#  Oscillating singular integral operators on compact Lie groups revisited

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.

In [24, Theorem 2’] Charles Fefferman has proved the weak (1,1) boundedness
for a class of oscillating singular integrals that includes the oscillating
spectral multipliers of the Euclidean Laplacian $\Delta,$ namely, operators of
the form

$T_{\theta}(-\Delta):=(1-\Delta)^{-\frac{n\theta}{4}}e^{i(1-\Delta)^{\frac{\theta}{2}}},\,0\leq\theta<1.$
(0.1)

The aim of this work is to extend Fefferman’s result to oscillating singular
integrals on any arbitrary compact Lie group. Applications to oscillating
spectral multipliers of the Laplace-Beltrami operator are also considered. The
proof of our main theorem illustrates the delicate relationship between the
condition on the kernel of the operator, its Fourier transform (defined in
terms of the representation theory of the group) and the microlocal/geometric
properties of the group.

###### Key words and phrases:

Calderón-Zygmund operator, Weak (1,1) inequality, Oscillating singular
integrals

###### 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
    2. 1.2 Fefferman’s approach
    3. 1.3 Compact Lie groups setting
  2. 2 Group Fourier transform on compact Lie groups
  3. 3 Proof of the main theorem
  4. 4 Examples
    1. 4.1 Oscillating integrals on the torus $\mathbb{T}^{n}$
    2. 4.2 Oscillating integrals on $\textnormal{SU}(2)\cong\mathbb{S}^{3}$
    3. 4.3 Application to oscillating multipliers

##  1\. Introduction

###  1.1. Outline

This work deals with the weak (1,1) boundedness of oscillating singular
integrals on compact Lie groups by extending to the non-commutative setting
the paradigm of the oscillating singular integrals introduced by Fefferman and
Stein in [24, 24], which among other things, generalises some classical
conditions introduced by Calderón and Zygmund [6] and Hörmander [35]. One of
the novelties of this work is that we make use of the geometric properties of
the group, of its microlocal analysis, and of its representation theory.

On compact Lie groups, oscillating singular integrals arose as generalisations
of the oscillating spectral multipliers, which are linear operators defined by
the spectral calculus via

$T_{\theta}(\mathcal{L}_{G}):=(1+\mathcal{L}_{G})^{-\frac{n\theta}{4}}e^{i(1+\mathcal{L}_{G})^{\frac{\theta}{2}}},\,0\leq\theta<1,$
(1.1)

where $\mathcal{L}_{G}:=-(X_{1}^{2}+\cdots X_{n}^{2})$ is the positive
Laplace-Beltrami operator on a compact Lie group $G.$

The problem of finding boundedness criteria for Fourier multipliers (left-
convolution operators) has been widely investigated for a long time. On a
compact Lie group $G,$ the problem has been considered for central operators
by N. Weiss [47], Coifman and Weiss [18], Cowling and Sikora [21] in the case
of $G=\textnormal{SU}(2),$ and Chen and Fan [15]. However, only symbol
criteria for general Fourier multipliers (and for pseudo-differential
operators) on compact Lie groups, making use of the difference structure of
the unitary dual $\widehat{G}$ of $G,$ were firstly proved by the second
author and Wirth in [39], and further generalisations were established in [8,
10, 11, 22] and in the works [9, 12, 26, 36] for the setting of graded Lie
groups. We refer the reader to [1, 10, 16] and [38] for an extensive list of
references on the subject as well as for a historical perspective, in
particular in the setting of central operators. In particular, the work [3]
introduces an interesting family of multipliers covering e.g. the Hörmander-
Mihlin condition.

Before presenting the contributions of this paper, let us review the classical
result due to Charles Fefferman which is of central interest for this work,
see [24, Page 23]. In the Euclidean setting, a convolution operator
$T:f\mapsto f\ast K,$ $f\in C^{\infty}_{0}(\mathbb{R}^{n}),$ with $K$ being a
distribution of compact support and locally integrable outside the origin, and
satisfying the conditions

$|\widehat{K}(\xi)|=O((1+|\xi|)^{-\frac{n\theta}{2}}),\quad 0\leq\theta<1,$
(1.2)

and

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

is called of oscillating type. Here,
$\widehat{K}:=\mathscr{F}_{\mathbb{R}^{n}}K$ denotes the Fourier transform of
the distribution $K.$ The $L^{p}$-properties for convolution operators of this
form were firstly studied by Hardy [32], Hirschman [34] and Wainger [46], in
the particular scenario of Fourier multipliers with symbols of the form

$\widehat{K}(\xi)={(1+|\xi|^{2})^{-\frac{n\theta}{4}}}{e^{i(1+|\xi|^{2})^{\frac{\theta}{2}}}},\,\,\quad
0<\theta<1.$

The corresponding operators are oscillating multipliers of the positive
Euclidean Laplacian $-\Delta=\mathcal{L}_{\mathbb{R}^{n}},$ namely, operators
of the form

$T_{\theta}(-\Delta):=(1-\Delta)^{-\frac{n\theta}{4}}e^{i(1-\Delta)^{\frac{\theta}{2}}},\,0\leq\theta<1.$
(1.4)

One of the fundamentals contributions to the subject, became with the work
[24] by Charles Fefferman where he established the weak (1,1) boundedness of a
class of oscillating singular integrals that includes the oscillating spectral
multipliers of the Laplacian (1.4). It is also important to mention that
Fefferman and Stein, when extending the theory of Hardy spaces to Euclidean
spaces of several variables, observed that the theory of oscillating singular
integrals in [24] also begets bounded operators from the Hardy space $H^{1}$
into $L^{1},$ see [25] for details.

###  1.2. Fefferman’s approach

Next, we review the fascinating technique developed by Fefferman in his Acta’s
paper [24] for the proof of the aforementioned weak (1,1) estimate. As it was
pointed out by Stein in [17, Page 1257], such a technique became a subject of
much wider interest, as it was adapted to various other problems e.g. by S.
Chanillo, M. Christ, Rubio de Francia, Seeger, Sogge and Stein [42], etc.

For this, let us fix a convolution operator $T$ with kernel $K$ satisfying
(1.2) and (1.3). To begin with he decomposed an arbitrary function $f\in
L^{1}(\mathbb{R}^{n}),$ for a fixed $\alpha>0,$ according to the Calderón-
Zygmund decomposition theorem, in the standard way, that is $f=g+b,$
$b:=\sum_{j}b_{j},$ where the $L^{2}$-norm of $g$ is bounded from above by
$\alpha,$ and the $b_{j}$’s are supported on disjoint dyadic cubes $I_{j}$’s
in such a way that the following two properties are satisfied

$\frac{1}{|I_{j}|}\smallint\limits_{I_{j}}|b_{j}(x)|dx\sim\alpha,\quad\smallint\limits_{I_{j}}b_{j}(x)dx=0.$
(1.5)

When estimating $|\\{x\in\mathbb{R}^{n}:|Tf(x)|>\alpha\\}|,$ he started with
the standard inequality

$|\\{x\in\mathbb{R}^{n}:|Tf(x)|>\alpha\\}|\leq|\\{x\in\mathbb{R}^{n}:|Tb(x)|>\alpha/2\\}|+|\\{x\in\mathbb{R}^{n}:|Tg(x)|>\alpha/2\\}|.$

Then, he observed that $T(g)$ can be estimated from above with the
$L^{1}$-norm of $f$ using the $L^{2}$-theory. As for the estimate of
$T(b_{j}),$ the main idea in Fefferman’s argument was the consideration of a
family of measurable sets $I_{j}^{*}$ such that,

$\textnormal{center}[I_{j}]=\textnormal{center}[I_{j}^{*}],\quad\textnormal{diameter}[I_{j}^{*}]\sim\textnormal{diameter}[I_{j}]^{1-\theta}.$
(1.6)

By making a suitable geometric reduction of the support of $K,$ he proved that
only cubes $I_{j}$ with $\textnormal{diam}[I_{j}]<1$ were significant when
estimating $T(b_{j}).$ Moreover, the contribution of $T(b_{j})$ outside the
set $I_{j}^{*}$ can be handled in view of the kernel condition (1.3).

About the critical contribution of $T(b_{j})$ in the set $I_{j}^{*}\setminus
I_{j},$ he introduced a nice replacement $\tilde{b}_{j}$ of $b_{j},$ and
another one $\tilde{b}=\sum_{j}\tilde{b}_{j}$ of $b,$ and he was able to prove
that

$\|T(\tilde{b})\|_{L^{2}}\lesssim\|(-\Delta)^{-\frac{n\theta}{4}}\tilde{b}\|_{L^{2}}\lesssim\alpha\|f\|_{L^{1}},$
(1.7)

by combining the $L^{2}$-theory, a suitable decomposition of the function
$(-\Delta)^{-\frac{n\theta}{4}}\tilde{b}=F_{1}+F_{2},$ and the Sobolev
inequality. Putting all these estimates together he proved the weak (1,1) type
of $T.$

###  1.3. Compact Lie groups setting

It is natural to investigate whether these results for oscillating singular
integrals on $\mathbb{R}^{n}$ extend to more general manifolds. In particular,
in view of the behaviour of the Fourier transform of the kernel (1.2) one can
analyse the problem in the setting of compact Lie groups where the group
Fourier transform is defined in terms of the representation theory of the
group.

