# Boundedness of oscillating singular integrals on Lie groups of polynomial
growth

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Michael Ruzhansky
Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete
Mathematics Ghent University, Belgium and School of Mathematical Sciences
Queen Mary University of London United Kingdom E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We investigate the boundedness of oscillating singular integrals on Lie groups
of polynomial growth in order to extend the classical oscillating conditions
due to Fefferman and Stein for the boundedness of oscillating convolution
operators. Kernel criteria are presented in terms of a fixed sub-Riemannian
structure on the group induced by a sub-Laplacian associated to a Hörmander
system of vector fields. In the case where the group is graded, kernel
criteria are presented in terms of the Fourier analysis associated to an
arbitrary Rockland operator.

###### Key words and phrases:

Calderón-Zygmund operator, Oscillating singular integrals, Lie group of
polynomial growth, Graded Lie groups

###### 2010 Mathematics Subject Classification:

35S30, 42B20; Secondary 42B37, 42B35

The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and
Partial Differential Equations and by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021). Michael
Ruzhansky is also supported by EPSRC grant EP/R003025/2.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline and historical remarks
    2. 1.2 The main result
  2. 2 Preliminaries
    1. 2.1 Sub-Laplacians on Lie groups of polynomial growth
    2. 2.2 Lie groups of Type I and group Plancherel formula
    3. 2.3 The spaces $H^{1}$ and $BMO$ on Lie groups of polynomial growth
    4. 2.4 Homogeneous and graded Lie groups
    5. 2.5 The spaces $H^{1}$ and $BMO$ on homogeneous Lie groups
  3. 3 Proof of the main theorem
  4. 4 Examples
    1. 4.1 Riemannian structures on compact Lie groups
    2. 4.2 Oscillating integrals on the torus $\mathbb{T}^{n}$
    3. 4.3 Oscillating integrals on $\textnormal{SU}(2)\cong\mathbb{S}^{3}$
    4. 4.4 Oscillating integrals on $\mathbb{R}^{n}$
    5. 4.5 Oscillating integrals on the Heisenberg group $\mathbb{H}_{n}$
    6. 4.6 Oscillating operators on Lie groups of polynomial growth and of type I

##  1\. Introduction

###  1.1. Outline and historical remarks

In this paper we study the boundedness of oscillating singular integrals on a
connected Lie group $G$ of polynomial growth. We address our investigation by
considering two settings.

  * •

First, we consider on $G$ a positive sub-Laplacian
$\mathcal{L}=-\sum_{i=1}^{k}X_{i}^{2}$ associated with a systems of vector
fields $X=\\{X_{i}\\}_{i=1}^{k}$ satisfying the Hörmander condition. Then, our
kernel criteria for the boundedness of operators are presented in terms of the
sub-Laplacian $\mathcal{L}$ and of the kernel of the operator.

  * •

Later, we consider the case where $G$ is a graded Lie group, but generalising
the sub-Laplacian context, the kernel criteria are presented in terms of the
Fourier analysis associated with an arbitrary Rockland operator $\mathcal{R}$.

In both cases, implicitly, our arguments make use of the hypoellipticity of
the sub-Laplacian $\mathcal{L}$ and of the Rockland operator $\mathcal{R}$ in
view of the Hörmander theorem of the sums of squares [33], and of the Helffer
and Nourrigat solution of the Rockland conjecture [29], respectively.

As a consequence of our investigation, further applications can be obtained
for many classes of Lie groups of interest in analysis and geometry e.g. the
Heisenberg group, the Cartan group, the Engel group, other stratified groups,
arbitrary compact Lie groups such as the torus $\mathbb{T}^{n},$
$\textnormal{SU}(2)\cong\mathbb{S}^{3},$ $\textnormal{SO}(3),$ graded Lie
groups, and general Lie groups of polynomial growth.

In analysis, the theory of singular integrals has a long tradition (see e.g.
Mihlin [36] and references therein). This class of operators was introduced in
the study of elliptic problems, for instance, its importance in the proof of
the celebrated Calderón theorem on the uniqueness of the Cauchy problem has
given testimony of its relevant applications to the theory of PDE, see [4] for
details. On the other hand, the program of Calderón and Zygmund (which started
with their memoir [5]) was dedicated to generalising several of the
fundamental results of the one-dimensional harmonic analysis to higher
dimensions (as the weak (1,1)-boundedness theorem for the Hilbert tranform due
to Kolmogorov, and its $L^{p}$-boundedness theorem due to M. Riesz). Being the
Hilbert transform, the fundamental one-dimensional example of a singular
integral, Calderón and Zygmund in [5] investigated convolution operators of
the form

$Tf(x)=\textnormal{p.v.}\smallint_{\mathbb{R}^{n}}K(x-y)f(y)dy,$ (1.1)

where the kernel $K,$ having some regularity properties, is homogeneous of
degree $-n,$ and satisfies the cancellation property
$\smallint_{|x|=1}K(x)d\sigma(x)=0.$ Here, $d\sigma$ is the surface density on
the unit sphere.

A well known condition for the boundedness properties of singular integrals is
the one due to Hörmander [32], where the distributional kernels $K$ are taken
satisfying a “smoothness condition” of the form

$\displaystyle[K]_{H_{\infty}}:=\sup_{0<R<1}\sup_{|y|<R}\smallint_{|x|\geq
2R}|K(x-y)-K(x)|dx<\infty.$ (1.2)

Indeed, the Hörmander condition in (1.2) together with the Fourier transform
condition $\widehat{K}\in L^{\infty}$ give sufficient conditions for the weak
(1,1) boundedness of $T$ (as in 1.1) and also for its $L^{p}$-boundendness in
the range $1<p<\infty$.

Other conditions generalising the one by Hörmander, arise in the analysis of
oscillating multipliers (see Hardy [27], Hirschman [30] and Wainger [46]). By
a suggestion of Stein, Fefferman in [18] has considered distributions
satisfying the “oscillating smoothness” condition

$[K]_{H_{\infty},\theta}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(x-y)-K(x)|dx<\infty.$ (1.3)

Indeed, roughly speaking, if $K$ satisfies (1.3) and its Fourier transform has
decay

$|\widehat{K}(\xi)|=O((1+|\xi|)^{-\frac{n\theta}{2}}),\quad 0\leq\theta<1,$
(1.4)

Fefferman’s theorem says that $T$ admits a bounded extension of weak (1,1)
type. Obviously, with $\theta=0,$ Fefferman’s condition agrees with the one in
(1.2) by Hörmander. Observe that Fefferman’s approach illustrates the delicate
relationship between the condition on $K$ and its Fourier transform
$\widehat{K}.$

Further extensions of the Calderón-Zygmund theory to non-commutative
structures were developed e.g. by Coifman and Weiss in [13] and by Coifman and
De Guzmán in [10] for spaces of homogeneous type just to mention a few.

###  1.2. The main result

In this work we prove that the Fefferman conditions for oscillating kernels
beget $L^{p}$-bounded operators not only in the Euclidean setting, but also in
general non-commutative structures, namely, connected Lie groups of polynomial
growth. In particular, for graded Lie groups (which are also Lie groups of
polynomial growth but with an extra structure of dilations on its graded Lie
algebra) we will use the existence of a general class of hypoelliptic
differential operators that in this context are called Rockland operators, see
e.g. Helfer and Nourrigat [29] or [21, Chapter IV] for details.

In order to present our main result (see Theorem 1.1 below), we introduce some
preliminary notions. If the Lie group is of type I (e.g. any compact Lie
group, any graded Lie group, or in general homogeneous Lie groups) we denote
by $\widehat{G}$ the unitary dual of $G,$ which consists of all equivalence
classes of unitary and irreducible representations on $G.$ In this setting,
$[\xi]$ denotes an equivalence class in the unitary dual $\widehat{G}$. The
representation spaces $H_{\xi}$ associated to
$\xi:G\rightarrow\mathcal{B}(H_{\xi})$ are separable Hilbert spaces (with
finite dimension in the case where $G$ is compact). Here,
$\mathcal{B}(H_{\xi})$ denotes the space of the bounded linear operators on
$H_{\xi}.$ The representation $\xi$ becomes operator-valued, and then, the
Fourier transform of a distribution $K$ on $G,$ is given by

$\widehat{K}(\xi)=\smallint_{G}K(x)\xi(x)^{*}dx:H_{\xi}\rightarrow H_{\xi}.$
(1.5)

As usually, $dx$ denotes the left-invariant Haar measure associated to $G.$

In the case of a graded Lie group, the unitary dual $\widehat{G}$ becomes to
be a purely continuous set, and by following the standard notation for the
Fourier analysis on graded Lie groups (taken from the formalism of
$C^{*}$-algebras) the equivalence classes of unitary representations will be
denoted by $[\pi]\in\widehat{G}$ (instead of $[\xi]$). Also, in the setting of
graded Lie groups the class $[\pi]$ is typically identified with a
representation $\pi$ of the class. The representation space $H_{\pi}$ is
$\pi$-a.e. a separable Hilbert space of infinite dimension. Nevertheless, for
the variety of examples presented in Section 4, e.g. in the case of compact
Lie groups, the notation $[\xi]\in\widehat{G}$ will be reserved.

On an arbitrary Lie group of polynomial growth, we will show (as in
Fefferman’s criterion above) that the Fourier transform condition (1.4) on the
kernel $K,$ can be replaced by the $L^{2}$-boundendess of the operator in
(1.9) below, and that the following conditions

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$ (1.6)

and

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$ (1.7)

can be used to study the $L^{p}$-boundedness of the corresponding singular
integral operator $Tf:=K\ast f.$ We have denoted by

$B(R)=\\{x\in G:|x|<R\\}$

the ball of radius $R>0$ centred at the identity and by
$|B(R)|:=\textnormal{Vol}(B(R))$ its Haar measure.

The distance $|\cdot|$ used in (1.6) or in (1.7) depends of the geometry of
the group $G$ under consideration. Indeed, for Lie groups of polynomial growth
endowed with a Hörmander sub-Laplacian the natural distance is the one defined
by using horizontal curves. On the another hand, if the group is graded we
chose a quasi-distance on the group compatible with its structure of
dilations.

We record that if $|\cdot|$ is the Carnot-Carathéodory distance on a Lie group
of polynomial growth $G,$ (see Coulhon, Saloff-Coste, and Varopoulos [14])
then there exist $d=d_{0}$ and $d^{\prime}=d_{\infty}$ such that

$\forall R\in[0,1],\,|B(x,R)|\sim R^{d_{0}},\,\forall R\geq 1,\,|B(x,R)|\sim
R^{d_{\infty}},$ (1.8)

where $B(x,R)$ is the ball of radius $R>0$ associated to the distance
$|\cdot|.$ Then $d_{0}$ and $d_{\infty}$ are called the local dimension and
the global dimension of the pair $(G,X),$ respectively.

It can be proved that the global dimension depends only of the structure of
the group and it is independent of the system $X,$ see Coulhon, Saloff-Coste,
and Varopoulos [14] for details. However, the local dimension $d_{0}$ depends
of both, of the group $G$ and of the vector fields system $X.$ One of the main
aspects of our work, is that our kernel criteria are presented in terms of the
local dimension $d_{0}.$

When asking how to generalise the condition (1.4) to the case of Lie groups of
polynomial growth, one can give a possible solution using the Fourier
transform of the group $G$ if it is a Lie group of type I, see Subsection 4.6
for details. Nevertheless, the hypothesis that the Lie group $G$ is of type
$I$ can be removed if we consider the group to be endowed with a sub-
Riemannian structure induced by a family of vector-fields
$\\{X_{j}\\}_{j=1}^{k},$ satisfying the Hörmander condition, that is, the
family of vector-fields $\\{X_{j}\\}_{j=1}^{k}$ and their iterated commutators
generate the Lie algebra $\mathfrak{g}\sim T_{e}G$ of $G$. In this general
setting, (1.4) can be extended to general Lie groups of polynomial growth just
by assuming that the operators

$(1+{\mathcal{L}})^{\frac{d_{0}\theta}{4}}T,\,T(1+{\mathcal{L}})^{\frac{d_{0}\theta}{4}}:L^{2}(G)\rightarrow
L^{2}(G),$ (1.9)

admit bounded extensions.

Now, when $G$ is a graded Lie group, the Euclidean condition (1.4) can be
extended by using the infinitesimal representation of any Rockland operator.
If $G$ is a stratified Lie group the prototype of Rockland operators are sub-
Laplacians $\mathcal{L}=-\sum_{i=1}^{k}X_{i}^{2}.$ However, Rockland operators
may have an arbitrary order. For instance, on the Heisenberg group
$\mathbb{H}_{n}\sim\mathbb{R}^{2n+1}$ instead of the positive sub-Laplacian
$\mathcal{L}=-\sum_{i=1}^{n}(X_{i}^{2}+Y_{i}^{2})$ we can choose e.g. the
positive Rockland operator $\mathcal{R}=\sum_{i=1}^{n}(X_{i}^{4}+Y_{i}^{4})$
of fourth order. For examples of Rockland operators with an arbitrary order we
refer the reader to Corollary 4.1.8 of [21].

