Spectral multipliers on Heisenberg–Reiter and related groups

Abstract

Let $L$ be a homogeneous sublaplacian on a $2$-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$ for $1 < p < \infty$ if $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. Under suitable assumptions on $G$ and $L$, here we show that a smoothness condition of order $s > d/2$ is sufficient. This extends to a larger class of $2$-step groups the results for the Heisenberg and related groups by Müller and Stein and by Hebisch and for the free group $N_{3,2}$ by Müller and the author.

1 Introduction

Let $L$ be a homogeneous sublaplacian on a stratified Lie group $G$ of homogeneous dimension $Q$. Since $L$ is a positive self-adjoint operator on $L^2(G)$, a functional calculus for $L$ is defined via the spectral theorem and, for all Borel functions $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$, the operator $F(L)$ is bounded on $L^2(G)$ whenever the “spectral multiplier” $F$ is bounded. As for the $L^p$-boundedness for $p \ne 2$ of $F(L)$, a sufficient condition in terms of smoothness properties of the multiplier $F$ is given by a theorem of Mihlin–Hörmander type due to Christ [4] and Mauceri and Meda [20]: the operator $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in ]1,\infty [$ whenever

$$\begin{aligned} \Vert F\Vert _{MW_2^s} \mathrel {:=}\sup _{t > 0} \Vert F(t \cdot ) \, \eta \Vert _{W_2^s} < \infty \end{aligned}$$

for some $s > Q/2$, where $W_2^s({\mathbb {R}})$ is the $L^2$ Sobolev space of fractional order $s$, and $\eta \in C^\infty _c(]0,\infty [)$ is a nontrivial auxiliary function.

A natural question that arises is whether the smoothness condition $s > Q/2$ is sharp. This is clearly true when $G$ is abelian, so $Q$ coincides with the topological dimension $d$ of $G$, and $L$ is essentially the Laplace operator on ${\mathbb {R}}^d$. Take, however, the smallest nonabelian example of a stratified group, that is, the Heisenberg group ${\mathrm {H}}_1$, which is defined by endowing ${\mathbb {R}}\times {\mathbb {R}}\times {\mathbb {R}}$ with the group law

$$\begin{aligned} (x,y,u) \cdot \left( x',y',u'\right) = \left( x+x',y+y',u+u'+\left( xy'-x'y\right) /2\right) \end{aligned}$$

(1)

and with the automorphic dilations

$$\begin{aligned} \delta _t(x,y,u) = \left( tx,ty,t^2 u\right) . \end{aligned}$$

(2)

${\mathrm {H}}_1$ is a $2$-step stratified group, and the homogeneous dimension of ${\mathrm {H}}_1$ is $4$. Nevertheless, a result by Müller and Stein [23] and Hebisch [12] shows that, for a homogeneous sublaplacian on ${\mathrm {H}}_1$, the smoothness condition on the multiplier can be pushed down to $s > d/2$, where $d = 3$ is the topological dimension of ${\mathrm {H}}_1$ (in [23], it is also proved that the condition $s > d/2$ is sharp). Such an improvement of the Christ–Mauceri–Meda theorem holds not only for ${\mathrm {H}}_1$, but for the larger class of Métivier groups (and for direct products of Métivier and abelian groups), and also for differential operators other than sublaplacians (see, e.g., [13, 17]); moreover, as shown subsequently by Cowling and Sikora [5] (see also [6]), the sharp result on ${\mathrm {H}}_1$ can be obtained by transplantation from an analogous result for a distinguished sublaplacian on the (nonstratified) group $\mathrm {SU}_2$ (which in turn improves, in the case of $\mathrm {SU}_2$, an extension of the Christ–Mauceri–Meda theorem to spaces of homogeneous type [1, 7, 11]). However, it is still an open question whether, for a general stratified Lie group (or even for a general $2$-step stratified group), the homogeneous dimension in the smoothness condition can be replaced by the topological dimension.

The aim of this paper is to extend the class of the $2$-step stratified groups and sublaplacians for which the smoothness condition in the multiplier theorem can be pushed down to half the topological dimension.

Take for instance the Heisenberg–Reiter group ${\mathrm {H}}_{{d_1},{d_2}}$ (cf. [27]), defined by endowing ${\mathbb {R}}^{{d_2}\times {d_1}} \times {\mathbb {R}}^{d_1}\times {\mathbb {R}}^{d_2}$ with the group law (1) and the automorphic dilations (2); here, however, ${\mathbb {R}}^{{d_2}\times {d_1}}$ is the set of the real ${d_2}\times {d_1}$ matrices, and the products $xy',x'y$ in (1) are interpreted in the sense of matrix multiplication. ${\mathrm {H}}_{{d_1},{d_2}}$ is a $2$-step stratified group of homogeneous dimension $Q = {d_1}{d_2}+ {d_1}+ 2{d_2}$ and topological dimension $d = {d_1}{d_2}+ {d_1}+ {d_2}$. Despite the formal similarity with ${\mathrm {H}}_1$, the group ${\mathrm {H}}_{{d_1},{d_2}}$ does not fall into the class of Métivier groups, unless ${d_2}= 1$ (in fact, ${\mathrm {H}}_{{d_1},1}$ is the $(2{d_1}+1)$-dimensional Heisenberg group ${\mathrm {H}}_{d_1}$). Nevertheless, the technique presented here allows one to handle the case ${d_2}> 1$ too.

Namely, let $X_{1,1},\dots ,X_{{d_2},{d_1}},Y_1,\dots ,Y_{d_1},U_1,\dots ,U_{d_2}$ be the left-invariant vector fields on ${\mathrm {H}}_{{d_1},{d_2}}$ extending the standard basis of ${\mathbb {R}}^{{d_2}\times {d_1}} \times {\mathbb {R}}^{d_1}\times {\mathbb {R}}^{d_2}$ at the identity, and define the homogeneous sublaplacian $L$ by

$$\begin{aligned} L = -\sum _{j=1}^{d_1}\sum _{k=1}^{d_2}X_{k,j}^2 - \sum _{j=1}^{d_1}Y_j^2. \end{aligned}$$

Then, a particular instance of our main result reads as follows.

Theorem 1

Suppose that a function $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ satisfies

$$\begin{aligned} \Vert F\Vert _{MW^s_2} < \infty \end{aligned}$$

for some $s > d/2$. Then, the operator $F(L)$ is of weak type $(1,1)$ and bounded on $L^p({\mathrm {H}}_{{d_1},{d_2}})$ for all $p \in ]1,\infty [$.

To the best of our knowledge, this result is new, at least in the case ${d_2}> {d_1}$. In fact, in the case ${d_2}\le {d_1}$, the extension described in [17] of the technique of [12, 13] would give the same result. However, the technique presented here is different, and yields the result irrespective of the parameters ${d_1},{d_2}$.

The left quotient of ${\mathrm {H}}_{{d_1},{d_2}}$ by the subgroup ${\mathbb {R}}^{{d_2}\times {d_1}} \times \{0\} \times \{0\}$ gives a homogeneous space diffeomorphic to ${\mathbb {R}}^{d_1}\times {\mathbb {R}}^{d_2}$, and the sublaplacian $L$ corresponds in the quotient to a Grushin operator. In recent joint works with Sikora [18] and Müller [14], we proved for these Grushin operators on ${\mathbb {R}}^{d_1}\times {\mathbb {R}}^{d_2}$ a sharp spectral multiplier theorem of Mihlin–Hörmander type, where the smoothness requirement is again half the topological dimension of the ambient space.

The proofs in [14, 18] rely heavily on properties of the eigenfunction expansions for the Hermite operators. Since a homogeneous sublaplacian on a $2$-step stratified group reduces to a Hermite operator in almost all irreducible unitary representations of the group, it is conceivable that an adaptation of the methods of [14, 18] may give an improvement to the multiplier theorem for $2$-step stratified groups, even outside of the Métivier setting. A first result in this direction is shown in [19], where the free $2$-step nilpotent Lie group $N_{3,2}$ on three generators is considered, and properties of Laguerre polynomials are exploited (somehow in the spirit of [21, 23, 24]). The argument presented here refines and extends the one in [19].

Theorem 1 above is just a particular case of the result presented here, and we refer the reader to the next section for a precise statement. We remark that the analog of Theorem 1 holds on ${\mathrm {H}}_{{d_1},{d_2}}$ when the sublaplacian $L$ has the more general form

$$\begin{aligned} L = -\sum _{j=1}^{d_1}\sum _{k,k'=0}^{d_2}a^j_{k,k'} X_{k,j} X_{k',j} \end{aligned}$$

(3)

where $X_{0,j} = Y_j$ and $(a^j_{k,k'})_{k,k'=0,\dots ,{d_2}}$ is a positive-definite symmetric matrix for all $j\in \{1,\dots ,{d_1}\}$. Other groups can be considered too, e.g., the complexification of a Heisenberg–Reiter group, or the quotient of the direct product of ${\mathrm {H}}_{1,3}$ and $N_{3,2}$ given by identifying the respective centers.

2 The general setting

Let $G$ be a connected, simply connected nilpotent Lie group of step $2$. Recall that, via exponential coordinates, $G$ may be identified with its Lie algebra $\mathfrak {g}$, that is, the tangent space of $G$ at the identity. In turn, $\mathfrak {g}$ may be identified with the Lie algebra of left-invariant vector fields on $G$. We refer to [9] for the basic definitions and further details.

Let $\mathfrak {g}$ be decomposed as ${\mathfrak {v}}\oplus {\mathfrak {z}}$, where ${\mathfrak {z}}$ is the center of $\mathfrak {g}$, and let $\langle \cdot , \cdot \rangle$ be an inner product on ${\mathfrak {v}}$. The sublaplacian $L$ associated with the inner product is defined by $L = -\sum \nolimits _j X_j^2$, where $\{X_j\}_j$ is any orthonormal basis of ${\mathfrak {v}}$. Note that, vice versa, by the Poincaré–Birkhoff–Witt theorem, any second-order operator $L$ of the form $-\sum \nolimits _j X_j^2$ for some basis $\{X_j\}_j$ of $\mathfrak {g}$ modulo ${\mathfrak {z}}$ determines uniquely a linear complement ${\mathfrak {v}}= {{\mathrm{{\mathrm {span}}}}}\{X_j\}_j$ of ${\mathfrak {z}}$ and an inner product on ${\mathfrak {v}}$ such that $\{X_j\}_j$ is orthonormal.

Let ${\mathfrak {z}}^*$ be the dual of $\mathfrak {z}$ and, for all $\eta \in {\mathfrak {z}}^*$, define $J_\eta$ as the linear endomorphism of ${\mathfrak {v}}$ such that $\eta ([z,z']) = \langle J_\eta z, z' \rangle$ for all $z,z'\in {\mathfrak {v}}$. Clearly, $J_\eta$ is skewadjoint with respect to the inner product; hence, $J_\eta ^2$ is self-adjoint and negative semidefinite, with even rank, for all $\eta \in {\mathfrak {z}}^*$. Set moreover ${\mathfrak {\dot{z}}}= {\mathfrak {z}}^* {\setminus } \{0\}$.

Assumption (A)

There exist integers $r_1,\dots ,r_{d_1}> 0$ and an orthogonal decomposition ${\mathfrak {v}}= {\mathfrak {v}}_1 \oplus \dots \oplus {\mathfrak {v}}_{d_1}$ such that, if $P_1,\dots ,P_{d_1}$ are the corresponding orthogonal projections, then $J_\eta P_j= P_jJ_\eta$ and $J^2_\eta P_j$ has rank $2r_j$ and a unique nonzero eigenvalue for all $\eta \in {\mathfrak {\dot{z}}}$ and all $j\in \{1,\dots ,{d_1}\}$.

Note that from Assumption $({\mathrm {A}})$  it follows that $J_\eta \ne 0$ for all $\eta \in {\mathfrak {\dot{z}}}$. Therefore $[{\mathfrak {v}},{\mathfrak {v}}] = {\mathfrak {z}}$, that is, the decomposition $\mathfrak {g} = {\mathfrak {v}}\oplus {\mathfrak {z}}$ is a stratification of $\mathfrak {g}$, and the sublaplacian $L$ is hypoelliptic.

In fact, $J_\eta$ has constant rank $2(r_1 + \dots + r_k)$ for all $\eta \in {\mathfrak {\dot{z}}}$. If $J_\eta$ is invertible for all $\eta \in {\mathfrak {\dot{z}}}$, then $G$ is a Métivier group, and if in particular $J_\eta ^2 = -|\eta |^2 {\mathrm {id}}_{{\mathfrak {v}}}$ for some inner product norm $|\cdot |$ on ${\mathfrak {z}}^*$, then $G$ is an H-type group. The main novelty of our Assumption $({\mathrm {A}})$ is that it allows $J_\eta$ to have a nonzero kernel when $\eta \in {\mathfrak {\dot{z}}}$, although the dimension of the kernel must be constant.