On a compact Lie group $G,$ an oscillating singular integral is a left-
invariant operator $T:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ with
right convolution kernel $K$ satisfying the estimate

$[K]_{H_{\infty,\theta}(G)}:=\sup_{R>0}\|\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx\|_{L^{\infty}(B(0,R),\,dy)}<\infty,$ (1.8)

for some $0\leq\theta<1.$ We have denoted by $|x|=d(x,e)$ the geodesic
distance from $x\in G$ to the neutral element $e\in G.$

In order to study the boundedness of oscillating singular integrals in the
non-commutative setting, one could adopt the argument of Fefferman and to
study its extension to compact Lie groups. Nevertheless, when extending such a
construction we found some obstructions related with the structure of the
group $G.$ More precisely, in Fefferman’s argument, the
$\tilde{b}_{j}^{\prime}s$ are defined by the convolution $b_{j}\ast\phi_{j},$
where roughly speaking, any $\phi_{j}$ is given by

$\phi_{j}(y):=\textnormal{diam}[I_{j}]^{-n/(1-\theta)}\phi(\textnormal{diam}[I_{j}]^{-1/(1-\theta)}\cdot
y),\quad\smallint\phi=1,\quad\textnormal{supp}[\phi]\subset\\{x:|x|\leq 1\\}.$

By taking into account the geometry of the group, we observe that there is not
a global action $\mathbb{R}^{+}\times G\rightarrow G$ of $\mathbb{R}^{+}$ into
$G,$ that allows us to define the elements $r\cdot y\in G,$ for $r>0$ and
$y\in G,$ in a compatible way with Fefferman’s argument. Of particular
interest are the dilations factors

$r=\textnormal{diam}[I_{j}]^{-1/(1-\theta)}>1.$

Indeed, only on small neighborhoods $\nu$ of $0\in\mathfrak{g},$ and
$\nu^{\prime}$ of the the neutral element $e\in G,$ the exponential mapping
$\textnormal{exp}:\nu\rightarrow\nu^{\prime}$ is a diffeomorphism, and a local
family of dilations $D_{r}:\nu\rightarrow\nu,$ with $0<r\leq 1,$ denoted by
$y\in\nu\mapsto D_{r}(y)=r\cdot y,$ can be defined via $r\cdot
y=\exp(r\exp^{-1}(y)).$ One can translate these dilations to small
neighborhoods of any point of the group by using the multiplication operation
on $G$, but, still, the case of dilations by factors $r>1$ cannot be covered
with this local construction.

However, inspired by the approach developed by Coifman and Weiss [20] and by
Coifman and De Guzmán [19], we observe that for the scenario of compact Lie
groups by taking the family of functions

$\phi_{j}(y)=\frac{1}{|B(e,R_{j})|}1_{B(e,R_{j})},\quad
R_{j}\sim\textnormal{diam}[I_{j}]^{1/(1-\theta)},$ (1.9)

will provide the necessary properties for extending Fefferman’s argument.

The main theorem of this work is the following. Here $\widehat{G}$ denotes the
unitary dual of $G$ and the elliptic weight $\langle\xi\rangle,$
$[\xi]\in\widehat{G},$ is defined in terms of the spectrum of the Laplacian on
$G,$ see Section 2 for details.

######  Theorem 1.1.

Let $G$ be a compact Lie group of dimension $n,$ and let
$T:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ be a left-invariant
operator with right-convolution kernel $K\in
L^{1}_{\textnormal{loc}}(G\setminus\\{e\\}).$ Let us consider that for
$0\leq\theta<1,$ $K$ satisfies the group Fourier transform condition

$\exists
C>0,\quad\forall[\xi]\in\widehat{G},\quad\|\widehat{K}(\xi)\|_{\textnormal{op}}\leq
C\langle\xi\rangle^{-\frac{n\theta}{2}},$ (1.10)

and the oscillating Hörmander condition

$[K]_{H_{\infty,\theta}(G)}:=\sup_{R>0}\sup_{|y|\leq
R}\smallint\limits_{|x|\geq 2R^{1-\theta}}|K(y^{-1}x)-K(x)|dx<\infty.$ (1.11)

Then $T$ admits an extension of weak $(1,1)$ type.

######  Remark 1.2.

It was proved in [13] that under the hypothesis in Theorem 1.1, $T$ is bounded
from the Hardy space $H^{1}(G)$ into $L^{1}(G).$ So, in the case of compact
Lie groups our main Theorem 1.1 together with the results in [13] provide a
complete perspective on the subject, related with the Hörmander condition in
(1.11) for all $0\leq\theta<1$. We observe that the case $\theta=1$ is outside
of the analysis in our approach. It is related to the wave operator for the
Laplace-Beltrami operator and in view of the weak (1,1) estimate for Fourier
integral operators by Tao [44], it is expected that the right decay condition
in (1.10) for $\theta=1$ could be the one with the order $-(n-1)/2$ (instead
of $-n/2$).

######  Remark 1.3.

We also observe that the relevant contribution of Theorem 1.1 is the case
where $0<\theta<1.$ Indeed, for $\theta=0,$ the Fourier transform condition in
(1.10) can be replaced for the $L^{2}$-boundedness of the operator $T,$ and
the condition in (1.11) is reduced to the  smoothness Hörmander condition
which can be analysed in the setting of the Calderón-Zygmund theory on spaces
of homogeneous type in the sense of Coifman and Weiss [20].

######  Remark 1.4.

One of the differences between the approach due to Fefferman in [24] and our
proof of Theorem 1.1 lies in our application of the $L^{2}$-boundedness of the
operator $(-\mathcal{L}_{G})^{z}$ when $\textnormal{Re}(z)<0,$ in view of the
Stein identity (see [43, Page 58])

$(-\mathcal{L}_{G})^{z}=\left(-\mathcal{L}_{G}+\textnormal{Proj}_{\textnormal{Ker}(-\mathcal{L}_{G})}\right)^{z}-\textnormal{Proj}_{\textnormal{Ker}(-\mathcal{L}_{G})}.$

See e.g. Remark 2.4 of [40]. Indeed, we have that $(-\mathcal{L}_{G})^{z}$ is
a pseudo-differential operator on $G$ of order $\textnormal{Re}(z)<0.$ Note
that, this argument does not apply in the case of $\mathbb{R}^{n}$ because the
spectrum of the Euclidean Laplacian is continuous and the negative powers of
$-\Delta_{\mathbb{R}^{n}}$ are not pseudo-differential operators, while the
powers of $1-\Delta_{\mathbb{R}^{n}}$ are pseudo-differential operators with
symbols in the Kohn-Nirenberg classes, see e.g. Taylor [45]. The boundedness
of the operator $(-\mathcal{L}_{G})^{z}$ on $L^{2}$ allows us to apply the
microlocal analysis of $G.$

######  Remark 1.5.

In view of the Hopf-Rinow theorem the exponential mapping on $G$ viewed as a
Riemannian manifold agrees with the exponential mapping on $G,$ considered as
a closed sub-group of (the linear group of unitary matrices)
$\textnormal{U}(N)$ for $N$ large enough. In particular on small neighborhoods
$\nu$ of $0\in\mathfrak{g},$ and $\nu^{\prime}$ of the the neutral element
$e\in G,$ the exponential mapping
$\textnormal{exp}:\nu\rightarrow\nu^{\prime}$ is a diffeomorphism. This
geometric property of the group will be involved in the proof of our main
Theorem 1.1.

######  Remark 1.6.

Clearly $G$ being a compact topological space is a homogeneous space in the
sense of Coifman and Weiss [20]. The geometric measure theory of the group
will be used due to the existence of Calderón-Zygmund type decompositions for
any $f\in L^{1}(G),$ see [20, Pages 73-74].

######  Remark 1.7.

We observe that using the theory of Coifman and De Guzmán [19], the second
author and Wirth in [39] have proved Hörmander-Mihlin criteria for Fourier
multipliers on compact Lie groups. In that setting, as expected, it was a non-
trivial fact that the operators satisfy Calderón-Zygmund type conditions
(corresponding to the case $\theta=0$ in Theorem 1.1).

######  Remark 1.8.

For limited-range versions of the Calderón-Zygmund theorem we refer the reader
to Baernstein and Sawyer [2], Carbery [7], Seeger [41], and Grafakos, Honzík,
Ryabogin [31]. We observe that the case in the Euclidean setting, the case
$\theta=0$ includes Fourier multipliers satisfying Hörmander-Mihlin
conditions. Improvements for the Hörmander condition and conditions of
Hörmander-Mihlin type in the Euclidean framework can be found in Grafakos
[30]. We refer the reader to [14] for the boundedness properties of
oscillating singular integrals on $\mathbb{R}^{n}.$

This paper is organised as follows. In the short Section 2 we present the
preliminaries on the Fourier analysis on compact Lie groups. Our main theorem
is proved in Section 3. Finally, some examples are given in the case of the
torus $\mathbb{T}^{n},$ $\textnormal{SU}(2)\cong\mathbb{S}^{3},$ and for
oscillating Fourier multipliers in Section 4.

##  2\. Group Fourier transform on compact Lie groups

In this section we present some preliminaries about the Fourier analysis on
compact Lie groups, for this we follow the book [39, Part III].

