present the general framework in which we shall work in the sequel. It is basically the same as that of [28], except for the fact that we shall not require that the graded group G be a free nilpotent Lie group. We shall briefly repeat the basic constructions for the ease of the reader.

Contractions
Let G be a graded, connected and simply connected Lie group with Lie algebra g, with gradation ( g j ); let pr j be the projection of g onto g j with kernel j ′ =j g j ′ , and define n := max{ j > 0 : g j = 0 }.
On G we introduce the dilations x → r · x, r ∈ R + = (0, ∞), adapted to the given gradation, i.e., such that r · x = r j x if x ∈ g j . We shall sometimes denote by ρ r the dilation by r. A linear subspace v of g is graded, i.e., v = j v ∩ g j , if and only if it is homogeneous, i.e., invariant under dilations. We say that a linear map from a graded subspace v of g to g is • homogeneous if it preserves the gradation, i.e., it maps v ∩ g j into g j for every j; • strictly sub-homogeneous if it maps v ∩ g j into j ′ <j g j ′ for every j; • strictly super-homogeneous if it maps v ∩ g j into j ′ >j g j ′ for every j. Now, let G be the quotient of G by a (not necessarily homogeneous) normal subgroup, and denote by π the corresponding projection; we shall assume that G is simply connected. Let i be the kernel of dπ, and observe that ker π = exp G i since G is simply connected. Then, define For s ∈ (0, ∞), we define i s := s −1 · i. The following result is basically a generalization of [28, Proposition 2, Lemma, and Corollary of § 2].
Proposition 2.1. The vector spaces i 0 and i ∞ are graded ideals of g and have the same dimension as i.
Proof. 1. It is clear that i 0 is a graded subspace of g; let (i 0,j = i 0 ∩ g j ) be its gradation. Take x ∈ g j1 for some j 1 and y ∈ i 0,j2 for some j 2 ; let us prove that [x, y] ∈ i 0,j1+j2 . Now, there is y ′ ∈ i such that y − y ′ ∈ j ′ <j2 g j ′ , so that [x, y] ∈ [x, y ′ ] + j ′ <j2 g j1+j ′ , whence [x, y] = pr j1+j2 ([x, y ′ ]) ∈ i 0,j1+j2 . By the arbitrariness of x and y, it follows that i 0 is a graded ideal. In the same way one proves that i ∞ is a graded ideal.
2. Now, let us define ψ 0,1 . Observe that, by induction, we may define a basis (e k ) of i and an increasing sequence (k j ) such that (e k ) k kj is a basis of i ∩ j ′ j g j ′ for every j. Let us prove that, for every j, (pr j (e k )) kj−1<k kj is a basis of i 0,j . Clearly, it will suffice to prove linear independence. Now, if (λ k ) kj−1<k kj is a family of real numbers such that k λ k pr j (e k ) = 0, then kj−1<k kj λ k e k ∈ i ∩ j ′ <j g j ′ . Hence, there is a family (λ k ) k kj−1 of real numbers such that kj−1<k kj whence λ k = 0 for every k = 1, . . . , k j . Then, we may simply define ψ 0,1 as the linear map such that for every j = 1, . . . , n and for every k = k j−1 + 1, . . . , k j . Then ψ 0,1 is strictly sub-homogeneous and showing that I + ψ 0,1 maps i 0 onto i bijectively. It is also clear that which tends to 0 as s → 0 + . In a similar way one constructs ψ ∞,1 and proves the corresponding properties. In particular, we see that i, i 0 , and i ∞ have the same dimension.
3. Let h 0 be a graded complement of i 0 in g. Since the mapping s → i s is continuous on [0, ∞] (with values in the Grassmannian of (dim i)-dimensional subspaces of g), it follows that h 0 is an algebraic complement of i r for some r > 0. Therefore, h 0 = (s −1 r) · h 0 is an algebraic complement of i s = (s −1 r) · i r for every s ∈ (0, ∞).
The assertions concerning h ∞ are proved in a similar way.
For s ∈ [0, ∞], consider the quotient Lie algebras g s = g/i s . Dilation of g by r > 0 induces an isomorphism between g s and g r −1 s ; in particular, g s is isomorphic to g 1 for every s ∈ (0, ∞), while g 0 and g ∞ need not be isomorphic with any other g s . We call g 0 and g ∞ the local and the global contractions of g 1 , respectively.
We fix once and for all two graded algebraic complements h 0 and h ∞ of i 0 and i ∞ , respectively. By Proposition 2.1, both h 0 and h ∞ are complementary to i s for all s ∈ R + . Definition 2.2. For s ∈ [0, ∞), let P 0,s be the projection of g onto h 0 with kernel i s and, for s ∈ (0, ∞], let P ∞,s be the projection of g onto h ∞ with kernel i s . Lemma 2.3. P 0,0 is homogeneous and, for every s ∈ R + , P 0,s − P 0,0 is strictly sub-homogeneous; in addition, P 0,rs = r −1 · P 0,s (r · ) for every r, s ∈ R + . Analogously, P ∞,∞ is homogeneous and, for every s ∈ R + , P ∞,s −P ∞,∞ is strictly super-homogeneous; in addition, P ∞,rs = r −1 · P ∞,s (r · ) for every r, s ∈ R + .
Each map P 0,s (resp. P ∞,s ) induces a Lie algebra structure on h 0 (resp. h ∞ ); we denote by [ · , · ] 0,s (resp. [ · , · ] ∞,s ) the corresponding Lie bracket. In other words, Notice that, for r, s ∈ R + , We use the Baker-Campbell-Hausdorff products induced by the Lie brackets in (2) to realize either , as the underlying manifold 3 of the group G s := G/ exp G i s . We call G 0 and G ∞ the local and the global contractions of G 1 , respectively. Notice that and the analogous formulae for products in G s . We denote by π s , s ∈ [0, ∞], the canonical projection of G onto G s and by dπ s : g → g s its differential. By an abuse of language, we shall keep the same notation whenever G s , or g s , is identified with either h 0 or h ∞ .
Since the ideals i 0 and i ∞ are graded, the corresponding quotients g 0 and g ∞ inherit a gradation and the corresponding dilations. These dilations coincide with the restriction to h 0 and h ∞ , respectively, of the dilations of g.
For s, r ∈ R + , it follows from (3) that dilation by r on either h 0 or h ∞ induces an isomorphism of G s onto G r −1 s for every s. Notice that the so-induced mappings G s → G r −1 s do not depend on the chosen identifications (cf. (ii) and (iv) of Proposition 2.4 below).
We denote by Q, Q 0 , and Q ∞ the homogeneous dimensions of G, G 0 , and G ∞ , respectively. (i) λ s is the unique linear mapping such that x−λ s (x) ∈ i s for every x ∈ h 0 ; in addition, λ s is invertible and its inverse λ −1 (iii) λ s − I is strictly super-homogeneous and λ −1 s − I is strictly sub-homogeneous; (iv) λ rs = r −1 · λ s (r · ) for every r > 0.
Proof. By the definition of P 0,s , P ∞,s , the two cosets x + i s and λ s (x) + i s coincide for x ∈ h 0 . This gives (i) and (ii); (iii) and (iv) follow directly from Lemma 2.3. To prove (v), we argue as in the proof of [28,Proposition 3]. Define h 0,j := g j ∩h 0 and h ∞,j := g j ∩h ∞ , and set s = 1. Since λ 1 is super-homogeneous we see that, for every k = 1, . . . , n, Summing up all these inequalities, we see that For what concerns (vi), fix a norm · on g and observe that there is a constant C 1 such that In principle, we shall privilege the realization of Gs on h 0 for s close to 0 and that on h∞ for s close to ∞. We prefer anyhow to keep the double realization for every s ∈ R + in order to avoid apparent discontinuities in s at some finite point on one hand, and a-priori quantifications of "closeness" to 0 or ∞ on the other.
for every x ∈ g. Further, by the quasi-sub-additivity of N , there is a constant C ′ > 0 such that, for Therefore, there is a constant C ′′ > 0 such that The second part is proved similarly.

Invariant Vector Fields
We now pass to the approximation of differential operators, following [28, § 4].
We say that f is log-homogeneous of degree d ∈ N if there are a homogeneous polynomial P of degree d on V and a homogeneous norm N such that f − P log N is homogeneous of degree d. 4 We say that a continuous linear operator X : Notice that, if X is a left-invariant differential operator under a homogeneous Lie group structure on V , then Xf = f * (Xδ 0 ) and X is homogeneous of order d if and only if the distribution Xδ 0 is homogeneous of degree −Q − d.
In addition, if f is a function of class C ∞ on V and M f is the operator of multiplication by f , then f is homogeneous of degree d if and only if M f is homogeneous of order −d.
As a consequence, if X is a homogeneous differential operator of order d and f is a homogeneous function of degree d ′ and of class C ∞ , then f X is a homogeneous differential operator of order d − d ′ .