If we fix a Rockland operator $\mathcal{R}$ and if $\tau$ is the right-
convolution kernel associated with $\mathcal{R},$ $\mathcal{R}f=f\ast\tau,$
then

$\pi(\mathcal{R}):=\widehat{\tau}(\pi)\equiv d\pi(\mathcal{R})$ (1.10)

denotes its Fourier transform, and the condition in (1.4) takes in the setting
of a graded Lie group
$G\cong\mathfrak{g}_{0}\times\mathfrak{g}_{1}\times\cdots\mathfrak{g}_{s}$ the
following form

$\sup_{\pi\in\widehat{G}}\|\widehat{K}(\pi)(1+\pi(\mathcal{R}))^{\frac{Q\theta}{2\nu}}\|_{\textnormal{op}},\,\sup_{\pi\in\widehat{G}}\|(1+\pi(\mathcal{R}))^{\frac{Q\theta}{2\nu}}\widehat{K}(\pi)\|_{\textnormal{op}}<\infty,$

where $\nu=\nu_{\mathcal{R}}>0$ denotes the homogeneity degree of
$\mathcal{R}.$ As before, $\widehat{K}(\pi)$ denotes the Fourier transform of
a distribution $K$ on $G.$ The main result of this work is the following
theorem.

######  Theorem 1.1.

Let $G$ be a Lie group of polynomial growth. Let $T$ be a convolution operator
with a distribution $K\in\mathscr{D}^{\prime}(G),$ and let $0\leq\theta<1$.
The following statements hold true.

  * •

Let $X=\\{X_{i}\\}_{i=1}^{k}$ be a Hörmander system of vector fields and let
$\mathcal{L}=-\sum_{j=1}^{k}X_{j}^{2}$ be the associated positive sub-
Laplacian. Let $|\cdot|$ be the Carnot-Carathéodory distance on $G$ associated
to $X,$ and let $d=d_{0}$ be the local dimension of $(G,X).$ Assume that

$(1+{\mathcal{L}})^{\frac{d_{0}\theta}{4}}T:C^{\infty}_{0}(G)\subset
L^{2}(G)\rightarrow L^{2}(G),$ (1.11)

admits a bounded extension. If additionally, $K$ satisfies the kernel
condition

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$ (1.12)

then $T:L^{\infty}(G)\rightarrow BMO(G)$ admits a bounded extension. On the
other hand, if

$T(1+{\mathcal{L}})^{\frac{d_{0}\theta}{4}}:C^{\infty}_{0}(G)\subset
L^{2}(G)\rightarrow L^{2}(G),$ (1.13)

is bounded and $K$ satisfies the kernel condition

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$ (1.14)

then $T:H^{1}(G)\rightarrow L^{1}(G)$ extends to a bounded operator.

  * •

Consider $G$ to be a graded Lie group (and hence non-compact), let $|\cdot|$
be a homogeneous quasi-norm on $G$ and let $Q$ be its homogeneous dimension.
Let $\mathcal{R}$ be a Rockland operator of homogeneous degree $\nu>0.$ Assume
that $K$ satisfies the estimate

$\sup_{\pi\in\widehat{G}}\|(1+\pi(\mathcal{R}))^{\frac{Q\theta}{2\nu}}\widehat{K}(\pi)\|_{\textnormal{op}}<\infty.$
(1.15)

If additionally, $K$ satisfies the kernel condition

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$ (1.16)

then $T:L^{\infty}(G)\rightarrow BMO(G)$ admits a bounded extension. On the
other hand, if one has the estimate

$\sup_{\pi\in\widehat{G}}\|\widehat{K}(\pi)(1+\pi(\mathcal{R}))^{\frac{Q\theta}{2\nu}}\|_{\textnormal{op}}<\infty,$
(1.17)

and the kernel condition

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$ (1.18)

then $T:H^{1}(G)\rightarrow L^{1}(G)$ extends to a bounded operator.

In any case above, $T$ admits a bounded extension on $L^{p}(G)$ for all $2\leq
p<\infty,$ provided that $K$ satisfies (1.11) and (1.12), or (1.15) and
(1.16), respectively. Moreover, $T$ admits a bounded extension on $L^{p}(G)$
for all $1<p\leq 2,$ provided that $K$ satisfies (1.13) and (1.14), or (1.17)
and (1.18), respectively.

Now, we briefly present some remarks related with Theorem 1.1.

######  Remark 1.2.

We note that the statement in Theorem 1.1 for $\theta=0$ follows from the
general theory of singular integrals developed by Coifman and Weiss [13] for
spaces of homogeneous type. So, the contribution in Theorem 1.1 corresponds to
the oscillating setting, namely, when $0<\theta<1.$

######  Remark 1.3.

In the context of $G=\mathbb{R}^{n},$ and of the positive Laplacian
$\mathcal{R}=\mathcal{L}=-\Delta_{x},$ the
$H^{1}(\mathbb{R}^{n})-L^{1}(\mathbb{R}^{n})$-boundedness result was proved in
the non-oscillating case $\theta=0$ by Suzuki in [43]. As mentioned above, in
the Euclidean case that the conditions $[K]_{\infty,\theta}<\infty$ and (1.4)
are enough to deduce the $H^{1}(\mathbb{R}^{n})-L^{1}(\mathbb{R}^{n})$
boundedness of the corresponding convolution operator $Tf=f\ast K$ were proved
by Fefferman and Stein in [20] when $0\leq\theta<1.$

######  Remark 1.4.

In the non-commutative case, the $L^{1}$-average conditions in Theorem 1.1 are
already new, even considering Lie groups of polynomial growth or the graded
case. Observe that in the setting of Lie groups of polynomial growth $G$ both
of the situations $d_{0}\leq d_{\infty}$ and $d_{\infty}\leq d_{0}$ are
equally probable. For instance, if $G$ is a simply connected nilpotent Lie
group, then $d_{0}\leq d_{\infty},$ and when $G$ is compact $d_{\infty}=0.$

An interesting fact is that starting with a Lie group $G$ where $d_{0}\leq
d_{\infty}$ we can always consider the product space
$G^{\prime}=\mathbb{T}^{d_{\infty}-d_{0}+1}\times G,$ which has local
dimension $d_{0}^{\prime}=(d_{\infty}-d_{0}+1)+d_{0}=d_{\infty}+1,$ global
dimension $d_{\infty}^{\prime}=0+d_{\infty}=d_{\infty}$ and of course
$d_{\infty}^{\prime}<d_{0}^{\prime}.$ Let us finalise this remark by observing
that in the case of a stratified nilpotent Lie group $G,$ $d_{0}=d_{\infty},$
see Coulhon, Saloff-Coste, and Varopoulos [14, Chapter VI]. Also, if $G$ is a
graded Lie group, then $d_{0}=d_{\infty}=Q,$ the homogeneous dimension of $G$.

######  Remark 1.5.

In our analysis the appearance of the local dimension $d_{0}$ is justified by
the estimates of the heat kernel $e^{-t\mathcal{L}}$ of the sub-Laplacian for
small time $t\rightarrow 0^{+}$, see Coulhon, Saloff-Coste, and Varopoulos
[14, Chapter VIII].

######  Remark 1.6.

For $\theta=0$ the conditions in (1.11) and (1.15) are reduced to the
$L^{2}$-boundedness of the operator $T,$ in view of the group Plancherel
theorem. For singular integrals on $\mathbb{R}^{n}$ the $L^{2}$-boundedness of
singular integrals (of possibly non-convolution type) was characterised by
David and Journé in their celebrated work [17].

######  Remark 1.7.

Fourier multipliers satisfying Marcinkiewicz, and Hörmander-Mihlin criteria in
the context of Lie groups and several spaces of homogeneous type have
Calderón-Zygmund type kernels. In the context of spectral multipliers of self-
adjoint operators, e.g., sub-Laplacians, or of other operators with heat
kernels satisfying Gaussian estimates, with general contexts that go beyond of
the objective of this paper we refer the reader e.g. to [1, 2, 9, 15, 12], the
book of Stein [42] and to the extensive list of references therein.

This paper is organised as follows. The paper ends with Section 4 where we
present some examples illustrating our main Theorem 1.1, however the notations
and the notions (of representation theory and of Lie theory) that we use are
contained in Section 2. In Section 3 we present the proof of Theorem 1.1.

##  2\. Preliminaries

###  2.1. Sub-Laplacians on Lie groups of polynomial growth

Let $G$ be a Lie group of polynomial growth with Lie algebra $\mathfrak{g}.$
Under the identification $\mathfrak{g}\simeq T_{e_{G}}G,$ where $e_{G}$ is the
identity element of $G,$ let us consider a system of $C^{\infty}$-vector
fields $X=\\{X_{1},\cdots,X_{k}\\}\in\mathfrak{g}$. For all
$I=(i_{1},\cdots,i_{\omega})\in\\{1,2,\cdots,k\\}^{\omega},$ of length
$\omega\geqslant 2,$ denote

$X_{I}:=[X_{i_{1}},[X_{i_{2}},\cdots[X_{i_{\omega-1}},X_{i_{\omega}}]\cdots]],$

and for $\omega=1,$ $I=(i),$ $X_{I}:=X_{i}.$ Let $V_{\omega}$ be the subspace
generated by the set $\\{X_{I}:|I|\leqslant\omega\\}.$ That $X$ satisfies the
Hörmander condition, means that there exists $\kappa^{\prime}\in\mathbb{N}$
such that $V_{\kappa^{\prime}}=\mathfrak{g}.$ Certainly, we consider the
smallest $\kappa^{\prime}$ with this property and we denote it by $\kappa$
which will be later called the step of the system $X.$ We also say that $X$
satisfies the Hörmander condition of order $\kappa.$ Note that the sum of
squares

$\mathcal{L}=-\sum_{i=1}^{k}X_{i}^{2},$

is a subelliptic operator which by following the usual nomenclature is called
the subelliptic Laplacian associated with the family $X.$ For short we refer
to $\mathcal{L}$ as the sub-Laplacian. In view of the Hörmander theorem on
sums of the squares of vector fields (see Hörmander [33]) it is a hypoelliptic
operator (i.e. if $\mathcal{L}u\in C^{\infty}(G)$ with
$u\in\mathscr{D}^{\prime}(G)$ then $u\in C^{\infty}(G),$ and also locally at
all points).

A central notion in the analysis of the sub-Laplacian is that of the Hausdorff
dimension, in this case, associated to $\mathcal{L}$. Indeed, for all $x\in
G,$ denote by $H_{x}^{\omega}G$ the subspace of the tangent space $T_{x}G$
generated by the $X_{i}$’s and all the Lie brackets

$[X_{j_{1}},X_{j_{2}}],[X_{j_{1}},[X_{j_{2}},X_{j_{3}}]],\cdots,[X_{j_{1}},[X_{j_{2}},[X_{j_{3}},\cdots,X_{j_{\omega}}]]],$

with $\omega\leqslant\kappa.$ The Hörmander condition can be stated as
$H_{x}^{\kappa}G=T_{x}G,$ $x\in G.$ We have the filtration

$H_{x}^{1}G\subset H_{x}^{2}G\subset H_{x}^{3}G\subset\cdots\subset
H_{x}^{\kappa-1}G\subset H_{x}^{\kappa}G=T_{x}G,\,\,x\in G.$

In our case, the dimension of every $H_{x}^{\omega}G$ does not depend on $x$
and we write $\dim H^{\omega}G:=\dim H_{x}^{\omega}G,$ for any $x\in G.$ So,
the Hausdorff dimension can be defined as (see e.g. [28, p. 6]),

$Q:=\dim(H^{1}G)+\sum_{i=1}^{\kappa-1}(i+1)(\dim H^{i+1}G-\dim H^{i}G).$ (2.1)

###  2.2. Lie groups of Type I and group Plancherel formula

Let $G$ be a second countable Lie group $G$ of type I and let $dx$ be its
left-invariant Haar measure. Let us record the notion of the unitary dual
$\widehat{G}$ of $G$. Let us denote by $\xi$ a strongly continuous, unitary
and irreducible representation of $G,$ this means that,

  * •

$\xi\in\textnormal{Hom}(G,\textnormal{U}(H_{\xi})),$ for some infinite-
dimensional Hilbert space $H_{\xi},$ i.e. $\xi(xy)=\xi(x)\xi(y)$ and for the
adjoint of $\xi(x),$ $\xi(x)^{*}=\xi(x^{-1}),$ for every $x,y\in G.$

  * •

The map $(x,v)\mapsto\xi(x)v,$ from $G\times H_{\xi}$ into $H_{\xi}$ is
continuous.