Let $G$ be a compact Lie group and let $dx$ be its normalised left-invariant
Haar measure. To define the group Fourier transform let us define the unitary
dual $\widehat{G}$ of $G.$ One reason for this is that the Fourier inversion
formula becomes a series over $\widehat{G}.$ 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 finite-
dimensional vector space $H_{\xi}\cong\mathbb{C}^{d_{\xi}},$ 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 $W_{\xi}=\emptyset.$

In view of the compactness of the group, any strongly continuous unitary
representation on $G$ is continuous. Let $\textnormal{Rep}(G)$ be the set of
unitary, continuous and irreducible representations of $G.$

One can define an equivalence relation on $\textnormal{Rep}(G)$ as follows:

$\xi_{1}\sim\xi_{2},\,\xi_{1},\xi_{2}\in\textnormal{Rep}(G)\,\,\Longleftrightarrow\,\exists
A\in\textnormal{End}(H_{\xi_{1}},H_{\xi_{2}}):\,\,\forall x\in
G,\,A\xi_{1}(x)A^{-1}=\xi_{2}(x).$

With respect to this equivalence relation the quotient

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

is the unitary dual of $G.$

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

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

The Fourier inversion formula

$f(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{{Tr}}[\xi(x)\widehat{f}(\xi)]$

holds if for example $f\in L^{1}(G).$ In this setting, the Plancherel theorem
states that $\mathscr{F}:L^{2}(G)\rightarrow L^{2}(\widehat{G})$ is an
isometry, where the inner product on $L^{2}(\widehat{G})$ is defined via

$(f,g)_{L^{2}(\widehat{G})}:=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{{Tr}}[\widehat{f}(\xi)\widehat{g}(\xi)^{*}].$
(2.1)

So, the Plancherel theorem takes the form

$\|f\|_{L^{2}(G)}^{2}=\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\widehat{f}(\xi)\|_{\textnormal{HS}}^{2}.$
(2.2)

We have denoted by $\|\cdot\|_{\textnormal{HS}}$ the usual Hilbert-Schmidt
norm on every representation space $H_{\xi}.$

######  Remark 2.1.

In the symbol condition (1.10),

$\textnormal{Spect}((1+\mathcal{L}_{G})^{\frac{1}{2}}):=\\{\langle\xi\rangle:[\xi]\in\widehat{G}\\},$

is the system of eigenvalues of the Bessel potential operator
$(1+\mathcal{L}_{G})^{\frac{1}{2}}$ associated to the positive Laplacian on
$G,$ which can be defined as follows. Let $X=\\{X_{1},\cdots,X_{n}\\}$ be an
orthonormal basis of its Lie algebra $\mathfrak{g}$ with respect to the unique
(up to by a constant factor) bi-invariant Riemannian metric
$g=(g_{x}(\cdot,\cdot))_{x\in G}$ on $G.$ The Laplace-Beltrami operator is
defined by (minus) the sum of squares

$\mathcal{L}_{G}=-\sum_{j=1}^{n}X_{j}^{2},$

and it is independent of the choice of the vector space basis $X$ of
$\mathfrak{g}.$ Note that $\mathcal{L}_{G}$ is an elliptic operator on $G,$
which admits a self-adjoint extension on $L^{2}(G,dx).$ Such an extension is
an elliptic operator and its spectrum is a discrete set
$\\{\lambda_{[\xi]}:[\xi]\in\widehat{G}\\}$ that can be enumerated by the
unitary dual $\widehat{G}$ of the group. By the spectral mapping theorem one
has that $\langle\xi\rangle=(1+\lambda_{[\xi]})^{\frac{1}{2}},$
$[\xi]\in\widehat{G}.$

######  Remark 2.2.

Observe that in view of the Plancherel theorem, the hypothesis (1.10) in
Theorem 1.1 implies that the operator
$(1+\mathcal{L}_{G})^{\frac{n\theta}{4}}T$ admits a bounded extension on
$L^{2}(G)$. Indeed, the Plancherel theorems implies that

$\|(1+\mathcal{L}_{G})^{\frac{n\theta}{4}}Tf\|_{L^{2}(G)}^{2}=\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\langle\xi\rangle^{\frac{n\theta}{2}}\widehat{K}(\xi)\widehat{f}(\xi)\|_{\textnormal{HS}}^{2}\leq\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\langle\xi\rangle^{\frac{n\theta}{2}}\widehat{K}(\xi)\|^{2}_{\textnormal{op}}\|\widehat{f}(\xi)\|_{\textnormal{HS}}^{2}$
(2.3)

and in view of (1.10) we have that

$\|(1+\mathcal{L}_{G})^{\frac{n\theta}{4}}Tf\|_{L^{2}(G)}^{2}\leq
C^{2}\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\widehat{f}(\xi)\|_{\textnormal{HS}}^{2}\lesssim\|f\|_{L^{2}(G)}^{2},$

proving the boundedness of $(1+\mathcal{L}_{G})^{\frac{n\theta}{4}}T$ on
$L^{2}(G).$

##  3\. Proof of the main theorem

In this section we are going to prove that for a singular integral operator
$T$ satisfying (1.2) and (1.11) there is a constant $C>0,$ such that

$\|Tf\|_{L^{1,\infty}(G)}:=\sup_{\alpha>0}\alpha|\\{x\in
G:|Tf(x)|>\alpha\\}|\leq C\|f\|_{L^{1}(G)},$ (3.1)

with $C$ independent of $f.$ The positive Laplacian on $G$ or on
$\mathbb{R}^{n}$ will be denoted by $\mathcal{L}_{G}$ and by
$\mathcal{L}_{\mathbb{R}^{n}},$ respectively. The Euclidean Fourier transform
on $\mathbb{R}^{n}$ of a function $V\in L^{1}(\mathbb{R}^{n}),$ will be
denoted by
$\mathscr{F}_{\mathbb{R}^{n}}V(\Xi)=\smallint_{\mathbb{R}^{n}}e^{i2\pi
X\cdot\Xi}V(X)dX,$ and it will be used at the end of this section.

For this fix $f\in L^{1}(G),$ and let us consider its Calderón-Zygmund
decomposition, see Coifman and Weiss [20, Pages 73-74]. So, for any
$\gamma,\alpha>0,$ such that

$\alpha\gamma>\frac{1}{|G|}\smallint\limits_{G}|f(x)|dx,$ (3.2)

we can have the decomposition

$f=g+b=g+\sum_{j}b_{j}$

where the following properties are satisfied:

  * (1)

$\|g\|_{L^{\infty}}\lesssim_{G}\gamma\alpha$ and
$\|g\|_{L^{1}}\lesssim_{G}\|f\|_{L^{1}}.$

  * (2)

The $b_{j}$’s are supported in open balls $I_{j}=B(x_{j},r_{j})$ where they
satisfy the cancellation property

$\smallint\limits_{I_{j}}b_{j}(x)dx=0.$ (3.3)

  * (3)

Any component $b_{j}$ satisfies the $L^{1}$-estimate

$\|b_{j}\|_{L^{1}}\lesssim_{G}(\gamma\alpha)|I_{j}|.$ (3.4)

  * (4)

The sequence $\\{|I_{j}|\\}_{j}\in\ell^{1}$ and

$\sum_{j}|I_{j}|\lesssim_{G}(\gamma\alpha)^{-1}\|f\|_{L^{1}}.$ (3.5)

  * (5)

$\|b\|_{L^{1}}\leq\sum_{j}\|b_{j}\|_{L^{1}}\lesssim_{G}\|f\|_{L^{1}}.$

  * (6)

There exists $M_{0}\in\mathbb{N},$ such that any point $x\in G$ belongs at
most to $M_{0}$ balls of the collection $I_{j}.$

First of all note that given an altitude $\alpha\gamma>0,$ the Calderón-
Zygmund decomposition above works in the case where
$\alpha\gamma>\frac{1}{|G|}\smallint|f(x)|dx.$ We have denoted by $|G|$ the
volume of the group $G$ without assuming that the Haar measure is normalised.

We want to emphasize that when proving the weak (1,1) boundedness of $T,$ one
has to estimate the quantity $|\\{x\in G:|Tf(x)|>\gamma\alpha\\}|$ only in the
case where the Calderón-Zymund decomposition in [20] is available, that is
when $\alpha\gamma>\frac{1}{|G|}\smallint|f(x)|dx.$ Indeed, when estimating
$|\\{x\in G:|Tf(x)|>\alpha\gamma\\}|$ at any altitude $\alpha\gamma$ with
$0<\alpha\gamma\leq\frac{1}{|G|}\smallint|f(x)|dx,$ one trivially has that

$|\\{x\in
G:|Tf(x)|>\alpha\gamma\\}|\leq|G|\leq\frac{1}{\alpha\gamma}\smallint\limits_{G}|f(x)|dx,$
(3.6)

as desired.