Finally, observe that, if an element X of the enveloping algebra of G is homogeneous of degree d, then the corresponding left-(or right-)invariant differential operator is homogeneous of order d, and conversely. Similar statements hold for G, G 0 , and G ∞ . Now, observe that h 0 and h ∞ are graded subspaces of g, so that also h 0 ∩ h ∞ is a graded subspace of g. Hence, we may complete a homogeneous basis of h 0 ∩ h ∞ to homogeneous bases of h 0 and h ∞ , and then complete the union of the two (which is a homogeneous basis of h 0 + h ∞ ) to a homogeneous basis of g. Consequently, we may state the following definition. Definition 2.6. We denote by ( X j ) j∈J a homogeneous basis of g such that there are two subsets J 0 and J ∞ of J such that ( X j ) j∈J0 is a basis of h 0 , while ( X j ) j∈J∞ is a basis of h ∞ . We denote by d j the degree of X j (as an element of the graded Lie algebra g, so that X j is homogeneous of order d j as a differential operator). Fixing coordinates on g associated with the basis ( X j ) j∈J , we denote by (∂ j ) j∈J the corresponding partial derivatives.
Define X s,j := dπ s ( X j ) for every j ∈ J and for every s ∈ [0, ∞], so that (r · ) * X s,j = r dj X r −1 s,j for every s ∈ [0, ∞], for every r > 0, and for every j ∈ J. Finally, fix a total ordering on J and define, for every γ ∈ N J , so that X γ is homogeneous of order d γ := j∈J γ j d j . Define ∂ γ and X γ s , for every s ∈ [0, ∞], in a similar way. To simplify the notation, we shall identify N J0 and N J∞ with subsets of N J ; when γ ∈ N J0 (resp. γ ∈ N J∞ ), we shall also write ∂ γ 0 (resp. ∂ γ ∞ ) instead of ∂ γ .
Proposition 2.7. For every γ ∈ N J there are two unique finite families (p 0,γ,γ ′ ) γ ′ ∈N J 0 and (p ′ 0,γ,γ ′ ) γ ′ ∈N J 0 of polynomials on h 0 such that, identifying G s and G 0 with h 0 for every s ∈ [0, ∞), In addition, p 0,γ,γ ′ and p ′ 0,γ,γ ′ are sums of homogeneous polynomials of degrees strictly greater than Analogously, there are two unique finite families In addition, p ∞,γ,γ ′ and p ′ ∞,γ,γ ′ are sums of homogeneous polynomials of degrees strictly smaller than d γ ′ − d γ . 5 In particular, lim Notice that, when γ ∈ N J0 , it may happen that p 0,γ,γ = 0 and p ′ 0,γ,γ = 0. Nonetheless, it is always true that both p 0,γ,γ and p ′ 0,γ,γ vanish at 0. For example, consider the case in which G is the free 2-step nilpotent Lie group on three generators X 1 , X 2 and X 3 (and the standard dilations), and define i as the vector space generated by [ X 1 , and G 1 is isomorphic to the three-dimensional Heisenberg group, while G 0 is isomorphic to R 3 . Fix coordinates (x 1 , x 2 , x 3 ) on G 1 corresponding to the basis (X 1,1 , X 1,2 , X 1,3 ), so that X 0,j = ∂ xj under the identification of G 0 and G 1 with h 0 . Then, simple computations show that for every s ∈ (0, ∞). Now, let us prove that the p 0,γ,γ ′ and the p ′ 0,γ,γ ′ are polynomials. Observe that ((X γ ′ s ) 0 ) γ ′ ∈N J 0 , j γ ′ j k is a basis of the space of distributions on h 0 supported at 0 and of order at most k, for every k ∈ N and for every s ∈ [0, ∞). Therefore, there are two families (S 0,γ ′ ) and ) are polynomials; therefore, it is easily seen that p 0,γ,γ ′ and p ′ 0,γ,γ ′ are polynomials. Finally, let us prove that p 0,γ,γ ′ and p ′ 0,γ,γ ′ are sums of homogeneous polynomials of degrees strictly greater than d γ ′ − d γ . Indeed, observe that the continuity of P 0,s and [ · , · ] 0,s in s at 0 shows that Since the X γ ′ s are pointwise linearly independent for every s and converge to the X γ ′ 0 , we must have The assertion follows in this case. The properties of the families (p ∞,γ,γ ′ ) and (p ′ ∞,γ,γ ′ ) are proved even more easily.

Moduli
Here we construct some control moduli on G and the G s , following [19, § 2.3]. Cf. also [9,30] for more details on 'weighted' control distances.
The proof is simple and is omitted.
In order to provide some more insight into the moduli | · | and | · | * , let us introduce some more notation. First, we define |x| ′ R as the greatest lower bound of the set of ε > 0 such that there are an absolutely continuous curve γ : [0, 1] → G and some measurable functions a j : [0, 1] → R such that a j ∞ ε for every j ∈ J, such that γ(0) = e and γ(1) = x, and such that for almost every t ∈ [0, 1]. Then, it is not hard to see that the following hold: • |x| * = |x| ′ R for every x ∈ G such that |x| * 1 or, equivalently, |x| ′ R 1; • |x| n * |x| ′ R |x| * for every x ∈ G such that |x| * 1 or, equivalently, |x| ′ R 1; • |x| * = max(|x|, |x| ′ R ) for every x ∈ G. In addition, if we denote by d R the (left-invariant) Riemannian distance associated with the (leftinvariant) Riemannian metric for which ( X j ) j∈J is an orthonormal basis, then |x| Consequently, | · | * is a reasonable compromise between a homogeneous norm (locally) and a Riemannian distance (globally). We define B s (r) (resp. B s, * (r)) as the set of x ∈ G s such that |x| s < r (resp. |x| s, * < r), for every r > 0.
One may prove that the moduli | · | s and | · | s, * can be defined in the same fashion of the moduli | · | and | · | * . We leave the details to the reader.
Proof. 1-4. These assertions follow from the corresponding ones of Proposition 2.9. 5. Fix z ∈ g and observe that, since | · | is proper, for every s ∈ [0, ∞] there is y s ∈ i s such that |z + y s | = |π s (z)| s . In particular, |z + y s | |z|, so that the set { y s : s ∈ [0, ∞] } is relatively compact in g. Then, fix s ′ ∈ [0, ∞] and observe that there is a sequence (s k ) of elements of [0, ∞] converging to s ′ such that lim k→∞ |π s k (z)| s k = lim inf s→s ′ |π s (z)| s . Notice that we may assume that (y s k ) converges to some y ′ in g, so that y ′ ∈ i s ′ . Therefore, |π s (z)| s , and observe that we may take y ′ s ′ k ∈ i s ′ k , for every k ∈ N, in such a way that the sequence (y ′ s ′ k ) converges to y s ′ . Therefore, whence the first assertion. The second assertion is proved similarly. 6. The assertion follows from Proposition 2.4 and from 4 and 5 above.
Definition 2.12. For every s ∈ [0, ∞], we define ν Gs as the unique Haar measure on G s such that ν Gs (B s (1)) = 1. We define D s , the volume growth of G s , in such a way that ν Gs (U k ) ≍ k Ds for k → ∞ an for every compact neighbourhood U of e (cf., for instance, [13, Theorem II.1]).
Corollary 2.13. The following hold: Notice that it may happen that either where H 1 is the three-dimensional Heisenberg group; denote by X, Y, T, U a basis of g such that [X, Y ] = T while the other commutators vanish, and endow G with coordinates such that ((z, t), u) corresponds to exp(Re zX + Im zY + tT + uU ); endow G with similar coordinates and define π((z, t), u) := (z, t + u). Define dilations on G so that X, Y, T, U have degrees 1, 1, 2, 3, respectively.
If, in the same example considered above, we choose dilations on G in such a way that X, Y, T, U have degrees 1, 1, 2, 1, respectively, then Finally, if we consider G, π, and G as the products of the ones in the preceding examples, then clearly In addition, denote by g s the Lie algebra of G s , and define inductively g s,[1] := g s and g s,[j+1] := [g s , g s, [j] ] for every j 1.
In the same way one proves that D ∞ D 1 .
2. Indeed, Proposition 2.11 and the above remarks imply that as r → +∞. The assertion follows.
3. This follows easily from the formula for D s used in 1.
Here is a simple result which will be useful later on. The proof, which is a simple modification of that of [37, VIII.1.1], is omitted. Lemma 2.14. For every s ∈ [0, ∞], for every p ∈ [1, ∞], for every f ∈ C 1 (G s ), and for every x ∈ G s , We conclude this subsection with some uniform estimates on the growth of the volume of the balls associated with the | · | s, * . Indeed, observe that the preceding facts prove that for every s ∈ [0, ∞] there is a constant C s > 0 such that ν Gs (B s, * (r)) C s r Ds for every r 1; however, we shall need to know that one may take the C s to be independent of s. Actually, we shall prove a finer result, showing how the growth of the volume of balls decreases as s approaches 0 or ∞.
for every s ∈ [0, ∞] and for every r 1. In addition, when G is stratified, so that Q 0 = D 0 and Q ∞ = D ∞ = D 1 by Corollary 2.13, one may take N 0 = Q ∞ − Q 0 .