  * •

For every $x\in G,$ and $W_{\xi}\subset H_{\xi},$ if $\xi(x)W_{\xi}\subset
W_{\xi},$ then $W_{\xi}=H_{\xi}$ or $W_{\xi}=\emptyset.$

Let $\textnormal{Rep}(G)$ be the set of unitary, strongly continuous and
irreducible representations of $G.$ The relation,

$\xi_{1}\sim\xi_{2}\textnormal{ if and only if, there exists
}A\in\textnormal{End}(H_{\xi_{1}},H_{\xi_{2}}),\textnormal{ such that
}A\xi_{1}(x)A^{-1}=\xi_{2}(x),$

for every $x\in G,$ is an equivalence relation and the unitary dual of $G,$
denoted by $\widehat{G}$ is defined via

$\widehat{G}:={\textnormal{Rep}(G)}/{\sim}.$

If additionally, $G$ is a Lie group of polynomial growth, for every
representation $\xi\in\widehat{G},$ $\xi:G\rightarrow\mathscr{B}({H}_{\xi}),$
we denote by ${H}_{\xi}^{\infty}$ the set of smooth vectors, that is, the
space of elements $v\in{H}_{\xi}$ such that the function $x\mapsto\xi(x)v,$
$x\in{G},$ is smooth.

The Fourier transform $\widehat{f}$ of a distribution
$f\in\mathscr{D}^{\prime}(G)$ is defined via,

$\widehat{f}(\xi)\equiv(\mathscr{F}f)(\xi):=\smallint\limits_{G}f(x)\xi(x)^{*}dx.$

If $f\in L^{1}(G),$ then for a.e. $\xi,$
$\widehat{f}(\xi)\in\mathscr{B}(H_{\xi}).$ Because $G$ is of type I, the
unitary dual $\widehat{G}$ admits a Borel structure $(\widehat{G},d\nu(\xi))$
called in this setting the Mackey-Borel structure. Indeed, by denoting
$\mathscr{B}^{2}(\mathcal{H}_{\xi})$ the class of Hilbert-Schmidt operators on
$\mathcal{H}_{\xi},$ one has the direct Hilbert space integral

$\mathscr{B}^{2}(\widehat{G}):=\smallint\limits_{\widehat{G}}^{\oplus}\mathscr{B}^{2}(H_{\xi})d\nu(\xi)\cong\smallint\limits_{\widehat{G}}^{\oplus}H_{\xi}\otimes\overline{H}_{\xi}d\nu(\xi).$
(2.2)

In particular, the Fourier transform
$\mathscr{F}:L^{1}(G)\rightarrow\mathscr{B}(\widehat{G}):=\smallint\limits_{\widehat{G}}^{\oplus}\mathscr{B}(H_{\xi})d\nu(\xi)$
is injective and one can reconstruct a function using the Fourier inversion
formula

$f(x)=\smallint_{[\xi]\in\widehat{G}}\textnormal{{Tr}}[\xi(x)\widehat{f}(\xi)]d\nu(\xi),\,\,f\in
L^{1}(G)\cap L^{2}(G).$

The Plancherel theorem states that
$\mathscr{F}:L^{2}(G)\rightarrow\mathscr{B}^{2}(\widehat{G})$ is an isometry,
where the inner product on $\mathscr{B}^{2}(\widehat{G})$ is defined via

$(f,g)_{\mathscr{B}^{2}(\widehat{G})}:=\smallint_{[\xi]\in\widehat{G}}\textnormal{{Tr}}[\widehat{f}(\xi)\widehat{g}(\xi)^{*}]d\nu(\xi).$
(2.3)

So, the Plancherel theorem takes the form

$\|f\|_{L^{2}(G)}^{2}=\smallint_{[\xi]\in\widehat{G}}\|\widehat{f}(\xi)\|_{\textnormal{HS}}^{2}d\nu(\xi).$
(2.4)

We have denoted by $\|\cdot\|_{\textnormal{HS}}$ the usual Hilbert-Schmidt
norm on $\xi$-a.e. representation space $\mathscr{B}^{2}(H_{\xi}).$

###  2.3. The spaces $H^{1}$ and $BMO$ on Lie groups of polynomial growth

Let $G$ be a Lie group of polynomial growth. In that follows we record the
definitions of the Hardy space and of the $BMO$ space on a Lie group of
polynomial growth (a Lie group of this type is unimodular, see [44, Page 3]).
For this, we follow ter Elst, Robinson and Zhu [44] and Coifman and Weiss
[13].

Let us consider a sub-Laplacian $\mathcal{L}=-(X_{1}^{2}+\cdots+X_{k}^{2})$ on
$G,$ where the system of vector fields $X=\\{X_{i}\\}_{i=1}^{k}$ satisfies the
Hörmander condition of step $\kappa$. For every point $g\in G,$ let us denote
$X_{g}=\\{X_{i,g}\\}_{i=1}^{k},$
$\mathcal{H}_{g}=\textnormal{span}\\{X_{g}\\}.$ We say that a curve
$\gamma:[0,1]\rightarrow G$ is horizontal if

$\dot{\gamma}(t)\in\mathcal{H}_{\gamma(t)},\textnormal{ for a.e. }t\in(0,1).$

The Carnot-Carathéodory distance associated to the sub-Riemannian structure
induced by $X,$ is defined by

$d_{s}(g_{0},g_{1}):=\inf_{\gamma\textnormal{ horizontal
}}\\{l(\gamma):=\smallint\limits_{0}^{1}|\dot{\gamma}(t)|dt:\,\,\gamma(0)=g_{0},\,\gamma(1)=g_{1}\\},\,\,g_{0},g_{1}\in
G.$

We will fix a subelliptic distance on $G,$ $|\cdot|,$ defined by the Carnot-
Carathéodory distance in the natural way: $|g|=d_{s}(g,e_{G}),$ where $e_{G}$
is the identity element of $G.$ As usual, the ball of radius $r>0,$ is defined
as

$B(x,r)=\\{y\in G:|y^{-1}x|<r\\}.$

An important geometric property of the family of balls associated to the
Carnot-Carathéodory distance arises when one wants to compute their volume.
Indeed, there exist $d=d_{0}$ and $d^{\prime}=d_{\infty}$ such that

$\forall R\in[0,1],\,|B(x,R)|\sim R^{d_{0}},\,\forall R\geq 1,\,|B(x,R)|\sim
R^{d_{\infty}},$ (2.5)

where $B(x,R)$ is the ball of radius $R>0$ associated to the distance
$|\cdot|.$ Then $d_{0}$ and $d_{\infty}$ are called the local dimension and
the global dimension of the pair $(G,X),$ respectively. One of the main
aspects of our work, is that our kernel criteria are presented in terms of the
local dimension $d_{0}.$ It is well known that $d_{0}$ depends of the
Hörmander system of vector fields $X$ and that $d_{\infty}$ only depends of
the group $G$.

The subelliptic $BMO$ space on $G,$ $\textnormal{BMO}^{\mathcal{L}}(G),$ is
the space of locally integrable functions $f$ satisfying

$\|f\|_{\textnormal{BMO}^{\mathcal{L}}(G)}:=\sup_{\mathbb{B}}\frac{1}{|\mathbb{B}|}\smallint\limits_{\mathbb{B}}|f(x)-f_{\mathbb{B}}|dx<\infty,\textnormal{
where
}f_{\mathbb{B}}:=\frac{1}{|\mathbb{B}|}\smallint\limits_{\mathbb{B}}f(x)dx,$

and $\mathbb{B}$ ranges over all balls $B(x_{0},r),$ with $(x_{0},r)\in
G\times(0,\infty).$ The subelliptic Hardy space
$\textnormal{H}^{1,\mathcal{L}}(G)$ will be defined via the atomic
decomposition: $f\in\textnormal{H}^{1,\mathcal{L}}(G)$ if and only if $f$ can
be expressed as

$f=\sum_{j=1}^{\infty}c_{j}a_{j},$

where $\\{c_{j}\\}_{j=1}^{\infty}$ is a sequence in $\ell^{1}(\mathbb{N}),$
and every function $a_{j}$ is an atom, i.e., $a_{j}$ is supported in some ball
$B_{j},$ ($a_{j}$ satisfies the cancellation property)

$\smallint\limits_{B_{j}}a_{j}(x)dx=0,$

and

$\|a_{j}\|_{L^{\infty}(G)}\leqslant\frac{1}{|B_{j}|}.$

The norm $\|f\|_{\textnormal{H}^{1,\mathcal{L}}(G)}$ is the infimum over all
possible series $\sum_{j=1}^{\infty}|c_{j}|.$ Furthermore
$\textnormal{BMO}^{\mathcal{L}}(G)$ is the topological dual of the Hardy space
$\textnormal{H}^{1,\mathcal{L}}(G),$ see Coifman and Weiss [13] or ter Elst,
Robinson and Zhu [44, Page 4]. This can be understood in the following sense:

  * (a).

If $\phi\in\textnormal{BMO}^{\mathcal{L}}(G),$ then

$\Phi:f\mapsto\smallint\limits_{G}f(x)\phi(x)dx,$

admits a bounded extension on $\textnormal{H}^{1,\mathcal{L}}(G).$

  * (b).

Conversely, every continuous linear functional $\Phi$ on
$\textnormal{H}^{1,\mathcal{L}}(G)$ arises as in (a) with a unique element
$\phi\in\textnormal{BMO}^{\mathcal{L}}(G).$

The norm of $\phi$ as a linear functional on
$\textnormal{H}^{1,\mathcal{L}}(G)$ is equivalent with the
$\textnormal{BMO}^{\mathcal{L}}(G)$-norm. Important properties of the
$\textnormal{BMO}^{\mathcal{L}}(G)$ and the
$\textnormal{H}^{1,\mathcal{L}}(G)$ norms are the following,

$\|f\|_{\textnormal{BMO}^{\mathcal{L}}(G)}=\sup_{\|g\|_{\textnormal{H}^{1,\mathcal{L}}(G)}=1}\left|\smallint\limits_{G}f(x)g(x)dx\right|,$
(2.6)
$\|g\|_{\textnormal{H}^{1,\mathcal{L}}(G)}=\sup_{\|f\|_{\textnormal{BMO}^{\mathcal{L}}(G)}=1}\left|\smallint\limits_{G}f(x)g(x)dx\right|.$
(2.7)

If we replace $\mathcal{L}$ by the Laplacian $\mathcal{L}_{G}$ in the
definitions above, we will write $BMO(G)$ and $H^{1}(G),$ defined by the
distance induced by the usual bi-invariant Riemannian metric on $G.$ The
subelliptic Fefferman-Stein interpolation theorem in this case can be stated
as follows (see Carbonaro, Mauceri and Meda [7]).

######  Theorem 2.1.

Let $G$ be a Lie group of polynomial growth. Let us consider a sub-Laplacian
$\mathcal{L}=-(X_{1}^{2}+\cdots+X_{k}^{2})$ on $G,$ where the system of vector
fields $X=\\{X_{i}\\}_{i=1}^{k}$ satisfies the Hörmander condition of step
$\kappa$. For every $\theta\in(0,1),$ we have,

  * •

If $p_{\theta}=\frac{2}{1-\theta},$ then
$(L^{2},\textnormal{BMO}^{\mathcal{L}})_{[\theta]}(G)=L^{p_{\theta}}(G).$

  * •

If $p_{\theta}=\frac{2}{2-\theta},$ then
$(\textnormal{H}^{1,\mathcal{L}},L^{2})_{[\theta]}(G)=L^{p_{\theta}}(G).$

From now, for the simplicity of the notation we will write $BMO(G)$ and
$H^{1}(G)$ for the corresponding $BMO^{\mathcal{L}}(H)$ and
$H^{1,\mathcal{L}}(G)$ associated to a sub-Laplacian $\mathcal{L}$,
respectively, even when these spaces can depend of the Hörmander system of
vector-fields.