So, by fixing $\alpha\gamma$ as in (3.2), note that in terms of $g,$ $b$ and
$f$ one has the trivial estimate

$\displaystyle|\\{x:|Tf(x)|>\alpha\\}|\leq|\\{x:|Tg(x)|>\alpha/2\\}|+|\\{x:|Tb(x)|>\alpha/2\\}|.$

The estimates $\|g\|_{L^{\infty}}\lesssim\gamma\alpha$ and
$\|g\|_{L^{1}}\lesssim\|f\|_{L^{1}},$ imply that

$\displaystyle\|g\|_{L^{2}}^{2}\lesssim\|g\|_{L^{\infty}}\|g\|_{L^{1}}\leq(\gamma\alpha)\|f\|_{L^{1}}.$

So, by applying the Chebishev inequality and the $L^{2}$-boundedness of $T,$
we have

$\displaystyle|\\{x:|Tg(x)|>\alpha/2\\}|\lesssim
2^{2}\alpha^{-2}\|Tg\|_{L^{2}}\leq(2\|T\|_{\mathscr{B}(L^{2})})^{2}\alpha^{-2}\|g\|_{L^{2}}^{2}$
$\displaystyle\leq(2\|T\|_{\mathscr{B}(L^{2})})^{2}\alpha^{-2}(\gamma\alpha)\|f\|_{L^{1}}\lesssim\|T\|_{\mathscr{B}(L^{2})}^{2}\gamma\alpha^{-1}\|f\|_{L^{1}}$
$\displaystyle\lesssim_{\gamma}\alpha^{-1}\|f\|_{L^{1}}.$

In what follows, let us denote $I^{*}=\bigcup I_{j}^{*},$ where
$I_{j}^{*}=B(x_{j},2r_{j}),$ and let us make use of the doubling condition in
order to have the estimate

$|I^{*}_{j}|\sim c^{n}|I_{j}|,$

from which follows that

$|I^{*}|\lesssim\sum_{j}|I_{j}^{*}|\lesssim\sum_{j}|I_{j}|\lesssim\gamma^{-1}\alpha^{-1}\|f\|_{L^{1}}.$

Consequently, we have the estimates

$|\\{x:|Tb(x)|>\alpha/2\\}|\leq|I^{*}|+|\\{x\in G\setminus
I^{*}:|Tb(x)|>\alpha/2\\}|$
$\leq(2\sqrt{n})^{n}\gamma^{-1}\alpha^{-1}\|f\|_{L^{1}}+|\\{x\in G\setminus
I^{*}:|Tb(x)|>\alpha/2\\}|.$
$\lesssim_{\gamma}\alpha^{-1}\|f\|_{L^{1}}+|\\{x\in G\setminus
I^{*}:|Tb(x)|>\alpha/2\\}|.$

So, to conclude the inequality (3.1) we have to prove that

$\sup_{\alpha>0}\alpha|\\{x\in G\setminus I^{*}:|Tb(x)|>\alpha/2\\}|\leq
C\|f\|_{L^{1}},$ (3.7)

with $C$ independent of $f.$ So, the proof of Theorem 1.1 consists of
estimating the term

$|\\{x\in G\setminus I^{*}:|Tb(x)|>\alpha/2\\}|.$

###### Proof of Theorem 1.1.

We start by considering only the case $0<\theta<1.$ Indeed, the statement for
$\theta=0$ in Theorem 1.1 follows from the fundamental theorem of singular
integrals due to Coifman and Weiss, see [20, Theorem 2.4, Page 74].

We shall return to the proof of (3.7) after the required geometric
constructions. First, let us assume that the diameter of the support of $K$ is
small and that it is centred at the identity $e\in G.$ We illustrate this step
in Figure 1 below.

Figure 1.

Indeed, because of the compactness of $\textnormal{supp}(K),$ we have that
$\textnormal{supp}(K)\subset B(a,R_{0}),$ for some $a\in G,$ where
$R_{0}\sim\frac{1}{2}\textnormal{diam}(\textnormal{supp}(K))$ is comparable
with the diameter of the support of $K.$ Then, the kernel
$K_{a}:=K(a^{-1}\cdot)$ has its support inside of the ball $B(e,R_{0}).$
Moreover, in view of the identity

$\displaystyle T_{a}f(x)$ $\displaystyle:=f\ast K_{a}(x)=\smallint
K(a^{-1}y^{-1}x)f(y)dy=\smallint K(z^{-1}x)f(za^{-1})dz$
$\displaystyle=T(f(\cdot a^{-1}))(x),$

it follows that
$\|T\|_{\mathscr{B}(L^{1},L^{1,\infty})}=\|T_{a}\|_{\mathscr{B}(L^{1},L^{1,\infty})}.$
Also, by the compactness of $G,$ for all $\varepsilon>0,$ there exists a
finite number of elements $x_{0}^{\varepsilon}=e_{G},x_{k}^{\varepsilon},$
$1\leqslant k\leqslant N_{0}(\varepsilon),$ in $G,$ and some smooth functions
$\chi_{k}^{\varepsilon}\in C^{\infty}(G,[0,1]),$ supported in
$B(e_{G},\frac{\varepsilon}{2}),$ such that

$G=\bigcup_{k=1}^{N_{0}(\varepsilon)}B(x_{k}^{\varepsilon},\varepsilon/4),\,\,\,\textnormal{
and
}\sum_{j=0}^{N_{0}(\varepsilon)}\chi_{k}^{\varepsilon}(x_{k}^{-1}x)=1,\,\,\,x\in
G.$ (3.8)

Then, let us decompose the operator $T$ into the sum
$T=\sum_{j=0}^{N_{0}(\varepsilon)}T_{j}^{\varepsilon},$ where any convolution
operator $T_{j}^{\varepsilon}$ is associated to the right-convolution kernel

$K_{j}^{\varepsilon}(x):=K(x)\chi_{k}^{\varepsilon}(x_{k}^{-1}x),\,x\in G.$

Nevertheless, the support of $K_{j}^{\varepsilon}$ is so small as
$\varepsilon$ is. Finally, one has that

$\|T\|_{\mathscr{B}(L^{1},L^{1,\infty})}\lesssim_{N_{0}(\varepsilon)}\sum_{j=1}^{N_{0}}\|T_{j}^{\varepsilon}\|_{\mathscr{B}(L^{1},L^{1,\infty})}\lesssim_{N_{0}(\varepsilon)}\max_{1\leq
j\leq N_{0}}\|T_{j}^{\varepsilon}\|_{\mathscr{B}(L^{1},L^{1,\infty})}.$ (3.9)

So, in view of this argument, from now, let us suppose that the diameter of
the support of $K$ is small, for instance, that

$\textnormal{diam}(\textnormal{supp}(K))<c/1000<1/1000$

where $0<c<1$ is small enough, and that $T$ is an arbitrary operator of the
list $\\{T_{j}^{\varepsilon}\\}_{1\leq j\leq N(\varepsilon)}$. In particular,
we can take $c$ small enough in order that we can guarantee that the
exponential mapping

$\omega=\textnormal{exp}:\nu\rightarrow\nu^{\prime},\,\,{B}(e,c)\subset\nu^{\prime},$
(3.10)

is a diffeomorphism between two small neighborhoods $\nu^{\prime}$ and $\nu$
of the identity element $e$ and of the origin
$0\in\mathfrak{g}\cong\mathbb{R}^{n},$ respectively.

For $\varepsilon>0$ define

$\phi(y,\varepsilon):=\frac{1}{|B(e,\varepsilon)|}1_{B(e,\varepsilon)}.$
(3.11)

Now, for any $j,$ define

$\phi_{j}(y):=\phi(y,2^{-\frac{1}{1-\theta}}\textnormal{diam}(I_{j})^{\frac{1}{1-\theta}}),\,\,\,$
(3.12) $\tilde{b}_{j}:=b_{j}(\cdot)*\phi_{j},\,\,$ (3.13)

and

$\tilde{b}:=\sum_{j}\tilde{b}_{j}.$ (3.14)

Note that

$Tb=\sum_{j}T{b}_{j}.$ (3.15)

It is important to mention that in (3.15) the sums on the right hand side only
runs over $j$ whith $\textnormal{diam}(I_{j})<c/1000.$ Indeed, for all $x\in
G\setminus I^{*},$ the property of the support
$\textnormal{diam}(\textnormal{supp}(K))<c/1000,$ implies that for all $j$
with $\textnormal{diam}(I_{j})\geq c/1000,$ we have that

$Tb_{j}=b_{j}\ast K=0.$

So we only require to analyse the case where
$\textnormal{diam}(I_{j})<c/1000.$ Indeed, for $x\in G\setminus I^{*},$ and
$j$ such that $\textnormal{diam}(I_{j})\geq c/1000,$

$b_{j}\ast K(x)=\smallint\limits_{I_{j}}K(y^{-1}x)b_{j}(y)dy.$

Because, in the integral above $x\in G\setminus I^{*}$ and $y\in I_{j},$

$|y^{-1}x|=\textnormal{dist}(x,y)>\textnormal{diam}(I_{j})>c/1000,$

we have that the element $y^{-1}x$ is not in the support of $K$ and then the
integral vanishes.

In Figure 2 we compare the size of $I_{j}$ with the size of the support of
$\phi_{j}.$

Figure 2.