In particular, for every ε > 0 there is a constant C ε > 0, independent of s, such that Notice that, when G is not stratified, then (the optimal) N 0 and N ∞ may be smaller or larger than Let F k be the Lie group whose Lie algebra has a basis X, Y 1 , . . . , Y k such that Y j+1 = [X, Y j ] for every j = 1, . . . , k − 1, while the other commutators vanish. Consider G := R × F k , with basis of the corresponding Lie algebra U, X, . Then, we may choose h 0 = X, Y 1 , . . . , Y k−1 , U and a neighbourhood of the identity Q := [−1, 1] k+1 (in the coordinates associated with the basis X, Y 1 , . . . , Y k−1 , U ). Then, the Baker-Campbell-Hausdorff formula shows that, for every s ∈ [0, ∞], where P h and R h are suitable polynomial mappings (independent of s). Integrating in (x, y) first and then in u, we see that Proof. 1. We consider only the case s ∈ [0, 1], since the case s ∈ [1, ∞] is completely analogous (or almost trivial when G is stratified, see 4 below). Define g [1] := g and, by induction, ) is a decreasing sequence of graded ideals of g (the lower central series). Notice that, arguing as in the proof of Proposition 2.1, one may prove that i s ∩ g [k] converges to some for every k ∈ N * , so that we may assume that h 0 = k V k . Analogously, observe that ; arguing as in the proof of Proposition 2.1, we then see that for every s ∈ (0, ∞).
Then, we may find a family (k j ) j∈J0 of positive integers and a homogeneous basis ( Y j ) j∈J0 of h 0 such that ( Y j ) kj =k is a basis of V k , for every k ∈ N * , and such that Y j has degree d j for every j ∈ J 0 . Choose, in addition, a homogeneous basis ( Y j ) j∈J k of W k for every k ∈ N * (to make the notation consistent, we assume that the J k , for k ∈ N, are mutually disjoint); we define k j := k and we denote by for every s ∈ [0, 1]. In addition, denoting by ν h0 the (fixed) Lebesgue measure on h 0 , again by 5 of Proposition 2.11 we see that there is a constant C 2 > 0 such that under the identification of G s with h 0 . Thanks to 3 of Proposition 2.11, it will then suffice to estimate ν h0 (Q ·G s h s ) for every h ∈ N * and for every s ∈ [0, 1]. 3. Now, observe that, arguing as in the proof of [5, Theorem 2 of Chapter II, § 6, No. 4], we see that, for every j 1 , . . . , j h ∈ J 0 , for every ℓ 1 , . . . , ℓ m with |ℓ 1 |, . . . , |ℓ m | 1 and for every m ∈ N * . Therefore, there is a constant for every s ∈ [0, 1] and for every h ∈ N * . Now, arguing by induction on Card( J ) Card(J 0 ), we see that the measure ν h0 (that is, with respect to any basis whose fundamental parallelotope has measure 1).
, the first assertion will be established if we prove that Then, by 1 we see that J ′ = J 0 , so that the assertion follows arguing as before.
4. Now, assume that G is stratified. Then, it is clear that g [k] = q k g q , so that the assertion for s ∈ [1, ∞] is trivial. Then, consider the preceding construction for s ∈ [0, 1] and observe that k j d j for every j ∈ J, with equality when j ∈ J 0 . Take J ′ as in 3.
Observe that we may construct, by induction on k = 1, . . . , n, mutually disjoint subsets is the basis of a graded complement of q<k g q ∩ h 0 in h 0 , for every k = 1, . . . , n. Define k ′ j := k for every j ∈ J ′ k and for every k = 1, . . . , n, and observe that d j k ′ j for every j ∈ J ′ thanks to Proposition 2.7. Furthermore, define Y J ′ j as the homogeneous for every j ∈ J ′ , and observe that ( Y J ′ j ) j∈J ′ is a basis of h 0 . In addition, arguing as in 3 above we see that , so that our assertion will be established if we prove that To prove this fact one may use Gauss elimination to the family (s k ′ j −δj Y (s) j ) (more precisely, to the matrix of the coordinates of the vectors Y (s) j with respect to the basis ( Y j ) j∈J0 ) and observe that, by homogeneity arguments, the resulting family is (linearly independent and) independent of s.

Estimates of the Heat Kernel
We now introduce the operators in which we shall be mainly interested. Fix a homogeneous leftinvariant differential operator L on G such that L + L * is a positive Rockland operator of degree δ; then, we define L s := dπ s ( L) for every s ∈ [0, ∞]. We shall sometimes write L instead of L 1 to simplify the notation. 6 Then, the operators L, L, and L s are weighted subcoercive, hence hypoelliptic (cf. [19,Theorem 2.3]).
Denote by ( h t ) t>0 the heat kernel of L, which we shall consider as a semigroup of measures on G. In addition, for every s ∈ [0, ∞] and for every t > 0, we shall define h s,t : for every r > 0, for every s ∈ [0, ∞], and for every t > 0.
We fix a Lebesgue measure on g and identify h t with its density. With the h s,t we shall be more careful, though. Indeed, for s ∈ (0, ∞) the group G s can be identified with both h 0 and h ∞ , and it is not possible to find Lebesgue measures on h 0 and h ∞ which induce the same measure on G s for all s ∈ (0, ∞). Therefore, we shall fix two Lebesgue measures on h 0 and h ∞ and define two densities h 0,s,t and h ∞,s,t of h s,t accordingly.
Precisely, for s ∈ [0, ∞), we define h 0,s,t as the density of (P 0,s ) * ( h t ) with respect to the fixed Lebesgue measure on h 0 ; in this way, h 0,s,t becomes the (density of) h s,t , under the identification of g s (hence of G s ) with h 0 given in Definition 2.2. Observe that, with these choices (and with a suitable Lebesgue measure on i 0 , independent of s), for every s ∈ [0, ∞), for every t > 0, and for every x ∈ h 0 .
Analogously, for s ∈ (0, ∞] we shall define h ∞,s,t as the density of (P ∞,s ) * ( h t ) with respect to the fixed Lebesgue measure on h ∞ . Similar remarks apply.
We now prove some uniform estimates on h 0,s,t and h ∞,s,t and their derivatives which cannot be derived from the general estimates for weighted subcoercive operators.
Theorem 3.1. Fix c > 0 and d ∈ R, and let X 0 and X ∞ be two homogeneous differential operators with continuous coefficients on h 0 and h ∞ , respectively, of order d. Then, for every k ∈ N there are two constants C, b > 0 (independent of s) such that for every s ∈ [0, ∞), for every x ∈ h ∞ , and for every t > cs δ , while for every s ∈ [0, ∞), for every x ∈ h 0 , and for every t ∈ (0, cs −δ ].
Proof. 1. Consider the first assertion; notice that we may reduce to the case in which 6 Notice that in [28] the operator L is only required to be Rockland; nonetheless, since we are interested in the corresponding heat kernels, additional restrictions have to be imposed.
for every x ∈ h ∞ , for every s ∈ [0, ∞), and for every t > 0. Therefore, Faà di Bruno's formula shows that for every s ∈ [0, ∞), for every t > 0, for every k ∈ N, and for every x ∈ h ∞ . In addition, observe that ∂ ℓ s ′ pr j •ψ ∞, 1 /s ′ is a (linear) polynomial of degree at most j − ℓ for every j = 2, . . . , n and for every ℓ = 1, . . . , j − 1, and is 0 otherwise. Therefore, there are C 1 , b > 0 such that for every (x, y) ∈ h ∞ ⊕ i s , for every t > 0, and for every s ∈ (0, ∞], with some abuses of notation. Therefore, Now, fix a norm · on g, and recall that ψ ∞, 1 /s ′ is strictly super-homogeneous, so that there are two constants C 2 , C ′ 2 > 0 such that for every y ∈ i ∞ with homogeneous components y 1 , . . . , y n , and for every s ′ ∈ [0, c 1 /δ ]. In addition, observe that all homogeneous norms on G are equivalent and that both h ∞ and i ∞ are homogeneous subspaces of g, so that there is a constant C 3 > 0 such that |z 1 + z 2 | C 3 (|z 1 | + |z 2 |) for every z 1 , z 2 ∈ g, and such that |x + y| Then, for every x ∈ h ∞ , for every y ∈ i ∞ and for every t > cs δ , Therefore, there is a constants C 5 > 0 such that , and for every t > cs δ .
