# Titchmarsh theorems on Damek-Ricci spaces via moduli of continuity of higher
order

Manoj Kumar  Manoj Kumar Department of Mathematics Indian Institute of Science
Bangalore, India<EMAIL_ADDRESS>,  Vishvesh Kumar  Vishvesh Kumar
Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent
University, Belgium<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-
<EMAIL_ADDRESS>
###### Abstract.

A classical theorem of Titchmarsh relates the $L^{2}$-Lipschitz functions and
decay of the Fourier transform of the functions. In this note, we prove the
Titchmarsh theorem for Damek-Ricci space (also known as harmonic $NA$ groups)
via moduli of continuity of higher orders. We also prove an analogue of
another Titchmarsh theorem which provides integrability properties of the
Fourier transform for functions in the Hölder Lipschitz spaces.

###### Key words and phrases:

Titchmarsh theorems, Damek-Ricci spaces, Harmonic $NA$ groups, Helgason
transform, Moduli of continuity, Generalized Lipschitz class

###### 2010 Mathematics Subject Classification:

Primary 43A85 Secondary 22E30

##  1\. Introduction

The classical Titchmarsh theorem [28, Theorem 85] characterizes the
$L^{2}$-Lipschitz functions in terms of certain decay of the Fourier transform
of the functions. It can be stated as follows: Let $\alpha\in(0,1)$ and $f\in
L^{2}(\mathbb{R}).$ Then, $\|\tau_{t}f-f\|_{2}\leq C_{1}t^{\alpha}$ for all
sufficiently small $t>0,$ if and only if

$\int_{|\xi|>\frac{1}{t}}|\widehat{f}(\xi)|^{2}\,d\xi\leq C_{2}t^{2\alpha},$

for all sufficiently small $t>0.$ Here, $\widehat{f}$ is the Fourier transform
of $f$ and $\tau_{t}$ is the translation operator.

The Titchmarsh theorem has been extensively studied in many different contexts
on various groups, for instances, the higher dimensional Euclidean spaces [31,
5], the Vilenkin groups [32], the special linear group of real matrices of
order two $SL_{2}(\mathbb{R})$ [33], the rank one symmetric spaces of non-
compact type [25, 15], the $p$-adic groups [26] and the compact homogeneous
manifolds [12]. In terms of the moduli of continuity, the theorem has been
explored on $\mathbb{R}$ [30, 13] and the rank one symmetric spaces [16]. See
[6, 17, 18] for some growth properties of the Fourier transform on certain
spaces via moduli of continuity.

Damek-Ricci spaces, also known as Harmonic $NA$ groups, are natural
generalizations of the Iwasawa $NA$ groups of the real rank one simple Lie
groups. Particularly, the rank one symmetric spaces of non-compact type form a
subclass of the Damek-Ricci spaces. In general, the Damek-Ricci spaces need
not be symmetric spaces.

Very recently, Titchmarsh type results for Damek-Ricci spaces were explored in
[23, 14]. In this note, we extend the classical Titchmarsh theorem to the
setting of Damek-Ricci spaces in terms of the moduli of continuity of higher
orders (see Corollary 3.5). We note here that it is new on the Damek-Ricci
spaces even in the case of moduli of continuity of order one.

[18, Theorem 1.5 (A)] provides certain decay properties of the Helgason
Fourier transform for functions in the generalized Besov spaces. We also prove
an analogue of this theorem on Damek-Ricci spaces.

Another Titchmarsh theorem [28, Theorem 84] provides the integrability
properties of the Fourier transform of functions belonging to the Hölder-
Lipschitz spaces. This theorem has been studied for various groups, for
examples, $SL_{2}(\mathbb{R})$ [33], the Euclidean space [5], the compact
homogeneous manifolds [12] and the Damek-Ricci spaces [23, 14]. We also prove
a generalization of this Titchmarsh theorem over Damek-Ricci spaces.

##  2\. Preliminaries

In this section, we recall basics required about the Damek-Ricci spaces and
moduli of continuity. Throughout the paper, we denote by $C,C_{1},C_{2},...$
constants whose values may vary from one line to the other.

###  2.1. Fourier analysis on Damek-Ricci spaces

For the details about the analysis and geometry of Damek-Ricci spaces and
associated Fourier analysis, one can refer to [8, 9, 10, 11, 1, 4, 7, 27, 21,
19, 2, 3, 22].

Let $\mathfrak{n}$ be a two-step real nilpotent Lie algebra, equipped with an
inner product $\langle\,,\,\rangle.$ Let $\mathfrak{z}$ denote the center of
$\mathfrak{n}$ and let $\mathfrak{v}$ denote the orthogonal complement of
$\mathfrak{z}$ in $\mathfrak{n}$ w.r.t. $\langle\,,\,\rangle.$ Suppose that
the dimensions of $\mathfrak{v}$ and $\mathfrak{z}$ are denoted by $m$ and $k$
respectively as real vector spaces. The Lie algebra $\mathfrak{n}$ is said to
be $H$-type algebra if for each $Z\in\mathfrak{z},$ the map
$J_{Z}:\mathfrak{v}\rightarrow\mathfrak{v}$ given by

$\langle J_{Z}X,Y\rangle=\langle
Z,[X,Y]\rangle,\,\,\,\,\,\,X,Y\in\mathfrak{v},\,Z\in\mathfrak{z},$

satisfies the condition $J_{Z}^{2}=-\|Z\|^{2}I_{\mathfrak{v}}.$ Here,
$I_{\mathfrak{v}}$ denotes the identity operator on $\mathfrak{v}.$ Kaplan
[19] proved that for $Z\in\mathfrak{z}$ with $\|Z\|=1$ one has
$J_{Z}^{2}=-I_{\mathfrak{v}}$; that is, $J_{Z}$ induces a complex structure on
$\mathfrak{v}.$ Therefore, $m=\dim(\mathfrak{v})$ is always even.