Now, going back to the analysis of (3.7), note that

$\displaystyle|\\{x\in G\setminus I^{*}:|Tb(x)|>\frac{\alpha}{2}\\}|$
$\displaystyle\leq|\\{x\in G\setminus
I^{*}:|Tb(x)-T\tilde{b}(x)|>\frac{\alpha}{4}\\}|+|\\{x\in G\setminus
I^{*}:|T\tilde{b}(x)|>\frac{\alpha}{4}\\}|$
$\displaystyle\leq\frac{4}{\alpha}\|T(b-\tilde{b})\|_{L^{1}(G\setminus
I^{*})}+|\\{x\in G\setminus I^{*}:|T\tilde{b}(x)|>\frac{\alpha}{4}\\}|,$

and let us take into account the estimate:

$\displaystyle\|T(b-\tilde{b})\|_{L^{1}(G\setminus I^{*})}$
$\displaystyle=\smallint\limits_{G\setminus I^{*}}|Tb(x)-T\tilde{b}(x)|dx$
$\displaystyle\leq\sum_{j}\smallint\limits_{G\setminus
I^{*}}|Tb_{j}(x)-T\tilde{b}_{j}(x)|dx.$

We are going to prove that $T\tilde{b}$ and $T\tilde{b}_{j}$ are good
replacements for $T{b}$ and $T{b}_{j},$ respectively, on the set $G\setminus
I^{*}.$ Observe that

$\displaystyle\smallint\limits_{G\setminus
I^{*}}|Tb_{j}(x)-T\tilde{b}_{j}(x)|dx=\smallint\limits_{G\setminus
I^{*}}|b_{j}\ast K(x)-\tilde{b}_{j}\ast K(x)|dx$
$\displaystyle=\smallint\limits_{G\setminus I^{*}}|b_{j}\ast
K(x)-[{b}_{j}\ast\phi_{j}\ast K](x)|dx$
$\displaystyle=\smallint\limits_{G\setminus
I^{*}}\left|\smallint\limits_{I_{j}}K(y^{-1}x)b_{j}(y)dy-\smallint\limits_{I_{j}}(\phi_{j}\ast
K)(y^{-1}x){b}_{j}(y)dy\right|dx$
$\displaystyle\leq\smallint\limits_{I_{j}}\smallint\limits_{G\setminus
I^{*}}|K(y^{-1}x)-\phi_{j}\ast K(y^{-1}x)|dx|b_{j}(y)|dy$
$\displaystyle\leq\smallint\limits_{I_{j}}\smallint\limits_{|z|>\textnormal{diam}(I_{j})}|K(z)-\phi_{j}\ast
K(z)|dz|b_{j}(y)|dy$
$\displaystyle\leq\smallint\limits_{I_{j}}\smallint\limits_{|z|>\textnormal{diam}(I_{j})}|K(z)-\phi_{j}\ast
K(z)|dz|b_{j}(y)|dy,$

where, in the last line we have used the changes of variables $x\mapsto
z=y^{-1}x,$ and then we observe that $|z|>\textnormal{diam}(I_{j})$ when $x\in
G\setminus I^{*}$ and $y\in I_{j}.$ Using that $\phi_{j}$ is supported in a
ball of radius

$R_{j}:=2^{-\frac{1}{1-\theta}}\textnormal{diam}(I_{j})^{\frac{1}{1-\theta}}$

and that $\|\phi_{j}\|_{L^{\infty}}\leq 1/|B(e,R_{j})|,$ we have that

$\displaystyle\smallint\limits_{|z|>\textnormal{diam}(I_{j})}|K(z)-\phi_{j}\ast
K(z)|dz$
$\displaystyle=\smallint\limits_{|z|>\textnormal{diam}(I_{j})}\left|K(z)\smallint\limits_{|y|<2^{-\frac{1}{1-\theta}}\textnormal{diam}(I_{j})^{\frac{1}{1-\theta}}}\phi_{j}(y)dy-\smallint\limits_{|y|<2^{-\frac{1}{1-\theta}}\textnormal{diam}(I_{j})^{\frac{1}{1-\theta}}}K(y^{-1}z)\phi_{j}(y)dy\right|dz$
$\displaystyle=\smallint\limits_{|z|>\textnormal{diam}(I_{j})}\left|\,\smallint\limits_{|y|<2^{-\frac{1}{1-\theta}}\textnormal{diam}(I_{j})^{\frac{1}{1-\theta}}}(K(z)-K(y^{-1}z))\phi_{j}(y)dy\right|dz$
$\displaystyle\leq\smallint\limits_{|y|<2^{-\frac{1}{1-\theta}}\textnormal{diam}(I_{j})^{\frac{1}{1-\theta}}}\smallint\limits_{|z|>\textnormal{diam}(I_{j})}|K(y^{-1}z)-K(z)|dz|\phi_{j}(y)|dy$
$\displaystyle\leq\frac{1}{|B(e,R_{j})|}\smallint\limits_{|y|<R_{j}}\smallint\limits_{|z|>2R_{j}^{(1-\theta)}}|K(y^{-1}z)-K(z)|dz\,dy$
$\displaystyle\lesssim\frac{1}{R_{j}^{n}}\smallint\limits_{|y|<R_{j}}\smallint\limits_{|z|>2R_{j}^{(1-\theta)}}|K(y^{-1}z)-K(z)|dz\,dy$
$\displaystyle\leq[K]_{H_{\infty,\theta}(G)},$

where we have used that $2R_{j}^{1-\theta}=\textnormal{diam}(I_{j}).$ Now, the
inequalities above allow us to finish the estimate of
$\|T(b-\tilde{b})\|_{L^{1}(G\setminus I^{*})}.$ Indeed,

$\displaystyle\|T(b-\tilde{b})\|_{L^{1}(G\setminus I^{*})}$
$\displaystyle\leq\sum_{j}\smallint\limits_{I_{j}}\smallint\limits_{G\setminus
I^{*}}|K(y^{-1}x)-\phi_{j}\ast K(y^{-1}x)|dx|b_{j}(y)|dy$
$\displaystyle\leq\sum_{j}\smallint\limits_{I_{j}}\smallint\limits_{|z|>\textnormal{diam}(I_{j})}|K(z)-\phi_{j}\ast
K(z)|dz|b_{j}(y)|dy$
$\displaystyle\lesssim[K]_{H_{\infty,\theta}(G)}\sum_{j}\smallint\limits_{I_{j}}|b_{j}(y)|dy\leq[K]_{H_{\infty,\theta}(G)}\|b\|_{L^{1}}$
$\displaystyle\lesssim[K]_{H_{\infty,\theta}(G)}\|f\|_{L^{1}}.$

Putting together the estimates above we deduce that

$\displaystyle|\\{x\in G\setminus
I^{*}:|Tb(x)|>\frac{\alpha}{2}\\}|\leq\frac{4}{\alpha}\|Tb-T\tilde{b}\|_{L^{1}(G\setminus
I^{*})}+|\\{x\in G\setminus I^{*}:|T\tilde{b}(x)|>\frac{\alpha}{4}\\}|$
$\displaystyle\lesssim\frac{4}{\alpha}[K]_{H_{\infty,\theta}(G)}\|f\|_{L^{1}}+|\\{x\in
G\setminus I^{*}:|T\tilde{b}(x)|>\frac{\alpha}{4}\\}|.$

Now, we will estimate the term in the right hand side of the previous
inequality. Indeed, note that

$|\\{x\in G\setminus I^{*}:|T\tilde{b}(x)|>\frac{\alpha}{4}\\}|\leq|\\{x\in
G\setminus I^{*}:|T\tilde{b}(x)|^{2}>\frac{\alpha^{2}}{16}\\}|$
$\leq\frac{16}{\alpha^{2}}\|T\tilde{b}\|_{L^{2}}^{2}.$

From now, let us consider the positive operator

$-\tilde{\Delta}:=1+\mathcal{L}_{G}.$ (3.16)

Now, using (1.2) we deduce that $T(-\tilde{\Delta})^{\frac{n\theta}{4}}$ is
bounded on $L^{2},$ (see Remark 2.2) and then

$\|T\tilde{b}\|^{2}_{L^{2}}\leq\|T(-\tilde{\Delta})^{\frac{n\theta}{4}}\|_{\mathscr{B}(L^{2})}^{2}\|(-\tilde{\Delta})^{-\frac{n\theta}{4}}\tilde{b}\|_{L^{2}}^{2}.$

In order to estimate the $L^{2}$-norm, let us use the following lemma whose
proof we postpone for a moment.

######  Lemma 3.1.

The function $F:=(-\tilde{\Delta})^{-\frac{n\theta}{4}}\tilde{b}$ can be
decomposed a the sums $F=F_{1}+F_{2},$ where $\|F_{2}\|_{L^{2}}^{2}\leq
C\alpha\gamma\|f\|_{L^{1}},$ and $F_{1}$ is also a sums of functions
$F_{1}^{j}$ with the following property.