Hence, there is a constant C 6 > 0 such that for every x ∈ h ∞ , for every s ∈ [0, ∞), and for every t > cs δ , so that 2. Consider, now, the second assertion. Observe that we may assume that X 0 = f ∂ α h0 for some α and some continuous homogeneous for every s ∈ [0, 1], for every t > 0, for every k ∈ N, and for every x ∈ h 0 . In addition, observe that ∂ ℓ s ′ pr j •ψ 0,s ′ is a (linear) polynomial of degree at most n and of homogeneous order at least j + ℓ for every j = 1, . . . , n − 1 and for every ℓ = 1, . . . , n − j, and is 0 otherwise. Therefore, there are for every (x, y) ∈ h 0 ⊕ i 0 , for every s ∈ [0, ∞) and for every t > 0 (cf. [19,Theorem 2.3]). Therefore, arguing as in 1 we see that for every x ∈ h 0 , for every s ∈ [0, ∞), and for every t > 0. Now, observe that there is a constant C 2 1 such that for every z ∈ g. In addition, observe that the linear mapping L s ′ : x + y → x + y + ψ 0,s ′ (y) is an automorphism of g for every s ′ ∈ [0, ∞), and that the mapping In particular, assuming that x + y = x + y for every (x, y) ∈ h 0 ⊕ i 0 for simplicity, for every (x, y) ∈ h 0 ⊕ i 0 , for every s ∈ [0, ∞), and for every t ∈ (0, cs −δ ]. Hence, there is a constant C 4 > 0 such that for every (x, y) ∈ h 0 ⊕ i 0 , for every s ∈ [0, 1], and for every t ∈ (0, cs −δ ], so that , and for every t ∈ (0, cs −δ ]. The proof is complete. Proof. Observe that [19, Theorem 2.3 (f)] implies that there are C and ω such that for every t > 0. Therefore, for every t > 0 and for every s ∈ [0, ∞].

Riesz Potentials
We keep the notation of the preceding section. Here we generalize the asymptotic study of the fundamental solutions made in [28] to the complex powers of L s . Notice first that, while the convolution kernels of L − α δ s (the Riesz potentials) are easily defined when Re α < Q ∞ , in order to define them also for Re α Q ∞ we shall need to argue by analytic continuation.
Proof. Fix s ∈ (0, ∞]. In addition, fix k 1 , k 2 ∈ N and observe that, if 0 < Re α < Q∞ δ , then for every x ∈ h ∞ . Taking into account Lemmas 7.5, 7.6, and 7.7, it will suffice to prove that the mapping α → P s,α,k2 extends to a meromorphic mapping on C with poles of order at most 1 at the elements of for every x ∈ G and for every t > 0. In addition, since ψ ∞,s is linear and strictly super-homogeneous, for every t > 0 and y ∈ i ∞ . As a consequence, the mapping t → ∂ γ ∞ h ∞,s,t −δ (0) extends to a mapping of class C ∞ on R. Let j Q∞+dγ b s,γ,j t j be its Taylor development at the origin. Now, fix N ∈ N and observe that, for Re α < N + 1, By the arbitrariness of N , it follows that the mapping α → +∞ 1 t α δ ∂ γ h∞ h ∞,s,t (0) dt t extends to a meromorphic mapping on C with poles of order at most 1 at every element of Q ∞ + d γ + N. Summing up all these facts, it follows that the mapping α → I ∞,s,α extends to a meromorphic mapping on C, with poles of order at most 1 at every element of Q ∞ + N. Finally, it is clear that I ∞,s,−k = L k s δ 0 . The case s = 0 is treated similarly.
For every α ∈ C such that α ∈ Q ∞ +N, we define I s,α as the distribution on G s induced by the distribution I ∞,s,α of Proposition 4.1 under the identification of G s with h ∞ (in other words, I s,α = (π s ) * (I ∞,s,α ) by an abuse of notation). We define I s,α , for α ∈ Q ∞ + N, as the 0-th order term of the Laurent expansion of the mapping α ′ → I s,α ′ at α. 7 We denote by I 0,s,α the distribution on h 0 induced by I s,α under the identification of G s with h 0 , for every s ∈ (0, ∞); I 0,0,α is defined as in Proposition 4.1. We define I 0,α := (π 0 ) * (I 0,0,α ), with the same abuse of notation used above. Proposition 4.3. For every s ∈ (0, ∞], for every r > 0, and for every α ∈ C, the following hold: 1. (r · ) * I s,α = r −α I r −1 s,α if I s, · is regular at α (in which case also I r −1 s, · is regular at α); 7 Notice that the mapping α ′ → I s,α ′ may be regular at α.
Analogous assertions hold for s = 0, replacing h ∞ with h 0 and Q ∞ with Q 0 .
Analogous statements hold for I 0,α , with the obvious modifications.
Proof. The first assertion for 0 < Re α < Q ∞ follows easily from the equality (r · ) * h s,t = h r −1 s,r δ t , which holds for every r > 0, for every s ∈ [0, ∞], and for every t > 0. The general statement then holds by holomorphy.
For what concerns the second assertion, take s ∈ (0, ∞] and a pole α of I s, · , so that, in particular, α ∈ Q ∞ + N. Then, for every α ′ = α in a neighbourhood of α, and for every r > 0 so that, taking the 0-th order term of the Laurent expansions of both sides of the equality at α, Now, with the notation of Proposition 4.1, it is easily seen that, chosen k 1 = 0 and k 2 = −Q ∞ + α + 1, By inspection of the proof of Proposition 4.1, it is easily seen that lim

Asymptotic Developments
We keep the notation of the preceding sections. We prove some asymptotic developments of the I α generalizing those proved in [28] for the fundamental solutions. Even though the procedure of [28] may be generalized to the present setting, we prefer to give a different proof, which is shorter and gives a little more insight into the meaning of the further terms of the development. We then present, under rather restrictive assumptions, another proof which describes quite explicitly the terms of the development.
Theorem 4.4. Take and α ∈ C and s ∈ R + . Then, the following hold: 1. there is a sequence of log-homogeneous functions (I ∞,α has degree −Q ∞ + α − k, and such that for every N ∈ N and for every γ there is a constant C N,γ > 0 such that, for every s ∈ [1, ∞), for every x ∈ h ∞ such that |x| s −1 ; the factor 1 + |log|s · x|| may be omitted if α ∈ Q ∞ + d γ + N; 2. there are a sequence (P α,k ) of homogeneous polynomials on h 0 and a sequence (I 0,α has degree −Q 0 + α + k, such that P 0,α,k has degree k, and such that for every N ∈ N and for every γ there is a constant C ′ N,γ > 0 such that, for every s ∈ (0, 1], for every x ∈ h 0 such that 0 = |x| s −1 ; the factor 1 + |log|s·x|| may be omitted if α ∈ Q 0 + d γ + N. Proof. 1. Define H ∞ (s ′ , t, x) := h ∞, 1 /s ′ ,t (x) for every s ′ ∈ [0, ∞), for every t > 0, and for every x ∈ h ∞ , to simplify the notation. Take α ∈ C such that 0 < Re α < Q ∞ , and observe that a Taylor expansion of H ∞ in the first variable gives Now, Lemma 7.6 implies that the mapping α → 1 for every s ′ ∈ [0, ∞), for every t > 0, and for every x ∈ h ∞ , so that for every x ∈ h ∞ and for every t > 0.
Therefore, Lemma 7.6 and the estimates of ∂ k 1 H ∞ (0, t, · ) provided in Theorem 3.1 show that, for 0 < Re α < Q ∞ + k, In addition, we also see that the mapping, initially defined for 0 < Re α < Q ∞ + k, dt t extends to a meromorphic function on C such that I Finally, assume that Re α < Q ∞ + N . Observe that there is a constant C > 0 such that for every s ′ ∈ [1, ∞] and for every x ∈ h ∞ , thanks to Proposition 2.11. Therefore, Theorem 3.1 and the preceding computations imply that R s,α,N is well-defined for x = 0 and that there are there are two constants C ′ > 0 and b > 0 such that, for every γ, Hence, The assertion follows for s fixed. In order to get uniform estimates for s ∈ [1, ∞), reduce to the case s = 1 by means of Proposition 4.3. Let us give some more details in the case in which I s, · has a pole at α. Indeed, for every s ∈ [1, ∞], I ∞,s,α − k<N s −k I (k) ∞,α equals where P 1,α is defined in Proposition 4.3, while P ′ α,k is a suitable homogeneous polynomial on h ∞ of degree −Q ∞ + α − k. Since the term s −α (s −1 · ) * I ∞,1,α − k<N I (k) ∞,α satisfies the estimates of the statement, all we need to prove is that P 1,α • π 1 − k<N P ′ α,k has degree at most −Q ∞ + α − N . One may prove this by expressing P 1,α and the P ′ α,k in terms of h s,t and its derivatives in s −1 . Nonetheless, since the above proof shows that I ∞,s,α − k<N s −k I (k) ∞,α satisfies the estimates of the statement (with constants depending on s), the same necessarily applies to s −α log s(s −1 ) * P 1,α • π 1 − k<N P ′ α,k , whence our claim.