###  2.4. Homogeneous and graded Lie groups

Let $G$ be a homogeneous Lie group. This means that $G$ is a connected and
simply connected Lie group whose Lie algebra $\mathfrak{g}$ is endowed with a
family of dilations $D_{r}^{\mathfrak{g}},$ $r>0,$ which are automorphisms on
$\mathfrak{g}$ satisfying the following two conditions:

  * •

For every $r>0,$ $D_{r}^{\mathfrak{g}}$ is a map of the form

$D_{r}^{\mathfrak{g}}=\textnormal{Exp}(\ln(r)A)$

for some diagonalisable linear operator
$A\equiv\textnormal{diag}[\nu_{1},\cdots,\nu_{n}]$ on $\mathfrak{g}.$

  * •

$\forall X,Y\in\mathfrak{g},$ and $r>0,$
$[D_{r}^{\mathfrak{g}}X,D_{r}^{\mathfrak{g}}Y]=D_{r}^{\mathfrak{g}}[X,Y].$

We call the eigenvalues of $A,$ $\nu_{1},\nu_{2},\cdots,\nu_{n},$ the
dilations weights or weights of $G$. The homogeneous dimension of a
homogeneous Lie group $G$ is given by

$Q=\textnormal{{Tr}}(A)=\nu_{1}+\cdots+\nu_{n}.$

The dilations $D_{r}^{\mathfrak{g}}$ of the Lie algebra $\mathfrak{g}$ induce
a family of maps on $G$ defined via,

$D_{r}:=\exp_{G}\circ D_{r}^{\mathfrak{g}}\circ\exp_{G}^{-1},\,\,r>0,$

where $\exp_{G}:\mathfrak{g}\rightarrow G$ is the usual exponential mapping
associated to the Lie group $G.$ We refer to the family $D_{r},$ $r>0,$ as
dilations on the group. If we write $rx=D_{r}(x),$ $x\in G,$ $r>0,$ then a
relation on the homogeneous structure of $G$ and the Haar measure $dx$ on $G$
is given by

$\smallint\limits_{G}(f\circ D_{r})(x)dx=r^{-Q}\smallint\limits_{G}f(x)dx.$

A Lie group is graded if its Lie algebra $\mathfrak{g}$ may be decomposed as
the sum of subspaces
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}\oplus\cdots\oplus\mathfrak{g}_{s}$
such that $[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subset\mathfrak{g}_{i+j},$ and
$\mathfrak{g}_{i+j}=\\{0\\}$ if $i+j>s.$ Examples of such groups are the
Heisenberg group $\mathbb{H}^{n}$ and more generally any stratified groups
where the Lie algebra $\mathfrak{g}$ is generated by $\mathfrak{g}_{1}$. Here,
$n$ is the topological dimension of $G,$ $n=n_{1}+\cdots+n_{s},$ where
$n_{k}=\mbox{dim}\mathfrak{g}_{k}.$

A Lie algebra admitting a family of dilations is nilpotent, and hence so is
its associated connected, simply connected Lie group.

A graded Lie group $G$ is a homogeneous Lie group equipped with a family of
weights $\nu_{j},$ all of them positive rational numbers. Let us observe that
if $\nu_{i}=\frac{a_{i}}{b_{i}}$ with $a_{i},b_{i}$ integer numbers, and $b$
is the least common multiple of the $b_{i}^{\prime}s,$ the family of dilations

$\mathbb{D}_{r}^{\mathfrak{g}}=\textnormal{Exp}(\ln(r^{b})A):\mathfrak{g}\rightarrow\mathfrak{g},$

have integer weights, $\nu_{i}=\frac{a_{i}b}{b_{i}}.$ So, in this paper we
always assume that the weights $\nu_{j},$ defining the family of dilations are
non-negative integer numbers which allow us to assume that the homogeneous
dimension $Q$ is a non-negative integer number. This is a natural context for
the study of Rockland operators (see Remark 4.1.4 of [21]).

###  2.5. The spaces $H^{1}$ and $BMO$ on homogeneous Lie groups

We will fix a homogeneous quasi-norm on $G,$ $|\cdot|.$ This means that
$|\cdot|$ is a non-negative function on $G,$ satisfying

$|x|=|x^{-1}|,\,\,\,r|x|=|D_{r}(x)|,\,\,\,\textnormal{ and }|x|=0\textnormal{
if and only if }x=e_{G},$ (2.8)

where $e_{G}$ is the identity element of $G.$ It satisfies a triangle
inequality with a constant: there exists a constant $\gamma\geq 1$ such that
$|xy|\leq\gamma(|x|+|y|).$ As usual, the ball of radius $r>0,$ is defined as

$B(x,r)=\\{y\in G:|y^{-1}x|<r\\}.$

Then $BMO(G)$ is the space of locally integrable functions $f$ satisfying

$\|f\|_{BMO(G)}:=\sup_{\mathbb{B}}\frac{1}{|\mathbb{B}|}\smallint\limits_{\mathbb{B}}|f(x)-f_{\mathbb{B}}|dx<\infty,\textnormal{
where
}f_{\mathbb{B}}:=\frac{1}{|\mathbb{B}|}\smallint\limits_{\mathbb{B}}f(x)dx,$

and $\mathbb{B}$ ranges over all balls $B(x_{0},r),$ with $(x_{0},r)\in
G\times(0,\infty).$ The Hardy space $H^{1}(G)$ will be defined via the atomic
decomposition. Indeed, $f\in H^{1}(G),$ if and only if, $f$ can be expressed
as $f=\sum_{j=1}^{\infty}c_{j}a_{j},$ where $\\{c_{j}\\}_{j=1}^{\infty}$ is a
sequence in $\ell^{1}(\mathbb{N}_{0}),$ and every function $a_{j}$ is an atom,
i.e., $a_{j}$ is supported in some ball $B_{j},$
$\smallint_{B_{j}}a_{j}(x)dx=0,$ and

$\|a_{j}\|_{L^{\infty}(G)}\leq\frac{1}{|B_{j}|}.$

The norm $\|f\|_{H^{1}(G)}$ is the infimum over all possible series
$\sum_{j=1}^{\infty}|c_{j}|.$ Furthermore $BMO(G)$ is the dual of $H^{1}(G),$
(see Folland and Stein [24]). This can be understood in the following sense:

  * (a).

If $\phi\in BMO(G),$ then $\Phi:f\mapsto\smallint\limits_{G}f(x)\phi(x)dx,$
admits a bounded extension on $H^{1}(G).$

  * (b).

Conversely, every continuous linear functional $\Phi$ on $H^{1}(G)$ arises as
in (a) with a unique element $\phi\in BMO(G).$

The norm of $\phi$ as a linear functional on $H^{1}(G)$ is equivalent with the
$BMO(G)$-norm. Important properties of the $BMO(G)$ and the $H^{1}(G)$ norms
are the following,

$\|f\|_{BMO(G)}=\sup_{\|g\|_{H^{1}(G)}=1}\left|\smallint\limits_{G}f(x)g(x)dx\right|,$
(2.9)
$\|g\|_{H^{1}(G)}=\sup_{\|f\|_{BMO(G)}=1}\left|\smallint\limits_{G}f(x)g(x)dx\right|.$
(2.10)

##  3\. Proof of the main theorem

Before presenting the proof of our main result (Theorem 1.1) we present some
auxiliary facts to be used in our further analysis.

######  Remark 3.1.

For the proof of the boundedness Theorem 1.1 let us use the equivalence of
norms (whose proof follows the lines of its Euclidean analogue, see
Duoandikoetxea [16, Proposition 6.15, Page 117])

$\|f\|_{BMO}\sim\sup_{\delta>0,\,x_{0}\in
G}\inf_{a\in\mathbb{C}}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|f(x)-a|dx,$
(3.1)

for any $f\in BMO(G).$ Indeed, by the definition of the $BMO$-norm,

$\displaystyle\|f\|_{BMO}=\sup_{\delta>0,\,x_{0}\in
G}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|f(x)-f_{B}|dx,\,f_{B}:=\frac{1}{|B(x_{0},\delta)|}\smallint
f(x)dx$

it is clear that

$\displaystyle\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|f(x)-a|dx\leq\|f\|_{BMO}.$

On the other hand, note that for any $a\in\mathbb{C},$ and all balls
$B=B(x_{0},\delta),$ the triangle inequality gives the estimate

$\displaystyle\smallint\limits_{B}|f(x)-f_{B}|dx\leq\smallint\limits_{B}|f(x)-a|dx+\smallint\limits_{B}|a-f_{B}|dx\leq
2\smallint\limits_{B}|f(x)-a|dx.$

Now, by dividing both sides by the Haar measure of $B,$ by taking the infimum
over $a\in\mathbb{C}$ and the supremun over $B$ we conclude that

$\displaystyle\|f\|_{BMO}\leq 2\sup_{\delta>0,\,x_{0}\in
G}\inf_{a\in\mathbb{C}}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|f(x)-a|dx,$

as desired.

######  Remark 3.2.

We record that if $G$ is a graded Lie group, a quasi-distance on $G$ is
symmetric (that is, $|x|=|x^{-1}|$ for any $x\in G$), see (2.8) or [21, Page
109]. On the other hand, the Carnot-Carathéodory distance on a Lie group of
polynomial growth $G$ is also symmetric in view of its $G$-invariance (see
Coulhon, Saloff-Coste and Varopoulos [14, Page 40]).

When estimating the action of singular integrals on functions supported in
small balls we require suitable versions of Sobolev inequalities. We record it
in the following remarks.

######  Remark 3.3 (Sobolev inequality on unimodular Lie groups).

Let $G$ be a unimodular Lie group and let $\mathcal{L}$ be a positive sub-
Laplacian associated to a Hörmander system of vector fields
$X=\\{X_{i}\\}_{i=1}^{k}$. For $0<p<\infty,$ the subelliptic $L^{p}$-Sobolev
space of order $s\in\mathbb{R},$ is defined by the family of distributions
$f\in\mathscr{D}^{\prime}(G)$ such that

$\|f\|_{L^{p,\mathcal{L}}_{s}(G)}:=\|(1+\mathcal{L})^{\frac{s}{2}}f\|_{L^{p}(G)}<\infty.$

In our further analysis we will make use of the following Sobolev inequality
(see e.g. Coulhon, Saloff-Coste and Varopoulos [14])

$\|f\|_{L^{p}(G)}\leq C\|(1+\mathcal{L})^{\frac{a}{2}}f\|_{L^{q}(G)},\quad
a=d_{0}\left(\frac{1}{q}-\frac{1}{p}\right),\,1<q<p<\infty,$ (3.2)

where $d_{0}$ is the local dimension of $(G,X).$ Note that (3.2) covers the
case where $G$ is of polynomial growth.

######  Remark 3.4 (Sobolev inequality on graded Lie groups).

Let $G$ be a graded Lie group and let $\mathcal{R}$ be a positive Rockland
operator of homogeneous degree $\nu>0.$ For $0<p<\infty,$ the $L^{p}$-Sobolev
space of order $s\in\mathbb{R},$ on $G$ is defined by the family of
distributions $f\in\mathscr{D}^{\prime}(G)$ such that

$\|f\|_{L^{p}_{s}(G)}:=\|(1+\mathcal{R})^{\frac{s}{\nu}}f\|_{L^{p}(G)}<\infty.$

In our further analysis we will make use of the following Sobolev inequality
on graded groups (see [23, Page 1674])

$\|f\|_{L^{p}(G)}\leq C\|(1+\mathcal{R})^{\frac{a}{\nu}}f\|_{L^{q}(G)},\quad
a=Q\left(\frac{1}{q}-\frac{1}{p}\right),\,1<q<p<\infty,$ (3.3)

where $Q$ is the homogeneous of the group.

######  Remark 3.5.

Note that the condition (1.15) and the Plancherel theorem imply the
$L^{2}$-boundedness of the operator $(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}T,$
since in terms of the group Fourier transform we have that

$(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}Tf(x)=\smallint_{\widehat{G}}\textnormal{Tr}[\pi(x)(1+\pi(\mathcal{R}))^{\frac{Q\theta}{2\nu}}\widehat{K}(\pi)\widehat{f}(\pi)]d\pi,\,f\in
C^{\infty}_{0}(G).$

Note that in the previous identity we have used that for any $s\in\mathbb{R},$

$\mathscr{F}[(1+\mathcal{R})^{\frac{s}{\nu}}\delta_{e}])(\pi)=(1+\pi({\mathcal{R}}))^{\frac{s}{\nu}},$

in view of the functional calculus for Rockland operators, see [21, Chapter
IV]. Here $\delta_{e}$ denotes the delta distribution at the identity $e\in
G.$ Observe that we have written $d\pi=d\nu(\pi)$ for the Plancherel measure
on $\widehat{G}.$ Similarly, the condition in (1.17) is equivalent to the
$L^{2}$-boundedness of the operator $T(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}.$

###### Proof of Theorem 1.1.

Let $e\in G$ be the identity element of $G$, and for any $z\in G,$ let
$B(z,\delta)$ be the ball of radius $\delta>0$ centred at $z.$ For simplicity
for $z=e$ we will denote

$B(\delta):=B(e,\delta),\,\quad|B(\delta)|:=\textnormal{Vol}(B(\delta)).$

We want to prove the boundedness inequality

$\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|Tf(x)-a|dx\leq
C\|f\|_{L^{\infty}(G)}.$ (3.4)

The equivalence in (3.1) provides a way to show that $Tf\in BMO$ without using
its average on $B(x_{0},\delta)$: it suffices to find a constant $a$ (that can
depend on $B(x_{0},\delta)$).