  * •

There exists $M_{0}\in\mathbb{N},$ and $A^{\prime}>0,$ such that
$F_{1}=\sum_{j:\textnormal{diam}(I_{j})<1}F_{1}^{j}$,
$\|F_{1}^{j}\|_{L^{2}}^{2}\leq A^{\prime}\alpha^{2}|I_{j}|,$ and for any
$x\in\mathbb{R}^{n},$ there at most $M_{0}$ values of $j$ such that
$F_{1}^{j}(x)\neq 0.$

Let us continue with the proof of Theorem 1.1. Using Lemma 3.1 and the
inequalities in (3.5) we have that

$\displaystyle\|T\tilde{b}\|_{L^{2}}^{2}$
$\displaystyle\lesssim\|F_{1}\|_{L^{2}}^{2}+\|F_{2}\|_{L^{2}}^{2}\lesssim\alpha\gamma\|f\|_{L^{1}}+\sum_{j}\|F_{1}^{j}\|_{L^{2}}^{2}$
$\displaystyle\lesssim\alpha\gamma\|f\|_{L^{1}}+\sum_{j}\alpha^{2}|I_{j}|\lesssim\alpha\gamma\|f\|_{L^{1}}+\alpha^{2}(\gamma\alpha)^{-1}\|f\|_{L^{1}}$
$\displaystyle=(\gamma^{-1}+\gamma)\alpha\|f\|_{L^{1}}.$

Consequently

$|\\{x\in G\setminus
I^{*}:|T\tilde{b}(x)|>\frac{\alpha}{4}\\}|\lesssim\frac{16}{\alpha^{2}}\|T\tilde{b}\|_{L^{2}}^{2}\lesssim(\gamma^{-1}+\gamma)\alpha^{-1}\|f\|_{L^{1}}.$

Thus, we have proved that

$|\\{x\in\mathbb{R}^{n}:|Tf(x)|>\alpha\\}|\leq
C_{\gamma,[K]_{H_{\infty,\theta}(G)}}\alpha^{-1}\|f\|_{L^{1}},$ (3.17)

with $C:=C_{\gamma,[K]_{H_{\infty,\theta}(G)}}$ independent of $f.$ Because
$\gamma$ is fixed we have proved the weak $(1,1)$ type of $T.$ Note that the
$L^{p}$-boundedness of $T$ can be deduced from Marcinkiewicz interpolation
theorem and the standard duality argument. Thus, the proof is complete once
proved the statement in Lemma 3.1. To do so, let us consider the right-
convolution kernel $k_{\theta}:=(-\tilde{\Delta})^{-\frac{n\theta}{4}}\delta$
of the operator $(-\tilde{\Delta})^{-\frac{n\theta}{4}}$ and let us split the
function $(-\tilde{\Delta})^{-\frac{n\theta}{4}}\tilde{b}(x)$ as follows:

$\displaystyle(-\tilde{\Delta})^{-\frac{n\theta}{4}}\tilde{b}(x)$
$\displaystyle=\sum_{j}(-\tilde{\Delta})^{-\frac{n\theta}{4}}\tilde{b}_{j}(x)=\sum_{j}\tilde{b}_{j}\ast
k_{\theta}(x)$ $\displaystyle=\sum_{j:x\in I_{j}}\tilde{b}_{j}\ast
k_{\theta}(x)+\sum_{j:x\nsim I_{j}}\tilde{b}_{j}\ast
k_{\theta}(x)=:G_{1}(x)+G_{2}(x),$

where $G_{1}(x):=\sum_{j:x\sim I_{j}}\tilde{b}_{j}\ast k_{\theta}(x)$ and
$G_{2}(x):=\sum_{j:x\nsim I_{j}}\tilde{b}_{j}\ast k_{\theta}(x).$ We have
denoted by $x\sim I_{j},$ if $x$ belongs to $I_{j}$ or to some
$I_{j^{\prime}}$ with non-empty intersection with $I_{j}.$ By the properties
of these sets there are at most $M_{0}$ sets $I_{j^{\prime}}$ such that
$I_{j}\cap I_{j^{\prime}}\neq\emptyset.$ Also, the notation $x\nsim I_{j}$
will be employed to define the opposite of the previous property.

Let us prove the estimate

$\|G_{2}\|_{L^{2}}^{2}\leq C\alpha\gamma\|f\|_{L^{1}}.$ (3.18)

Observe that

$\displaystyle\|G_{2}\|_{L^{1}}$
$\displaystyle=\smallint\limits_{G}|\sum_{j:x\nsim
I_{j}}{b}_{j}\ast\phi_{j}\ast k_{\theta}(x)|dx\leq\sum_{j:x\nsim
I_{j}}\smallint\limits_{G}|{b}_{j}\ast\phi_{j}\ast k_{\theta}(x)|dx$
$\displaystyle\leq\sum_{j}\|{b}_{j}\ast\phi_{j}\ast
k_{\theta}\|_{L^{1}}\leq\sum_{j}\|{b}_{j}\|_{L^{1}}\|\phi_{j}\ast
k_{\theta}\|_{L^{1}}.$

Note that $(-\tilde{\Delta})^{-\frac{n\theta}{4}}$ is a pseudo-differential
operator of order $-n\theta/2,$ and consequently, its kernel satisfies the
estimate

$|k_{\theta}(x)|\leq
C|x|^{{-(-\frac{n\theta}{2}+n)}}\lesssim|x|^{-{n(1-\frac{\theta}{2})}},\,x\in
G\setminus\\{e\\}.$

The condition $0<\theta<1,$ implies that $k_{\theta}$ is an integrable
distribution and then

$\|\phi_{j}\ast
k_{\theta}\|_{L^{1}}\lesssim\|\phi_{j}\|_{L^{1}}\|k_{\theta}\|_{L^{1}}=\|k_{\theta}\|_{L^{1}}<\infty.$

So, we have that

$\displaystyle\|G_{2}\|_{L^{1}}\lesssim\sum_{j}\|b_{j}\|_{L^{1}}\lesssim\|f\|_{L^{1}}.$

So, for the proof of (3.18), and in view of the inequality
$\|G_{2}\|^{2}_{L^{2}}\leq\|G_{2}\|_{L^{1}}\|G_{2}\|_{L^{\infty}}$ is enough
to show that $\|G_{2}\|_{L^{\infty}}\lesssim\alpha\gamma\|f\|_{L^{1}}.$ To do
this, let us consider $j$ such that $x\nsim I_{j}.$ Since $\tilde{b}_{j}\ast
k_{\theta}={b}_{j}\ast\phi_{j}\ast k_{\theta},$ one has that

$\displaystyle|{b}_{j}\ast\phi_{j}\ast k_{\theta}(x)|$
$\displaystyle\leq\smallint\limits_{I_{j}}|\phi_{j}\ast
k_{\theta}(y^{-1}x)||b_{j}(y)|dy\leq\sup_{y\in I_{j}}|\phi_{j}\ast
k_{\theta}(y^{-1}x)|\smallint\limits_{I_{j}}|b_{j}(y)|dy$
$\displaystyle=\sup_{y\in I_{j}}|\phi_{j}\ast
k_{\theta}(y^{-1}x)||I_{j}|\times\frac{1}{|I_{j}|}\smallint\limits_{I_{j}}|b_{j}(y)|dy$

To continue, we follow as in [24, Page 26] the observation of Fefferman, that
in view of the property $x\nsim I_{j},$ we have that $\phi_{j}\ast
k_{\theta}(y^{-1}x)$ is essentially constant over the ball
$I_{j}=B(x_{j},r_{j})$ and we can estimate

$\sup_{y\in I_{j}}|\phi_{j}\ast
k_{\theta}(y^{-1}x)||I_{j}|\lesssim\smallint\limits_{I_{j}}|\phi_{j}\ast
k_{\theta}(y^{\prime-1}x)|dy^{\prime}.$ (3.19)

On the other hand, observe that the positivity of the kernel $k_{\theta},$ and
of $\phi_{j}$ leads to

$\displaystyle\smallint\limits_{I_{j}}|\phi_{j}\ast
k_{\theta}(y^{\prime-1}x)|dy^{\prime}\frac{1}{|I_{j}|}\smallint\limits_{I_{j}}|b_{j}(y)|dy$
$\displaystyle=\smallint\limits_{G}|\phi_{j}\ast
k_{\theta}(y^{\prime-1}x)|\left(\frac{1}{|I_{j}|}\smallint\limits_{I_{j}}|b_{j}(y)|dy\right)1_{I_{j}}(y^{\prime})dy^{\prime}$
$\displaystyle=\left(\frac{1}{|I_{j}|}\smallint\limits_{I_{j}}|b_{j}(y)|dy\times
1_{I_{j}}\right)\ast\phi_{j}\ast k_{\theta}(x).$

Consequently, we have

$\displaystyle|G_{2}(x)|$ $\displaystyle\leq\sum_{j:x\nsim
I_{j}}|\tilde{b}_{j}\ast k_{\theta}(x)|\lesssim\sum_{j:x\nsim
I_{j}}\left(\frac{1}{|I_{j}|}\smallint\limits_{I_{j}}|b_{j}(y)|dy\times
1_{I_{j}}\right)\ast\phi_{j}\ast k_{\theta}(x)$
$\displaystyle\lesssim\sum_{j:x\nsim I_{j}}\gamma\alpha\times
1_{I_{j}}\ast\phi_{j}\ast k_{\theta}(x)=\smallint\limits_{G}\sum_{j:x\nsim
I_{j}}\gamma\alpha\times 1_{I_{j}}\ast\phi_{j}(z)k_{\theta}(z^{-1}x)dz$
$\displaystyle\lesssim\gamma\alpha\|k_{\theta}\|_{L^{1}}\left\|\sum_{j:x\nsim
I_{j}}1_{I_{j}}\ast\phi_{j}\right\|_{L^{\infty}}.$

By observing that the supports of the functions $1_{I_{j}}\ast\phi_{j}$’s have
bounded overlaps we have that $\left\|\sum_{j:x\nsim
I_{j}}1_{I_{j}}\ast\phi_{j}\right\|_{L^{\infty}}<\infty,$ and that
$\|G_{2}\|_{L^{\infty}}\lesssim\gamma\alpha.$