2. Define H 0 (s, t, x) := h 0,s,t (x) for every s ∈ [0, ∞), for every t > 0, and for every x ∈ h 0 , to simplify the notation. Take α ∈ C such that 0 < Re α < Q 0 , and observe that a Taylor expansion of H 0 in the first variable gives Now, by means of Lemma 7.7 we see that the mapping α → 1 dt t extends to a meromorphic function on C with values in E(G). In addition, as in 1 one may prove that for every x ∈ h 0 and for every t > 0. Therefore, making use of the estimates of ∂ k 1 H 0 (0, t, · ) provided in Theorem 3.1, we see that initially defined for Re α < Q 0 − k, extends to a meromorphic function on C. In addition, we also see that the mapping, initially defined for −k < Re α < Q 0 − k, extends to a meromorphic mapping on C. 8 Let us prove that I (k) 0,α is homogeneous of degree −Q 0 + α + k for α in the domain of holomorphy of I (k) 0,α . By analyticity, we may reduce to prove this fact for −k < Re α < Q 0 − k, in which case so that the assertion is easily established. Log-homogeneity holds at the poles of I |π s (x)| s for every x ∈ h 0 , and observe that |x| ′ > 0 for every non-zero x ∈ h 0 , and that there is a constant C > 0 such that for every x ∈ h 0 such that |x| 1 (cf. Proposition 2.11). In addition, Theorem 3.1 and the preceding computations imply that there are there are two constants C ′ > 0 and b > 0 such that so that R s,α,N (x) is well defined for x = 0. In addition, The assertion follows for s fixed. In order to get uniform estimates for s ∈ (0, 1], reduce to the case s = 1 by means of Proposition 4.3 (argue as in 1). 8 Using the estimates of H 0 provided in Theorem 3.1, it is not hard to see that 1 0 t α δ ∂ k Observe that, with the same techniques used to prove [28,Theorem 2], one may prove the following result.
Corollary 4.5. Take s ∈ (0, ∞), γ ∈ N J , and α ∈ C such that Re α d γ . Then, for every p, q ∈ (1, ∞) such that , convolution on the right with X γ s I s,α induces a bounded operator from L p (G s ) into L q (G s ).
Notice that, if convolution on the right with X γ s I s,α induces a bounded operator T s from L p (G s ) into L q (G s ) for some p, q ∈ (1, ∞) and for some s ∈ (0, ∞), then . Indeed, take r > 0 and f ∈ L p (G r −1 s ), and define ρ r (x) := r · x for every x ∈ G r −1 s . Then, where T r −1 s is given by convolution on the right with X γ r −1 s I r −1 s,α . Now, identify G s ′ with h 0 for every s ′ ∈ [0, s]. Observe that, denoting by ν h0 the fixed Lebesgue measure on h 0 , we have ν G s ′ = a 0,s ′ ν h0 for some a 0,s ′ > 0; in addition, the mapping s ′ → a 0,s ′ is continuous on [0, s] thanks to 5 of Proposition 2.11. Therefore, there is a constant C > 0 such that for every f ∈ L p (h 0 ) and for every r 1. Now, T r −1 s f converges pointwise to T 0 f as r → +∞ for every f ∈ C ∞ c (h 0 ) with vanishing moments of all orders, 9 so that for every such f .
The other inequality is proved similarly.
Proof. We shall briefly indicate the procedure employed in [28], for the sake of completeness. When p = q, observe that X γ s I s,α belongs to weak L r for every r ∈ (1, ∞) such that thanks to Theorem 4.4. Then, arguing as in the proof of [10, Proposition 1.19], we see that weak L r convolves L p (G s ) into L q (G s ) for 1 p − 1 q = 1 r ′ and p, q, r ∈ (1, ∞). When p = q, take τ ∈ C ∞ c (G s ) so that τ equals 1 in a neighbourhood of e. We shall prove that τ X γ s I s,α convolves L p (G s ) into itself; one may prove analogously that also (1 − τ )X γ s I s,α convolves L p (G s ) into itself and conclude the proof. Now, Proposition 2.7 and Theorem 4.4 show that τ X γ s I s,α equals τ X γ 0 I 0,α up to an integrable function, under the identification of G s with G 0 through h 0 ; consequently, it will suffice to show that τ X γ 0 I 0,α convolves L p (G s ) into itself (with respect to the convolution of G s ). Now, it is clear that there is a constant C > 0 such that for every x ∈ G s and for every γ ′ with length at most 1, thanks to 6 of Proposition 2.11. By Lemma 2.14 we then see that we may take C in such a way that for every x, y ∈ G s such that |x| s > 2|y| s > 0. In addition, since X γ 0 I 0,α is a homogeneous distribution of degree −Q 0 + α − d γ , it is clear that X γ 0 I 0,α has zero mean on the unit sphere (relative to | · | 0 ) when Im α = 0. Similar remarks apply to (X γ 0 I 0,α ) * . Taking into account [11, Lemma of Chapter III, § 3.1], it is not hard to see that we may apply [11, Theorem of Chapter III, § 4.3], so that τ X γ 0 I 0,α convolves L p (G s ) into itself for every p ∈ (1, ∞).
Remark 4.6. Observe that, if G = R n and L = ∆ 2 − ∆, then it is not hard to prove that 2 )δ 0 and I ∆ α is the kernel of (−∆) − α 2 defined by analytic continuation, for every α ∈ C. Then, observe that, for α ∈ (0, ∞) \ (n + N), I ∆ α and J α keep a constant sign (in particular, they vanish nowhere), so that when α > n. Hence, the polynomials appearing in the local expansion of I α in Theorem 4.4 cannot be omitted, in general.
Theorem 4.7. Take α ∈ C and s ∈ R + . Then, the following hold: 1. assume that G 1 = G ∞ (under the identification through h ∞ ) as Lie groups and that [L 1 , L ∞ ] = 0. Let d ∞ be the least degree of the non-zero homogeneous components of L 1 − L ∞ . Then, for every N ∈ N and for every γ there is a constant C N,γ > 0 such that, for every s ∈ [1, ∞), for every x ∈ h ∞ such that |x| s −1 ; the factor 1 + log|s · x| may be omitted if α ∈ Q ∞ + N; 2. assume that G 1 = G 0 (under the identification through h 0 ) as Lie groups and that [L 1 , L 0 ] = 0. Let d 0 be the greatest degree of non-zero homogeneous components of L 1 − L 0 . Then, there is a sequence (P α,k ) of homogeneous polynomials on G 0 such that P α,k has degree k for every k ∈ N, and such that for every N ∈ N and for every γ there is a constant C N,γ > 0 such that, for every s ∈ (0, 1], for every x ∈ h 0 such that 0 = |x| s −1 ; the factor 1 + log|s · x| may be omitted if α ∈ Q 0 + N.
Let us make some examples. Take a 2-step nilpotent Lie group G and a hypoelliptic sub-Laplacian L thereon. Then, we may endow G with the structure of a stratified group in such a way that L = L ∞ + L ′ , where L ∞ and L ′ are homogeneous sums of squares of degrees 2 and 4, respectively. By means of the construction described in the introduction, we may choose a 2-step stratified group G and a sub-Laplacian L in such a way that G s = G ∞ as Lie groups 10 and L s = L ∞ + s −2 L ′ for every s ∈ (0, ∞]. Thus, in this case the first part of Theorem 4.7 applies. If, in the preceding example, we define L 1 = L k ∞ + L ′ for some k 3, then, applying an analogous construction, we get G s = G 0 as Lie groups and L s = L k ∞ + s 2(k−1) L ′ for every s ∈ [0, ∞), so that the second part of Theorem 4.7 applies.
Proof. 1. Assume that G 1 = G ∞ as Lie groups and that [L 1 , L ∞ ] = 0. Define, for every t > 0, for every s ∈ (0, ∞] and for every θ ∈ [0, 1], h (θ) s,t ) t is a semi-group under convolution and that the mapping θ → h (θ) for every t > 0, for every s ∈ (0, ∞], and for every θ ∈ [0, 1]. Now, Proposition 2.11 and Theorem 3.1 imply that for every γ and for every k ∈ N there are two constants C, b > 0 such that for every s ∈ [1, ∞], for every t 1, for every θ ∈ [0, 1], and for every x ∈ G s . Now, take α ∈ C such that 0 < Re α < Q ∞ , and observe that a Taylor expansion of h The proof then proceeds as that of Theorem 4.4.
The case in which G s = G 0 and [L s , L 0 ] = 0 is treated similarly.

Spectral Measures and Multipliers
In this section, we assume that L is Rockland and formally self-adjoint, but not necessarily positive. Then L 2 s is weighted subcoercive, so that (L s ) is a weighted subcoercive system in the sense of [18,19]. Then, the operator L s , considered as an unbounded operator on L 2 (G s ) with initial domain C ∞ c (G s ), is essentially self-adjoint (cf. [19,Proposition 3.2]). We shall then denote by σ(L s ) the corresponding spectrum. Now, if m : σ(L s ) → C is bounded and Borel measurable, then there is a unique distribution K Ls (m) on G s such that m(L s )ϕ = ϕ * K Ls (m) for every ϕ ∈ C ∞ c (G s ) (cf. [19,Subsection 3.2]). In addition, there is a unique positive Radon measure β Ls on σ(L s ) such that K Ls extends to an isometry of L 2 (β Ls ) into L 2 (G s ) (cf. [19,Theorem 3.10]).