A connected and simply connected Lie group $N$ is said to be $H$-type if its
Lie algebra is an $H$-type algebra. Since $\mathfrak{n}$ is nilpotent, it
follows that the exponential map is a global diffeomorphism. Hence, the
elements of $N=\exp{\mathfrak{n}}$ can be parametrized by $(X,Z)$, for
$X\in\mathfrak{v}$ and $Z\in\mathfrak{z}.$ From the Campbell-Baker-Hausdorff
formula, the multiplication on $N$ is given by

$(X,Z)(X^{\prime},Z^{\prime})=\left(X+X^{\prime},Z+Z^{\prime}+\frac{1}{2}[X,X^{\prime}]\right).$

Note that the group $A=\mathbb{R}_{+}^{*}$ acts on $N$ by nonisotropic
dilations: $(X,Y)\mapsto(a^{\frac{1}{2}}X,aZ).$ Therefore, by setting
$\dim(\mathfrak{z})=k,$ the homogeneous dimension of $N$ is given by
$Q=\frac{m}{2}+k.$ At times, we also use symbol $\rho$ for $\frac{Q}{2}.$
Hence, $\dim(\mathfrak{s})=m+k+1,$ denoted by $d.$

Let $S=N\ltimes A$ be the semidirect product of $N$ with $A$ under the
aforementioned action. Therefore, the group multiplication on $S$ is defined
by

$(X,Z,a)(X^{\prime},Z^{\prime},a^{\prime})=\left(X+a^{\frac{1}{2}}X^{\prime},Z+aZ^{\prime}+\frac{1}{2}a^{\frac{1}{2}}[X,X^{\prime}],aa^{\prime}\right).$

Then, $S$ is a solvable (connected and simply connected) Lie group with Lie
algebra $\mathfrak{s}=\mathfrak{z}\oplus\mathfrak{v}\oplus\mathbb{R}$ and Lie
bracket

$[(X,Z,\ell),(X^{\prime},Z^{\prime},\ell^{\prime})]=\left(\frac{1}{2}\ell
X^{\prime}-\frac{1}{2}\ell^{\prime}X,\ell
Z^{\prime}-\ell^{\prime}Z+[X,X]^{\prime},0\right).$

The group $S$ is equipped with the left-invariant Riemannian metric induced by

$\langle(X,Z,\ell),(X^{\prime},Z^{\prime},\ell^{\prime})\rangle=\langle
X,X^{\prime}\rangle+\langle Z,Z^{\prime}\rangle+\ell\ell^{\prime}$

on $\mathfrak{s}.$ The associated left Haar measure $dx$ on the group $S$ is
given by $a^{-Q-1}dX\,dZ\,da=a^{-Q-1}dn\,da,$ where $dX,\,dZ$ and $da$ are the
Lebesgue measures on $\mathfrak{v},\mathfrak{z}$ and $\mathbb{R}_{+}^{*},$
respectively. The elements of $A$ will be identified with $a_{t}=e^{t},$
$t\in\mathbb{R}.$ We will also write any element $s\in S$ as $na_{t}$ by
writing $S=NA.$ In particular, any element $a_{t}\in A$ can be thought as an
element of $S$ by writing $a_{t}=e_{N}a_{t},$ where $e_{N}$ is the identity
element of $N.$ The group $S$ can be realized as the unit ball
$B(\mathfrak{s})$ in $\mathfrak{s}$ using the Cayley transform $C:S\rightarrow
B(\mathfrak{s})$ (see [1]).

To define the Helgason Fourier transform on the group $S,$ we need to describe
the notion of the Poisson kernel ([4]). The Poisson Kernel
$\mathcal{P}:S\times N\rightarrow\mathbb{R}$ is defined by
$\mathcal{P}(na_{t},n^{\prime})=P_{a_{t}}(n^{\prime-1}n),$ where

$P_{a_{t}}(n)=P_{a_{t}}(X,Z)=Ca_{t}^{Q}\left(\left(a_{t}+\frac{|X|^{2}}{4}\right)^{2}+|Z|^{2}\right)^{-Q},\,\,\,\,n=(X,Z)\in
N\mbox{ and }t\in\mathbb{R}.$

The value of $C$ is appropriately chosen so that $\int_{N}P_{a}(n)dn=1$ and
$P_{1}(n)\leq 1.$ Now, we list some useful properties of the Poisson kernel;
see [21, 27, 4]. For $\lambda\in\mathbb{C},$ the complex power of the Poisson
kernel is given by

$\mathcal{P}_{\lambda}(x,n)=\mathcal{P}(x,n)^{\frac{1}{2}-\frac{i\lambda}{Q}}.$

It is known ([27, 4]) that for each fixed $x\in S,$
$\mathcal{P}_{\lambda}(x,\cdot)\in L^{p}(N)$ for $1\leq p\leq\infty$ if
$\lambda=i\gamma_{p}\rho,$ where $\gamma_{p}=\frac{2}{p}-1.$ An important
feature of the Poisson kernel $\mathcal{P}_{\lambda}(x,n)$ is that it is
constant on the hypersurfaces
$H_{n,a_{t}}=\\{n\sigma(a_{t}n^{\prime}):\,n^{\prime}\in N\\}.$ Here, $\sigma$
denotes the geodesic inversion on $S;$ see [7].