It remains only to prove that
$\|G_{1}\|_{L^{2}}^{2}\lesssim\alpha\|f\|_{L^{1}}.$ Let us define

$G^{j}(x):=\begin{cases}{b}_{j}\ast\phi_{j}\ast k_{\theta}(x),&\text{ }x\in
I_{j},\\\ 0,&\,\text{otherwise. }\end{cases}$ (3.20)

Then $G_{1}=\sum_{j}G^{j}$ and in view of the finite overlapping of the balls
$I_{j}$’s, there is $M_{0}\in\mathbb{N},$ such that for any $x\in G,$
$G^{j}(x)\neq 0,$ for at most $M_{0}$ values of $j.$ Therefore, we have that

$\displaystyle\smallint_{G}|G_{1}(x)|^{2}dx$ $\displaystyle\leq
M_{0}\sum_{j}\smallint_{G}|G^{j}(x)|^{2}dx=\sum_{j}\smallint_{I_{j}}|{b}_{j}\ast\phi_{j}\ast
k_{\theta}(x)|^{2}dx$
$\displaystyle\leq\sum_{j}\|{b}_{j}\|_{L^{1}}^{2}\|\phi_{j}\ast
k_{\theta}\|_{L^{2}}^{2}\leq\sum_{j}\alpha^{2}\gamma^{2}|I_{j}|^{2}\|\phi_{j}\ast
k_{\theta}\|_{L^{2}}^{2}.$

Because our argument is purely local let us use the diffeomorphism
$\omega=\textnormal{exp}:\nu\rightarrow\nu^{\prime}$ in (3.10). Because
${B}(e,c)\subset\nu^{\prime},$ we can estimate the norm $\|\phi_{j}\ast
k_{\theta}\|_{L^{2}}$ in a small neighbourhood $\nu^{\prime}$ of the identity
element $e\in G.$ Note that for any $j$ with
$\textnormal{diam}(I_{j})<c/1000,$ we have the inclusion
$B(e,R_{j})\subset\nu^{\prime}.$ So, we have

$\|\phi_{j}\ast
k_{\theta}\|_{L^{2}}^{2}=\|(1+\mathcal{L}_{G})^{-\frac{n\theta}{4}}\phi_{j}\|^{2}_{L^{2}(\nu^{\prime})}\asymp\|(1+\mathcal{L}_{\mathbb{R}^{n}})^{-\frac{n\theta}{4}}{\phi}^{\prime}_{j}\|_{L^{2}(\nu)}^{2},$

where

$\phi_{j}^{\prime}=\phi_{j}\circ\omega=\frac{1}{B(e,R_{j})}1_{B(e,R_{j})}\circ\omega,\,\,\textnormal{supp}(\phi_{j}^{\prime})\subset\nu.$

Note that $(1+\mathcal{L}_{\mathbb{R}^{n}})^{-\frac{n\theta}{4}}$ is bounded
on $L^{2},$ and clearly $\phi_{j}^{\prime}\in L^{2},$ so that
$\phi_{j}^{\prime}\in\textnormal{Dom}((1+\mathcal{L}_{\mathbb{R}^{n}})^{-\frac{n\theta}{4}}).$
Also,

$\textnormal{supp}[\phi_{j}^{\prime}]\subset{B}(0,R_{j}^{\prime}),\,R_{j}^{\prime}\sim
R_{j}.$ (3.21)

Define for any $j,$

$\psi_{j}^{\prime}(X):=|B(0,R_{j}^{\prime})|\phi_{j}^{\prime}(R_{j}^{\prime}X),\,X\in\nu.$

Then, one has the identity

$\phi_{j}^{\prime}(X)=\frac{1}{|B(0,R_{j}^{\prime})|}\psi_{j}^{\prime}\left(\frac{X}{R_{j}^{\prime}}\right),\,X\in\nu.$

Observe that $|B(0,R_{j}^{\prime})|=v_{n}{R_{j}^{\prime}}^{n},$ where $v_{n}$
is the volume of the unite ball $B(0,1).$ Also, note that
$\|\psi_{j}^{\prime}\|_{L^{\infty}}=1,$ and
$\textnormal{supp}(\psi_{j}^{\prime})\subset B(0,1).$ Using the Plancherel
theorem, we have that

$\displaystyle\|(1+\mathcal{L}_{\mathbb{R}^{n}})^{-\frac{n\theta}{4}}{\phi}_{j}^{\prime}\|_{L^{2}(\nu)}$
$\displaystyle=\|(1+|\eta|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}[\phi_{j}^{\prime}](\eta)\|_{L^{2}(\nu)}$
$\displaystyle=\|(1+|\eta|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}\left[\frac{1}{|B(0,R_{j}^{\prime})|}\psi_{j}^{\prime}\left(\frac{\cdot}{R_{j}^{\prime}}\right)\right]\left(\eta\right)\|_{L^{2}(\nu)}$
$\displaystyle=(1/v_{n})\|(1+|\eta|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}\left[\frac{1}{{R_{j}^{\prime}}^{n}}\psi_{j}^{\prime}\left(\frac{\cdot}{R_{j}^{\prime}}\right)\right]\left(\eta\right)\|_{L^{2}(\nu)}$
$\displaystyle=(1/v_{n})\|(1+|\eta|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}[\psi_{j}^{\prime}]\left(R_{j}^{\prime}\eta\right)\|_{L^{2}(\nu)}.$

Observe that

$\displaystyle\|(1+|\eta|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}[\psi_{j}^{\prime}]\left(R_{j}\eta\right)\|_{L^{2}(\nu)}^{2}$
$\displaystyle=\smallint\limits|(1+|\eta|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}[\psi_{j}^{\prime}]\left(R_{j}\eta\right)|^{2}d\eta$
$\displaystyle={R^{\prime}_{j}}^{-n}\smallint\limits|(1+|{R^{\prime}_{j}}^{-1}z|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}[\psi_{j}^{\prime}](z)|^{2}dz$
$\displaystyle={R^{\prime}_{j}}^{-n}{R^{\prime}_{j}}^{n\theta}\smallint\limits|(R_{j}^{\prime}+|z|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}[\psi_{j}^{\prime}](z)|^{2}dz$
$\displaystyle\leq{R^{\prime}_{j}}^{-n}{R^{\prime}_{j}}^{n\theta}\smallint\limits|(|z|^{2})^{-\frac{n\theta}{4}}\mathscr{F}_{\mathbb{R}^{n}}[\psi_{j}^{\prime}](z)|^{2}dz$
$\displaystyle\asymp{R^{\prime}_{j}}^{-n}{R^{\prime}_{j}}^{n\theta}\smallint\limits|(-\Delta)^{-\frac{n\theta}{4}}\psi_{j}^{\prime}(x)|^{2}dx$
$\displaystyle=R_{j}^{-n(1-\theta)}\|(-\Delta)^{-\frac{n\theta}{4}}\psi_{j}^{\prime}\|_{L^{2}}^{2}.$

Note that for $\varepsilon>0$ small enough, the support of any
$x\mapsto\chi_{k}^{\varepsilon}(x_{k}^{-1}x)$ is inside of $\nu^{\prime}.$ In
consequence, we can estimate

$\displaystyle\|(-\Delta)^{-\frac{n\theta}{4}}\psi_{j}^{\prime}\|_{L^{2}}$
$\displaystyle\asymp\|(-\mathcal{L}_{G})^{-\frac{n\theta}{4}}\psi_{j}^{\prime}\circ\textnormal{exp}\|_{L^{2}}\leq\sum_{j=0}^{N_{0}(\varepsilon)}\|(-\mathcal{L}_{G})^{-\frac{n\theta}{4}}[\psi_{j}^{\prime}\circ\textnormal{exp}\cdot\chi_{k}^{\varepsilon}(x_{k}^{-1}\cdot)]\|_{L^{2}}$
$\displaystyle\lesssim\sum_{j=0}^{N_{0}(\varepsilon)}\|\psi_{j}^{\prime}\circ\textnormal{exp}\cdot\chi_{k}^{\varepsilon}(x_{k}^{-1}\cdot)\|_{L^{2}}\asymp\|\psi_{j}^{\prime}\circ\textnormal{exp}\|_{L^{2}}\asymp\|\psi_{j}^{\prime}\|_{L^{2}},$

where we have used the $L^{2}$-boundedness of the operator
$(-\mathcal{L}_{G})^{-\frac{n\theta}{4}}$ in view of the Stein identity (see
[43, Page 58])

$(-\mathcal{L}_{G})^{-\frac{n\theta}{4}}=\left(-\mathcal{L}_{G}+\textnormal{Proj}_{\textnormal{Ker}(-\mathcal{L}_{G})}\right)^{-\frac{n\theta}{4}}-\textnormal{Proj}_{\textnormal{Ker}(-\mathcal{L}_{G})},$

and of Remark 2.4 of [40]. Indeed, we have that
$(-\mathcal{L}_{G})^{-\frac{n\theta}{4}}$ is a pseudo-differential operator on
$G$ of order $-n\theta/2$ and hence bounded on $L^{2}(G).$ Note that, this
argument does not apply in the case of $\mathbb{R}^{n}$ because the spectrum
of the Euclidean Laplacian is continuous and the negative powers of
$-\Delta_{\mathbb{R}^{n}}$ are not pseudo-differential operators while the
powers of $1-\Delta_{\mathbb{R}^{n}}$ are pseudo-differential operators with
symbols in the Kohn-Nirenberg classes, see e.g. Taylor [45].