The fact that $J_\eta$ has constant rank for $\eta \in {\mathfrak {\dot{z}}}$ depends only on the algebraic structure of $G$. What depends on the inner product, that is, on the sublaplacian $L$, are the values and multiplicities of the eigenvalues of the $J_\eta$. The above Assumption $({\mathrm {A}})$ asks for a sort of simultaneous diagonalizability of the $J_\eta$.

Under our Assumption $({\mathrm {A}})$ on the group $G$ and the sublaplacian $L$, we are able to prove the following multiplier theorem.

Theorem 2

Suppose that a function $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ satisfies

$$\begin{aligned} \Vert F \Vert _{MW_2^s} < \infty \end{aligned}$$

for some $s > (\dim G)/2$. Then, the operator $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in ]1,\infty [$.

The previously mentioned Heisenberg–Reiter groups ${\mathrm {H}}_{{d_1},{d_2}}$ satisfy Assumption $({\mathrm {A}})$, where the inner product is determined by the sublaplacian (3), and the orthogonal decomposition of the first layer is given by the natural isomorphism ${\mathbb {R}}^{{d_2}\times {d_1}} \times {\mathbb {R}}^{{d_1}} \cong ({\mathbb {R}}^{{d_2}} \times {\mathbb {R}})^{d_1}$. Other examples are the free $2$-step nilpotent Lie group $N_{3,2}$ on $3$ generators, considered in [19], and its complexification $N_{3,2}^{\mathbb {C}}$. Moreover, if $G_1$ and $G_2$ satisfy Assumption $({\mathrm {A}})$, and their centers have the same dimension, then the quotient of $G_1 \times G_2$ given by any linear identification of the centers satisfy Assumption $({\mathrm {A}})$. Note that the direct product $G_1 \times G_2$ itself does not satisfy Assumption $({\mathrm {A}})$, but an adaptation of the argument presented here allows one to consider that case too. We postpone to the end of this paper a more detailed discussion of these remarks.

From now on, unless otherwise specified, we assume that $G$ and $L$ are a $2$-step stratified group and a homogeneous sublaplacian on $G$ satisfying Assumption $({\mathrm {A}})$. Since $L$ is a left-invariant operator, so is any operator of the form $F(L)$. Let ${{\mathrm{{\mathcal {K}}}}}_{F(L)}$ denote the convolution kernel of $F(L)$. As shown, e.g., by [17, Theorem 4.6], the previous theorem is a consequence of the following estimate.

Proposition 3

For all $s > (\dim G)/2$, there exists a weight $w_s : G \rightarrow [1,\infty [$ such that $w_s^{-1} \in L^2(G)$ and, for all compact sets $K \subseteq {\mathbb {R}}$ and for all functions $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ with ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$,

$$\begin{aligned} \Vert w_s \, {{\mathrm{{\mathcal {K}}}}}_{F(L)}\Vert _2 \le C_{K,s} \Vert F\Vert _{W_2^s}; \end{aligned}$$

(4)

in particular,

$$\begin{aligned} \Vert {{\mathrm{{\mathcal {K}}}}}_{F(L)}\Vert _1 \le C_{K,s} \Vert F\Vert _{W_2^s}. \end{aligned}$$

(5)

The rest of the paper, except for the last section, is devoted to the proof of this estimate.

3 The joint functional calculus

Let ${d_2}= \dim {\mathfrak {z}}$, and let $U_1,\dots ,U_{d_2}$ be any basis of the center ${\mathfrak {z}}$. Let moreover the “partial sublaplacian” $L_j$ be defined as $L_j= - \sum \nolimits _l X^2_{j,l}$, where $\{X_{j,l}\}_l$ is any orthonormal basis of ${\mathfrak {v}}_j$, for all $j\in \{1,\dots ,{d_1}\}$; in particular $L = L_1 + \dots + L_{d_1}$. Then, the left-invariant differential operators

$$\begin{aligned} L_1,\dots ,L_{d_1},-iU_1,\dots ,-iU_{d_2}\end{aligned}$$

(6)

are essentially self-adjoint and commute strongly; hence, they admit a joint functional calculus (see, e.g., [16]). Therefore, if ${\mathbf {L}}$ and ${\mathbf {U}}$ denote the “vectors of operators” $(L_1,\dots ,L_{d_1})$ and $(-iU_1,\dots ,-iU_{d_2})$, and if we identify ${\mathfrak {z}}^*$ with ${\mathbb {R}}^{d_2}$ via the dual basis of $U_1,\dots ,U_n$, then, for all bounded Borel functions $H : {\mathbb {R}}^{d_1}\times {\mathfrak {z}}^* \rightarrow {\mathbb {C}}$, the operator $H({\mathbf {L}},{\mathbf {U}})$ is defined and bounded on $L^2(G)$. Moreover, $H({\mathbf {L}},{\mathbf {U}})$ is left-invariant, and we can express its convolution kernel ${{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}$ in terms of Laguerre functions.

Namely, for all $n,k \in {\mathbb {N}}$, let

$$\begin{aligned} L_n^{(k)}(t) = \frac{t^{-k} e^t}{n!} \left( \frac{d}{dt} \right) ^n \left( t^{k+n} e^{-t}\right) \end{aligned}$$

be the $n$-th Laguerre polynomial of type $k$, and define

$$\begin{aligned} {\mathcal {L}}_n^{(k)}(t) = (-1)^n e^{-t} L_n^{(k)}(2t). \end{aligned}$$

Note that, by Assumption $({\mathrm {A}})$, for all $\eta \in {\mathfrak {\dot{z}}}$ and $j\in \{1,\dots ,{d_1}\}$,

$$\begin{aligned} J_\eta ^2 P_j = -\left( b_j^\eta \right) ^2 P_j^\eta \end{aligned}$$

for some orthogonal projection $P_j^\eta$ of rank $2r_j$ and some $b_j^\eta > 0$. Set moreover

$$\begin{aligned} \bar{P}_j^\eta = P_j- P_j^\eta . \end{aligned}$$

Modulo reordering the ${\mathfrak {v}}_j$ in the decomposition of ${\mathfrak {v}}$, we may suppose that there exists ${\tilde{d_1}}\in \{0,\dots ,{d_1}\}$ such that $\dim {\mathfrak {v}}_j> 2r_j$ if $j\le {\tilde{d_1}}$, and $\dim {\mathfrak {v}}_j= 2r_j$ if $j > {\tilde{d_1}}$. In particular, $\bar{P}^\eta _j= 0$ and $P^\eta _j= P_j$ for all $j> {\tilde{d_1}}$ and $\eta \in {\mathfrak {\dot{z}}}$. We will also use the abbreviations $r = (r_1,\dots ,r_{d_1}),\,{\mathbb {R}}^r = {\mathbb {R}}^{r_1} \times \dots \times {\mathbb {R}}^{r_{d_1}},\,{\mathbb {N}}^r = {\mathbb {N}}^{r_1} \times \dots \times {\mathbb {N}}^{r_{d_1}},\,|r| = r_1 + \dots + r_{d_1}$. Moreover $\langle \cdot , \cdot \rangle$ will also denote the duality pairing $\mathfrak {z}^* \times \mathfrak {z} \rightarrow {\mathbb {R}}$.

Proposition 4

Let $H : {\mathbb {R}}^{{d_1}} \times {\mathfrak {z}}^* \rightarrow {\mathbb {C}}$ be in the Schwartz class, and set

$$\begin{aligned} m\left( n,\mu ,\eta \right)&= H\left( (2n_1+r_1) b_1^\eta + \mu _1, \dots ,\left( 2n_{{\tilde{d_1}}} +r_{{\tilde{d_1}}}\right) b_{{\tilde{d_1}}}^\eta +\mu _{{\tilde{d_1}}},\right. \nonumber \\&\left. \left( 2n_{{\tilde{d_1}}+1} +r_{{\tilde{d_1}}+1}\right) b_{{\tilde{d_1}}+1}^\eta ,\dots ,\left( 2n_{d_1}+r_{d_1}\right) b_{d_1}^\eta , \eta \right) \end{aligned}$$

(7)

for all $n \in {\mathbb {N}}^{d_1},\,\mu \in {\mathbb {R}}^{{\tilde{d_1}}},\,\eta \in {\mathfrak {\dot{z}}}$. Then, for all $(z,u) \in G$,

$$\begin{aligned} {{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}(z,u)&= \frac{2^{|r|}}{(2\pi )^{\dim G}} \int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{{\mathfrak {v}}} \sum _{n \in {\mathbb {N}}^{d_1}} m\left( n,\left( |\bar{P}^\eta _1 \xi |^2,\dots ,|\bar{P}^\eta _{{\tilde{d_1}}} \xi |^2\right) ,\eta \right) \nonumber \\&\times \left[ \prod _{j=1}^{d_1}{\mathcal {L}}_{n_j}^{(r_j-1)}\left( |P^\eta _j\xi |^2 /b^\eta _j\right) \right] \, e^{i \langle \xi , z \rangle } \, e^{i \langle \eta , u \rangle } \,d\xi \,d\eta . \end{aligned}$$

(8)

Proof

For all $\eta \in {\mathfrak {\dot{z}}}$ and $j\in \{1,\dots ,{d_1}\}$, let $E^\eta _{j,1},\bar{E}^\eta _{j,1},\dots ,E^\eta _{j,r_j}, \bar{E}^\eta _{j,r_j}$ be an orthonormal basis of the range of $P^\eta _j$ such that

$$\begin{aligned} J_\eta E^\eta _{j,l} = b^\eta _j\bar{E}^\eta _{j,l}, \qquad J_\eta \bar{E}^\eta _{j,l} = - b^\eta _jE^\eta _{j,l}, \qquad \text {for}\,l=1,\dots ,r_j. \end{aligned}$$

Hence, for all $z \in {\mathfrak {v}},\,\eta \in {\mathfrak {\dot{z}}}$, and $j\in \{1,\dots ,{d_1}\}$, we can write

$$\begin{aligned} P^\eta _jz = \sum _{l=1}^{r_j} \left( z^\eta _{j,l} E^\eta _j+ \bar{z}^\eta _{j,l} \bar{E}^\eta _{j,l}\right) \end{aligned}$$

for some uniquely determined $z^\eta _{j,l},\bar{z}^\eta _{j,l} \in {\mathbb {R}}$; set then $z^\eta _j= (z^\eta _{j,1},\dots ,z^\eta _{j,r_j}),\,\bar{z}^\eta _j= (\bar{z}^\eta _{j,1},\dots ,\bar{z}^\eta _{j,r_j})$, and moreover $z^\eta = (z^\eta _1,\dots ,z^\eta _{d_1})$ and $\bar{z}^\eta = (\bar{z}^\eta _1,\dots ,\bar{z}^\eta _{d_1})$.

For all $\eta \in {\mathfrak {\dot{z}}}$ and all $\rho \in \ker J_\eta$, an irreducible unitary representation $\pi _{\eta ,\rho }$ of $G$ on $L^2({\mathbb {R}}^r)$ is defined by

$$\begin{aligned} \pi _{\eta ,\rho }(z,u) \phi (v) = e^{i \langle \eta , u \rangle } e^{i \langle \rho , \bar{P}^\eta z \rangle } e^{i \sum _{j= 1}^{d_1}b_j^\eta \langle v_j+ z^\eta _j/2, \bar{z}^\eta _j\rangle } \phi (z^\eta + v) \end{aligned}$$

for all $(z,u) \in G,\,v \in {\mathbb {R}}^r,\,\phi \in L^2({\mathbb {R}}^r)$, where $\bar{P}^\eta = \bar{P}^\eta _1 + \dots + \bar{P}^\eta _{{\tilde{d_1}}}$ is the orthogonal projection onto $\ker J_\eta$. Following, e.g., [2, §2], one can see that these representations are sufficient to write the Plancherel formula for the group Fourier transform of $G$, and the corresponding Fourier inversion formula:

$$\begin{aligned} f(z,u) = (2\pi )^{|r|-\dim G} \int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{\ker J_\eta } {{\mathrm{{\mathrm {tr}}}}}(\pi _{\eta ,\rho }(z,u) \, \pi _{\eta ,\rho }(f)) \, \prod _{j=1}^{d_1}\left( b_j^\eta \right) ^{r_j} \, d\rho \, d\eta \end{aligned}$$

(9)

for all $f : G \rightarrow {\mathbb {C}}$ in the Schwartz class and all $(z,u) \in G$, where $\pi _{\eta ,\rho }(f) = \int _{G} f(g) \, \pi _{\eta ,\rho }(g^{-1}) \,dg$.