For the proof, argue as in [32, § 2], using the estimates for the heat kernel associated with L 2 s provided in Theorem 3.1.
Therefore, by means of the spectral calculus we see that K Lrs (m) = (r −1 · ) * K Ls (m(r δ · )) for every bounded Borel measurable function m : R → C. In addition, if m ∈ S(R), then, identifying K Lrs (m) and K Ls (m) with their densities with respect to ν Grs and ν Gs , respectively, so that β Lrs = ν Gs (B s (r))(r δ · ) * β Ls by the arbitrariness of m. In addition, it is easily seen that K L s ′ (m)(e) converges to K Ls (m)(e) as s ′ → s (see also Lemma 5.4 below). Thanks to Lemma 5.1 and the preceding remarks, this is sufficient to prove that β L s ′ converges vaguely to β Ls as s ′ → s.

Asymptotic Developments
Definition 5.3. For every m ∈ S(R) and for every s ∈ [0, ∞], we denote by K 0,s (m) (for s = ∞) and K ∞,s (m) (for s = 0) the densities of the measures corresponding to K Ls (m) on h 0 and h ∞ , respectively, under the usual identifications.
are of class C ∞ .
Proof. We prove only the first assertion. Observe that K Ls = (π s ) * • K L ; since K L ∈ L(S(R); S( G)) by [18,Proposition 4.2.1], it will suffice to prove that the mapping [0, ∞) ∋ s → (P 0,s ) * ∈ L(S( g); S(h 0 )) is of class C ∞ . Now, for every s ∈ [0, ∞), denote by L s the automorphism x + y → x + y + ψ 0,s (y) of (the vector space) g ∼ = h 0 ⊕ i 0 , and observe that L s depends polynomially on s, so that we may define L s for every s ∈ R. With this modification, it is readily verified that L s is still a measurepreserving automorphism of g for every s ∈ R, since ψ 0,s is strictly sub-homogeneous. Then, observe that P 0,s = P 0,0 • L −1 s for every s ∈ [0, ∞); since (P 0,0 ) * ∈ L(S( g); S(h 0 )), it will suffice to prove that the mapping R ∋ s → (L −1 s ) * ∈ L(S( g)) is of class C ∞ . However, this last assertion is an easy consequence of the fact that the mapping R ∋ s → L s ∈ L( g) is of class C ∞ .
Definition 5.5. For every k ∈ N, for every s ∈ [0, ∞), and for every m ∈ S(R), define Lemma 5.6. Take k ∈ N and m ∈ S(R). Then, for every s ∈ [0, ∞) and for every r > 0, Proof. The assertion follows from Proposition 5.2 by differentiation.
Definition 5.7. Take r ∈ R, and define M r (R * ) as the set of m ∈ C ∞ (R * ) such that for every k ∈ N there is a constant C k > 0 d k dλ k m(λ) C k |λ| r−k for every λ ∈ R * . We endow M r (R * ) with the corresponding semi-norms.
Definition 5.8. Take r ∈ R ∪ { ∞ } and s ∈ [0, ∞]. We define S ′ r (G s ) as the dual of the set S r (G s ) of ϕ ∈ S(G s ) such that Gs ϕ(x)P (x) dx = 0 for every polynomial P such that P (x) = O(|x| r s ) for x → ∞; we thus identify S ′ r (G s ) with the quotient of S ′ (G s ) by the set of the polynomials as above. Similar definitions replacing G s with h 0 or h ∞ , and | · | s with | · |.
Observe that, if s ∈ (0, ∞], then S r (G s ) = S r (h ∞ ) under the identification of G s with h ∞ (cf. 6 of Proposition 2.11).
We define CZ r (h ∞ ) in a similar way. For every s ∈ (0, ∞) we define CZ r (G s ) as the set of K ∈ S ′ −Q∞−r (G s ) such that there are K 0 ∈ E ′ (G s ) + S(G s ) and K ∞ ∈ C ∞ (G s ) ∩ S ′ −Q∞−r (G s ) such that K = K 0 + K ∞ , and such that the distributions on h 0 and h ∞ corresponding to K 0 and K ∞ belong to CZ r (h 0 ) and CZ r (h ∞ ), respectively. We endow CZ r (G s ) with the corresponding topology.
Finally, we denote by ν R+ the Haar measure on (R + , · ) such that Proposition 5.10. Take r ∈ R, a set B and a bounded family (ϕ t,b ) t∈(0,∞),b∈B of elements of S r (h 0 ). Then, the mapping t → t −r (t · ) * ϕ t,b ∈ S ′ −Q0−r (h 0 ) is ν R+ -integrable and the set of as b runs through B, is bounded in CZ r (h 0 ). In addition, K b has a representative K b , for every b ∈ B, such that for every α there is a constant C α > 0 such that for every x ∈ h 0 \ { 0 }, and for every b ∈ B; the factor 1 + |log|x|| may be omitted if In addition, if L is a bounded subset of C Next, take α such that d α > −Q 0 − r, and observe that there is a constant C 3,α > 0 such that for every x ∈ h 0 , for every t > 0 and for every b ∈ B. Then, fix a non-zero x ∈ h 0 , and observe that Taking into account all the preceding inequalities, we see that the mapping t → t −r (t · ) * ϕ t,b ∈ S ′ −Q0−r (h 0 ) is ν R+ -integrable and that the set of K b , as b runs through B, is bounded in CZ r (h 0 ). 2. Keep the notation of 1, and denote by P t,b,j the homogeneous component of P t,b of degree j, for every j = 0, . . . , −Q 0 + [−r]; define P ′ t,b := j<−Q0−r P t,b,j . Then, the arguments of 1 show that . In addition, arguing as in 1 we see that, for every α, Corollary 5.11. Take s ∈ (0, ∞), r ∈ R, a set B, and a family (ϕ t,b ) t>0,b∈B such that ϕ t,b ∈ S r (G st ) for every t > 0 and for every b ∈ B, and such that for every k ∈ N there is a constant C k > 0 such that for every γ such that d γ k, for every b ∈ B, for every t > 0, and for every x ∈ G st . Then, the mapping t → t −r (t · ) * ϕ t,b ∈ S ′ −Q∞−r (G s ) is ν R+ -integrable for every b ∈ B, and the set of as b runs through B, is bounded in CZ r (G s ).
Proof. By an abuse of notation, we shall identify G s ′ with h 0 if s ′ ∈ (0, 1) and with h ∞ if s ′ ∈ (1, ∞). In addition we shall identify the measures ϕ t,b ν Gst with its density ϕ t,b with respect to the fixed Lebesgue measure of h 0 , for t < s −1 , or to the fixed Lebesgue measure of h ∞ , for t > s −1 . Then, ϕ t,b differs from ϕ t,b • π st by a multiplicative constant which stays bounded as t runs through R + .
Observe first that, using Proposition 2.4 and 6 of Proposition 2.11, it is not difficult to show that there is a constant C > 0 such that 1 + |x| s ′ C(1 + |x| 1 /n ) for every x ∈ h 0 and for every s ′ ∈ (0, 1), while for every x ∈ h ∞ and for every s ′ ∈ (1, ∞). Hence, the set of ϕ t,b , as t runs through (0, s −1 ) and b runs through B, is bounded in S(h 0 ), while the set of ϕ t,b , as t runs through (s −1 , ∞) and b runs through B, is bounded in S(h ∞ ). Consequently, Proposition 5.10 and its proof imply that the dt t are well-defined elements of S ′ (G s ) and stay bounded in CZ r (h 0 ); analogously, the are well-defined elements of S ′ −Q0−r (G s ), and stay bounded in CZ r (h ∞ ). It will then suffice to prove that K b,0 equals a Schwartz function in a neighbourhood of ∞, and that (every representative of) K b,∞ is of class C ∞ on the whole of G s (with the required boundedness).
On the one hand, take k 1 and α, and observe that there is a constant C k,α > 0 such that for every x ∈ h 0 , for every t ∈ (0, s −1 ), and for every b ∈ B. Then, for every non-zero x ∈ h 0 and for every b ∈ B. By the arbitrariness of k and α, it follows that the (1 − τ )K b,0 stay in a bounded subset of S(h 0 ) as b runs through B, where τ is an element of C ∞ c (h 0 ) which equals 1 on a neighbourhood of 0.
On the other hand, denote by P t,b,j the homogeneous component of degree j of the Taylor series of ϕ t,b ∈ S(h ∞ ) about 0, for every t > 0, for every b ∈ B, and for every j ∈ N.
is a well-defined homogeneous polynomial of degree k, while for every α there is a constant C ′ k,α > 0 such that for every x ∈ h ∞ , and for every b ∈ B. Define so that K b,∞ is a well-defined representative of K b,∞ (under the identification of G s with h ∞ ) by the proof of Proposition 5.10. Then, for every k ∈ N, the . By the arbitrariness of k, it follows that the K ∞,b stay bounded in C ∞ (h ∞ ).