The first step of the proof is to reduce the estimate to the case where
$x_{0}=e,$ that is

$\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|Tf(x)-a|dx\leq
C\|f\|_{L^{\infty}(G)}.$ (3.5)

For this, let us use the left-invariance of $T.$ Indeed, observe that the
identities

$\displaystyle\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|Tf(x)-a|dx$
$\displaystyle=\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|Tf(x_{0}z)-a|dz$
$\displaystyle=\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|T(f(x_{0}\cdot))(z)-a|dz$

prove the equivalence between (3.4) and (3.5). So, in order to prove (3.5) let
us fix a function $f\in L^{\infty}(G),$ and let us consider the decomposition

$f=f_{1}+f_{2},\quad\,f_{1}=f\cdot 1_{\\{x\in G:|x|\leq
2\delta^{1-\theta}\\}},\quad f_{2}=f\cdot 1_{\\{x\in
G:|x|>2\delta^{1-\theta}\\}},$

in the case where $0<\delta<1.$ This is the relevant case. Indeed, if
$\delta>1,$ we take $c>0$ small enough so that $c\delta^{1-\theta}<\delta+1.$
Then, define $f_{1}(x)=f(x)1_{\\{x\in G:|x|\leq c\delta^{1-\theta}\\}}.$ Note
that $Tf_{2}=0,$ where $f_{2}=f-f_{1}.$ The $L^{2}$-boundedness of $T$ implies
that

$\frac{1}{|B(\delta)|}|Tf(x)|dx=\frac{1}{|B(\delta)|}|Tf_{1}(x)|dx\lesssim\delta^{-\frac{Q}{2}}\|Tf_{1}\|_{L^{2}}\lesssim\delta^{-\frac{Q}{2}}\delta^{Q/2}\|f\|_{L^{\infty}}.$

Then we have that

$\sup_{\delta>1}\inf_{a\in\mathbb{C}}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|Tf(x)-a|dx\leq
C\|f\|_{L^{\infty}(G)}.$ (3.6)

Note that a similar analysis can be applied to the adjoint $T^{*}$ of $T$ in
order to get the estimate

$\sup_{\delta>1}\inf_{a\in\mathbb{C}}\frac{1}{|B(x_{0},\delta)|}\smallint\limits_{B(x_{0},\delta)}|T^{*}f(x)-a|dx\leq
C\|f\|_{L^{\infty}(G)}.$ (3.7)

Indeed, by the duality argument the $L^{2}$-boundedness of $T$ and $T^{*}$ are
equivalent facts.

We illustrate the decomposition for the case $\delta\in(0,1)$ of the domain of
$f$ in the case of the $2D$-torus $G=\mathbb{T}^{2}$ in Figure 1.

Figure 1.

So, for any $0<\delta<1$ we are going to prove the estimate

$\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|Tf_{1}(x)|dx\lesssim\|f\|_{L^{\infty}(G)},$
(3.8)

if one of the conditions (1.11) or (1.15) hold true, and the estimate

$\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|T^{*}f_{1}(x)|dx\lesssim\|f\|_{L^{\infty}(G)},$
(3.9)

in the case where one of the conditions (1.13) or (1.17) remain valid, with
$T^{*}$ being the $L^{2}$-adjoint of $T.$ For this we will analyse separately
two cases: the first one, is the case where $G$ is a Lie group of polynomial
growth endowed with a Hörmander system of vector fields
$X=\\{X_{i}\\}_{i=1}^{k},$ while in the second one $G$ is a graded Lie group
of homogeneous dimension $Q=\nu_{1}+\cdots+\nu_{n}$ where a Rockland operator
$\mathcal{R}$ is fixed.

  * •

Case 1. Let us assume that $G$ is a Lie group of of polynomial growth and let

$\mathcal{L}=-\sum_{i=1}^{k}X_{i}^{2}$

be a positive sub-Laplacian associated with a Hörmander system of vector
fields $X=\\{X_{i}\\}_{i=1}^{k}$. Let $d=d_{0}$ be the local dimension of
$(G,X).$

    * –

Case 1a. Assume that $(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}T$ is bounded on
$L^{2}.$ Using the Sobolev inequality (see Remark 3.3) we have

$\displaystyle\|Tf_{1}\|_{L^{p}(G)}=\|(1+\mathcal{L})^{-\frac{d_{0}\theta}{4}}(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}Tf_{1}\|_{L^{p}(G)}\leq
C\|(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}Tf_{1}\|_{L^{2}(G)}$

with $\frac{1}{p}=\frac{1-\theta}{2}.$ Now, using that the operator
$(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}T$ is bounded on $L^{2}(G),$ and that
the group $G$ satisfies the global doubling condition, we have that

$\displaystyle\|Tf_{1}\|_{L^{p}(G)}$
$\displaystyle\lesssim\|f_{1}\|_{L^{2}(G)}\leq\|f\|_{L^{\infty}(G)}|B(e,2\delta^{1-\theta})|^{\frac{1}{2}}\lesssim\|f\|_{L^{\infty}(G)}|B(\delta^{1-\theta})|^{\frac{1}{2}}$
$\displaystyle\lesssim\|f\|_{L^{\infty}(G)}|B(\delta)|^{\frac{1-\theta}{2}}.$

In consequence

$\displaystyle\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|Tf_{1}(x)|dx$
$\displaystyle\lesssim\frac{1}{|B(\delta)|}\left(\smallint\limits_{B(\delta)}|Tf_{1}(x)|^{p}dx\right)^{\frac{1}{p}}|B(\delta)|^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq|B(\delta)|^{-1}\|Tf_{1}\|_{L^{p}(G)}|B(\delta)|^{\frac{(p-1)}{p}}.$

Observing that $\frac{p-1}{p}=1-\frac{1}{p}=\frac{1+\theta}{2},$ we have

$\displaystyle\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|Tf_{1}(x)|dx$
$\displaystyle\lesssim|B(\delta)|^{-1}|B(\delta)|^{\frac{(1-\theta)}{2}}|B(\delta)|^{\frac{(p-1)}{p}}\|f\|_{L^{\infty}(G)}$
$\displaystyle\lesssim|B(\delta)|^{-1+\frac{(1-\theta)}{2}+\frac{(1+\theta)}{2}}\|f\|_{L^{\infty}(G)}=\|f\|_{L^{\infty}(G)}.$

    * –

Case 1b. Now, assume that $T(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}$ is
bounded on $L^{2}(G).$ Observe also that, by following the argument in Case 1a
above we also can prove the following estimate for the adjoint operator
$T^{*}$ of $T,$

$\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|T^{*}f_{1}(x)|dx\lesssim\|f\|_{L^{\infty}(G)},$
(3.10)

indeed, such a estimate is independent of the kernel conditions. We start by
observing that

$\displaystyle\|T^{*}f_{1}\|_{L^{p}(G)}=\|(1+\mathcal{L})^{-\frac{d_{0}\theta}{4}}(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}T^{*}f_{1}\|_{L^{p}(G)}\leq
C\|(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}T^{*}f_{1}\|_{L^{2}(G)},$

in view of the Sobolev inequality. Note that we have the identity of operators

$(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}T^{*}=[T(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}]^{*}.$

Since $T(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}$ is bounded on $L^{2}(G),$ we
also have that its adjoint $(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}T^{*}$ is
bounded on $L^{2}(G).$ In consequence we have the estimate

$\displaystyle\|(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}T^{*}f_{1}\|_{L^{2}(G)}\leq
C\|f_{1}\|_{L^{2}(G)}.$

So, we can proceed as in Case 1a in order to get the estimate in (3.10).

  * •

Case 2. Let us assume that $G$ is a graded Lie group and let $\mathcal{R}$ be
a positive Rockland operator of homogeneous degree $\nu>0.$ Using the same
approach like in Case 1, we can estimate

$\displaystyle\|Tf_{1}\|_{L^{p}(G)}=\|(1+\mathcal{R})^{-\frac{Q\theta}{2\nu}}(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}Tf_{1}\|_{L^{p}(G)}\leq
C\|(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}Tf_{1}\|_{L^{2}(G)},\,$

where we have used the Sobolev inequality on graded groups (see Remark 3.4),
with $\frac{1}{p}=\frac{1-\theta}{2}.$ Now, if we use the $L^{2}$-boundedness
of the operator $(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}T,$ we can proceed as
in Case 1 in order to get (3.8) by observing that for all $\delta>0,$
$|B(\delta)|\sim\delta^{Q}$. Also, if on the other hand, we have the
$L^{2}$-boundedness of the operator $T(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}$
we can conclude the estimate in (3.9), by observing again that for all
$\delta>0,$ $|B(\delta)|\sim\delta^{Q},$ and the estimate

$\displaystyle\|T^{*}f_{1}\|_{L^{p}(G)}$
$\displaystyle=\|(1+\mathcal{R})^{-\frac{Q\theta}{2\nu}}(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}T^{*}f_{1}\|_{L^{p}(G)}\leq
C\|[T(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}]^{*}f_{1}\|_{L^{2}(G)}$
$\displaystyle\leq
C\|[T(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}]^{*}\|_{\mathcal{B}(L^{2})}\|f_{1}\|_{L^{2}(G)}$
$\displaystyle=C\|T(1+\mathcal{R})^{\frac{Q\theta}{2\nu}}\|_{\mathcal{B}(L^{2})}\|f_{1}\|_{L^{2}(G)}\lesssim\|f_{1}\|_{L^{2}(G)}.$

Now, let us apply the $BMO$-property in Remark 3.1. Again, we will consider
two cases.

  * •

Case A. Let us assume that the kernel of $T$ satisfies the estimate

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\frac{1}{|B(R)|}\smallint\limits_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$

and one of the conditions (1.11) or (1.15) hold true. Define

$a_{\delta}:=\smallint\limits_{G}K(y^{-1})f_{2}(y)dy.$

Then,

$\displaystyle|Tf_{2}(x)-a_{\delta}|\leq\smallint\limits_{|y|\geq
2\delta^{1-\theta}}|K(y^{-1}x)-K(y^{-1})||f_{2}(y)|dy=\smallint\limits_{|y^{-1}|\geq
2\delta^{1-\theta}}|K(yx)-K(y)||f(y^{-1})|dy.$

Now, using Remark 3.2 we have the property $|x|=|x^{-1}|$ for any $x\in G$ and
consequently

$\displaystyle\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|Tf_{2}(x)-a_{\delta}|dx$
$\displaystyle\lesssim\frac{1}{|B(\delta)|}\smallint\limits_{|x|\leq\delta}\smallint\limits_{|y^{-1}|\geq
2\delta^{1-\theta}}|K(yx)-K(y)||f(y^{-1})|dy\,dx$
$\displaystyle\leq\frac{1}{|B(\delta)|}\smallint\limits_{|x^{-1}|\leq\delta}\smallint\limits_{|y^{-1}|\geq
2\delta^{1-\theta}}|K(yx^{-1})-K(y)|dydx\|f\|_{L^{\infty}(G)}$
$\displaystyle=\frac{1}{|B(\delta)|}\smallint\limits_{|x|\leq\delta}\smallint\limits_{|y|\geq
2\delta^{1-\theta}}|K(yx^{-1})-K(y)|dydx\|f\|_{L^{\infty}(G)}$
$\displaystyle=\frac{1}{|B(\delta)|}\smallint\limits_{|y|\leq\delta}\smallint\limits_{|x|\geq
2\delta^{1-\theta}}|K(xy^{-1})-K(x)|dydx\|f\|_{L^{\infty}(G)}\lesssim[K]_{\infty,\theta}\|f\|_{L^{\infty}(G)}.$

So, we can deduce the estimates

$\displaystyle\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|Tf(x)-a_{\delta}|dx$
$\displaystyle\leq\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|Tf_{1}(x)|dx$
$\displaystyle+\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|Tf_{2}(x)-a_{\delta}|dx\lesssim\|f\|_{L^{\infty}},$

for any $\delta\in(0,1).$ Then, by combining the previous analysis with (3.6)
we get the inequality

$\displaystyle\|Tf\|_{BMO}\sim\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|Tf(x)-a|dx\lesssim\|f\|_{L^{\infty}(G)}.$

  * •

Case B. Now, let us consider the case where $T$ satisfies the kernel condition

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\frac{1}{|B(R)|}\smallint\limits_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$

and when one of the conditions (1.13) or (1.17) remain valid. We have the
estimate in (3.9), that is

$\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|T^{*}f_{1}(x)|dx\lesssim\|f\|_{L^{\infty}(G)}.$

Also, by replacing the previous analysis with $T^{*}$ instead of $T,$ we have
the inequality

$\displaystyle\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|T^{*}f_{2}(x)-a_{\delta}|dx$
$\displaystyle\lesssim\frac{1}{|B(\delta)|}\smallint\limits_{|y|\leq\delta}\smallint\limits_{|x|\geq
2\delta^{1-\theta}}|K^{*}(xy^{-1})-K^{*}(x)|dydx\|f\|_{L^{\infty}(G)},$

where $K^{*}$ is the right-convolution kernel associated to $T^{*}.$ Using the
identity of distributions $K^{*}(x)=\overline{K}(x^{-1}),$ $x\in G,$ we get

$\displaystyle\frac{1}{|B(\delta)|}\smallint\limits_{|y|\leq\delta}\smallint\limits_{|x|\geq
2\delta^{1-\theta}}|K^{*}(xy^{-1})-K^{*}(x)|dydx$
$\displaystyle=\frac{1}{|B(\delta)|}\smallint\limits_{|y|\leq\delta}\smallint\limits_{|x|\geq
2\delta^{1-\theta}}|K(yx^{-1})-K(x^{-1})|dydx$
$\displaystyle=\frac{1}{|B(\delta)|}\smallint\limits_{|y^{-1}|\leq\delta}\smallint\limits_{|x^{-1}|\geq
2\delta^{1-\theta}}|K(y^{-1}x)-K(x)|dydx$
$\displaystyle=\frac{1}{|B(\delta)|}\smallint\limits_{|y|\leq\delta}\smallint\limits_{|x|\geq
2\delta^{1-\theta}}|K(y^{-1}x)-K(x)|dydx$
$\displaystyle\lesssim[K]_{H_{\infty,\theta}}^{\prime}.$

So, we can deduce the estimate

$\displaystyle\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|T^{*}f(x)-a_{\delta}|dx$
$\displaystyle\leq\frac{1}{|B(\delta)|}\smallint\limits_{B(\delta)}|T^{*}f_{1}(x)|dx$
$\displaystyle+\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|T^{*}f_{2}(x)-a_{\delta}|dx\lesssim\|f\|_{L^{\infty}}.$

Using the analysis above for $0<\delta<1$ and the inequality in (3.7) we get

$\displaystyle\|T^{*}f\|_{BMO}\sim\sup_{\delta>0}\inf_{a\in\mathbb{C}}\frac{1}{|B(\delta)|}\smallint_{B(\delta)}|T^{*}f(x)-a|dx\lesssim\|f\|_{L^{\infty}(G)}.$

So, we have proved the boundedness of $T^{*}$ from $L^{\infty}(G)$ into
$BMO(G),$ and hence, by the duality argument, we deduce that $T$ is bounded
from $H^{1}(G)$ into $L^{1}(G).$

The proof of Theorem 1.1 is complete. ∎

##  4\. Examples

We end this paper by illustrating our main Theorem 1.1 with some (possibly
well known) examples (with the exception of Subsection 4.6). Applications of
Theorem 1.1 in the context of graded Lie groups can be found in the second
part of this work, namely, in [8]. We start with the case of a compact Lie
group endowed with its standard Riemannian structure.