Fix $\eta \in {\mathfrak {\dot{z}}}$ and $\rho \in \ker J_\eta$. The operators (6) are represented in $\pi _{\eta ,\rho }$ as

$$\begin{aligned} d\pi _{\eta ,\rho }(L_j) = - \varDelta _{v_j}^2 + \left( b_j^\eta \right) ^2 |v_j|^2 + |P_j\rho |^2, \qquad d\pi _{\eta ,\rho }(-iU_k) = \eta _k, \end{aligned}$$

(10)

for all $j\in \{1,\dots ,{d_1}\}$ and $k\in \{1,\dots ,{d_2}\}$, where $v_j\in {\mathbb {R}}^{r_j}$ denotes the $j$-th component of $v \in {\mathbb {R}}^r$, and $\varDelta _{v_j}$ denotes the corresponding partial Laplacian. Let $h_\ell$ denote the $\ell$-th Hermite function, that is,

$$\begin{aligned} h_\ell (t) = (-1)^\ell (\ell ! \, 2^\ell \sqrt{\pi })^{-1/2} e^{t^2/2} \left( \frac{d}{dt}\right) ^\ell e^{-t^2}, \end{aligned}$$

and, for all $\omega \in {\mathbb {N}}^r$, define $\tilde{h}_{\eta ,\omega } : {\mathbb {R}}^r \rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \tilde{h}_{\eta ,\omega } = \tilde{h}_{\eta ,\omega ,1} \otimes \dots \otimes \tilde{h}_{\eta ,\omega ,{d_1}}, \qquad \tilde{h}_{\eta ,\omega ,j}(v_j) = \left( b_j^\eta \right) ^{r_j/4} \prod _{l=1}^{r_j} h_{\omega _{j,l}}\left( \left( b_j^\eta \right) ^{1/2} v_{j,l}\right) \!, \end{aligned}$$

for all $j\in \{1,\dots ,{d_1}\}$, where $\omega _{j,l}$ and $v_{j,l}$ denote the $l$-th components of $\omega _j\in {\mathbb {N}}^{r_j}$ and $v_j\in {\mathbb {R}}^{r_j}$. Then, $\{\tilde{h}_{\eta ,\omega }\}_{\omega \in {\mathbb {N}}^r}$ is a complete orthonormal system for $L^2({\mathbb {R}}^r)$, made of joint eigenfunctions of the operators (10). In fact,

$$\begin{aligned} d\pi _{\eta ,\rho }(L_j) \tilde{h}_{\eta ,\omega }&= \left( \left( 2|\omega _j|+r_j\right) b_j^\eta + |P_j\rho |^2\right) \, \tilde{h}_{\eta ,\omega }, \nonumber \\ d\pi _{\eta ,\rho }(-iU_k) \tilde{h}_{\eta ,\omega }&= \eta _k\, \tilde{h}_{\eta ,\omega }, \end{aligned}$$

(11)

where $|\omega _j| = \omega _{j,1} + \dots + \omega _{j,r_j}$; it should be observed that $P_j\rho = 0$ if $j> {\tilde{d_1}}$.

Since $H : {\mathbb {R}}^{d_1}\times {\mathfrak {z}}^* \rightarrow {\mathbb {C}}$ is in the Schwartz class, ${{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})} : G \rightarrow {\mathbb {C}}$ is in the Schwartz class too (see [3, Theorem 5.2] or [15, §4.2]). Moreover,

$$\begin{aligned} \pi _{\eta ,\rho }\left( {{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}\right) \tilde{h}_{\eta ,\omega } = m\left( \left( |\omega _1|,\dots ,|\omega _{d_1}|\right) ,\left( |P_1 \rho |^2,\dots ,|P_{{\tilde{d_1}}} \rho |^2\right) ,\eta \right) \tilde{h}_{\eta ,\omega } \end{aligned}$$

by (11) and [22, Proposition 1.1]; hence, if $\varphi _{\eta ,\rho ,\omega }(z,u) = \langle \pi _{\eta ,\rho }(z,u) \tilde{h}_{\eta ,\omega }, \tilde{h}_{\eta ,\omega } \rangle$ is the corresponding diagonal matrix coefficient of $\pi _{\eta ,\rho }$, then

$$\begin{aligned} \langle \pi _{\eta ,\rho }(z,u) \, \pi _{\eta ,\rho }\left( {{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}\right) \tilde{h}_{\eta ,\omega }, \tilde{h}_{\eta ,\omega } \rangle = m\left( (|\omega _j|)_{j\le {d_1}},\left( |P_j\rho |^2\right) _{j\le {\tilde{d_1}}},\eta \right) \, \varphi _{\eta ,\rho ,\omega }(z,u). \end{aligned}$$

Therefore, (9) gives that

$$\begin{aligned}&{{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}(z,u) \nonumber \\&\quad = (2\pi )^{|r|-\dim G} \int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{\ker J_\eta } \sum _{n \in {\mathbb {N}}^{d_1}} m\left( n,\left( |P_j\rho |^2\right) _{j\le {\tilde{d_1}}},\eta \right) \, \psi _{\eta ,\rho ,n}(z,u) \prod _{j=1}^{d_1}\left( b_j^\eta \right) ^{r_j} \, d\rho \, d\eta , \nonumber \\ \end{aligned}$$

(12)

where

$$\begin{aligned} \psi _{\eta ,\rho ,n}(z,u) = \sum _{\begin{array}{c} \omega \in {\mathbb {N}}^r \\ |\omega _1| = n_1,\dots ,|\omega _{d_1}| = n_{d_1} \end{array}} \varphi _{\eta ,\rho ,\omega }(z,u). \end{aligned}$$

On the other hand,

$$\begin{aligned} \varphi _{\eta ,\rho ,\omega }(z,u)&= e^{i \langle \eta , u \rangle } e^{i \langle \rho , \bar{P}^\eta z \rangle } \prod _{j=1}^{d_1}\prod _{l=1}^{r_j} \Biggl [ \left( b_j^\eta \right) ^{1/2} \\&\times \int \limits _{\mathbb {R}}e^{i b_j^\eta s \bar{z}^\eta _{j,l}} h_{\omega _{j,l}}\left( \left( b_j^\eta \right) ^{1/2} (s+z^\eta _{j,l}/2)\right) h_{\omega _{j,l}}\left( \left( b_j^\eta \right) ^{1/2} (s-z^\eta _{j,l}/2)\right) \,ds \Biggr ]. \end{aligned}$$

The last integral is essentially the Fourier–Wigner transform of a pair of Hermite functions, whose bidimensional Fourier transform is a Fourier–Wigner transform too [10, formula (1.90)]. The parity properties of the Hermite functions then yield

$$\begin{aligned}&\varphi _{\eta ,\rho ,\omega }(z,u) = e^{i \langle \eta , u \rangle } e^{i \langle \rho , \bar{P}^\eta z \rangle } \prod _{j=1}^{d_1}\prod _{l=1}^j\Biggl [ \frac{(-1)^{\omega _{j,l}}}{\pi \, b_j^\eta } \int \limits _{{\mathbb {R}}\times {\mathbb {R}}} e^{i \theta _1 z^\eta _{j,l}} e^{i \theta _2 \bar{z}^\eta _{j,l}} \\&\quad \times \int \limits _{\mathbb {R}}e^{it \left( 2\theta _1/ \left( b_j^\eta \right) ^{1/2}\right) } \, h_{\omega _{j,l}}\left( t+\theta _2/ \left( b_j^\eta \right) ^{1/2}\right) \,h_{\omega _{j,l}}\left( t-\theta _2/\left( b_j^\eta \right) ^{1/2}\right) \,dt \,d\theta _1 \,d\theta _2 \Biggr ]. \end{aligned}$$

Since the Fourier–Wigner transform of a pair of Hermite functions can be expressed in terms of Laguerre polynomials (see [10, Theorem 1.104] or [26, Theorem 1.3.4]), we obtain that

$$\begin{aligned} \varphi _{\eta ,\rho ,\omega }(z,u)&= \frac{e^{i \langle \eta , u \rangle } e^{i \langle \rho , \bar{P}^\eta z \rangle }}{\pi ^{|r|}} \int \limits _{{\mathbb {R}}^r \times {\mathbb {R}}^r} e^{i \langle \zeta _1, z^\eta \rangle } e^{i \langle \zeta _2, \bar{z}^\eta \rangle } \\&\times \prod _{j=1}^{d_1}\Biggl [ \left( b_j^\eta \right) ^{-r_j} \prod _{l=1}^{r_j} {\mathcal {L}}_{\omega _{j,l}}^{(0)}\left( \left( \zeta _{1,j,l}^2 + \zeta _{2,j,l}^2\right) /b^\eta _j\right) \Biggr ] \,d\zeta _1 \,d\zeta _2 \end{aligned}$$

Consequently, for all $n \in {\mathbb {N}}^{d_1}$,

$$\begin{aligned} \psi _{\eta ,\rho ,n}(z,u)&= \frac{e^{i \langle \eta , u \rangle } e^{i \langle \rho , \bar{P}^\eta z \rangle }}{\pi ^{|r|}} \int \limits _{{\mathbb {R}}^r \times {\mathbb {R}}^r} e^{i \langle \zeta _1, z^\eta \rangle } e^{i \langle \zeta _2, \bar{z}^\eta \rangle } \nonumber \\&\times \prod _{j=1}^{d_1}\Biggl [ \left( b_j^\eta \right) ^{-r_j} {\mathcal {L}}_{n_j}^{(r_j-1)}\left( \left( |\zeta _{1,j}|^2 + |\zeta _{2,j}|^2\right) /b^\eta _j\right) \Biggr ] \,d\zeta _1 \,d\zeta _2 \end{aligned}$$

(13)

[9, §10.12, formula (41)]. The conclusion then follows by plugging (13) into (12) and performing a change of variable by rotation in the inner integrals. $\square$

4 A weighted Plancherel estimate

Proposition 4 expresses the convolution kernel ${{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}$ as the inverse Fourier transform of a function of the multiplier $H$. Due to the properties of the Fourier transform, it is not unreasonable to think that multiplying the kernel by a polynomial weight might correspond to taking derivatives of the multiplier. As a matter of fact, the presence of the Laguerre expansion leads us to consider both “discrete” and “continuous” derivatives of the reparametrization $m : {\mathbb {N}}^{d_1}\times {\mathbb {R}}^{{\tilde{d_1}}} \times {\mathfrak {\dot{z}}}\rightarrow {\mathbb {C}}$ of the multiplier $H$ given by (7).

For convenience, set ${\mathcal {L}}_n^{(k)} = 0$ for all $n < 0$. From the properties of Laguerre polynomials (see, e.g., [9, §10.12]), one can easily derive the following identities.

Lemma 5

For all $k,n,m \in {\mathbb {N}}$ and $t \in {\mathbb {R}}$,

$$\begin{aligned} {\mathcal {L}}_n^{(k)}(t)&= {\mathcal {L}}_{n-1}^{(k+1)}(t) + {\mathcal {L}}_n^{(k+1)}(t), \end{aligned}$$

(14)

$$\begin{aligned} \frac{d}{dt} {\mathcal {L}}_n^{(k)}(t)&= {\mathcal {L}}_{n-1}^{(k+1)}(t) - {\mathcal {L}}_{n}^{(k+1)}(t),\end{aligned}$$

(15)

$$\begin{aligned} \int \limits _{0}^{\infty } {\mathcal {L}}_n^{(k)}(t) \, {\mathcal {L}}_m^{(k)}(t) \, t^k \,dt&= {\left\{ \begin{array}{ll} \frac{(n+k)!}{2^{k+1} n!} &{}\quad \text {if }\,n=m,\\ 0 &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

(16)

Let $e_1,\dots ,e_{d_1}$ denote the standard basis of ${\mathbb {R}}^{d_1}$. We introduce some operators on functions $f : {\mathbb {N}}^{d_1}\times {\mathbb {R}}^{{\tilde{d_1}}} \times {\mathfrak {\dot{z}}}\rightarrow {\mathbb {C}}$:

$$\begin{aligned} \tau _jf(n,\mu ,\eta )&= f(n+e_j,\mu ,\eta ), \\ \delta _jf(n,\mu ,\eta )&= f(n+e_j,\mu ,\eta ) - f(n,\mu ,\eta ), \\ \partial _{\mu _l} f(n,\mu ,\eta )&= \frac{\partial }{\partial \mu _l} f(n,\mu ,\eta ), \\ \partial _{\eta _k} f(n,\mu ,\eta )&= \frac{\partial }{\partial \eta _k} f(n,\mu ,\eta ) \end{aligned}$$

for all $j\in \{1,\dots ,{d_1}\},\,l \in \{1,\dots ,{\tilde{d_1}}\},\,k\in \{1,\dots ,{d_2}\}$.

For all $h \in {\mathbb {N}}$ and all multiindices $\alpha \in {\mathbb {N}}^h$, we denote by $|\alpha |$ the length $\alpha _1+\dots +\alpha _h$ of $\alpha$. Inequalities between multiindices, such as $\alpha \le \alpha '$, shall be interpreted componentwise. Set moreover $(\alpha )_+ = ((\alpha _1)_+,\dots ,(\alpha _h)_+)$, where $(\ell )_+ = \max \{\ell ,0\}$.