Theorem 5.12. Take r ∈ R. Then, the following hold: • for every k ∈ N, the continuous linear mapping K (k) • for every k ∈ N, the continuous linear mapping K • for every s ∈ (0, ∞), the continuous linear mapping K Ls : C ∞ c (R * ) → CZ rδ (G s ) induces a unique continuous linear mapping K Ls : M r (R * ) → CZ rδ (G s ) such that, if F is a bounded filter on M r (R * ) which converges pointwise to some m in M r (R * ), then K Ls (F) converges to K Ls (m) in S ′ −Q∞−rδ (G s ).
In addition, let M be a bounded subset of M r (R * ), and take τ 0 ∈ C ∞ c (h 0 ) and τ ∞ ∈ C ∞ c (h ∞ ) such that τ 0 and τ ∞ equal 1 in a neighbourhood of 0. Then, the following hold: 11 • for every N ∈ N there is a bounded family (K 0,m,N,s ) m∈M,s∈(0,1] of elements of CZ rδ−N (h 0 ) such that , for every m ∈ M and for every s ∈ (0, 1]; , for every m ∈ M and for every s ∈ [1, ∞). Proof. Let M be a bounded subset of M r (R * ) and fix a positive function ϕ ∈ C ∞ c (R * ) such that ∞ 0 ϕ(y δ · λ) dy y = 1 for every λ ∈ R * . Let us prove that the family (y δr m(y −δ · )ϕ) y>0,m∈M is bounded in S(R). Indeed, take h and observe that there is a constant C h > 0 such that for every λ ∈ R * , for every m ∈ M , and for every p = 0, . . . , h. Then, for every λ ∈ R * , for every y > 0, and for every m ∈ M , whence the assertion. Next, let us prove that K ∞,s (m ′ )) has all vanishing moments for m ′ ∈ C ∞ c (R * ). It will suffice to prove our assertion for k = 0, hence for K Ls (m ′ ). Now, for every h ∈ N we have m ′ h : Since every polynomial is L h s -harmonic for sufficiently large h (use Proposition 2.7 or observe that a similar property applies to L by homogeneity arguments), the assertion follows by (sesquilinear) transposition.
Therefore, for every k ∈ N, the family (y δr K y −rδ (y · ) * K Lsy (m(y −δ · )ϕ) dy y are well defined and stay in a bounded subset of K rδ (G s ). Therefore, the so-defined linear mappings ∞,∞ , and K Ls are continuous; in addition, by Proposition 5.2 and Lemma 5.6 and the choice of ϕ, they agree with their previous definition on S(R), S(R), and M r (R * ) ∩ ℓ ∞ (R * ), respectively. Now, if F is a filter on M which converges to some m 0 pointwise on R * , then y δr F(y −δ · )ϕ converges pointwise to y δr m 0 (y −δ · )ϕ, hence in S(R). As a consequence, also K (k) The analogous assertions concerning K Ls , for s ∈ (0, ∞), and K (k) ∞,∞ , for k ∈ N, are proved similarly. The first three assertions of the statement are therefore established. Now, observe that, for every N ∈ N, for y ∈ (s −1 , ∞). In addition, the as y runs through (s −1 , ∞). In addition observe that, if V is a finite-dimensional vector space, F is a closed subset of V with non-empty interior, and P is a linearly independent finite set of polynomials on V , then the mapping as m runs through M and s is fixed. In order to establish uniform boundedness for general s as in the statement, it suffices to reduce to the case s = 1, taking into account Proposition 5.2 and Lemma 5.6. The proof is therefore complete.

Multiplier Theorems
Here we shall repeat the arguments of [18, § 4.1] in order to provide a multiplier theorem for the operators L s , which will imply some sort of continuity for the mapping s → K Ls (m) for more general m. Even though the following results hold when L is self-adjoint, in order to avoid some technical issues we shall assume that L is positive.
In this section, when µ is a measure on G s which is absolutely continuous with respect to the Haar measure, we shall write µ L p (νG s ) to denote the L p norm of its density with respect to ν Gs , p ∈ [1, ∞].
for every x ∈ R, endowed with the corresponding topology. We denote by Recall that ν R+ is a Haar measure on the multiplicative group R + . Proposition 5.14. For every r > 0, for every γ, and for every α 1 , α 2 0 such that α 2 > α 1 there is a constant C > 0 such that r], and for every s ∈ [0, ∞]. If, in addition, β L1 has a density with respect to ν R+ bounded by min[( · ) Q 0/δ , ( · ) Q∞ /δ ], then we may take C in such a way that • define for every ℓ ∈ Z; • replace the references to [19, 2.3 (e) Theorem 5.15. Take N 0 and N ∞ as in Proposition 2.15. In addition, take a non-zero ψ ∈ C ∞ c (R + ) and α > 0, and for every s ∈ [0, ∞] denote by M α,s the space of m ∈ L 1 loc (R + ) such that m Mα,s := sup Finally, assume that β L1 has a density with respect to ν R+ bounded by min[( · ) Q 0/δ , ( · ) Q∞ /δ ]; define M ′ α,s as the space of m ∈ L 1 loc (R + ) such that is finite. Then, M α,s may be replaced by M ′ α,s in the previous assertions. Taking into account Theorem 6.4, this generalizes [1] for higher-order operators, and also [32,Theorem 2] for quasi-homogeneous sums of even powers of left-invariant vector fields on a homogeneous group, with N 0 = D 1 − D 0 at least when these powers are all equal. Notice that the proofs of [1,Theorem] and [32,Theorem 2], which are based on the property of finite speed of propagation of the wave equation, cannot be extended to the present setting.
Proof. 1. We shall denote by M s the space M α,s under the first set of assumptions, and the space M ′ α,s under the second set of assumptions. Notice that we may assume that ψ is positive and chosen in such a way that j∈Z ψ(2 −δj · λ) = 1 for every λ > 0. Fix ε ∈ (0, α). Then, Propositions 5.14 and 2.15 imply that there is p 0 > 1 such that for every γ there is a constant C γ > 0 such that for every s ∈ [0, ∞], for every m ∈ M s , for every γ, and for every j ∈ Z. Now, since m is the sum of the series j∈Z ψ(2 −δj · )m pointwise on (0, ∞), and since the partial sums of that series are uniformly bounded, we see that in the space of (right) convolutors of L 2 (G s ), for every s ∈ [0, ∞] and for every m ∈ M s ; in particular, in S ′ (G s ).
Let us first prove that the sum converges in L 1 loc (G s \ { e }). Indeed, take a compact subset L of G s \ { e }, and observe that j∈Z which is finite for every s ∈ [0, ∞] and for every m ∈ M s with m Ms 1, since ν G 2 −j s B 2 −j s (2 j ) ≍ 2 jQ∞ as j → −∞ for fixed s = 0 thanks to 6 of Proposition 2.11, while ν G0 B 0 (2 j ) = 2 jQ0 for every j ∈ Z.
Notice that the regularity threshold in Theorem 5.15 is not optimal, in general. We shall now present an improvement of Theorem 5.15, under more restrictive hypotheses, in the spirit of [14,15,20]. Let us briefly recall the notion of capacity introduced in [18,20]; we shall present it in a slightly simpler way in the setting of 2-step stratified groups.
Definition 5.16. Let G ′ be a 2-step stratified group with Lie algebra g ′ ; let (g ′ 1 , g ′ 2 ) be the stratification of g ′ and take h ∈ { 0, . . . , dim g ′ 2 }. Endow g ′ with a scalar product. Then, we say that G ′ is h-capacious if there is a linearly independent family X 1 , . . . , X h of elements of g ′ 1 and a linearly independent family T 1 , . . . , T h of elements of g ′ 2 such that for every X ∈ g ′ 1 and for every T ∈ g ′ 2 . For instance, if G ′ is the product of a finite family of Métivier or abelian groups, then G ′ is dim[G ′ , G ′ ]capacious (cf. [20,Proposition 3.9]), so that the following result applies (with a suitable choice of G) when L 1 has the form j∈J1 (iX j ) α , where α ∈ 2N * and (X j ) is a family of left-invariant vector fields on G 1 which generates its Lie algebra.
Notice that, when G ′ is an H-type group and L 1 = L ′ 1 − j T 2 j , where L ′ 1 is the standard (homogeneous) sub-Laplacian and the T j stay in the centre of g 1 , then Theorem 5.17 is a consequence of [26, Corollary 2.4].
Theorem 5.17. Assume that G is a 2-step stratified group and that G ∞ is h-capacious for some h ∈ N.
In addition, take a non-zero ψ ∈ C ∞ c (R + ) and α > 0, and for every s ∈ [0, ∞] denote by M α,s the space of m ∈ L 1 loc (R + ) such that is finite. Then, there is a constant C > 0 such that m(L s ) L(L p (Gs)) C m Ms,α for every s ∈ [0, ∞] and for every m ∈ M s,α .