Continuing with the proof, since $\|\psi_{j}\|_{L^{\infty}}=1$ and $\psi_{j}$
is supported in the unit ball, we have that $\|\psi_{j}\|_{L^{2}}\leq 1.$
Consequently,

$\|\phi_{j}\ast
k_{\theta}\|_{L^{2}}^{2}=\|(1+\mathcal{L}_{G})^{-\frac{n\theta}{4}}\phi_{j}\|_{L^{2}(\nu^{\prime})}\asymp\|(1+\mathcal{L}_{\mathbb{R}^{n}})^{-\frac{n\theta}{4}}{\phi}^{\prime}_{j}\|_{L^{2}(\nu)}\lesssim
R_{j}^{-n(1-\theta)}\sim|I_{j}|^{-1}.$

Finally, we deduce that

$\displaystyle\smallint_{G}|G_{1}(x)|^{2}dx$
$\displaystyle\leq\sum_{j}\alpha^{2}\gamma^{2}|I_{j}|^{2}\|\phi_{j}\ast
k_{\theta}\|_{L^{2}}^{2}\lesssim\sum_{j}\alpha^{2}\gamma^{2}|I_{j}|^{2}|I_{j}|^{-1}=\alpha^{2}\gamma^{2}\sum_{j}|I_{j}|$
$\displaystyle\lesssim\alpha\gamma\|f\|_{L^{1}}\lesssim_{\gamma}\alpha\|f\|_{L^{1}},$

in view of (3.5). Consequently, we can take $F_{2}:=G_{2},$ $F_{1}:=G_{1}$ and
$F_{1}^{j}:=G_{1}^{j}.$ Thus, the proof of Lemma 3.1 is complete as well as
the proof of Theorem 1.1. ∎

##  4\. Examples

In this section we give some applications of Theorem 1.1 on the torus, on
$\textnormal{SU}(2)\cong\mathbb{S}^{3}$ and for oscillating multipliers. We
refer the reader to [13, Section 4] for a variety of examples on the
boundedness of oscillating singular integrals on Lie groups of polynomial
growth.

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

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

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

$\widehat{K}(\ell):=\smallint_{\mathbb{T}^{n}}e_{-\ell}(x)K(x)dx,\,\,\ell\in\mathbb{Z}^{n},$
(4.1)

the Fourier transform condition (1.10) becomes equivalent to the estimate

$\forall\ell\in\mathbb{Z}^{n},\,\,|\widehat{K}(\ell)|\leq
C\langle\ell\rangle^{-\frac{n\theta}{2}},\,\
\langle\ell\rangle:=(1+4\pi^{2}|\ell|^{2})^{\frac{1}{2}}\sim|\ell|:=\sqrt{\ell_{1}^{2}+\cdots+\ell_{n}^{2}}.$
(4.2)

Note that the kernel condition (1.11) takes the form

$[K]_{H_{\infty,\theta}}=\sup_{R>0}\sup_{|y|<R}\smallint\limits_{|x|\geq
2R^{1-\theta}}|K(x-y)-K(x)|dxdy<\infty.$ (4.3)

In view of Theorem 1.1, a convolution operator $T$ associated to a convolution
kernel $K,$ satisfying the Fourier transform condition (4.2) and the
smoothness condition (4.3) is of weak (1,1) type, that is
$T:L^{1}(\mathbb{T}^{n})\rightarrow L^{1,\infty}(\mathbb{T}^{n})$ is bounded.

###  4.2. 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 [39]) and let us define

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

where $X_{3}=[X_{1},X_{2}].$ The system $X=\\{X_{1},X_{2},X_{3}\\}$ is an
orthonormal basis of the Lie algebra $\mathfrak{su}(2)$ of
$\textnormal{SU}(2),$ and its positivie Laplacian is given by

$\mathcal{L}_{\textnormal{SU}(2)}=-X_{1}^{2}-X_{2}^{2}-X_{3}^{2}=-\partial_{0}^{2}-\frac{1}{2}[\partial_{+}\partial_{-}+\partial_{-}\partial_{+}].$
(4.4)

We record that the unitary dual of $\textnormal{SU}(2)$ (see [39]) can be
identified as

$\widehat{\textnormal{SU}}(2)\equiv\\{[t_{l}]:2l\in\mathbb{N},d_{l}:=\dim
t_{l}=(2l+1)\\}\sim\frac{1}{2}\mathbb{N}.$ (4.5)

There are explicit formulae for $t_{l}$ as functions of Euler angles in terms
of the so-called Legendre-Jacobi polynomials, see [39]. Again, by following
e.g. [39], 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 (1.10)
takes the form

$\exists
C>0,\,\forall\ell\in\frac{1}{2}\mathbb{N},\quad\|\widehat{K}(\ell)\|_{\textnormal{op}}\leq
C(1+\ell(\ell+1))^{-\frac{3\theta}{4}}\sim(1+\ell)^{-\frac{3\theta}{2}},$
(4.6)

where $\widehat{K}(\ell)=\smallint_{\textnormal{SU}(2)}K(Z)t_{\ell}(Z)^{*}dZ,$
is the Fourier transform of the group of $K.$ In view of Theorem 1.1, if $K$
satisfies (4.6) and the kernel condition

$[K]_{H_{\infty,\theta}}:=\sup_{R>0}\sup_{|Y|<R}\smallint\limits_{|X|\geq
2R^{1-\theta}}|K(Y^{-1}X)-K(X)|dXdY<\infty,$ (4.7)

then $T$ is of weak (1,1) type, that is
$T:L^{1}(\textnormal{SU}(2))\rightarrow L^{1,\infty}(\textnormal{SU}(2))$
extends to a bounded operator.

###  4.3. Application to oscillating multipliers

Let us illustrate our main Theorem 1.1 in the context of oscillating Fourier
multipliers. On $G,$ the prototype of a oscillating singular integral is the
spectral multiplier of $\mathcal{L}_{G},$ given by

$T_{\theta}(\mathcal{L}_{G}):=(1+\mathcal{L}_{G})^{-\frac{n\theta}{4}}e^{i(1+\mathcal{L}_{G})^{\frac{\theta}{2}}},\,\,0\leq\theta<1.$
(4.8)

Indeed, $T_{\theta}(\mathcal{L}_{G})$ is a convolution operator with right-
convolution kernel $K=K_{\theta}$ whose Fourier transform is given by

$\widehat{K}(\xi)=\langle\xi\rangle^{-\frac{n\theta}{2}}{e^{i\langle\xi\rangle^{\theta}}},\,\langle\xi\rangle:=(1+\lambda_{[\xi]})^{\frac{1}{2}},\,[\xi]\in\widehat{G}.$

Let $d(x,y)$ be the geodesic distance on $G$ induced by the Riemannian metric
$g$. It was proved by Chen and Fan in [15] that $K(x)$ behaves essentially as
$c_{n}d(x,e)^{-n}e^{ic_{n}^{\prime}d(x,e)^{\theta^{\prime}}},$ where
$\theta^{\prime}=\frac{\theta}{\theta-1},$ and consequently that

$|\nabla K(x)|=|(X_{1}K(x),\cdots,X_{n}K(x))|\lesssim
d(x,e)^{-n-1+\theta^{\prime}},$

from which it is well known that $K$ satisfies (1.11), that is,

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

justifying the term ‘oscillating’ for this family of multipliers. In view of
Theorem 1.1, $T_{\theta}(\mathcal{L}_{G})$ extend to an operator of weak (1,1)
type.

######  Remark 4.1.

By following the analysis of pseudo-differential operators under local
coordinates systems, one can prove that for $0\leq\theta<\frac{1}{2},$ the
operator $T_{\theta}(\mathcal{L}_{G})$ belongs to the Hörmander class
$\textnormal{Op}(S^{m}_{1-\theta,\theta})$ on $G,$ with the order
$m=-\frac{n\theta}{2},$ see e.g the book of M. Taylor [45] for details.

By microlocalising the Fefferman weak (1,1) estimate in [24], we deduce that
for any $0\leq\theta<\frac{1}{2},$ $T_{\theta}(\mathcal{L}_{G})$ extends to an
operator of weak $(1,1)$ type. This is consequence of the fact that for all
$0\leq\theta<\frac{1}{2},$ the class
$\textnormal{Op}(S^{m}_{1-\theta,\theta})$ is invariant under changes of
coordinates. However, the range $\frac{1}{2}\leq\theta<1$ can be covered by
our Theorem 1.1.

We end this remark by observing that the boundedness of (4.8) from the Hardy
space $H^{1}(G)$ into $L^{1}(G)$ has been proved by Chen and Fan in [15] in
the complete range $0\leq\theta<1$.

######  Remark 4.2.

Other generalisations on compact Lie groups of estimates for oscillating
integrals to pseudo-differential operators have been considered in the papers
[8, 22] and [40]. We refer the reader to [10, Chapter V] for Fourier transform
conditions of Fourier multipliers of weak $(1,1)$ type on compact Lie groups
in subelliptic settings.

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