A function $\varPsi : {\mathfrak {\dot{z}}}\!\times \! {\mathfrak {v}}\!\rightarrow \! {\mathbb {C}}$ will be called multihomogeneous if there exist $h_0,h_1,\dots ,h_{d_1}\!\in \! {\mathbb {R}}$ such that

$$\begin{aligned} \varPsi \Biggl (\lambda _0 \eta ,\sum _{j=1}^{d_1}\lambda _jP_j\xi \Biggr ) = \lambda _0^{h_0} \lambda _1^{h_1} \dots \lambda _{d_1}^{h_{d_1}} \varPsi (\eta ,\xi ) \end{aligned}$$

for all $\eta \in {\mathfrak {\dot{z}}},\,\xi \in {\mathfrak {v}},\,\lambda _0,\lambda _1,\dots ,\lambda _{d_1}\in ]0,\infty [$; the homogeneity degrees $h_0,h_1,\dots ,h_{d_1}$ of $\varPsi$ will also be denoted as $\deg _{\mathfrak {z}}\varPsi , \deg _{{\mathfrak {v}}_1} \varPsi , \dots , \deg _{{\mathfrak {v}}_{d_1}} \varPsi$. Note that, if $\varPsi$ is multihomogeneous and continuous, then $\deg _{{\mathfrak {v}}_j} \varPsi \ge 0$ for all $j\in \{1,\dots ,{d_1}\}$.

Proposition 6

Let $H : {\mathbb {R}}^{d_1}\times {\mathfrak {z}}^* \rightarrow {\mathbb {C}}$ be smooth and compactly supported in ${\mathbb {R}}^{d_1}\times {\mathfrak {\dot{z}}}$, and let $m(n,\mu ,\eta )$ be defined by (7). For all $\alpha \in {\mathbb {N}}^{d_2}$,

$$\begin{aligned} u^\alpha \, {{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}(z,u)&= \sum _{\iota \in I_\alpha } \int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{{\mathfrak {v}}} \sum _{n \in {\mathbb {N}}^{d_1}} \partial _\eta ^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \delta ^{\beta ^\iota } m\left( n,\left( |\bar{P}^\eta _j\xi |^2\right) _{j\le {\tilde{d_1}}},\eta \right) \\&\times \varPsi _\iota (\eta ,\xi ) \, \Biggl [ \prod _{j=1}^{d_1}{\mathcal {L}}^{(r_j-1 + \beta ^\iota _j)}_{n_j}\left( |P^\eta _j\xi |^2/b^\eta _j\right) \Biggr ] \, e^{i\langle \xi ,z \rangle } \, e^{i\langle \eta ,u\rangle } \,d\xi \,d\eta , \end{aligned}$$

for almost all $(z,u) \in G$, where $I_\alpha$ is a finite set and, for all $\iota \in I_\alpha$,

• $\gamma ^\iota \in {\mathbb {N}}^{d_2},\,\theta ^\iota \in {\mathbb {N}}^{{\tilde{d_1}}},\,\beta ^\iota \in {\mathbb {N}}^{d_1},\,\gamma ^\iota \le \alpha$,

• $\varPsi _\iota = \varPsi _{\iota ,0} \varPsi _{\iota ,1} \dots \varPsi _{\iota ,{d_1}}$, where $\varPsi _{\iota ,j} : {\mathfrak {\dot{z}}}\times {\mathfrak {v}}\rightarrow {\mathbb {C}}$ is smooth and multihomogeneous for all $j\in \{0,\dots ,{d_1}\}$,

• $\deg _{\mathfrak {z}}\varPsi _\iota = |\gamma ^\iota | - |\alpha | - |\beta ^\iota |$ and $\deg _{{\mathfrak {v}}_j} \varPsi _\iota = 2\beta ^\iota _j+ 2\theta ^\iota _j$ for all $j\in \{1,\dots ,{d_1}\}$,

• for all $j\in \{1,\dots ,{d_1}\},\,\varPsi _{\iota ,j}(\eta ,\xi )$ is a product of factors of the form $|P_j^\eta \xi |^2$ or $\partial _{\eta _k} |P^\eta _j\xi |^2$ for $k\in \{1,\dots ,{d_2}\}$,

• $|\gamma ^\iota | + |\theta ^\iota | + |\beta ^\iota | + \sum \nolimits _{j=1}^{{d_1}} (\beta ^\iota _j- (\deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,j}) /2)_+ \le |\alpha |$.

Proof

By Proposition 4 and the properties of the Fourier transform, we are reduced to proving that, for all $\alpha \in {\mathbb {N}}^{{d_2}},\,\eta \in {\mathfrak {\dot{z}}},\,\xi \in {\mathfrak {v}}$,

$$\begin{aligned}&\left( \frac{\partial }{\partial \eta }\right) ^\alpha \sum _{n \in {\mathbb {N}}^{d_1}} m\left( n,\left( |\bar{P}^\eta _j\xi |^2\right) _{j\le {\tilde{d_1}}},\eta \right) \, \prod _{j=1}^{d_1}{\mathcal {L}}_{n_j}^{(r_j-1)}\left( |P^\eta _j\xi |^2 /b^\eta _j\right) \\&\quad = \sum _{\iota \in I_\alpha } \sum _{n \in {\mathbb {N}}^{d_1}} \partial _\eta ^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \delta ^{\beta ^\iota } m\left( n,\left( |\bar{P}^\eta _j\xi |^2\right) _{j\le {\tilde{d_1}}},\eta \right) \, \varPsi _\iota (\eta ,\xi ) \, \prod _{j=1}^{d_1}{\mathcal {L}}^{(r_j-1 + \beta ^\iota _j)}_{n_j}\left( |P^\eta _j\xi |^2/b^\eta _j\right) \!, \end{aligned}$$

where $I_\alpha ,\,\gamma ^\iota ,\,\theta ^\iota ,\,\beta ^\iota ,\,\varPsi _\iota$ are as in the above statement.

This is easily proved by induction on $|\alpha |$. For $|\alpha | = 0$, it is trivially verified. For the inductive step, one applies Leibniz’ rule and exploits the following observations:

• when a derivative $\partial _{\eta _k}$ hits a Laguerre function, by the identity (15) and summation by parts, the type of the Laguerre function is increased by $1$, as well as the corresponding component of $\beta ^\iota$;

• for all $j\in \{1,\dots ,{d_1}\},\,b^\eta _j= \sqrt{{{\mathrm{{\mathrm {tr}}}}}(-J_\eta ^2 P_j)/(2r_j)}$ is a smooth function of $\eta \in {\mathfrak {\dot{z}}}$, homogeneous of degree $1$;

• for all $j\in \{1,\dots ,{d_1}\},\,P^\eta _j= -J_\eta ^2 P_j/(b^\eta _j)^2$ is a smooth function of $\eta \in {\mathfrak {\dot{z}}}$, homogeneous of degree $0$, and in fact it is constant if $j> {\tilde{d_1}}$;

• for all $j\in \{1,\dots ,{\tilde{d_1}}\},\,|P^\eta _j\xi |^2 = \langle P^\eta _jP_j\xi , P_j\xi \rangle$ is a smooth bihomogeneous function of $(\eta ,P_j\xi ) \in {\mathfrak {\dot{z}}}\times {\mathfrak {v}}_j$ of bidegree $(0,2)$, and moreover

  $$\begin{aligned}&|\bar{P}^\eta _j\xi |^2 = |P_j\xi |^2 - |P^\eta _j\xi |^2, \qquad \partial _{\eta _k} |\bar{P}^\eta _j\xi |^2 = - \partial _{\eta _k} |P^\eta _j\xi |^2, \\&\partial _{\eta _k} \left( |P^\eta _j\xi |^2/b^\eta _j\right) = |P^\eta _j\xi |^2 \partial _{\eta _k} \left( 1/b^\eta _j\right) + \left( \partial _{\eta _k} |P^\eta _j\xi |^2\right) /b^\eta _j\end{aligned}$$

  for all $k\in \{1,\dots ,{d_2}\}$.

The conclusion follows. $\square$

Note that, for all $j\!\in \{1,\dots ,{d_1}\},\,\mu \!\in {\mathbb {R}}^{{\tilde{d_1}}},\,\eta \in {\mathfrak {\dot{z}}}$, the quantities $\tau _jf(\cdot ,\mu ,\eta ),\,\delta _jf(\cdot ,\mu ,\eta )$ depend only on $f(\cdot ,\mu ,\eta )$; in other words, $\tau _j$ and $\delta _j$ can be considered as operators on functions ${\mathbb {N}}^{d_1}\rightarrow {\mathbb {C}}$.

The following lemma exploits the orthogonality properties (16) of the Laguerre functions, together with (14), and shows that a mismatch between the type of the Laguerre function and the exponent of the weight attached to the measure may be turned in some cases into discrete differentiation.

Lemma 7

For all $h,k \in {\mathbb {N}}^{d_1}$ and all compactly supported $f : {\mathbb {N}}^{d_1}\rightarrow {\mathbb {C}}$,

$$\begin{aligned}&\int \limits _{]0,\infty [^{d_1}} \Bigl | \sum _{n \in {\mathbb {N}}^{d_1}} f(n) \, \prod _{j=1}^{d_1}{\mathcal {L}}_{n_j}^{(k_j)}(t_j) \Bigr |^2 \,t^h \,dt \\&\quad \le C_{h,k} \sum _{n \in {\mathbb {N}}^{d_1}} |\delta ^{(k-h)_+} f(n)|^2 \, \prod _{j=1}^{d_1}(1+ n_j)^{h_j+2(k_j-h_j)_+}. \end{aligned}$$

Proof

Via an inductive argument, we may reduce to the case ${d_1}= 1$.

Note that, if $f$ is compactly supported, then $\tau ^l f$ is null for all sufficiently large $l \in {\mathbb {N}}$. Hence, the operator $1+\tau$, when restricted to the set of compactly supported functions, is invertible, with inverse given by

$$\begin{aligned} (1+\tau )^{-1} f = \sum _{l \in {\mathbb {N}}} (-1)^l \tau ^l f. \end{aligned}$$

Then by (14), we deduce that, for all $k \in {\mathbb {N}}$,

$$\begin{aligned} \sum _{n \in {\mathbb {N}}} f(n) \, {\mathcal {L}}_n^{(k)}(t)&= \sum _{n \in {\mathbb {N}}} (1+\tau ) f(n) \, {\mathcal {L}}_n^{(k+1)}(t), \\ \sum _{n \in {\mathbb {N}}} f(n) \, {\mathcal {L}}_n^{(k+1)}(t)&= \sum _{n \in {\mathbb {N}}} (1+\tau )^{-1} f(n) \, {\mathcal {L}}_n^{(k)}(t), \end{aligned}$$

and consequently, for all $h,k \in {\mathbb {N}}$,

$$\begin{aligned} \sum _{n \in {\mathbb {N}}} f(n) \, {\mathcal {L}}_n^{(k)}(t) = \sum _{n \in {\mathbb {N}}} (1+\tau )^{h-k} f(n) \, {\mathcal {L}}_n^{(h)}(t) \end{aligned}$$

Thus, the orthogonality properties (16) of the Laguerre functions give us that

$$\begin{aligned} \int \limits _0^\infty \Bigl | \sum _{n \in {\mathbb {N}}} f(n) \, {\mathcal {L}}_n^{(k)}(t) \Bigr |^2 \,t^h \,dt \le C_{h,k} \sum _{n \in {\mathbb {N}}} |(1+\tau )^{h-k} f(n)|^2 \, \langle n \rangle ^{h}, \end{aligned}$$

where $\langle n \rangle = 1+n$.

In the case $h \ge k,\,(1+\tau )^{h-k}$ is given by the finite sum

$$\begin{aligned} (1+\tau )^{h-k} = \sum _{\ell = 0}^{h-k} \left( {\begin{array}{c}h-k\\ \ell \end{array}}\right) \tau ^\ell , \end{aligned}$$

and the conclusion follows immediately by the triangular inequality.

In the case $h < k$, instead, since $\delta = \tau - 1$, from the identity $1-\tau ^2 = (1-\tau )(1+\tau )$, we deduce that

$$\begin{aligned} (1+\tau )^{h-k} = (-\delta )^{k-h} (1-\tau ^2)^{h-k} = (-1)^{k-h} \sum _{\ell \ge 0} \left( {\begin{array}{c}\ell + k - h - 1\\ \ell \end{array}}\right) \delta ^{k-h} \tau ^{2\ell }, \end{aligned}$$

hence

$$\begin{aligned} \sum _{n \in {\mathbb {N}}} |(1+\tau )^{h-k} f(n)|^2 \, \langle n \rangle ^{h}&= \sum _{n \in {\mathbb {N}}} \Biggl |\sum _{\ell \ge 0} \left( {\begin{array}{c}\ell + k - h - 1\\ \ell \end{array}}\right) \delta ^{k-h} f(n+2\ell )\Biggr |^2 \, \langle n \rangle ^{h} \\&\le C_{h,k} \sum _{n \in {\mathbb {N}}} \Biggl |\sum _{\ell \ge n} \langle \ell \rangle ^{k - h - 1} \delta ^{k-h} f(\ell )\Biggr |^2 \, \langle n \rangle ^{h} \\&\le C_{h,k} \sum _{n \in {\mathbb {N}}} \langle n \rangle ^{-1/2} \sum _{\ell \ge n} |\langle \ell \rangle ^{k - h - 1/4} \delta ^{k-h} f(\ell )|^2 \, \langle n \rangle ^{h} \\&\le C_{h,k} \sum _{\ell \in {\mathbb {N}}} \langle \ell \rangle ^{2k - 2h - 1/2} |\delta ^{k-h} f(\ell )|^2 \sum _{n = 0}^\ell \langle n \rangle ^{h-1/2} \\&\le C_{h,k} \sum _{\ell \in {\mathbb {N}}} \langle \ell \rangle ^{2k - h} |\delta ^{k-h} f(\ell )|^2, \end{aligned}$$

by the Cauchy–Schwarz inequality, and we are done. $\square$

Let $|\cdot |$ denote any Euclidean norm on $\mathfrak {z}^*$. The previous lemma, together with Plancherel’s formula for the Fourier transform, yields the following $L^2$-estimate.