Let $\Delta_{S}$ denote the Laplace-Beltrami operator on $S.$ Then, for every
fixed $n\in N,$ the function $\mathcal{P}_{\lambda}(x,n)$ is an eigenfunction
of $\Delta_{S}$ with eigenvalue $-(\lambda^{2}+\frac{Q^{2}}{4});$ see [4]. The
Helgason Fourier transform of a measurable function $f$ on $S$ is given by

$\widetilde{f}(\lambda,n)=\int_{S}f(x)\,\mathcal{P}_{\lambda}(x,n)dx$

provided that the integral exists.

For $f\in C_{c}^{\infty}(S),$ the Fourier inversion formula [4, Theorem 4.4]
is given by

$f(x)=C\int_{\mathbb{R}}\int_{N}\widetilde{f}(\lambda,n)\,\mathcal{P}_{-\lambda}(x,n)|c(\lambda)|^{-2}\,d\lambda
dn.$

The Plancherel theorem [4, Theorem 5.1] states that the Helgason Fourier
transform extends to an isometry from $L^{2}(S)$ onto the space
$L^{2}(\mathbb{R}_{+}\times N,|c(\lambda)|^{-2}d\lambda dn).$ For precise
value of the constants; see [4]. The function $|c(\lambda)|$ satisfies:

$\displaystyle|c(\lambda)|^{-2}\asymp\begin{cases}\lambda^{2}\quad&\mbox{ if
}\lambda\in[0,1]\\\ \lambda^{d-1}\quad&\mbox{ if }\lambda>1\end{cases}.$ (1)

See [29, Theorem 1.14] and [27, Lemma 4.8].

Let $e$ denote the identity element of $S$ and let $\tilde{\mu}$ be the metric
induced by the canonical left invariant Riemannian structure on $S.$ A
function $f$ on $S$ is said to be radial if for all $x,y\in S,$ $f(x)=f(y)$
whenever $\tilde{\mu}(x,e)=\tilde{\mu}(y,e).$ Note that the radial functions
on $S$ can be identified with the functions $f=f(r)$ of the geodesic distance
$r=\tilde{\mu}(x,e)\in[0,\infty).$ Clearly, $\tilde{\mu}(a_{t},e)=|t|$ for
$t\in\mathbb{R},$ where we have identified $a_{t}$ with $e_{N}a_{t}\in S:=NA$
with $e_{N}$ being the identity of $N.$ For any radial function $f,$ sometimes
we use the notation $f(a_{t})=f(t).$ The elementary spherical function
$\phi_{\lambda}(x)$ is given by

$\phi_{\lambda}(x):=\int_{N}\mathcal{P}_{\lambda}(x,n)\mathcal{P}_{-\lambda}(x,n)\,dn.$

Note that the function $\phi_{\lambda}$ is a radial eigenfunction of the
Laplace-Beltrami operator $\Delta_{S}$ with eigenvalue
$-(\lambda^{2}+\frac{Q^{2}}{4});$ see [1, 4]. Also,
$\phi_{\lambda}(x)=\phi_{-\lambda}(x),\,\,\phi_{\lambda}(x)=\phi_{\lambda}(x^{-1})$
and $\phi_{\lambda}(e)=1.$ In [1], the authors proved that the radial part (in
geodesic polar coordinates) of the Laplace-Beltrami operator $\Delta_{S}$
given by

$\textnormal{rad}\,\Delta_{S}=\frac{\partial^{2}}{\partial
t}+\left\\{\frac{m+l}{2}\coth{\frac{t}{2}}+\frac{k}{2}\tanh{\frac{t}{2}}\right\\}\frac{\partial}{\partial
t},$

is (by substituting $r=\frac{t}{2}$) equal to
$\frac{1}{4}\mathcal{L}_{\alpha,\beta}$ with indices $\alpha=\frac{m+l+1}{2}$
and $\beta=\frac{l-1}{2}.$ Here, $\mathcal{L}_{\alpha,\beta}$ is the Jacobi
operator and Koornwinder [24] treated it in detail. It is worth pointing out
that we are in the ideal condition of the Jacobi analysis with
$\alpha>\beta>\frac{-1}{2}.$ To be more precise, the Jacobi functions
$\phi_{\lambda}^{\alpha,\beta}$ are related to the elementary spherical
functions $\phi_{\lambda}$ by
$\phi_{\lambda}(t)=\phi_{2\lambda}^{\alpha,\beta}(\frac{t}{2});$ see [1].
Hence, we have the following important lemma; see [25].