The second part of Theorem 5.15 can be proved with the same techniques also under the assumptions of Theorem 5.17. We leave the details to the reader.

Quasi-Homogeneous Operators
We shall now investigate further the properties of the Plancherel measures β Ls in some specific situations: following [32], we shall prove that, when L s is 'quasi-homogeneous' in a suitable sense, then β Ls has a density of class C ∞ with respect to ν R+ , with complete and almost explicit asymptotic expansions at 0 and at ∞.
In addition to the assumptions of Sections 2 and 4, we assume now that there is a finite family ( L ℓ ) ℓ∈L of self-adjoint, positive, homogeneous, left-invariant differential operators on G with the same degree δ such that L = ℓ∈L L ℓ . We also assume that G 1 is endowed with the structure of a homogeneous group of homogeneous dimension Q, and that dπ 1 ( L ℓ ) is homogeneous of degree δ ℓ for every ℓ ∈ L.
Before proceeding further, let us make an example.
Example 6.1. Let (X ′ ℓ ) ℓ∈L be a (finite) generating family of homogeneous elements of the Lie algebra of G 1 , and define L 1 = ℓ∈L (iX ′ ℓ ) α ℓ , where α ℓ ∈ 2N * for every ℓ ∈ L. In addition, let G be the free nilpotent group with L generators and the same step as G; denote by ( X ′ ℓ ) ℓ∈L the generators of its Lie algebra. We endow G with the unique gradation for which X ′ ℓ is homogeneous of degree ℓ ′ =ℓ α ℓ ′ for every ℓ ∈ L. Let π 1 : G → G 1 be the unique homomorphism of Lie groups such that that dπ 1 ( X ′ ℓ ) = X ′ ℓ for every ℓ ∈ L. In this context, we may define L ℓ := (i X ′ ℓ ) α ℓ , δ := ℓ∈L α ℓ , and δ ℓ := d ′ ℓ α ℓ , where d ′ ℓ is the degree of X ′ ℓ , for every ℓ ∈ L. Now, for every θ ∈ (0, π] define Σ θ := e x+iy : x ∈ R, y ∈] − θ, θ[ , and for every a ∈ C L define L a := ℓ∈L a ℓ L ℓ ; the reader may easily verify that L a + L * a = L Re a is a positive Rockland operator for every a ∈ Σ L π /2 . We define L s,a := dπ s ( L a ) for every a ∈ Σ L π /2 and for every s ∈ [0, ∞]; observe that L s,a is weighted subcoercive, so that we may denote by (h s,a,t ) t>0 its heat kernel. In addition, we define t · a := (t δ ℓ a ℓ ) ℓ for every a ∈ C L and for every t ∈ C \ R − ; we still denote by ra the multiplication of a by the scalar r for every a ∈ C L and for every r ∈ C. Proposition 6.2. Denote by Ω the set of (t, a) ∈ C × C L such that ta ∈ Σ L π /2 , and observe that h 1,t,a is defined for every (t, a) ∈ Ω. In addition, the following hold: • the mapping Ω ∋ (t, a) → h 1,t,a ∈ C ∞ (G 1 ) is holomorphic; • h 1,t,ra = h 1,rt,a for every a ∈ C L and for every r, t ∈ C such that (t, ra), (rt, a) ∈ Ω; • h 1,t,r·a (e) = r −Q h 1,t,a (e) for every a ∈ C L and for every r, t ∈ C such that (t, r · a), (t, a) ∈ Ω.
Proof. Let us prove that, for every p ∈ N and for every (t, a) ∈ Ω, dom(L p 1,ta ) is the space W p of f ∈ L 2 (G 1 ) such that X γ 1 f ∈ L 2 (G 1 ) for every γ such that d γ δp, endowed with the topology induced by the hilbertian norm f → dγ δp X γ 1 f 2 2 1 /2 . On the one hand, arguing as in the proof of Corollary 4.5, we see that X γ 1 (I + L p 1,ta ) −1 induces a bounded operator on L 2 (G 1 ) for every such γ, so that dom(L p 1,ta ) embeds continuously into W p . On the other hand, it is easily seen that C ∞ c (G 1 ), which is contained (and dense) in dom(L p 1,ta ), is contained and dense in W p , whence the asserted equality. Now, it is clear that, if f ∈ W 1 , then the mapping Ω ∋ (t, a) → L 1,ta f ∈ L 2 (G 1 ) is holomorphic, so that (L 1,ta ) (t,a)∈Ω is an analytic family of type (A) in the sense of [17] (more precisely, the restriction of (L 1,ta ) (t,a)∈Ω to every complex line is an analytic family of type (A)). In addition, L 1,ta is weighted subcoercive thanks to the preceding remarks, so that it is the generator of a holomorphic semi-group by [9,Theorem 8.2]. Therefore, [17, Theorem and 2.6 of Chapter 9] implies that the mapping Ω ∋ (t, a) → e −L1,ta ∈ L(L 2 (G 1 )) is holomorphic. 12 Therefore, taking the derivatives in t we see that, for every p ∈ N, the mapping Ω ∋ (t, a) → L p 1,ta e −L1,ta ∈ L(L 2 (G 1 )) is holomorphic, so that the mapping Ω ∋ (t, a) → e −L1,ta ∈ L(L 2 (G 1 ); W p ) 12 First apply [17, Theorem and 2.6 of Chapter 9] to the intersection of every complex line with Ω, and then recall that a mapping from Ω into the Banach space L(L 2 (G 1 )) is holomorphic if and only if it is holomorphic on every line. is holomorphic. By the arbitrariness of p, this implies that the mapping Ω ∋ (t, a) → e −L1,ta ∈ L(L 2 (G 1 ); W ∞ ) is holomorphic, where W ∞ is the intersection of the W p , endowed with the corresponding topology. Since L * 1,ta = L ta , and since Ω is conjugate-symmetric, by (sesquilinear) transposition we see that the mapping Ω ∋ (t, a) → e −L1,ta ∈ L(W −∞ ; L 2 (G 1 )) is holomorphic, where W −∞ is the strong dual of W ∞ . 13 Finally, arguing again as above we see that the mapping Ω ∋ (t, a) → e −L1,ta ∈ L(W −∞ ; W ∞ ) is holomorphic. Now, the Sobolev embeddings easily show that the inclusion W ∞ ⊆ C ∞ (G 1 ) is continuous, so that the canonical mapping L(W −∞ ; W ∞ ) → L(E ′ (G 1 ); C ∞ (G 1 )) is continuous; now, observe that L(E ′ (G 1 ); C ∞ (G 1 )) is canonically isomorphic to C ∞ (G 1 × G 1 ) by the Schwartz's kernel theorem (cf. [34,Proposition 50.5]). Therefore, the mapping Ω ∋ (t, a) → h 1,t,a ∈ C ∞ (G 1 ) is holomorphic. The second assertion is trivial, while, for what concerns the third one, just observe that (ρ G1 r ) * L 1,ta = L r·(ta),1 for every (t, a) ∈ Ω and for every r > 0, where ρ G1 r denotes the dilation by r in G 1 (not to be confused with the mapping r · : G 1 → G r −1 of the preceding sections); the general assertion follows by holomorphy. Corollary 6.3. Take a ∈ R L + . Then, there is ε > 0 such that the mapping t → h 1,t,a (e) extends to a holomorphic mapping H a : Σπ /2+ε → C. In addition, for every k ∈ N there is a constant C k > 0 such that, for every t ∈ R * , The proof is similar to that of [32,Lemma 4] and is omitted.
Theorem 6.4. Take a ∈ R L + . Then, β L1,a has a density f a of class C ∞ with respect to ν R+ . In addition, there are two constants C 0 , C ∞ > 0 such that, for every k ∈ N, as λ → +∞, where x k := x(x − 1) · · · (x − k + 1) for every x ∈ R.
In particular, in this situation we may apply the second part of Theorem 5.15, thus extending [32, Theorem 2], which corresponds to the case α ℓ = 2 for every ℓ ∈ L in the situation of Example 6.1.
Proof. Observe that, with the notation of Corollary 6. for every t > 0, so that F (e −ε · β L1,a )(t) = H a (ε + it) 13 In principle we should endow L(W −∞ ; L 2 (G 1 )) with the topology of uniform convergence on the equicontinuous subsets of W −∞ , instead of the topology of bounded convergence. However, W ∞ is a reflexive Fréchet space since it is isomorphic to a closed subspace of the reflexive Fréchet space L 2 (G 1 ) N dim G 1 (cf. [ where d is the minimum degree of the non-zero homogeneous elements of V , for every x ∈ V and for every t > 0. Therefore, there is a continuous semi-norm ρ γ ′ on H ε,η (V ) such that where D is the maximum degree of the non-zero homogeneous elements of V , for every x ∈ V and for every t 1. The assertion follows easily.