Proposition 8

Under the hypotheses of Proposition 6, for all $\alpha \in {\mathbb {N}}^{d_2}$,

$$\begin{aligned}&\int \limits _{G} |u^\alpha \, {{\mathrm{{\mathcal {K}}}}}_{H({\mathbf {L}},{\mathbf {U}})}(z,u)|^2 \,dz\,du \le C_\alpha \sum _{\iota \in \tilde{I}_\alpha } \int _{{\mathfrak {\dot{z}}}} \int \limits _{[0,\infty [^{{\tilde{d_1}}}} \sum _{n \in {\mathbb {N}}^{d_1}} |\partial _\eta ^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \delta ^{\beta ^\iota } m(n,\mu ,\eta )|^2 \nonumber \\&\quad \times |\eta |^{2|\gamma ^\iota |-2|\alpha |-2|\beta ^\iota |+|a^\iota |+{d_1}} \, (1+n_1)^{a^\iota _1} \dots (1+n_{d_1})^{a^\iota _{d_1}} \, d\sigma _\iota (\mu ) \,d\eta , \end{aligned}$$

(17)

where $\tilde{I}_\alpha$ is a finite set and, for all $\iota \in \tilde{I}_\alpha$,

• $\gamma ^\iota \in {\mathbb {N}}^{d_2},\,\theta ^\iota \in {\mathbb {N}}^{{\tilde{d_1}}},\,a^\iota , \beta ^\iota \in {\mathbb {N}}^{d_1}$,

• $\gamma ^\iota \le \alpha ,\,|\gamma ^\iota | + |\theta ^\iota | + |\beta ^\iota | \le |\alpha |$,

• $\sigma _\iota$ is a regular Borel measure on $[0,\infty [^{{\tilde{d_1}}}$.

Proof

Note that, for all $j\in \{1,\dots ,{d_1}\}$,

$$\begin{aligned} \partial _{\eta _k} \left( |P^\eta _j\xi |^2\right) = 2 \left\langle \left( \partial _{\eta _k} P^\eta _j\right) P_j\xi , P^\eta _j\xi \right\rangle \le C|\eta |^{-1} |P^\eta _j\xi | |P_j\xi |; \end{aligned}$$

consequently, if $\varPsi _\iota ,\varPsi _{\iota ,j},\gamma ^\iota ,\theta ^\iota ,\beta ^\iota$ are as in the statement of Proposition 6, then

$$\begin{aligned} |\varPsi _{\iota ,j}(\eta ,\xi )|^2 \le C_\iota |\eta |^{2\deg _{\mathfrak {z}}\varPsi _{\iota ,j}} |P^\eta _j\xi |^{\deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,j}} |P_j\xi |^{\deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,j}} \end{aligned}$$

for all $j\in \{1,\dots ,{d_1}\}$, hence

$$\begin{aligned} |\varPsi _\iota (\eta ,\xi )|^2&\le C_\iota |\eta |^{2\deg _{\mathfrak {z}}\varPsi _\iota } \prod _{j=1}^{{d_1}} |P^\eta _j\xi |^{\deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,j}} |P_j\xi |^{2\deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,0} + \deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,j}} \\&\le C_\iota |\eta |^{2|\gamma ^\iota |-2|\alpha |-2|\beta ^\iota |} \prod _{j=1}^{{d_1}} \sum _{h_j= (\deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,j})/2}^{2\theta ^\iota _j+ 2\beta ^\iota _j} |P^\eta _j\xi |^{2h_j} |\bar{P}^\eta _j\xi |^{4\theta ^\iota _j+ 4\beta ^\iota _j-2h_j}, \end{aligned}$$

and moreover, for all $h \in {\mathbb {N}}^{d_1}$, if $h_j \ge (\deg _{{\mathfrak {v}}_j} \varPsi _{\iota ,j})/2$ for all $j \in \{1,\dots ,{d_1}\}$, then

$$\begin{aligned} |\gamma ^\iota | + |\theta ^\iota | + |\beta ^\iota | + \sum _{j=1}^{{\tilde{d_1}}} \left( \beta ^\iota _j- h_j\right) _+ \le |\alpha |. \end{aligned}$$

By Proposition 6, Plancherel’s formula and the triangular inequality, we then obtain that the left-hand side of (17) is majorized by a finite sum of terms of the form

$$\begin{aligned}&\int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{{\mathfrak {v}}} \Biggl | \sum _{n \in {\mathbb {N}}^{d_1}} \partial _\eta ^{\gamma } \partial _\mu ^{\theta } \delta ^{\beta } m\left( n,\left( |\bar{P}^\eta _j\xi |^2\right) _{j\le {\tilde{d_1}}},\eta \right) \, \prod _{j=1}^{d_1}{\mathcal {L}}^{(r_j-1+\beta _j)}_{n_j}\left( |P^\eta _j\xi |^2/b^\eta _j\right) \Biggr |^2 \nonumber \\&\quad \times |\eta |^{2|\gamma |-2|\alpha |-2|\beta |} \, \prod _{j=1}^{d_1}|P^\eta _j\xi |^{2h_j} \prod _{j=1}^{{\tilde{d_1}}} |\bar{P}^\eta _j\xi |^{2k_j} \,d\xi \,d\eta , \end{aligned}$$

(18)

where $\gamma \in {\mathbb {N}}^{d_2},\,\theta ,k \in {\mathbb {N}}^{{\tilde{d_1}}},\,\beta ,h \in {\mathbb {N}}^{d_1}$ and $|\gamma |+|\theta |+|\beta + (\beta -h)_+| \le |\alpha |$. Simple changes of variables (rotation, polar coordinates and rescaling) allow one to rewrite (18) as a constant times

$$\begin{aligned}&\int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{]0,\infty [^{{\tilde{d_1}}}} \int \limits _{]0,\infty [^{{d_1}}} \Biggl | \sum _{n \in {\mathbb {N}}^{d_1}} \partial _\eta ^{\gamma } \partial _\mu ^{\theta } \delta ^{\beta } m(n,\mu ,\eta ) \, \prod _{j=1}^{d_1}{\mathcal {L}}^{(r_j-1+\beta _j)}_{n_j}(t_j) \Biggr |^2 \, \prod _{j=1}^{d_1}t_j^{r_j-1+h_j} \,dt \\&\quad \times |\eta |^{2|\gamma |-2|\alpha |-2|\beta |} \prod _{j=1}^{d_1}\left( b^\eta _j\right) ^{h_j+r_j} \prod _{j=1}^{{\tilde{d_1}}} \mu _j^{k_j+(\dim {\mathfrak {v}}_j-2r_j)/2} \,\frac{d\mu }{\mu _1 \cdots \mu _{{\tilde{d_1}}}} \,d\eta . \end{aligned}$$

By exploiting the fact that the $b^\eta _j$ are smooth functions of $\eta \in {\mathfrak {\dot{z}}}$, homogeneous of degree $1$ (see the proof of Proposition 6), and applying Lemma 7 to the inner integral, the last quantity is majorized by

$$\begin{aligned}&C \int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{]0,\infty [^{{\tilde{d_1}}}} \sum _{n \in {\mathbb {N}}^{d_1}} | \partial _\eta ^{\gamma } \partial _\mu ^{\theta } \delta ^{\beta +(\beta - h)_+} m(n,\mu ,\eta ) |^2 \, \prod _{j=1}^{d_1}(1+ n_j)^{r_j-1 + h_j+2(\beta _j-h_j)_+} \\&\quad \times |\eta |^{2|\gamma |-2|\alpha |-2|\beta |+ |h|+|r|} \prod _{j=1}^{{\tilde{d_1}}} \mu _j^{k_j+(\dim {\mathfrak {v}}_j-2r_j)/2} \,\frac{d\mu }{\mu _1 \dots \mu _{{\tilde{d_1}}}} \,d\eta , \end{aligned}$$

and since the exponents $k_j+(\dim {\mathfrak {v}}_j-2r_j)/2$ are strictly positive, while

$$\begin{aligned} -2|\beta | + |h| + |r| = -2|\beta +(\beta -h)_+| + \sum _{j=1}^{d_1}\left( r_j-1+h_j+2(\beta _j-h_j)_+\right) + {d_1}\end{aligned}$$

and $|\gamma |+|\theta |+|\beta + (\beta -h)_+| \le |\alpha |$, the conclusion follows by suitably renaming the multiindices. $\square$

5 From discrete to continuous

Via the fundamental theorem of integral calculus, finite differences can be estimated by continuous derivatives. The next lemma is a multivariate analog of [19, Lemma 6], and we omit the proof (see also [18, Lemma 7]).

Lemma 9

Let $f : {\mathbb {N}}^{d_1}\rightarrow {\mathbb {C}}$ have a smooth extension $\tilde{f} : [0,\infty [^{d_1}\rightarrow {\mathbb {C}}$, and let $\beta \in {\mathbb {N}}^{d_1}$. Then,

$$\begin{aligned} \delta ^\beta f(n) = \int \limits _{J_\beta } \partial ^\beta \tilde{f}(n+s) \,d\nu _\beta (s) \end{aligned}$$

for all $n \in {\mathbb {N}}$, where $J_\beta = \prod _{j=1}^{d_1}[0,\beta _j]$, and $\nu _\beta$ is a Borel probability measure on $J_\beta$. In particular,

$$\begin{aligned} |\delta ^\beta f(n)|^2 \le \int \limits _{J_\beta } |\partial ^\beta \tilde{f}(n+s)|^2 \,d\nu _\beta (s) \end{aligned}$$

for all $n \in {\mathbb {N}}^{d_1}$.

We give now a simplified version of the right-hand side of (17), in the case we restrict to the functional calculus of $L$ alone. In order to avoid issues of divergent series, it is, however, convenient at first to truncate the multiplier along the spectrum of ${\mathbf {U}}$.

Lemma 10

Let $\chi \in C^\infty _c({\mathbb {R}})$ be supported in $[1/2,2 ],\,K \subseteq {\mathbb {R}}$ be compact and $M \in ]0, \infty [$. If $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ is smooth and supported in $K$, and $F_M : {\mathbb {R}}\times {\mathfrak {z}}^* \rightarrow {\mathbb {C}}$ is given by

$$\begin{aligned} F_M(\lambda ,\eta ) = F(\lambda ) \, \chi (|\eta |/M), \end{aligned}$$

then, for all $r \in [0, \infty [$,

$$\begin{aligned} \int \limits _{G} | |u|^r \, {{\mathrm{{\mathcal {K}}}}}_{F_M(L,{\mathbf {U}})}(z,u) |^2 \,dz \,du \le C_{K,\chi ,r} \, M^{{d_2}-2r} \Vert F\Vert _{W_2^r}^2. \end{aligned}$$

Proof

We may restrict to the case $r \in {\mathbb {N}}$, the other cases being recovered a posteriori by interpolation. Hence, we need to prove that

$$\begin{aligned} \int \limits _{G} | u^\alpha \, {{\mathrm{{\mathcal {K}}}}}_{F_M(L,{\mathbf {U}})}(z,u) |^2 \,dz \,du \le C_{K,\chi ,\alpha } \, M^{{d_2}-2|\alpha |} \Vert F\Vert _{W_2^{|\alpha |}}^2 \end{aligned}$$

(19)

for all $\alpha \in {\mathbb {N}}^{{d_1}}$. On the other hand, if $m$ is defined by

$$\begin{aligned} m(n,\mu ,\eta ) = F\Biggl ( \sum _{j=1}^{d_1}b^\eta _j\langle n_j\rangle _j+ |\mu |_\varSigma \Biggr ) \, \chi (|\eta |/M), \end{aligned}$$

(20)

where $\langle \ell \rangle _{j} = 2\ell + r_j$ and $|\mu |_\varSigma = \sum \nolimits _{j=1}^{\tilde{d_1}}\mu _j$, then the left-hand side of (19) is majorized by the right-hand side of (17), and we are reduced to proving that

$$\begin{aligned}&\sum _{n \in {\mathbb {N}}^{d_1}} \int \limits _{{\mathfrak {\dot{z}}}} \int \limits _{[0,\infty [^{{\tilde{d_1}}}} |\partial _\eta ^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \delta ^{\beta ^\iota } m(n,\mu ,\eta )|^2 \, |\eta |^{2|\gamma ^\iota |-2|\alpha |-2|\beta ^\iota |+|a^\iota |+{d_1}} \nonumber \\&\quad \times (1+n_1)^{a^\iota _1} \dots (1+n_{d_1})^{a^\iota _{d_1}} \, d\sigma _\iota (\mu ) \,d\eta \le C_{K,\chi ,\alpha } \, M^{{d_2}-2|\alpha |} \Vert F\Vert _{W_2^{|\alpha |}}^2 \end{aligned}$$

(21)

for all $\iota \in \tilde{I}_\alpha$, where $\tilde{I}_\alpha ,\,\gamma ^\iota ,\,\theta ^\iota ,\,\beta ^\iota ,\,a^\iota ,\,\sigma _\iota$ are as in Proposition 8.