######  Lemma 2.1.

Let $t,\lambda\in\mathbb{R}_{+}.$ Then,

  * •

$|\phi_{\lambda}(t)|\leq 1.$

  * •

$|1-\phi_{\lambda}(t)|\leq\frac{t^{2}}{2}\Big{(}4\lambda^{2}+\frac{Q^{2}}{4}\Big{)}.$

  * •

There exists a constant $C>0$ such that $|1-\phi_{\lambda}(t)|\geq C$ for
$\lambda t\geq 1.$

Let $\sigma_{t}$ denote the normalized surface measure induced by the left
invariant Riemannian metric on the geodesic sphere $S_{t}=\\{y\in
S:\tilde{\mu}(y,e)=t\\}$ of radius $t$. Then, $\sigma_{t}$ is a nonnegative
radial measure. The spherical mean operator $M_{t}$ is given by
$M_{t}f:=f*\sigma_{t}$ for a suitable function $f$ on $S.$ Note that
$M_{t}f(x)=\mathcal{R}(f^{x})(t).$ Here, $f^{x}$ is the right translation of
function $f$ by $x$ and $\mathcal{R}$ is the radialization operator given by

$\mathcal{R}f(x)=\int_{S_{\nu}}f(y)\,d\sigma_{\nu}(y)$

for a suitable function $f$ and $\nu=r(x)=\tilde{\mu}(C(x),0),$ where $C$ is
the Cayley transform. It is easy to check that $\mathcal{R}f$ is a radial
function. Also, $\mathcal{R}f=f$ for any radial function $f.$ Thus, for a
radial function $f,$ $M_{t}f$ is the usual translation of $f$ by $t.$ The
spherical mean operator $M_{t}$ is a bounded linear operator on $L^{p}(S)$ and

$\displaystyle\widetilde{M_{t}f}(\lambda,n)=\widetilde{f}(\lambda,n)\phi_{\lambda}(t)$
(2)

for a suitable function $f$ on $S.$ Further, $M_{t}f$ converges to $f$ as
$t\rightarrow 0.$ See [21].

###  2.2. Moduli of continuity of higher orders

Let $\omega$ be a mapping from $I\subset\mathbb{R}$ to the set $[0,\infty).$
The map $\omega$ is said to be almost increasing if there exists a constant
$C\geq 1$ such that $\omega(t)\leq C\omega(s)$ whenever $t\leq s$ and $t,s\in
I.$ The map $w$ is said to be almost decreasing if there exists a constant
$C\geq 1$ such that $\omega(t)\leq C\omega(s)$ whenever $t\geq s$ and $t,s\in
I.$

Let $\delta_{0}>0$ and $k\in\mathbb{R}_{+}.$ A continuous function
$\omega_{k}:[0,\delta_{0}]\rightarrow\mathbb{R}_{+}$ is said to be a $k$th
order modulus of continuity if $\omega_{k}(0)=0,$ $\omega_{k}(t)$ is almost
increasing on $t\in[0,\delta_{0}]$ and $\frac{\omega_{k}(t)}{t^{k}}$ is almost
decreasing on $t\in[0,\delta_{0}].$ Note that if $\omega$ is a $k$th-order
modulus of continuity then $\omega$ is also an $m$th-order modulus of
continuity for all $m\geq k.$

We say that the $k$th-order modulus of continuity $\omega_{k}$ belongs to the
Zygmund class $\mathscr{Z}^{0}$ if there exists a constant $C>0$ such that

$\int_{0}^{t}\frac{\omega_{k}(s)}{s}\,ds\leq C\omega_{k}(t),\
t\in[0,\delta_{0}].$

We say that the $k$th-order modulus of continuity $\omega_{k}$ belongs to the
Zygmund class $\mathscr{Z}_{k}$ if there exists a constant $C$ such that

$\int_{t}^{\delta_{0}}\frac{\omega_{k}(s)}{s^{1+k}}\,ds\leq
C\frac{\omega_{k}(t)}{t^{k}},\ t\in[0,\delta_{0}].$

The class $\mathscr{Z}^{0}\cap\mathscr{Z}_{k}$ is called the Zygmund-Bari-
Stechkin class. Some classical examples in the Zygmund-Bari-Stechkin class are
$t^{\alpha},\ t^{\alpha}\left(\ln{\frac{1}{t}}\right)^{\gamma}$ and
$t^{\alpha}\left(\ln\ln\frac{1}{t}\right)^{\gamma},$ where $\alpha\in(0,k)$
and $\gamma\in\mathbb{R}.$ For more details on Zygmund classes, see [20].

The crucial behaviour of the functions discussed above is near zero. In
Theorem 3.4, we also need to do certain estimations over
$[\delta_{0},\infty)$. Note that there is no loss of generality of the results
in assuming certain restrictions, as in Theorem 3.4, on $\omega_{k}$ over the
interval $[\delta_{0},\infty).$

##  3\. Titchmarsh theorems on Damek-Ricci spaces

In this section, we present the main result of this paper that provides a
description of generalised Lipschitz class functions in terms of the Helgason
Fourier transform. We also prove the decay properties of the Helgason Fourier
transform for the functions in the generalized Besov spaces. Lastly, we
discuss certain integrability properties of the Helgason Fourier transform for
the functions in the Hölder Lipschitz spaces.

Now, we begin with the following definition of generalised Lipschitz class.
For that, let us denote the harmonic $NA$ group by $S$ and the $k$th-order
modulus of continuity by $\omega_{k}$.

######  Definition 3.1.

A function $f\in L^{2}(S)$ is said to be in the generalised Lipschitz class
$\mathrm{Lip}(\omega_{k})$ if there exists a positive constant $C$ such that
for all sufficiently small $t\in(0,1)$ we have $\|M_{t}f-f\|_{2}\leq
C\omega_{k}(t).$

The following theorem provides a growth property of image of the generalised
Lipschitz functions under the Helgason Fourier transform.