###  4.1. Riemannian structures on compact Lie groups

Let $G$ be a compact Lie group. Assume $G$ to be endowed with its unique (up
to a constant) bi-invariant Riemannian metric. Let $\mathcal{L}_{G}$ be the
positive Laplace-Beltrami operator on $G.$ For an orthonormal basis
$\\{X_{i}\\}_{i=1}^{n},$ $n=\dim(G),$ with respect to the aforementioned
Riemannian metric, we have

$\mathcal{L}_{G}=-(X_{1}^{2}+\cdots+X_{n}^{2}).$

If we replace the sub-Laplacian $\mathcal{L}$ by the Laplace-Beltrami operator
$\mathcal{L}_{G},$ the corresponding Carnot-Carathéodory distance associated
with $\mathcal{L}$ can be replaced by the standard geodesic distance on $G,$
and in this context $d_{0}$ in Theorem 1.1 is the topological dimension $n.$

The $L^{2}$-spectrum of the Laplace-Beltrami operator
$\textnormal{Spect}(\mathcal{L}_{G})=\\{\lambda_{[\xi]}\\}_{[\xi]\in\widehat{G}}$
can be enumerated by using the unitary dual of $G$, and the Fourier transform
of its right-convolution kernel takes the form

$\widehat{\mathcal{L}}_{G}(\xi)=\lambda_{[\xi]}I_{H_{\xi}},\,[\xi]\in\widehat{G},$
(4.1)

where $I_{H_{\xi}}$ is the identity operator on any representation space
$H_{\xi}.$ The condition in (1.11) is equivalent to the following estimate

$\sup_{[\xi]\in\widehat{G}}\|\langle\xi\rangle^{\frac{n\theta}{2}}\widehat{K}(\xi)\|_{\textnormal{op}}<\infty,\,\,\
\langle\xi\rangle:=(1+\lambda_{[\xi]})^{\frac{1}{2}}.$ (4.2)

Observe that (4.2) is equivalent to the existence of a finite constant $C>0,$
such that

$\forall[\xi]\in\widehat{G},\,\,\|\widehat{K}(\xi)\|_{\textnormal{op}}\leq
C\langle\xi\rangle^{-\frac{n\theta}{2}},\,\
\langle\xi\rangle:=(1+\lambda_{[\xi]})^{\frac{1}{2}}.$ (4.3)

In view of Theorem 1.1, a convolution operator $T$ (defined via $Tf=f\ast K,$
$f\in C^{\infty}(G)$) associated to a right-convolution kernel $K,$ satisfying
(4.3) and the kernel condition

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$ (4.4)

admits a bounded extension from $L^{\infty}(G)$ into $BMO(G).$ Moreover, if
$K$ satisfies (4.3) and the kernel condition

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$ (4.5)

then $T$ admits a bounded extension from $H^{1}(G)$ into $L^{1}(G).$

Naturally, the family of compact Lie groups includes torus of arbitrary
dimension. In view of the simplicity of the representation theory on the torus
we separately analyse it in the following subsection.

###  4.2. Oscillating integrals on the torus $\mathbb{T}^{n}$

Let $G=\mathbb{T}^{n}\equiv[0,1)^{n},$ $0\sim 1,$ be the $n$-dimensional torus
$\mathbb{R}^{n}/\mathbb{Z}^{n}.$ In this case we have the identification
$\widehat{\mathbb{T}}^{n}=\\{e_{\ell}\\}_{\ell\in\mathbb{Z}^{n}}\sim\mathbb{Z}^{n}$
for the unitary dual of the torus. We have denoted $e_{\ell}$ to the
exponential function $e_{\ell}(x)=e^{i2\pi\ell\cdot x},$
$x=(x_{1},\cdots,x_{n})\in\mathbb{T}^{n}.$

The spectrum of the Laplace-Beltrami operator
$\mathcal{L}_{\mathbb{T}^{n}}=-\sum_{i=1}^{n}\partial_{x_{i}}^{2}$ is given by

$\textnormal{Spect}(\mathcal{L}_{\mathbb{T}^{n}})=\\{4\pi^{2}\ell^{2},\,\ell\in\mathbb{Z}^{n}\\}.$
(4.6)

So, all the representations of the torus are one-dimensional and (4.1) takes
the form

$\widehat{\mathcal{L}}_{\mathbb{T}^{n}}(\ell)=4\pi^{2}\ell^{2},\,\,\ell\in\mathbb{Z}^{n}.$
(4.7)

In terms of the Fourier transform of a distribution $K$ on the torus

$\widehat{K}(\ell):=\smallint_{\mathbb{T}^{n}}e_{-\ell}(x)K(x)dx,\,\,\ell\in\mathbb{Z}^{n},$
(4.8)

the Fourier transform condition (4.3) becomes equivalent to the estimate

$\forall\ell\in\mathbb{Z}^{n},\,\,|\widehat{K}(\ell)|\leq
C\langle\ell\rangle^{-\frac{n\theta}{2}},\,\
\langle\ell\rangle:=(1+4\pi^{2}|\ell|^{2})^{\frac{1}{2}}\sim|\ell|:=\sqrt{\ell_{1}^{2}+\cdots+\ell_{n}^{2}}.$
(4.9)

The commutativity of the torus makes (4.4) and (4.5) equivalent to the
following kernel condition

$[K]_{H_{\infty,\theta}}=[K]_{H_{\infty,\theta}}^{\prime}=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(x-y)-K(x)|dxdy<\infty.$ (4.10)

In view of Theorem 1.1, a convolution operator $T$ associated to a convolution
kernel $K,$ satisfying the Fourier transform condition (4.9) and the
smoothness condition (4.10) admits a bounded extension from
$L^{\infty}(\mathbb{T}^{n})$ into $BMO(\mathbb{T}^{n}),$ and from
$H^{1}(\mathbb{T}^{n})$ into $L^{1}(\mathbb{T}^{n}).$

###  4.3. Oscillating integrals on $\textnormal{SU}(2)\cong\mathbb{S}^{3}$

Let us consider the compact Lie group of complex unitary $2\times 2$-matrices

$\textnormal{SU}(2)=\\{X=[X_{ij}]_{i,j=1}^{2}\in\mathbb{C}^{2\times
2}:X^{*}=X^{-1}\\},\,X^{*}:=\overline{X}^{t}=[\overline{X_{ji}}]_{i,j=1}^{2}.$

Let us consider the left-invariant first-order differential operators

$\partial_{+},\partial_{-},\partial_{0}:C^{\infty}(\textnormal{SU}(2))\rightarrow
C^{\infty}(\textnormal{SU}(2)),$

called creation, annihilation, and neutral operators respectively, (see
Definition 11.5.10 of [38]) and let us define

$X_{1}=-\frac{i}{2}(\partial_{-}+\partial_{+}),\,X_{2}=\frac{1}{2}(\partial_{-}-\partial_{+}),\,X_{3}=-i\partial_{0},$

where $X_{3}=[X_{1},X_{2}],$ based on the commutation relations
$[\partial_{0},\partial_{+}]=\partial_{+},$
$[\partial_{-},\partial_{0}]=\partial_{-},$ and
$[\partial_{+},\partial_{-}]=2\partial_{0},$ the system $X=\\{X_{1},X_{2}\\}$
satisfies the Hörmander condition at step $\kappa=2,$ and the Hausdorff
dimension defined by the control distance associated to the sub-Laplacian
$\mathcal{L}_{sub}=-X_{1}^{2}-X_{2}^{2}$ is $Q=4.$ By the compactness of
$\textnormal{SU}(2),$ we have that $d_{0}=Q=4$ and $d_{\infty}=0.$

We record that the unitary dual of $\textnormal{SU}(2)$ (see [38]) can be
identified as

$\widehat{\textnormal{SU}}(2)\equiv\\{[t_{l}]:2l\in\mathbb{N},d_{l}:=\dim
t_{l}=(2l+1)\\}\sim\frac{1}{2}\mathbb{N}.$ (4.11)

There are explicit formulae for $t_{l}$ as functions of Euler angles in terms
of the so-called Legendre-Jacobi polynomials, see [38].

It was proved e.g. in [39], that the spectrum of the sub-Laplacian
$\mathcal{L}_{sub}$ can be indexed by the sequence

$\ell(\ell+1)-m^{2},\,\,-\ell\leq m\leq\ell,\,m,\ell\in\frac{1}{2}\mathbb{N},$

and then that $\widehat{\mathcal{L}}_{sub}(\ell)$ is given by the
$(2\ell+1)\times(2\ell+1)$-diagonal matrix

$\widehat{\mathcal{L}}_{sub}(\ell)=\textnormal{diag}[(\ell(\ell+1)-m^{2})]_{-\ell\leq
m\leq\ell,m\in\frac{1}{2}\mathbb{N}\,},\,\,\ell\in\frac{1}{2}\mathbb{N}.$
(4.12)

Since $d_{0}=Q=4,$ we have that $\frac{d_{0}\theta}{4}=\theta,$ and the
condition in (1.11) takes the form

$\sup_{\ell\in\frac{1}{2}\mathbb{N}}\|\textnormal{diag}[(1+\ell(\ell+1)-m^{2})^{{{{\theta}}}}]_{-\ell\leq
m\leq\ell,m\in\frac{1}{2}\mathbb{N}\,}\times\widehat{K}(\ell)\|_{\textnormal{op}}<\infty,$
(4.13)

where
$\widehat{K}(\ell)=\smallint_{\textnormal{SU}(2)}K(Z)t_{\ell}(Z)^{-1}dZ,$ is
the Fourier transform of the group of $K.$ In view of Theorem 1.1, if $K$
satisfies (4.13) and the kernel condition

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\sup_{|Y|<R}\smallint\limits_{|X|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$ (4.14)

then $T:L^{\infty}(\textnormal{SU}(2))\rightarrow BMO(\textnormal{SU}(2))$
admits a bounded extension. On the other hand, if $T$ satisfies the estimate

$\sup_{\ell\in\frac{1}{2}\mathbb{N}}\|\widehat{K}(\ell)\times\textnormal{diag}[(1+\ell(\ell+1)-m^{2})^{{{{\theta}}}}]_{-\ell\leq
m\leq\ell,m\in\frac{1}{2}\mathbb{N}\,}\|_{\textnormal{op}}<\infty,$ (4.15)

and the kernel condition

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\sup_{|Y|<R}\smallint\limits_{|X|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$ (4.16)

then $T:H^{1}(\textnormal{SU}(2))\rightarrow L^{1}(\textnormal{SU}(2))$
extends to a bounded operator. We observe that in the kernel conditions (4.14)
and (4.16) the sets $\\{Y:|Y|<R\\}$ and $\\{X:|X|\geq 2R^{1-\theta}\\},$ and
also the space $(H^{1}(\textnormal{SU}(2)))^{\prime}=BMO(\textnormal{SU}(2))$
are defined in terms of the Carnot-Carathéodory distance associated to the
system of vector fields $\\{X_{1},X_{2}\\}.$

######  Remark 4.1.

On the other hand, the Carnot-Carathéodory distance associated to the vector
fields $\\{X_{1},X_{2},X_{3}\\}$ agrees with the geodesic distance, and
(minus) the sums of squares