Note that the right-hand side of (20) makes sense for all $n \in {\mathbb {R}}^{d_1}$ and defines a smooth extension of $m$, which we still denote by $m$ by a slight abuse of notation. Hence, by Lemma 9,

$$\begin{aligned} |\partial _\eta ^{\gamma _\iota } \partial _\mu ^{\theta ^\iota } \delta ^{\beta ^\iota } m(n,\mu ,\eta )|^2 \le \int \limits _{J_\iota } |\partial _{\eta }^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \partial _n^{\beta ^\iota } m(n+s,\mu ,\eta )|^2 \,d\nu _\iota (s), \end{aligned}$$

(22)

where $J_\iota = \prod _{j=1}^{d_1}[0,\beta ^\iota _j]$ and $\nu _\iota$ is a suitable probability measure on $J_\iota$. Moreover, the measure $\sigma _\iota$ in (21) is finite on compacta, and the right-hand side of (22) vanishes when $|\mu |_\varSigma > \max K$, because ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$. Consequently, (21) will be proved if we show that

$$\begin{aligned}&\sum _{n \in {\mathbb {N}}^{d_1}} \int \limits _{{\mathfrak {\dot{z}}}} |\partial _\eta ^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \partial _n^{\beta ^\iota } m(n+s,\mu ,\eta )|^2 \, |\eta |^{2|\gamma ^\iota |-2|\alpha |-2|\beta ^\iota |+|a^\iota |+{d_1}} \nonumber \\&\quad \times (1+ n_1)^{a^\iota _1} \dots (1+n_{d_1})^{a^\iota _{d_1}} \,d\eta \le C_{K,\chi ,\alpha } \, M^{{d_2}-2|\alpha |} \Vert F\Vert _{W_2^{|\alpha |}}^2 \end{aligned}$$

(23)

for all $s \in J_\iota$ and $\mu \in [0,\max K ]^{{\tilde{d_1}}}$, uniformly in $s$ and $\mu$.

As observed in the proof of Proposition 6, the $b_j^\eta$ are positive, smooth functions of $\eta \in {\mathfrak {\dot{z}}}$, homogeneous of degree $1$; therefore, for all $n \in {\mathbb {N}}^{d_1},\,j\in \{1,\dots ,{d_1}\},\,\eta \in {\mathfrak {\dot{z}}}, s \in [0,\infty [^{d_1},\,\mu \in [0,\infty [^{{\tilde{d_1}}}$,

$$\begin{aligned} |\eta | (1+ n_j) \sim b_j^\eta \langle n_j\rangle _j\le \sum _{l=1}^{d_1}b^\eta _l \langle n_l + s_l \rangle _l + |\mu |_\varSigma , \end{aligned}$$

(24)

and the last quantity is bounded by the constant $\max K$ whenever $(n+s,\mu ,\eta ) \in {{\mathrm{{\mathrm {supp}}}}}m$, because ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$. Hence, the factors $|\eta | (1+ n_j)$ in the left-hand side of (23) can be discarded, that is, we are reduced to proving (23) in the case $a^\iota = 0$.

From (20), it follows immediately that

$$\begin{aligned} \partial _\mu ^{\theta ^\iota } \partial _n^{\beta ^\iota } m(n,\mu ,\eta ) = F^{(|\theta ^\iota |+|\beta ^\iota |)}\Biggl ( \sum _{j=1}^{d_1}b^\eta _j\langle n_j\rangle _j+ |\mu |_\varSigma \Biggr ) \, \chi (|\eta |/M) \prod _{j=1}^{d_1}(2b_j^\eta )^{\beta ^\iota _j} \end{aligned}$$

and then it is easily proved inductively that

$$\begin{aligned} \partial _\eta ^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \partial _n^{\beta ^\iota } m(n,\mu ,\eta )&= \sum _{\begin{array}{c} \upsilon \in {\mathbb {N}}^{d_1}\\ |\upsilon | \le |\gamma ^\iota | \end{array}} \sum _{q=0}^{|\gamma ^\iota | - |\upsilon |} F^{(|\theta ^\iota |+|\beta ^\iota |+|\upsilon |)}\Biggl (\sum _{j=1}^{d_1}b_j^\eta \langle n_j\rangle _j+ |\mu |_\varSigma \Biggr ) \\&\times \varPsi _{\iota ,\upsilon ,q}(\eta ) \, M^{-q} \, \chi ^{(q)}(|\eta |/M) \prod _{j=1}^{d_1}\langle n_j \rangle _j^{\upsilon _j} \end{aligned}$$

where $\varPsi _{\iota ,\upsilon ,q} : {\mathfrak {\dot{z}}}\rightarrow {\mathbb {R}}$ is smooth and homogeneous of degree $|\beta ^\iota | + |\upsilon | + q - |\gamma ^\iota |$. By exploiting again (24) and the fact that ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$, we can majorize the factors $\langle n_j \rangle _j$ in the right-hand side by $|\eta |^{-1} \sim M^{-1}$ and obtain that

$$\begin{aligned}&|\partial _\eta ^{\gamma ^\iota } \partial _\mu ^{\theta ^\iota } \partial _n^{\beta ^\iota } m(n,\mu ,\eta )|^2 \le C_{K,\chi ,\alpha } M^{2|\beta ^\iota | - 2|\gamma ^\iota |} \tilde{\chi }(|\eta |/M) \\&\quad \times \sum _{v=0}^{|\gamma ^\iota |} \Biggl |F^{(|\beta ^\iota |+|\theta ^\iota |+v)}\Biggl (\sum _{j=1}^{d_1}b_j^\eta \langle n_j\rangle _j+ |\mu |_\varSigma \Biggr )\Biggr |^2, \end{aligned}$$

where $\tilde{\chi }$ is the characteristic function of $[1/2,2 ]$. Hence, the left-hand side of (23), when $a^\iota = 0$, is majorized by

$$\begin{aligned}&C_{K,\chi ,\alpha } M^{{d_1}-2|\alpha |} \\&\quad \times \sum _{v=0}^{|\gamma ^\iota |} \int _{{\mathfrak {\dot{z}}}} \sum _{n\in {\mathbb {N}}^{d_1}} \Biggl |F^{(|\beta ^\iota |+|\theta ^\iota |+v)}\Biggl (\sum _{j=1}^{d_1}b_j^\eta \langle n_j+s_j\rangle _j+ |\mu |_\varSigma \Biggr )\Biggr |^2 \, \tilde{\chi }(|\eta |/M) \,d\eta . \end{aligned}$$

Let $S$ denote the unit sphere in ${\mathfrak {z}}^*$. By passing to polar coordinates and exploiting the homogeneity of the $b_j^\eta$, the integral in the above formula is majorized by

$$\begin{aligned}&C \int \limits _{S} \int \limits _0^\infty \sum _{n\in {\mathbb {N}}^{d_1}} \Biggl |F^{(|\beta ^\iota |+|\theta ^\iota |+v)}\Biggl (\rho \sum _{j=1}^{d_1}b_j^\omega \langle n_j+s_j\rangle _j+ |\mu |_\varSigma \Biggr )\Biggr |^2 \, \tilde{\chi }(\rho /M) \rho ^{{d_2}} \,\frac{d\rho }{\rho } \,d\omega \nonumber \\&\quad \le C M^{d_2}\int \limits _0^\infty |F^{(|\beta ^\iota |+|\theta ^\iota |+v)}(\rho + |\mu |_\varSigma )|^2 \int \limits _S \sum _{n \in {\mathbb {N}}^{d_1}} \tilde{\chi }(\rho /(M \langle n \rangle _{\omega ,s})) \,d\omega \,\frac{d\rho }{\rho } \end{aligned}$$

(25)

where $\langle n \rangle _{\omega ,s} = \sum \nolimits _{j=1}^{d_1}b_j^\omega \langle n_j+s_j\rangle _j\sim 1+|n|$ uniformly in $\omega \in S$ and $s \in J_\iota$. Since $\tilde{\chi }(\rho /(M \langle n \rangle _{\omega ,s}))$ vanishes unless $\langle n \rangle _{\omega ,s} \sim \rho /M$, the sum in the right-hand side of (25) has at most $C_\iota (\rho /M)^{d_1}$ nonvanishing summands, and the integral on $S$ is majorized by $C_\iota (\rho /M)^{d_1}$. In conclusion, the left-hand side of (23) is majorized by

$$\begin{aligned}&C_{K,\chi ,\alpha } M^{{d_2}-2|\alpha |} \sum _{v=0}^{|\gamma ^\iota |} \int \limits _0^\infty |F^{(|\beta ^\iota |+|\theta ^\iota |+v)}(\rho + |\mu |_\varSigma )|^2 \, \rho ^{{d_1}-1} \,d\rho \\&\quad \le C_{K,\chi ,\alpha } M^{{d_2}-2|\alpha |} \Vert F\Vert _{W_2^{|\alpha |}}^2, \end{aligned}$$

because $d_1 \ge 1,\,{{\mathrm{{\mathrm {supp}}}}}F \subseteq K$ and $|\beta ^\iota | + |\theta ^\iota | + |\gamma ^\iota | \le |\alpha |$, and we are done. $\square$

Proposition 11

Let $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ be smooth and such that ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$ for some compact set $K \subseteq {\mathbb {R}}$. For all $r \in [0, {d_2}/2 [$,

$$\begin{aligned} \int \limits _{G} \left| (1+|u|)^r \, {{\mathrm{{\mathcal {K}}}}}_{F(L)}(z,u) \right| ^2 \,dz \,du \le C_{K,r} \Vert F\Vert _{W_2^r}^2. \end{aligned}$$

Proof

Take $\chi \in C^\infty _c(]0,\infty [)$ such that ${{\mathrm{{\mathrm {supp}}}}}\chi \subseteq [1/2,2 ]$ and $\sum \nolimits _{k \in {\mathbb {Z}}} \chi (2^{-k} t) = 1$ for all $t \in ]0,\infty [$. If $F_M$ is defined for all $M \in ]0,\infty [$ as in Lemma 10, then ${{\mathrm{{\mathcal {K}}}}}_{F_M(L,{\mathbf {U}})}$ is given by the right-hand side of (8), where $m$ is defined by (20), and moreover,

$$\begin{aligned} \sum _{j=1}^{d_1}b^\eta _j\langle n_j\rangle _j+ |\mu |_\varSigma \ge C^{-1} |\eta | \end{aligned}$$

for all $\eta \in {\mathfrak {\dot{z}}},\,\mu \in [0,\infty [^{\tilde{d_1}}$ and $n \in {\mathbb {N}}^{d_1}$, therefore $F_M(L,{\mathbf {U}}) = 0$ whenever $M > 2C\max K$. Hence, if $k_K \in {\mathbb {Z}}$ is sufficiently large so that $2^{k_K} > 2 C \max K$, then

$$\begin{aligned} F(L) = \sum _{k \in {\mathbb {Z}}, \, k \le k_K} F_{2^{k}}(L,{\mathbf {U}}) \end{aligned}$$

(with convergence in the strong sense). Consequently, an estimate for ${{\mathrm{{\mathcal {K}}}}}_{F(L)}$ can be obtained, via Minkowski’s inequality, by summing the corresponding estimates for ${{\mathrm{{\mathcal {K}}}}}_{F_{2^{k}}}(L,{\mathbf {U}})$ given by Lemma 10. If $r < d/2$, then the series $\sum \nolimits _{k \le k_K} (2^{k})^{{d_2}/2-r}$ converges, thus

$$\begin{aligned} \int \limits _{G} \left| |u|^r \, {{\mathrm{{\mathcal {K}}}}}_{F(L)}(z,u) \right| ^2 \,dz \,du \le C_{K,r} \Vert F\Vert _{W_2^r}^2. \end{aligned}$$

The conclusion follows by combining the last inequality with the corresponding one for $r = 0$. $\square$

Let $|\cdot |_\delta$ be a $\delta _t$-homogeneous norm on $G$; take, e.g., $|(z,u)|_\delta = |z| + |u|^{1/2}$. Interpolation then allows us to improve the standard weighted estimate for a homogeneous sublaplacian on a stratified group.