######  Theorem 3.2.

Let $\omega_{k}$ be a $k$th-order modulus of continuity. If
$f\in\mathrm{Lip}(\omega_{k})$ then there exists a positive constant $C$ such
that for all sufficiently small $t\in(0,1),$

$\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq
Ct^{d-1}\omega_{k}(t)^{2}.$

###### Proof.

Using the Plancherel theorem and (2) we have

$\displaystyle\|M_{t}f-f\|_{2}^{2}=$
$\displaystyle\int_{\mathbb{R}_{+}}\int_{N}|\widetilde{(M_{t}f-f)}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}_{+}}\int_{N}|1-\phi_{\lambda}(a_{t})|^{2}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda.$

Since $f\in\mathrm{Lip}(\omega_{k}),$ it follows that

$\displaystyle\int_{\mathbb{R}_{+}}\int_{N}|1-\phi_{\lambda}(a_{t})|^{2}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda\leq
C^{2}\omega_{k}(t)^{2}.$ (3)

Now,

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda=$
$\displaystyle
t^{d-1}\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\frac{1}{t^{d-1}}\,d\lambda$
$\displaystyle\leq$ $\displaystyle
t^{d-1}\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\lambda^{d-1}\,d\lambda.$

Since $\frac{1}{t}>1,$ then using (1) we obtain

$\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq
Ct^{d-1}\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn|c(\lambda)|^{-2}\,d\lambda.$

Using Lemma 2.1 we have

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq$
$\displaystyle\frac{Ct^{d-1}}{C_{1}^{2}}\int_{\frac{1}{t}}^{\infty}|1-\phi_{\lambda}(a_{t})|^{2}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn|c(\lambda)|^{-2}\,d\lambda$
$\displaystyle\leq$
$\displaystyle\frac{Ct^{d-1}}{C_{1}^{2}}\int_{0}^{\infty}|1-\phi_{\lambda}(a_{t})|^{2}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn|c(\lambda)|^{-2}\,d\lambda.$

Hence, by (3) we have

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq$
$\displaystyle C_{2}t^{d-1}\omega_{k}(t)^{2}.\qed$

We need the following lemma to prove a converse to the previous theorem.

######  Lemma 3.3.

Let $\omega_{k}$ belongs to the Zygmund class $\mathscr{Z}^{0}$ and $f\in
L^{2}(S).$ The following are equivalent.

  1. (i)

There exists a positive constant $C$ such that

$\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda\leq
C\omega_{k}(t)^{2},\ t\in(0,1).$

  2. (ii)

There exists a positive constant $C$ such that

$\int_{\frac{1}{t}}^{\frac{2}{t}}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda\leq
C\omega_{k}(t)^{2},\ t\in(0,1).$

###### Proof.

(i) implies (ii) is clear. Now, assume that (ii) holds. Then, for $i\geq 0,$
we get

$\int_{\frac{2^{i}}{t}}^{\frac{2^{i+1}}{t}}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda\leq
C\omega_{k}\left(\frac{t}{2^{i}}\right)^{2}.$

Therefore,

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda=$
$\displaystyle\sum_{i=0}^{\infty}\int_{\frac{2^{i}}{t}}^{\frac{2^{i+1}}{t}}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda$
$\displaystyle\leq$ $\displaystyle
C\sum_{i=0}^{\infty}\omega_{k}\left(\frac{t}{2^{i}}\right)^{2}.$

Let $\mu>0$ be such that $\mu<m(\omega_{k}).$ Here, $m(\omega_{k})$ is the
lower MO index [20, Pg. 31]. Since $\omega_{k}$ belongs to the Zygmund class
$\mathscr{Z}^{0},$ it follows by [20, Theorem 2.10] that
$\frac{\omega_{k}(t)}{t^{\mu}}$ is almost increasing. Since
$\frac{t}{2^{i}}\leq t,\ i\geq 0,$ it follows by the definition of almost
increasing that

$\frac{\omega_{k}(\frac{t}{2^{i}})}{(\frac{t}{2^{i}})^{\mu}}\leq
C_{1}\frac{\omega_{k}(t)}{t^{\mu}}.$

Therefore, we have

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda\leq$
$\displaystyle
C_{2}\sum_{i=0}^{\infty}\left(\left(\frac{t}{2^{i}}\right)^{\mu}\frac{\omega_{k}(t)}{t^{\mu}}\right)^{2}$
$\displaystyle=$ $\displaystyle
C_{2}\omega_{k}(t)^{2}\sum_{i=0}^{\infty}\left(\frac{1}{2^{i}}\right)^{2\mu}.$

Hence, we have

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda\leq$
$\displaystyle C_{3}\omega_{k}(t)^{2}.\qed$

The following theorem is a converse to Theorem 3.2 under more assumptions.