$\mathcal{L}_{\textnormal{SU}(2)}=-X_{1}^{2}-X_{2}^{2}-X_{3}^{2}=-\partial_{0}^{2}-\frac{1}{2}[\partial_{+}\partial_{-}+\partial_{-}\partial_{+}],$
(4.17)

is the positive Laplacian on $\textnormal{SU}(2).$ Again, by following e.g.
[38], the spectrum of the positive Laplacian
$\mathcal{L}_{\textnormal{SU}(2)}$ can be indexed by the sequence

$\lambda_{\ell}:=\ell(\ell+1),\quad\ell\in\frac{1}{2}\mathbb{N}.$

Because $n=\dim(\textnormal{SU}(2))=3,$ the Fourier transform condition (4.3)
takes the form

$\exists
C>0,\,\forall\ell\in\frac{1}{2}\mathbb{N},\quad\|\widehat{K}(\ell)\|_{\textnormal{op}}\leq
C(1+\ell(\ell+1))^{-\frac{3\theta}{4}}\sim(1+\ell)^{-\frac{3\theta}{2}}.$
(4.18)

In view of Theorem 1.1, if $K$ satisfies (4.18) and the kernel condition

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\sup_{|Y|<R}\smallint\limits_{|X|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$ (4.19)

then $T:L^{\infty}(\textnormal{SU}(2))\rightarrow BMO(\textnormal{SU}(2))$
admits a bounded extension. On the other hand, if $T$ satisfies (4.18) and the
kernel condition

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\sup_{|Y|<R}\smallint\limits_{|X|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$ (4.20)

then $T:H^{1}(\textnormal{SU}(2))\rightarrow L^{1}(\textnormal{SU}(2))$
extends to a bounded operator. Note that the conditions (4.19) and (4.20), the
sets $\\{Y:|Y|<R\\}$ and $\\{X:|X|\geq 2R^{1-\theta}\\},$ and also the space
$(H^{1}(\textnormal{SU}(2)))^{\prime}=BMO(\textnormal{SU}(2))$ are defined in
terms of geodesic distance.

Now, we are going to present some applications of Theorem 1.1 to (the non-
compact case of) graded Lie groups. We start with the Euclidean case.

###  4.4. Oscillating integrals on $\mathbb{R}^{n}$

The unitary dual of $\mathbb{R}^{n}$ admits the identification
$\widehat{\mathbb{R}}^{n}=\\{e_{\xi}:\xi\in\mathbb{R}^{n}\\}\sim\mathbb{R}^{n},$
where $e_{\xi}(x):=e^{i2\pi x\cdot\xi}.$ The Fourier transform of a
distribution $K$ is given by

$\widehat{K}(\xi)=\smallint_{\mathbb{R}^{n}}e_{-\xi}(x)K(x)dx,$

for any $\xi\in\mathbb{R}^{n}.$ The model Rockland operator $\mathcal{R}$ on
$\mathbb{R}^{n}$ is the positive Laplacian
$\mathcal{R}=-\Delta_{x}:=-\partial_{x_{1}}^{2}-\cdots-\partial_{x_{n}}^{2},$
which is homogeneous of degree $\nu=2.$ In this case (1.10) becomes to be

$e_{\xi}(-\Delta_{x})=4\pi^{2}|\xi|^{2},\,\quad\xi\in\mathbb{R}^{n}.$

The Fourier transform condition in (1.15) takes the form

$\sup_{\xi\in\mathbb{R}^{n}}|e_{\xi}\left((1-\Delta_{x})^{\frac{n\theta}{4}}\right)\widehat{K}(\xi)|=\sup_{\xi\in\mathbb{R}^{n}}|(1+4\pi^{2}|\xi|^{2})^{\frac{n\theta}{4}}\widehat{K}(\xi)|<\infty.$
(4.21)

Note that the condition in (4.21) is equivalent to the existence of $C>0$
satisfying that

$\forall\xi\in\mathbb{R}^{n},\,|\widehat{K}(\xi)|\leq
C|(1+4\pi^{2}|\xi|^{2})^{-\frac{n\theta}{4}}\sim
C(1+|\xi|)^{-\frac{n\theta}{2}}.$ (4.22)

The commutativity of $\mathbb{R}^{n}$ makes (1.16) and (1.18) equivalent to
the following kernel condition

$[K]_{H_{\infty,\theta}}=[K]_{H_{\infty,\theta}}^{\prime}=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(x-y)-K(x)|dx<\infty.$ (4.23)

In view of Theorem 1.1, a convolution operator $T$ associated to a convolution
kernel $K,$ satisfying the Fourier transform condition (4.22) and the
smoothness condition (4.23) admits a bounded extension from
$L^{\infty}(\mathbb{R}^{n})$ into $BMO(\mathbb{R}^{n}),$ and from
$H^{1}(\mathbb{R}^{n})$ into $L^{1}(\mathbb{R}^{n}).$

###  4.5. Oscillating integrals on the Heisenberg group $\mathbb{H}_{n}$

Let us consider the Heisenberg group $\mathbb{H}_{n}$ as the manifold
$\mathbb{R}^{2n+1}$ endowed of the product

$(x,y,t)\cdot(x^{\prime},y^{\prime},z^{\prime})=(x+x^{\prime},y+y^{\prime},t+t^{\prime}+\frac{1}{2}(xy^{\prime}-x^{\prime}y)),\,(x,y,z),(x^{\prime},y^{\prime},z^{\prime})\in\mathbb{H}_{n}.$
(4.24)

The Lie algebra $\mathfrak{h}_{n}=\textnormal{Lie}(\mathbb{H}_{n})$ is spanned
by the vector-fields

$X_{j}=\partial_{x_{j}}-\frac{y_{j}}{2}\partial_{t},\,\quad
Y_{j}=\partial_{x_{j}}+\frac{x_{j}}{2}\partial_{t},\,1\leq j\leq n,\,\quad
T=\partial_{t}.$ (4.25)

The canonical relations are $[X_{j},Y_{j}]=T.$ The positive sub-Laplacian
$\mathcal{L}_{sub}$ on the Heisenberg group

$\mathcal{L}_{sub}:=-\sum_{j=1}^{n}X_{j}^{2}+Y_{j}^{2}=-\sum_{j=1}^{n}\left(\partial_{x_{j}}-\frac{y_{j}}{2}\partial_{t}\right)^{2}+\left(\partial_{x_{j}}+\frac{x_{j}}{2}\partial_{t}\right)^{2},$

is a Rockland operator of homogeneous degree $\nu=2$ and the homogeneous
dimension of $\mathbb{H}_{n}$ is $Q=2n+2.$ We observe that if $\mathbb{H}_{n}$
is endowed with the Carnot-Carathéodory distance $|\cdot|$ then the local
dimension $d_{0}$ agrees with the global dimension $d_{\infty}$ and
$d_{0}=d_{\infty}=Q=2n+2.$ Moreover, on any nilpotent stratified group ones
always has that $d_{0}=d_{\infty}.$

The unitary dual $\widehat{\mathbb{H}}_{n}$ of $\mathbb{H}_{n}$ admits the
identification

$\widehat{\mathbb{H}}_{n}=\\{\pi_{\lambda}:\lambda\neq
0,\,\lambda\in\mathbb{R}\\}\sim\mathbb{R}^{*}:=\mathbb{R}\setminus\\{0\\}.$
(4.26)

In (4.26), the family $\pi_{\lambda},$ $\lambda\in\mathbb{R}^{*},$ are the
Schrödinger representations, which are the unitary operators on
$L^{2}(\mathbb{R}^{n})$ defined via

$\pi_{\lambda}(x,y,z)h(u)=e^{i\lambda(t+\frac{1}{2}x\cdot
y)}e^{i\sqrt{\lambda}yu}h(u+\sqrt{|\lambda|}x),\,\sqrt{\lambda}:=\frac{\lambda}{|\lambda|}\sqrt{|\lambda|},\,\lambda\in\mathbb{R}^{*}.$

In the Heisenberg group setting, for $\mathcal{R}=\mathcal{L},$ (1.10) takes
the form of the dilated harmonic oscillator

$\pi_{\lambda}(\mathcal{L}_{sub})=|\lambda|(-\Delta_{u}+|u|^{2}),\,\Delta_{u}:=\sum_{j=1}^{n}\partial_{u_{j}}^{2},\,u=(u_{1},\cdots,u_{n})\in\mathbb{R}^{n}.$

For a distribution $K$ on $G=\mathbb{H}_{n},$ the condition in (1.15) for the
Fourier transform of $K,$

$\widehat{K}(\lambda)=\smallint_{\mathbb{H}_{n}}K(x,y,t)\pi_{\lambda}(x,y,t)^{-1}d(x,y,t):L^{2}(\mathbb{R}^{n})\rightarrow
L^{2}(\mathbb{R}^{n}),$

becomes equivalent to the following Fourier transform condition

$\sup_{\lambda\in\mathbb{R}^{*}}\|\pi_{\lambda}(1+\mathcal{L})^{\frac{Q\theta}{2\nu}}\widehat{K}(\lambda)\|_{\textnormal{op}}=\sup_{\lambda\in\mathbb{R}^{*}}\|\left(1+|\lambda|(-\Delta_{u}+|u|^{2})\right)^{\frac{(n+1)\theta}{2}}\widehat{K}(\lambda)\|_{\textnormal{op}}<\infty.$
(4.27)

In view of Theorem 1.1, a convolution operator $T$ (defined via $Tf=f\ast K,$
$f\in C^{\infty}(\mathbb{H}_{n})$) associated to a right-convolution kernel
$K,$ satisfying (4.27) and the kernel condition

$\sup_{0<R<1}\sup_{\\{Y\in\mathbb{H}_{n}:|Y|<R\\}}\smallint\limits_{\\{X\in\mathbb{H}_{n}:|X|\geq
2R^{1-\theta}\\}}|K(xy^{-1})-K(x)|dx<\infty,$ (4.28)

admits a bounded extension from $L^{\infty}(\mathbb{H}_{n})$ into
$BMO(\mathbb{H}_{n}).$ Moreover, if $K$ satisfies that

$\sup_{\lambda\in\mathbb{R}^{*}}\|\widehat{K}(\lambda)\pi_{\lambda}(1+\mathcal{L})^{\frac{Q\theta}{2\nu}}\|_{\textnormal{op}}=\sup_{\lambda\in\mathbb{R}^{*}}\|\widehat{K}(\lambda)\left(1+|\lambda|(-\Delta_{u}+|u|^{2})\right)^{\frac{(n+1)\theta}{2}}\|_{\textnormal{op}}<\infty,$
(4.29)

and the kernel condition

$\sup_{0<R<1}\sup_{\\{Y\in\mathbb{H}_{n}:|Y|<R\\}}\smallint\limits_{\\{X\in\mathbb{H}_{n}:|X|\geq
2R^{1-\theta}\\}}|K(y^{-1}x)-K(x)|dx<\infty,$ (4.30)

then $T$ admits a bounded extension from the Hardy space
$H^{1}(\mathbb{H}_{n})$ into $L^{1}(\mathbb{H}_{n}).$

###  4.6. Oscillating operators on Lie groups of polynomial growth and of type
I

Note that Theorem 1.1 has been proved for any Lie group of polynomial growth.
However, if we add the hypothesis that $G$ is also of type I, an explicit
condition can be given in terms of the Fourier analysis associated to the sub-
Laplacian in order to express the $L^{2}$-boundedness hypothesis in (1.11).
However, to do this we require some aspects of the functional calculus for the
sub-Laplacian $\mathcal{L}.$ We record it in the following remark.

######  Remark 4.2 (Functional calculus for the sub-Laplacian revisited).

Let $\mathcal{L}=-\sum_{i=1}^{k}X_{i}^{2}$ be a positive sub-Laplacian
associated to a Hörmander system of vector-fields $\\{X_{i}\\}_{i=1}^{k}.$
Then $\mathcal{L}$ is formally self-adjoint as an element of the universal
enveloping algebra $\mathfrak{u}(\mathfrak{g}).$ Moreover, for a.e. $\xi,$
$\widehat{\mathcal{L}}(\xi)=\sum_{i=1}^{k}d\xi(X_{k})^{2}$ is essentially
self-adjoint on $H_{\xi}$ and we keep the same notation for its self-adjoint
extension. Let $E$ and $E_{\xi}$ be the spectral measures of $\mathcal{L}$ and
$\widehat{\mathcal{L}}(\xi),$ respectively.