Proposition 12

Let $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ be smooth and such that ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$ for some compact set $K \subseteq {\mathbb {R}}$. For all $r \in [0, {d_2}/2 [,\,\alpha \ge 0$ and $\beta > \alpha + r$,

$$\begin{aligned} \int \limits _{G} \left| (1+|(z,u)|_\delta )^\alpha \, (1+|u|)^r \, {{\mathrm{{\mathcal {K}}}}}_{F(L)}(z,u) \right| ^2 \,dz \,du \le C_{K,\alpha ,\beta ,r} \Vert F\Vert _{W_2^\beta }^2. \end{aligned}$$

(26)

Proof

Note that $1+|u| \le C (1+|(z,u)|_\delta )^2$. Hence, in the case $\alpha \ge 0,\,\beta > \alpha + 2r$, the inequality (26) follows by the mentioned standard estimate (see [21, Lemma 1.2] or [17, Theorem 2.7]). On the other hand, if $\alpha = 0$ and $\beta \ge r$, then (26) is given by Proposition 11. The full range of $\alpha$ and $\beta$ is then obtained by interpolation. $\square$

We can finally prove the crucial estimate.

Proof of Proposition 3

Take $r \in ](\dim G)/2 + {d_2}/2-s,{d_2}/2 [$. Then,

$$\begin{aligned} s-r > (\dim G)/2 + {d_2}/2 - 2r = (\dim {\mathfrak {v}})/2 + {d_2}-2r, \end{aligned}$$

hence we can find $\alpha _1 > (\dim {\mathfrak {v}})/2$ and $\alpha _2 > {d_2}-2r$ such that $s - r > \alpha _1 + \alpha _2$. Set $w_s(z,u) = (1+|(z,u)|_\delta )^\alpha \, (1+|u|)^r$. The $L^2$-estimate (4) then follows from Proposition 12. On the other hand, for all $(z,u) \in G$,

$$\begin{aligned} w_s^{-2}(z,u) \le C_s (1+|z|)^{-2\alpha _1} \, (1+|u|)^{-\alpha _2 - 2r}, \end{aligned}$$

and the right-hand side is integrable over $G \cong \mathfrak {v} \times \mathfrak {z}$ since $2\alpha _1 > \dim {\mathfrak {v}}$ and $\alpha _2 + 2r > {d_2}= \dim {\mathfrak {z}}$. Therefore, $w_s^{-1} \in L^2(G)$, and the $L^1$-estimate (5) follows from (4) and Hölder’s inequality. $\square$

6 Remarks on the validity of the assumption and direct products

In this section, we do no longer suppose that $G$ and $L$ are a $2$-step stratified Lie group and a sublaplacian satisfying Assumption $({\mathrm {A}})$.

As observed in Sect. 2, a necessary condition for the validity of Assumption $({\mathrm {A}})$ is that the skewadjoint endomorphism $J_\eta$ of the first layer $\mathfrak {v}$ has constant rank for $\eta$ ranging in ${\mathfrak {\dot{z}}}= {\mathfrak {z}}^* {\setminus } \{0\}$. Here, we show that this condition is also sufficient when the rank is minimal.

Proposition 13

Let $G$ be a $2$-step nilpotent Lie group, with Lie algebra $\mathfrak {g} = \mathfrak {v} \oplus \mathfrak {z}$, and let $\langle \cdot ,\cdot \rangle$ be an inner product on $\mathfrak {v}$. Suppose that the skewadjoint endomorphism $J_\eta$ of $\mathfrak {v}$ has rank $2$ for all $\eta \in {\mathfrak {\dot{z}}}$. Then, $G$ satisfies Assumption $({\mathrm {A}})$ with the sublaplacian $L$ associated to the given inner product, and also with any other sublaplacian associated to an inner product on a complement of $\mathfrak {z}$.

Let moreover $G_{\mathbb {C}}$ be the complexification of $G$, considered as a real $2$-step group, with Lie algebra $\mathfrak {g}_{\mathbb {C}}= \mathfrak {v}_{\mathbb {C}}\oplus \mathfrak {z}_{\mathbb {C}}$, and let $\mathfrak {v}_{\mathbb {C}}$ be endowed with the real inner product induced by the inner product on $\mathfrak {v}$. Then, $G_{\mathbb {C}}$, with the sublaplacian associated to the given inner product, satisfies Assumption $({\mathrm {A}})$.

Proof

From the normal form for skewadjoint endomorphisms, it follows immediately that, if $J_\eta$ has rank $2$, then $J_\eta ^2$ has exactly one nonzero eigenvalue, and Assumption $({\mathrm {A}})$ is trivially verified. Moreover, if $\mathfrak {v}$ is identified with $\mathfrak {g}/\mathfrak {z}$, then $\ker J_\eta$ corresponds to the subspace

$$\begin{aligned} N_\eta = \left\{ x + \mathfrak {z} \,:\,x \in \mathfrak {g} \text { and } \eta ([x,x']) = 0 \text { for all } x' \in \mathfrak {g}\right\} \end{aligned}$$

of $\mathfrak {g}/\mathfrak {z}$; hence, the rank condition on $J_\eta$ can be rephrased by saying that $N_\eta$ has codimension $2$ for all $\eta \in {\mathfrak {\dot{z}}}$, and this condition does not depend on the sublaplacian $L$ chosen on $G$.

Let $R(J_\eta )$ denote the range of $J_\eta$. We show now that, for all $\eta ,\eta ' \in {\mathfrak {\dot{z}}}$, the intersection $R(J_\eta ) \cap R(J_{\eta '})$ is nontrivial. If it were trivial, since $J_{\eta +\eta '} = J_\eta + J_\eta '$, we would have $\ker J_{\eta +\eta '} = \ker J_\eta \cap \ker J_{\eta '}$, hence

$$\begin{aligned} R(J_{\eta +\eta '}) = (\ker J_{\eta +\eta '})^\perp = R(J_\eta ) \oplus R(J_{\eta '}), \end{aligned}$$

thus $J_{\eta +\eta '}$ would have rank $4$, contradiction.

Consider now the complexification $\mathfrak {g}_{\mathbb {C}}= \mathfrak {g} \oplus i \mathfrak {g}$. Via the linear identifications $\mathfrak {g}_{\mathbb {C}}= \mathfrak {g} \times \mathfrak {g},\,\mathfrak {z}^*_{\mathbb {C}}= \mathfrak {z}^* \times \mathfrak {z}^*,\,\mathfrak {v}_{\mathbb {C}}= \mathfrak {v} \times \mathfrak {v}$, the skewsymmetric endomorphism $\tilde{J}_\eta$ of the first layer $\mathfrak {v}_{\mathbb {C}}$ corresponding to the element $\eta =(\eta _R,\eta _I) \in \mathfrak {z}_{\mathbb {C}}^*$ is given by

$$\begin{aligned} \tilde{J}_\eta (x_R,x_I) = \left( J_{\eta _R} x_R + J_{\eta _I} x_I, J_{\eta _I} x_R - J_{\eta _R} x_I\right) . \end{aligned}$$

(27)

Take now $\eta = (\eta _R,\eta _I) \in {\mathfrak {\dot{z}}}_{\mathbb {C}}$; we want to show that $\tilde{J}_\eta ^2$ has rank $4$ and a unique nonzero eigenvalue. We distinguish several cases.

If $\eta _I = 0$, then $\tilde{J}_\eta = J_{\eta _R} \times (-J_{\eta _R})$, hence $\tilde{J}_\eta ^2 = J_{\eta _R}^2 \times J_{\eta _R}^2$ satisfies the condition. The same argument gives the conclusion in the case $\eta _R = 0$.

If both $\eta _R,\eta _I \in {\mathfrak {\dot{z}}}$, then $R(J_{\eta _R}) \cap R(J_{\eta _I}) \ne 0$, hence $\dim (R(J_{\eta _R}) \cap R(J_{\eta _I}))$ is either $2$ or $1$. In the first case, $R(J_{\eta _R}) = R(J_{\eta _I})$, so $J_{\eta _R}$ and $J_{\eta _I}$ commute and (27) implies that

$$\begin{aligned} \tilde{J}_\eta ^2 = \left( J_{\eta _R}^2 + J_{\eta _I}^2\right) \times \left( J_{\eta _R}^2 + J_{\eta _I}^2\right) ; \end{aligned}$$

since $J_{\eta _R}^2$ and $J_{\eta _I}^2$ are negative multiples of the same orthogonal projection, the conclusion follows.

Suppose now that $R(J_{\eta _R}) \cap R(J_{\eta _I}) = {\mathbb {R}}x$ for some unit vector $x \in \mathfrak {v}$, and set $y_R = J_{\eta _R}x,\,y_I = J_{\eta _I}x,\,b_R = |y_R|,\,b_I = |y_I|$; in particular, $J_{\eta _R}^2 x = -b_R^2 x$ and $J_{\eta _I}^2 x = -b_I^2 x$. Since $J_{\eta _R}$ and $J_{\eta _I}$ are skewadjoint and of rank $2$, necessarily $J_{\eta _R} x, J_{\eta _I}x \in x^\perp$ and $J_{\eta _R}(x^\perp ) = J_{\eta _I}(x^\perp ) = {\mathbb {R}}x$, therefore $J_{\eta _R} J_{\eta _I} x$ and $J_{\eta _I} J_{\eta _R} x$ are both multiples of $x$; on the other hand,

$$\begin{aligned} \langle J_{\eta _R} J_{\eta _I} x, x \rangle = - \langle J_{\eta _I} x, J_{\eta _R} x \rangle = \langle x, J_{\eta _I} J_{\eta _R} x \rangle , \end{aligned}$$

hence $J_{\eta _R} J_{\eta _I} x = J_{\eta _I} J_{\eta _R} x$. This identity, together with (27), allows us easily to show that

$$\begin{aligned} \tilde{J}_\eta (x,0)&= (y_R,y_I), \qquad \tilde{J}_\eta (y_R,y_I) = -(b_R^2 + b_I^2) (x, 0),\\ \tilde{J}_\eta (0,x)&= (y_I,-y_R), \qquad \tilde{J}_\eta (y_I,-y_R) = -(b_R^2 + b_I^2) (0, x). \end{aligned}$$

Note that $b_R^2 + b_I^2$ is the squared norm of both $(y_R,y_I)$ and $(y_I,-y_R)$. Hence, we would be done if we knew that $R(\tilde{J}_\mu )$ coincides with the linear span $W$ of $(x,0),\,(0,x),\,(y_R,y_I),\,(y_I,-y_R)$.

In fact, we just need to show that $R(\tilde{J}_\eta )$ is contained in $W$, or equivalently, that $W^\perp$ is contained in $\ker \tilde{J}_\eta$. On the other hand, if $v = (v_R,v_I) \in W^\perp$, then $v_R,v_I \in x^\perp$ and moreover

$$\begin{aligned} \langle v_R, y_R \rangle + \langle v_I, y_I \rangle = 0, \qquad \langle v_R, y_I \rangle - \langle v_I, y_R \rangle = 0, \end{aligned}$$

hence $J_{\eta _R} v_R, J_{\eta _R} v_I, J_{\eta _I} v_R, J_{\eta _I} v_I \in {\mathbb {R}}x$, and

$$\begin{aligned} \langle J_{\eta _R} v_R, x \rangle&= - \langle v_R, y_R \rangle = \langle v_I, y_I \rangle = -\langle J_{\eta _I} v_I, x \rangle , \\ \langle J_{\eta _I} v_R, x \rangle&= - \langle v_R, y_I \rangle = - \langle v_I, y_R \rangle = \langle J_{\eta _R} v_I, x \rangle , \end{aligned}$$

therefore $J_{\eta _R} v_R = -J_{\eta _I} v_I$ and $J_{\eta _I} v_R = J_{\eta _R} v_I$, from which it follows immediately that $\tilde{J}_\eta (v_R,v_I) = 0$. $\square$

The next proposition shows how groups and sublaplacians satisfying Assumption $({\mathrm {A}})$ may be “glued together”, so to give a higher-dimensional group and a sublaplacian that satisfy Assumption $({\mathrm {A}})$ too.

Proposition 14

Suppose that, for $j=1,2$, the sublaplacian $L_j$ on the $2$-step stratified Lie group $G_j$ satisfies Assumption $({\mathrm {A}})$. Suppose further that the centers of $G_1$ and $G_2$ have the same dimension. Let $G$ be the quotient of $G_1 \times G_2$ given by any linear identification of the respective centers, and let $L = L_1^\sharp + L_2^\sharp$, where $L_j^\sharp$ is the pushforward of $L_j$ to $G$. Then, the sublaplacian $L$ on the group $G$ satisfies Assumption $({\mathrm {A}})$.

Proof

Let $\mathfrak {g}_j$ be the Lie algebra of $G_j$, and let ${\mathfrak {v}}_j$ and $\langle \cdot , \cdot \rangle _j$ be the linear complement of the center ${\mathfrak {z}}_j$ and the inner product on ${\mathfrak {v}}_j$ determined by the sublaplacian $L_j$; denote moreover by $J_{j,\eta }$ the skewadjoint endomorphism of ${\mathfrak {v}}_j$ determined by $\eta \in {\mathfrak {z}}_j^*$.