######  Theorem 3.4.

Let $k\leq 2$ and assume that $\omega_{k}$ belongs to the Zygmund classes
$\mathscr{Z}^{0}$ and $\mathscr{Z}_{k}.$ Let $\omega_{k}(t)$ be bounded below
by a positive number on interval $[\delta_{0},\infty)$ and let
$\frac{\omega_{k}(t)^{2}}{t^{5}}\in L^{1}([\delta_{0},\infty)).$ For a
function $f\in L^{2}(S),$ if there exists a positive constant $C$ such that
for all sufficiently small $t\in(0,1),$ we have

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq
Ct^{d-1}\omega_{k}(t)^{2},$ (4)

then $f\in\mathrm{Lip}(\omega_{k}).$

###### Proof.

Using (4) we have

$\int_{\frac{1}{t}}^{\frac{2}{t}}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda\leq\left(\frac{2}{t}\right)^{d-1}\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq
C_{1}\omega_{k}(t)^{2}.$

Then, by Lemma 3.3 we have

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda\leq$
$\displaystyle C_{2}\omega_{k}(t)^{2}.$ (5)

As in the proof of Theorem 3.2, using the Plancherel theorem and (2) we get

$\displaystyle\|M_{t}f-f\|_{2}^{2}=$
$\displaystyle\int_{\mathbb{R}_{+}}\int_{N}|1-\phi_{\lambda}(a_{t})|^{2}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda=I_{1}+I_{2}.$
(6)

Here,

$I_{1}=\int_{0}^{\frac{1}{t}}\int_{N}|1-\phi_{\lambda}(a_{t})|^{2}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda$

and

$I_{2}=\int_{\frac{1}{t}}^{\infty}\int_{N}|1-\phi_{\lambda}(a_{t})|^{2}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda.$

First, we estimate $I_{2}.$ Since $\frac{1}{t}>1,$ using (1) and Lemma 2.1 we
have

$I_{2}\leq
2^{2}C\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\lambda^{d-1}\,dn\,d\lambda.$

Thus, applying (5) we get

$\displaystyle I_{2}\leq 4C_{3}\omega_{k}(t)^{2}.$ (7)

Now, we estimate $I_{1}.$ Using Lemma 2.1 we have

$\displaystyle I_{1}\leq$
$\displaystyle\int_{0}^{\frac{1}{t}}\int_{N}\left(\frac{t^{2}}{2}\left(4\lambda^{2}+\frac{Q^{2}}{4}\right)\right)^{2}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda$
$\displaystyle\leq$ $\displaystyle
C_{4}t^{4}\int_{0}^{\frac{1}{t}}\int_{N}\lambda^{4}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda$
$\displaystyle+C_{5}t^{4}\int_{0}^{\frac{1}{t}}\int_{N}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda.$

By the Plancherel theorem in the second term, we get

$I_{1}\leq
C_{4}t^{4}\int_{0}^{\frac{1}{t}}\int_{N}\lambda^{4}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda+C_{6}t^{4}.$

Denote
$\psi(s)=\int_{s}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}|c(\lambda)|^{-2}\,dn\,d\lambda,\
s\in\mathbb{R}_{+}.$ By the Plancherel theorem, $\psi(s)$ is bounded on
$\mathbb{R}_{+}.$ Then, using the assumption that $\omega_{k}(s)$ is bounded
below by a positive number on $[\delta_{0},\infty),$ and (5), we have
$\psi(s)\leq C_{3}\omega_{k}\left(\frac{1}{s}\right)^{2},\
s\in\mathbb{R}_{+}.$ Therefore,

$\displaystyle I_{1}\leq$ $\displaystyle
C_{4}t^{4}\int_{0}^{\frac{1}{t}}\lambda^{4}(-\psi^{\prime}(\lambda))\,d\lambda+C_{6}t^{4}$
$\displaystyle=$ $\displaystyle
C_{4}t^{4}\left(-\frac{1}{t^{4}}\psi\left(\frac{1}{t}\right)+\int_{0}^{\frac{1}{t}}4\lambda^{3}\psi(\lambda)\,d\lambda\right)+C_{6}t^{4}$
$\displaystyle\leq$ $\displaystyle
C_{4}t^{4}\int_{0}^{\frac{1}{t}}4\lambda^{3}\psi(\lambda)\,d\lambda+C_{6}t^{4}$
$\displaystyle\leq$ $\displaystyle
C_{7}t^{4}\int_{0}^{\frac{1}{t}}\lambda^{3}\omega_{k}\left(\frac{1}{\lambda}\right)^{2}\,d\lambda+C_{6}t^{4}$
$\displaystyle=$ $\displaystyle
C_{7}t^{4}\int_{t}^{\infty}\frac{\omega_{k}(\lambda)^{2}}{\lambda^{5}}\,d\lambda+C_{6}t^{4}.$

Using the assumption $\frac{\omega_{k}(t)^{2}}{t^{5}}\in
L^{1}([\delta_{0},\infty)),$ we get

$\displaystyle I_{1}\leq$ $\displaystyle
C_{7}t^{4}\int_{t}^{\delta_{0}}\frac{\omega_{k}(\lambda)^{2}}{\lambda^{5}}\,d\lambda+C_{8}t^{4}.$

Since $k\leq 2$ then $4-k\geq k.$ Therefore,
$\frac{w_{k}(\lambda)}{\lambda^{4-k}}$ is almost decreasing. Then, we have

$\displaystyle I_{1}\leq$ $\displaystyle
C_{7}t^{4}\frac{\omega_{k}(t)}{t^{4-k}}\int_{t}^{\delta_{0}}\frac{\omega_{k}(\lambda)}{\lambda^{k+1}}\,d\lambda+C_{8}t^{4}.$

Using the assumption that $\omega_{k}$ belongs to the Zygmund class
$\mathscr{Z}_{k}$ in the first term and using the the fact that $k\leq 2$
implies that $\frac{w_{k}(t)}{t^{2}}$ is almost decreasing which implies that
$t^{2}\leq C\omega_{k}(t)$ in the second term, it follows that

$\displaystyle I_{1}\leq$ $\displaystyle C_{9}\omega_{k}(t)^{2}.$ (8)

Hence, the result follows form (6), (8) and (7). ∎

The following result is the main result of the paper and it is a direct
consequence of Theorem 3.2 and Theorem 3.4.