Let $B$ be a Borel subset $B\subset\mathbb{R}$ and let
$E(B):L^{2}(G)\rightarrow L^{2}(G)$ and $E_{\xi}(B):H_{\xi}\rightarrow
H_{\xi}$ be the corresponding projections. In particular $E(B)$ is a left-
invariant operator. The group Fourier transform of its convolution kernel
$E(B)\delta$ is

$\mathscr{F}[E(B)\delta](\xi)=E_{\xi}(B),\,[\xi]\in\widehat{G}.$ (4.31)

One can prove (4.31) by following the argument in [21, Page 182]. Indeed, if
$F_{\xi}:B\mapsto\mathscr{F}[E(B)\delta](\xi),$ then it is not difficult to
prove that $F_{\xi}$ is a spectral resolution on $H_{\xi}.$ Note that if
$\phi$ is a measurable function on $G,$ then the spectral multiplier
$\phi(\mathcal{L})$ is defined via

$\phi(\mathcal{L}):=\smallint\limits_{0}^{\infty}\phi(\lambda)dE(\lambda).$

Note that $\textnormal{Dom}(\phi(\mathcal{L}))=\\{f\in
L^{2}(G):\smallint\limits_{0}^{\infty}|\phi(\lambda)|^{2}d\|E(\lambda)f\|_{L^{2}(G)}^{2}<\infty\\}.$
On the other hand, if $\phi(\mathcal{L})\delta$ is the right-convolution
kernel of $\phi(\mathcal{L}),$ then for any $f\in C^{\infty}_{0}(G),$

$\mathscr{F}[\phi(\mathcal{L})f](\xi)=\mathscr{F}[\phi(\mathcal{L})\delta](\xi)\widehat{f}(\xi)=\mathscr{F}[\smallint\limits_{0}^{\infty}\phi(\lambda)dE(\lambda)f](\xi)=\smallint\limits_{0}^{\infty}\phi(\lambda)dF_{\xi}(\lambda)\widehat{f}(\xi),$
(4.32)

with $\phi=1_{B}$ the characteristic function of $B.$ Note that (4.32) also
holds for a finite linear combination of characteristic functions, and then,
passing through the limit carefully, for any $\phi\in L^{\infty}(G),$ with
$f\in L^{2}(G)$ or $\phi(\lambda)=\lambda$ for $f\in C^{\infty}_{0}(G).$

Consequently,

$\displaystyle\mathscr{F}[\mathcal{L}f](\xi)=\smallint\limits_{0}^{\infty}\lambda
dF_{\xi}(\lambda)\widehat{f}(\xi)=\mathscr{F}[\smallint\limits_{0}^{\infty}\lambda
dE(\lambda)f](\xi)=\widehat{\mathcal{L}}(\xi)\widehat{f}(\xi),\,f\in
C^{\infty}_{0}(G),$

where $\widehat{\mathcal{L}}(\xi):=\mathscr{F}[\mathcal{L}\delta](\xi),$
$[\xi]\in\widehat{G}.$ In view of the Dixmier-Malliavin theorem (see Theorem
1.7.8 of [21]), the space of smooth vectors $H_{\xi}^{\infty}$ is spanned by
elements of the form $\widehat{f}(\xi)v,$ $v\in H_{\xi}^{\infty},$ (in the
sense that any vector in $H_{\xi}^{\infty}$ is a finite linear combination of
vectors of the form $\widehat{f}(\xi)v$), and we have that

$\widehat{\mathcal{L}}(\xi)v=\smallint\limits_{0}^{\infty}\lambda
dF_{\xi}(\lambda)v,\,v\in H_{\xi}^{\infty}.$

In view of the density of $H_{\xi}^{\infty}$ in $H_{\xi}$ (see Theorem 1.7.7
of [21]) and the uniqueness of the spectral measure, we have that
$F_{\xi}=E_{\xi}$ proving (4.31).

Note that one of the fundamental consequences of (4.31), is that for a
measurable function $\phi$ on $\mathbb{R},$ one has the equality of operators

$\mathscr{F}[\phi(\mathcal{L})\delta](\xi)=\mathscr{F}(\smallint\limits_{0}^{\infty}\phi(\lambda)dE(\lambda)\delta)(\xi)=\smallint\limits_{0}^{\infty}\phi(\lambda)dE_{\xi}(\lambda)=\phi(\widehat{\mathcal{L}}(\xi)),$
(4.33)

on the class of smooth vectors $v\in H_{\xi}^{\infty},$ such that
$\smallint\limits_{0}^{\infty}|\phi(\lambda)|^{2}d\|E_{\xi}(\lambda)v\|_{H_{\xi}}^{2}<\infty.$
In particular, for any $s\in\mathbb{R},$ in the same class of smooth vectors
one has the equality
$\mathscr{F}[(1+\mathcal{L})^{\frac{s}{2}}\delta]=(1+\widehat{\mathcal{L}}(\xi))^{\frac{s}{2}}.$

######  Remark 4.3.

Now we discuss the hypothesis (1.11) when the group $G$ is of polynomial
growth and of type I. In view of (4.33), via the Fourier transform we have

$(1+\mathcal{L})^{\frac{d_{0}\theta}{4}}Tf(x)=\smallint_{\widehat{G}}\textnormal{Tr}[\xi(x)(1+\widehat{\mathcal{L}}(\xi))^{\frac{d_{0}\theta}{4}}\widehat{K}(\xi)\widehat{f}(\xi)]d\nu(\xi),\,f\in
C^{\infty}_{0}(G).$

So, in view of the Plancherel formula, the estimate

$\sup_{[\xi]\in\widehat{G}}\|(1+\widehat{\mathcal{L}}(\xi))^{\frac{d_{0}\theta}{4}}\widehat{K}(\xi)\|_{\textnormal{op}}<\infty,$
(4.34)

is equivalent to the condition (1.11). In view of Theorem 1.1, the Fourier
transform condition (4.34) on $K$ together with oscillating condition

$[K]_{H_{\infty,\theta}}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(xy^{-1})-K(x)|dx<\infty,$ (4.35)

imply the boundedness of $T$ from $L^{\infty}(G)$ into $BMO(G).$ Moreover, if
$K$ satisfies the following estimate

$\sup_{[\xi]\in\widehat{G}}\|\widehat{K}(\xi)(1+\widehat{\mathcal{L}}(\xi))^{\frac{d_{0}\theta}{4}}\|_{\textnormal{op}}<\infty,$
(4.36)

and the kernel condition

$[K]_{H_{\infty,\theta}}^{\prime}:=\sup_{0<R<1}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty,$ (4.37)

then $T:H^{1}(G)\rightarrow L^{1}(G)$ extends to a bounded operator.

Conflict of interests statement. On behalf of all authors, the corresponding
author states that there is no conflict of interest.

## References

  * [1] Alexopoulos, G. Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc., 120, 973–979, (1994).
  * [2] Anker, J. P. $L^{p}$-Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math., 132, 597–628, (1990).
  * [3] Baernstein, A., Sawyer, E. T. Embedding and multiplier theorems for $H^{p}(R^{n})$, Mem. Am. Math. Soc. 53(318), iv–82, (1985).
  * [4] Calderón, A. P. Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math., 80, 16–36, (1958).
  * [5] Calderón, A. P., Zygmund, A. On the existence of certain singular integrals, Acta Math., 88, 85–139, (1952).
  * [6] Carbery, A. Variants of the Calderón-Zygmund theory for $L^{p}$-spaces, Rev. Mat. Iberoamericana, 2(4), 381–396, (1986).
  * [7] Carbonaro, A., Mauceri, G., Meda, S. $H^{1}$ and $BMO$ for certain locally doubling metric measure spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., (5)8, 543–582, (2009).
  * [8] Cardona, D. Ruzhansky, M. Björk-Sjölin condition for strongly singular convolution operators on graded Lie groups, to appear in Math. Z. arXiv:2205.03456.
  * [9] Chen, P., Ouhabaz, E. M., Sikora, A., Yan, L. Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means, arXiv:1202.4052.
  * [10] Coifman, R. R., De Guzmán, M. Singular integrals and multipliers on homogeneous spaces. Rev. un. Mat. Argentina, 137–143, (1970).
  * [11] Coifman, R., Weiss, G. Analyse harmonique non-commutative sur certains espaces homogénes. (French) Étude de certaines intégrales singuliéres. Lecture Notes in Mathematics, Vol. 242. Springer-Verlag, Berlin-New York, 1971. v+160 pp.
  * [12] Coifman, R., Weiss, G. Central multiplier theorems for compact Lie groups. Bull. Am. Math. Soc., 80, 124–126, (1973).
  * [13] Coifman, R. R., Weiss, G. Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569–645, (1977).
  * [14] Coulhon, T., Saloff-Coste, L., Varopoulos, N. Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, 100. Cambridge University Press, 1993.
  * [15] Cowling, M., Sikora, A. A spectral multiplier theorem for a sublaplacian on SU(2), Math. Z., 238, 1–36, (2001).
  * [16] Duoandikoetxea, J. Fourier Analysis. 29, American Mathematical Society, Providence, (2000).
  * [17] David, G., Journé, J. L. A boundedness criterion for generalized Calderón-Zygmund operators. Annals Math., 120(2), 371–397, (1984).
  * [18] Fefferman, C. Inequalities for strongly singular integral operators, Acta Math. 24, 9–36, (1970).
  * [19] Fefferman, C. $L^{p}$-bounds for pseudo-differential operators, Israel J. Math., 14, 413–417, (1973).
  * [20] Fefferman, C., Stein, E. $H^{p}$ spaces of several variables, Acta Math., 129, 137-193, (1972).
  * [21] Fischer V., Ruzhansky M., Quantization on nilpotent Lie groups, Progress in Mathematics, Vol. 314, Birkhauser, 2016. xiii+557pp.
  * [22] Fischer V., Ruzhansky M. Fourier multipliers on graded Lie groups, Colloq. Math., 165, 1–30, (2021).
  * [23] Fischer V., Ruzhansky M. Sobolev spaces on graded groups, Ann. Inst. Fourier (Grenoble), 67, 1671–1723, (2017).
  * [24] Folland, G., Stein, E. Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, N.J., 1982.
  * [25] Grafakos, L., Stockdale, C. B. A limited-range Calderón-Zygmund theorem. Bull. Hellenic Math. Soc., 63, 54–63, (2019).
  * [26] Grafakos, L. Some remarks on the Miklhin-Hörmander and Marcinkiewicz multiplier theorems: a short historical account and a recent improvement. J. Geom. Anal., to appear in a special volume in honor of E. M. Stein.
  * [27] Hardy, G. H. A theorem concerning Taylor’s series, Quart. J. Pure Appl. Math., 44, 147–160, (1913).
  * [28] Hassannezhad, A., Kokarev, G. Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci., XVI(4), 1049–1092, (2016).
  * [29] Helffer, B. Nourrigat, J. Caracterisation des operateurs hypoelliptiques homogenes invariants a gauche sur un groupe de Lie nilpotent gradue, Comm. Partial Differential Equations, 4(8), 899–958, (1979).
  * [30] Hirschman, I. I., Multiplier transformations I, Duke Math. J., 222–242, (1956).
  * [31] Hong Q., Hu G., Ruzhansky M., Fourier multipliers for Hardy spaces on graded Lie groups, arXiv:2101.07499.
  * [32] Hörmander, L. Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104, 93–139, (1960).
  * [33] Hörmander, L. Hypoelliptic second order differential equations, Acta Math., 119, 147–171, (1967).
  * [34] Hörmander, L. The Analysis of the linear partial differential operators, Vol. III-IV. Springer-Verlag, (1985).
  * [35] Martini, A. Ricci, F. Tolomeo, L. Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth. J. Funct. Anal. 277(6), 1603–1638, (2019).
  * [36] Mihlin, S. G., Singular integral equations, Uspehi Mat. Nauk, Vol. 3, So. 25, 1948, 29–112; New York Univ., Courant Inst. Math. Sci., (1963). Amer. Math. Sac. translation Vol. 24, (1950).
  * [37] M. Ruzhansky, J. Wirth. $L^{p}$ Fourier multipliers on compact Lie groups, Math. Z., 280, 621–642, (2015).
  * [38] Ruzhansky, M., Turunen, V. Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics Birkhäuser-Verlag, Basel, 2010.
  * [39] Ruzhansky M., Turunen V., Wirth J. Hörmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity, J. Fourier Anal. Appl. 20, 476-499, (2014).
  * [40] Ruzhansky M., Yessirkegenov N., Hypoelliptic functional inequalities. arXiv:1805.01064.
  * [41] Seeger, A. Some inequalities for singular convolution operators in Lp-spaces, Trans. Amer. Math. Soc. 308(1), 259–272, (1988).
  * [42] Stein, E. M. Topics in Harmonic Analysis Related to the Littlewood–Paley Theory, vol. 63 of Annals of Mathematics Studies. Princeton University Press, Princeton, (1970).
  * [43] Suzuki, S. The Calderón–Zygmund Theorem with an $L^{1}$ Mean Hörmander Condition, J. Fourier Anal. Appl. 27(10), (2021).
  * [44] Elst, ter, A. F. M., Robinson, D. W., Zhu, Y. Hardy spaces on Lie groups of polynomial growth. (CASA-report; Vol. 0431). Technische Universiteit Eindhoven, (2004).
  * [45] Watson, D. K. Weighted estimates for singular integrals via Fourier transform estimates, Duke Math. J. 60(2), 389–399, (1990).
  * [46] Wainger, S. Special trigonometric series in $k$-dimensions, Mem. Amer. Math. Soc. 59, (1965).