The linear identification of the centers of $G_1$ and $G_2$ corresponds to a linear isomorphism $\phi : \mathfrak {z}_1 \rightarrow \mathfrak {z}_2$, and the Lie algebra $\mathfrak {g}$ of the quotient $G$ can be identified with $\mathfrak {v}_1 \times \mathfrak {v}_2 \times \mathfrak {z}_2$, with Lie bracket

$$\begin{aligned} \left[ (v_1,v_2,z),(v_1',v_2',z')\right] = \left( 0,0,\phi \left( \left[ v_1,v_1'\right] \right) + \left[ v_2,v_2'\right] \right) . \end{aligned}$$

Then, the sublaplacian $L$ on $G$ corresponds to the inner product $\langle \cdot , \cdot \rangle$ on $\mathfrak {v}_1 \times \mathfrak {v}_2$ defined by

$$\begin{aligned} \langle (v_1,v_2) , (v_1',v_2') \rangle = \langle v_1, v_1' \rangle _1 + \langle v_2, v_2' \rangle _2. \end{aligned}$$

In particular, if $\phi ^* : \mathfrak {z}_2^* \rightarrow \mathfrak {z}_1^*$ denotes the adjoint map of $\phi : \mathfrak {z}_1 \rightarrow \mathfrak {z}_2$, then it is easily checked that the skewadjoint endomorphism of the first layer $\mathfrak {v}_1 \times \mathfrak {v}_2$ of $\mathfrak {g}$ corresponding to an element $\eta$ of the dual $\mathfrak {z}_2^*$ of the center of $\mathfrak {g}$ is given by $J_\eta = J_{1,\phi ^* \eta } \times J_{2,\eta }$. Hence, the orthogonal decomposition of $\mathfrak {v}_1 \times \mathfrak {v}_2$ giving the “simultaneous diagonalization” of the $J_\eta$ for all $\eta \in {\mathfrak {\dot{z}}}_2$ (in the sense of Sect. 2) is simply obtained by juxtaposing the corresponding orthogonal decompositions of $\mathfrak {v}_1$ and $\mathfrak {v}_2$. $\square$

Note that the direct product $G_1 \times G_2$ itself need not satisfy Assumption $({\mathrm {A}})$, even if the factors $G_1$ and $G_2$ do. However, a functional-analytic argument, as in [24, §4], can be used to deal with that case.

The key step in our proof of Theorem 2 is the weighted $L^2$-estimate (4) of Proposition 3. Let us now turn the conclusion of Proposition 3 into an assumption on a homogeneous sublaplacian $L$ on a stratified group $G$.

Assumption $(\hbox {B}_{t})$. For all $s > t$, there exist a weight $w_s : G \rightarrow [1,\infty [$ such that $w_s^{-1} \in L^2(G)$ and, for all compact sets $K \subseteq {\mathbb {R}}$ and all Borel functions $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ with ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$,

$$\begin{aligned} \Vert w_s \, {{\mathrm{{\mathcal {K}}}}}_{F(L)} \Vert _{L^2(G)} \le C_{K,s} \Vert F\Vert _{W_2^s({\mathbb {R}})}. \end{aligned}$$

(28)

Our Proposition 3 can then be rephrased by saying that Assumption $({\mathrm {A}})$ implies Assumption $({\mathrm {B}}_{t})$ for $t = (\dim G)/2$. Note, on the other hand, that Assumption $({\mathrm {B}}_{t})$ makes sense for homogeneous sublaplacians on stratified groups $G$ of step other than $2$. In fact, every homogeneous sublaplacian on a stratified group of homogeneous dimension $Q$ satisfies Assumption $({\mathrm {B}}_{t})$ for $t = Q/2$, by [21, Lemma 1.2] (suitably extended so to admit multipliers that do not vanish in a neighborhood of the origin of ${\mathbb {R}}$; see, e.g., [24, Lemma 3.1] for the $1$-dimensional case, and [17, Theorem 2.7] for the higher-dimensional case).

Differently from Assumption $({\mathrm {A}})$, the new Assumption $({\mathrm {B}}_{t})$ “behaves well” under direct products.

Proposition 15

For $j=1,\dots ,n$, let $L_j$ be a homogeneous sublaplacian on a stratified Lie group $G_j$ satisfying Assumption $({\mathrm {B}}_{t_j})$ for some $t_j > 0$. Let $G = G_1 \times \dots \times G_n$ and $L = L_1^\sharp + \dots + L_n^\sharp$, where $L_j^\sharp$ is the pushforward to $G$ of the operator $L_j$. Then, the sublaplacian $L$ on $G$ satisfies Assumption $({\mathrm {B}}_{t})$, where $t = t_1+\dots +t_n$.

Proof

Take $s > t$. Then, we can choose $s_1,\dots ,s_n$ such that $s_1 > t_1,\dots ,s_n> t_n$ and $s = s_1+\dots +s_n$. Let then $w_{j,s_j} : G_j \rightarrow [1,\infty [$ be the weight corresponding to $s_j$ given by Assumption $({\mathrm {B}}_{t_j})$ on $G_j$ and $L_j$, for $j=1,\dots ,n$. In particular, $w_{j,s_j}^{-1} \in L^2(G_j)$ and, for all $\phi \in C^\infty _c({\mathbb {R}})$, the map $F \mapsto {{\mathrm{{\mathcal {K}}}}}_{(\phi F)(L_j)}$ is a bounded linear map of Hilbert spaces $W_2^{s_j}({\mathbb {R}}) \rightarrow L^2(G_j,w_{j,s_j}^2(x_j) \,dx_j)$, where $dx_j$ denotes the Haar measure on $G_j$.

The operators $L_1^\sharp ,\dots ,L_n^\sharp$ are essentially self-adjoint and commute strongly, that is, they admit a joint spectral resolution and a joint functional calculus on $L^2(G)$, and moreover, for all bounded Borel functions $F_1,\dots ,F_n : {\mathbb {R}}\rightarrow {\mathbb {C}}$,

$$\begin{aligned} {{\mathrm{{\mathcal {K}}}}}_{(F_1 \otimes \dots \otimes F_n)(L_1^\sharp ,\dots ,L_n^\sharp )} = {{\mathrm{{\mathcal {K}}}}}_{F_1(L_1)} \otimes \dots \otimes {{\mathrm{{\mathcal {K}}}}}_{F_n(L_n)} \end{aligned}$$

[16, Corollary 5.5]. Hence, for all $\phi _1,\dots ,\phi _n \in C^\infty _c({\mathbb {R}})$, if $\phi = \phi _1 \otimes \dots \otimes \phi _n$, then the map $H \mapsto {{\mathrm{{\mathcal {K}}}}}_{(\phi H)(L_1^\sharp ,\dots ,L_n^\sharp )}$ is the tensor product of the maps $F_j \mapsto {{\mathrm{{\mathcal {K}}}}}_{(\phi _j F_j)(L_j)}$. Since these maps are bounded $W_2^{s_j}({\mathbb {R}}) \rightarrow L^2(G_j,w_{j,s_j}^2(x_j) \,dx_j)$, the map $H \mapsto {{\mathrm{{\mathcal {K}}}}}_{(\phi H)(L_1^\sharp ,\dots ,L_n^\sharp )}$ is bounded $S_2^{(s_1,\dots ,s_n)}W({\mathbb {R}}^n) \rightarrow L^2(G,w_s^2(x) \,dx)$, where $S^{(s_1,\dots ,s_n)}_2 W({\mathbb {R}}^n) = W_2^{s_1}({\mathbb {R}}) \otimes \dots \otimes W_2^{s_n}({\mathbb {R}})$ is the $L^2$ Sobolev space with dominating mixed smoothness [25] of order $(s_1,\dots ,s_n)$, and $w_s = w_{1,s_1} \otimes \dots \otimes w_{n,s_n}$ is the product weight on $G$. In particular, for all compact sets $K \subseteq {\mathbb {R}}$, if we choose the cutoffs $\phi _j \in C^\infty _c({\mathbb {R}})$ so that $\phi _j|_K = 1$, then we deduce that, for all $H : {\mathbb {R}}^n \rightarrow {\mathbb {C}}$ with ${{\mathrm{{\mathrm {supp}}}}}H \subseteq K^n$,

$$\begin{aligned} \Vert w_s \, {{\mathrm{{\mathcal {K}}}}}_{H(L_1^\sharp ,\dots ,L_n^\sharp )} \Vert _{L^2(G)} \le C_{K,s} \Vert H\Vert _{S^{(s_1,\dots ,s_n)}_2 W({\mathbb {R}}^n)}. \end{aligned}$$

(cf. [17, Proposition 5.2]). Since

$$\begin{aligned} \Vert f\Vert _{S^{(s_1,\dots ,s_n)}_2 W({\mathbb {R}}^n)}^2&\sim \int \limits _{{\mathbb {R}}^n} |\hat{f}(\xi )|^2 (1+|\xi _1|)^{2s_1} \dots (1+|\xi _n|)^{2s_n} \,d\xi \\&\le \int \limits _{{\mathbb {R}}^n} |\hat{f}(\xi )|^2 (1+|\xi |)^{2s_1+\dots +2s_n} \,d\xi \sim \Vert f\Vert ^2_{W_2^{s}({\mathbb {R}}^n)}, \end{aligned}$$

where $\hat{f}$ denotes the Euclidean Fourier transform of $f$, we see immediately that the estimate

$$\begin{aligned} \Vert w_s \, {{\mathrm{{\mathcal {K}}}}}_{H(L_1^\sharp ,\dots ,L_n^\sharp )} \Vert _{L^2(G)} \le C_{K,s_1,\dots ,s_n} \Vert H\Vert _{W_2^{s}({\mathbb {R}}^n)}, \end{aligned}$$

(29)

holds true whenever $K \subseteq {\mathbb {R}}$ is compact and $H : {\mathbb {R}}^n \rightarrow {\mathbb {C}}$ is supported in $K^n$.

Take now a compact set $K \subseteq {\mathbb {R}}$ and choose a smooth cutoff $\eta _K \in C^\infty _c({\mathbb {R}})$ such that $\eta _K|_{[0,\max K ]} = 1$. Let $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ be such that ${{\mathrm{{\mathrm {supp}}}}}F \subseteq K$, and define $H : {\mathbb {R}}^n \rightarrow {\mathbb {C}}$ by

$$\begin{aligned} H(\lambda _1,\dots ,\lambda _n) = F(\lambda _1 + \dots + \lambda _n) \, \eta _K(\lambda _1) \dots \, \eta _K(\lambda _n) \end{aligned}$$

for all $(\lambda _1,\dots ,\lambda _n) \in {\mathbb {R}}^n$. Then, ${{\mathrm{{\mathrm {supp}}}}}H \subseteq ({{\mathrm{{\mathrm {supp}}}}}\eta _K)^n$, and

$$\begin{aligned} F(\lambda _1 + \dots + \lambda _n) = H(\lambda _1,\dots ,\lambda _n) \end{aligned}$$

for all $(\lambda _1,\dots ,\lambda _n) \in [0,\infty [^n$. Since the operators $L_1,\dots ,L_n$ are nonnegative, the joint spectrum of $L_1^\sharp ,\dots ,L_n^\sharp$ is contained in $[0,\infty [^n$, hence

$$\begin{aligned} F(L) = F\left( L_1^\sharp + \dots + L_n^\sharp \right) = H\left( L_1^\sharp ,\dots ,L_n^\sharp \right) . \end{aligned}$$

Consequently, by (29) and the smoothness of the map $(\lambda _1,\dots ,\lambda _n) \mapsto \lambda _1+\dots +\lambda _n$, we obtain that

$$\begin{aligned} \Vert w_s {{\mathrm{{\mathcal {K}}}}}_{F(L)} \Vert _{L^2(G)} \le C_{K,s} \Vert H\Vert _{W_2^s({\mathbb {R}}^n)} \le C_{K,s} \Vert F\Vert _{W_2^s({\mathbb {R}})}. \end{aligned}$$

Since clearly $w_s^{-1} = w_{1,s_1}^{-1} \otimes \dots \otimes w_{n,s_n}^{-1} \in L^2(G)$, we are done. $\square$

The previous results, together with the known weighted estimates for abelian [24, Lemma 3.1] and Métivier [12, 13, 17] groups, then yield the following extension of Theorem 2.

Theorem 16

For $j=1,\dots ,n$, suppose that $L_j$ is a homogeneous sublaplacian on a stratified Lie group $G_j$. Suppose further that, for each $j \in \{1,\dots ,n\}$, at least one of the following conditions holds:

• $G_j$ and $L_j$ satisfy Assumption $({\mathrm {A}})$;

• $G_j$ is a Métivier group;

• $G_j$ is abelian.

Let $G = G_1 \times \dots \times G_n$ and $L = L_1^\sharp + \dots + L_n^\sharp$, as in Proposition 15. If $F : {\mathbb {R}}\rightarrow {\mathbb {C}}$ satisfies

$$\begin{aligned} \Vert F\Vert _{MW_2^s} < \infty \end{aligned}$$

for some $s > (\dim G)/2$, then $F(L)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in ]1,\infty [$.