######  Corollary 3.5.

Let $k\leq 2$ and assume that $\omega_{k}$ belongs to the Zygmund classes
$\mathscr{Z}^{0}$ and $\mathscr{Z}_{k}.$ Let $\omega_{k}(t)$ be bounded below
by a positive number on interval $[\delta_{0},\infty)$ and let
$\frac{\omega_{k}(t)^{2}}{t^{5}}\in L^{1}([\delta_{0},\infty)).$ A function
$f\in L^{2}(S)$ belongs to the generalised Lipschitz class
$\mathrm{Lip}(\omega_{k})$ if and only if there exists a positive constant $C$
such that for all sufficiently small $t\in(0,1),$ we have

$\displaystyle\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq
Ct^{d-1}\omega_{k}(t)^{2}.$

If we take $\omega_{2}(t)=t^{\alpha}\left(\ln{\frac{1}{t}}\right)^{\gamma},$
where $\alpha\in(0,2),\gamma\in\mathbb{R}$ and $t\in[0,\delta_{0}],$ and
$\omega_{2}$ is constant on $[\delta_{0},\infty)$, then as a special case of
Corollary 3.5 we obtain the following result.

######  Corollary 3.6.

Let $\alpha\in(0,2),\gamma\in\mathbb{R}$ and $f\in L^{2}(S).$ Then, we have

$\|M_{t}f-f\|_{2}\leq C_{1}t^{\alpha}\left(\ln{\frac{1}{t}}\right)^{\gamma},$

for all sufficiently small $t\in(0,1)$ if and only if

$\int_{\frac{1}{t}}^{\infty}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,d\lambda\leq
C_{2}t^{2\alpha+d-1}\left(\ln{\frac{1}{t}}\right)^{2\gamma},$

for all sufficiently small $t\in(0,1).$

The following result is an analogue of [18, Theorem 1.5 (A)] which gives decay
properties of the Helgason Fourier transform for functions in the generalized
Besov spaces.

######  Theorem 3.7.

Let $\alpha>0$ and $f\in L^{2}(S).$ If

$\displaystyle\int_{0}^{1}\left(\frac{\|M_{t}f-f\|_{2}}{t^{\alpha}}\right)^{2}\,\frac{dt}{t}<\infty,$
(9)

then

$\displaystyle\int_{0}^{\infty}\int_{\lambda=t}^{2t}\int_{N}\left(\frac{|\widetilde{f}(\lambda,n)|}{\lambda^{-\alpha}}\right)^{2}\,dn\,|c(\lambda)|^{-2}d\lambda\,\frac{dt}{t}<\infty.$
(10)

###### Proof.

Let $f\in L^{2}(S)$ be such that (9) holds. We decompose

$\int_{0}^{\infty}\int_{\lambda=t}^{2t}\int_{N}\left(\frac{|\widetilde{f}(\lambda,n)|}{\lambda^{-\alpha}}\right)^{2}\,dn\,|c(\lambda)|^{-2}d\lambda\,\frac{dt}{t}=I_{1}+I_{2},$

where

$I_{1}=\int_{0}^{\frac{1}{2}}\int_{\lambda=t}^{2t}\int_{N}\left(\frac{|\widetilde{f}(\lambda,n)|}{\lambda^{-\alpha}}\right)^{2}\,dn\,|c(\lambda)|^{-2}d\lambda\,\frac{dt}{t}$

and

$I_{2}=\int_{\frac{1}{2}}^{\infty}\int_{\lambda=t}^{2t}\int_{N}\left(\frac{|\widetilde{f}(\lambda,n)|}{\lambda^{-\alpha}}\right)^{2}\,dn\,|c(\lambda)|^{-2}d\lambda\,\frac{dt}{t}.$

First, we estimate $I_{1}.$ Using the Plancherel theorem we have

$\displaystyle I_{1}\leq$ $\displaystyle
C_{1}\int_{0}^{\frac{1}{2}}t^{2\alpha}\int_{\lambda=t}^{2t}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,|c(\lambda)|^{-2}d\lambda\,\frac{dt}{t}$
$\displaystyle\leq$ $\displaystyle
C_{2}\|f\|_{2}^{2}\int_{0}^{\frac{1}{2}}t^{2\alpha}\,\frac{dt}{t}<\infty.$

Now, we estimate $I_{2}.$ By the change of variables $t=\frac{1}{2s}$ we
obtain

$\displaystyle I_{2}=$
$\displaystyle\int_{0}^{1}\int_{\lambda=\frac{1}{2s}}^{\frac{1}{s}}\int_{N}\left(\frac{|\widetilde{f}(\lambda,n)|}{\lambda^{-\alpha}}\right)^{2}\,dn\,|c(\lambda)|^{-2}d\lambda\,\frac{ds}{s}$
$\displaystyle\leq$ $\displaystyle
C_{3}\int_{0}^{1}\frac{1}{s^{2\alpha}}\int_{\lambda=\frac{1}{2s}}^{\frac{1}{s}}\int_{N}|\widetilde{f}(\lambda,n)|^{2}\,dn\,|c(\lambda)|^{-2}d\lambda\,\frac{ds}{s}.$

Since $s\lambda\leq 1,$ it follows by [21, Theorem 4.7 (a)] that

$\displaystyle I_{2}\leq$ $\displaystyle
C_{4}\int_{0}^{1}\frac{1}{s^{2\alpha}}\|M_{s}f-f\|_{2}^{2}\,\frac{ds}{s}.$

Therefore, by (9), $I_{2}$ is finite. Hence (10) holds. ∎

Now, we deal with the analogue of the [28, Theorem 37]. This provides an
improvement of the well known Hausdorff-Young inequality. For $q=p^{\prime},$
the next theorem is also proved in [23].

Let $\alpha\in(0,1]$ and $p\geq 1.$ A function $f\in L^{p}(S)$ is said to be
in the Hölder-Lipschitz space $Lip(\alpha;p)$ if there exists a positive
constant $C$ such that for all sufficiently small $t\in(0,1)$ we have
$\|M_{t}f-f\|_{p}\leq Ct^{\alpha}.$

######  Theorem 3.8.

Let $\alpha\in(0,1],$ $p\in(1,2]$ and $q\in[p,p^{\prime}].$ Assume that $f\in
Lip(\alpha;p)$ and define

$F(\lambda):=\left(\int_{N}|\widetilde{f}(\lambda+i\gamma_{q}\rho,n)|^{q}\,dn\right)^{\frac{1}{q}}.$

Then, $F\in L^{\beta}(\mathbb{R}_{+},|c(\lambda)|^{-2}\,d\lambda),$ for any
$\beta\in\left(\frac{dp^{\prime}}{d+\alpha p^{\prime}},p^{\prime}\right].$

###### Proof.

Using [21, Theorem 4.7] we get

$\int_{\mathbb{R}}\min\\{1,(\lambda
t)^{2p^{\prime}}\\}F(\lambda)^{p^{\prime}}|c(\lambda)|^{-2}\,d\lambda\leq
C\|M_{t}f-f\|_{p}^{p^{\prime}}.$

Then, for small enough $t$ we have

$\int_{|\lambda|<\frac{1}{t}}\lambda^{2p^{\prime}}F(\lambda)^{p^{\prime}}|c(\lambda)|^{-2}\,d\lambda\leq
C\left(\frac{\|M_{t}f-f\|_{p}}{t^{2}}\right)^{p^{\prime}}\leq
C_{1}t^{(\alpha-2)p^{\prime}}.$ (11)

For $s=\frac{1}{t}$ and $\beta<p^{\prime},$ define

$\phi(s):=\int_{1}^{s}\lambda^{2\beta}F(\lambda)^{\beta}|c(\lambda)|^{-2}\,d\lambda.$

By the Hölder inequality with
$\frac{\beta}{p^{\prime}}+\left(1-\frac{\beta}{p^{\prime}}\right)=1$ and using
(1), we get

$\displaystyle\phi(s)$ $\displaystyle\leq
C_{2}\left(\int_{1}^{s}\lambda^{2p^{\prime}}F(\lambda)^{p^{\prime}}|c(\lambda)|^{-2}\,d\lambda\right)^{\frac{\beta}{p^{\prime}}}\left(\int_{1}^{s}|c(\lambda)|^{-2}\,d\lambda\right)^{1-\frac{\beta}{p^{\prime}}}$
$\displaystyle\leq
C_{2}\left(\int_{1}^{s}\lambda^{2p^{\prime}}F(\lambda)^{p^{\prime}}|c(\lambda)|^{-2}d\lambda\right)^{\frac{\beta}{p^{\prime}}}\left(\int_{1}^{s}\lambda^{d-1}\,d\lambda\right)^{1-\frac{\beta}{p^{\prime}}}.$

Using (11) we have

$\phi(s)\leq
C_{2}s^{(2-\alpha)\beta+d\left(1-\frac{\beta}{p^{\prime}}\right)}.$

Hence, using the integration by parts, we obtain

$\displaystyle\int_{1}^{s}F(\lambda)^{\beta}|c(\lambda)|^{-2}\,d\lambda$
$\displaystyle=\int_{1}^{s}\lambda^{-2\beta}\,\phi^{\prime}(\lambda)\,d\lambda$
$\displaystyle=s^{-2\beta}\phi(s)+2\beta\int_{1}^{s}\lambda^{-2\beta-1}\phi(\lambda)\,d\lambda$
$\displaystyle\leq
C_{2}s^{-\alpha\beta+d\left(1-\frac{\beta}{p^{\prime}}\right)}+C_{3}\int_{1}^{s}\lambda^{-1-\alpha\beta+d\left(1-\frac{\beta}{p^{\prime}}\right)}\,d\lambda$
$\displaystyle=O(1)$

provided that $-\alpha\beta+d\left(1-\frac{\beta}{p^{\prime}}\right)<0,$ that
is, $\beta>\frac{dp^{\prime}}{d+\alpha p^{\prime}}.$ ∎

## Acknowledgment

Manoj Kumar is supported by the NBHM post-doctoral fellowship with Ref.
number: 0204/3/2021/R&D-II/7356. Vishvesh Kumar and Michael Ruzhansky are
supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial
Differential Equations, the Methusalem programme of the Ghent University
Special Research Fund (BOF) (Grant number 01M01021) and by FWO Senior Research
Grant G011522N. MR is also supported by EPSRC grant EP/R003025/2.

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