11institutetext: R. Duduchava22institutetext: The University of Georgia,
Institute of Mathematics, IV, 77a Merab Kostava St, Tbilisi 0171, Georgia.
Tel.: +995-555333855
22email<EMAIL_ADDRESS>33institutetext: A.Razmadze Mathematical Institute,
Tbilisi State University, Tamarashvili str. 6, Tbilisi 0177, Georgia

# CONVOLUTION EQUATIONS ON THE LIE GROUP $G=(-1,1)$

Roland Duduchava

(Received: date / Accepted: date)

###### Abstract

The interval $G=(-1,1)$ turns into a Lie group under the group operation
$x\circ y:=(x+y)(1+xy)^{-1},\qquad x,y\in G$. This enables definition of the
invariant measure $dG(x):=(1-x^{2})^{-1}dx$ and the Fourier transformation
${\mathscr{F}}\hskip-2.84526pt_{G}$ on the interval $G$ and, as a consequence,
we can consider Fourier convolution operators
$W^{0}_{G,a}:={\mathscr{F}}\hskip-2.84526pt_{G}^{-1}a{\mathscr{F}}\hskip-2.84526pt_{G}$
on $G$. This class of convolutions includes celebrated Prandtl, Tricomi and
Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary
order with the natural weighted derivative
$\mathfrak{D}_{G}u(x)=(1-x^{2})u^{\prime}(x)$, $x\in G$. Equations are solved
in the scale of Bessel potential $\mathbb{H}^{s}_{p}(G,dG(x))$, $1\leqslant
p\leqslant\infty$, and Hölder-Zygmound $\mathbb{Z}^{\nu}(G)$,
$0<\mu,\nu<\infty$ spaces, adapted to the group $G$. Boundedness of
convolution operators (the problem of multipliers) is discussed. The symbol
$a(\xi)$, $\xi\in\mathbb{R}$, of a convolution equation $W^{0}_{G,a}u=f$
defines solvability: the equation is uniquely solvable if and only if the
symbol $a$ is elliptic. The solution is written explicitly with the help of
the inverse symbol.

We touch shortly the multidimensional analogue-the Lie group $G^{n}$.

###### Keywords:

Lie group Fourier transformation Convolution equation Prandtl equation Tricomi
equation Lavrentjev-Bitsadze equation

###### MSC:

MSC 45A05 45E10 43A25 42A45

###### Contents

  1. 1 Lie groups $G$ and $G^{n}$
  2. 2 Function spaces on $G$
  3. 3 Convolutions and multipliers on $G$
  4. 4 Convolution integro-differential equation
  5. 5 Prandtl equation
  6. 6 Tricomi equation
  7. 7 Lavrentjev-Bitsadze equation

## Introduction

The present investigation is inspired by papers of V.E. Petrov Pe06a ; Pe06b ;
SP (20), where the author applied the finite interval Fourier transformation

$\displaystyle({\mathscr{F}}\hskip-2.84526pt_{G}v)(\xi):=\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}\frac{v(y)dy}{1-y^{2}},\qquad\xi\in\mathbb{R}$
(1)

to the investigation of convolution equations of the following type

$\displaystyle
c_{0}u(x)+\int_{-1}^{1}k\left(\frac{x-y}{1-xy}\right)\frac{v(y)dy}{1-y^{2}}=h(x),\qquad
x\in G:=(-1,1).$ (2)

(2) becomes convolution if the interval $G=(-1,1)$ is endowed with the group
operation $x\circ y:=\displaystyle\frac{x+y}{1+xy}$, making $G$ a Lie group
(see details in § 1 below).

To the class of convolution equations (2) belong the celebrated Prandtl
equation

$\displaystyle{\boldsymbol{P}}u(x)=\frac{c_{0}u(x)}{1-x^{2}}+\frac{c_{1}}{\pi}\int_{-1}^{1}\displaystyle\frac{u^{\prime}(y)dy}{y-x}=f(x),\qquad
x\in G$ (3)

(actually a special case of the Prandtl equation for a parabolic wing chord),
singular Tricomi equation

$\displaystyle{\boldsymbol{T}}v(x)=c_{0}v(x)+\frac{c_{1}}{\pi}\int_{-1}^{1}\displaystyle\frac{v(y)dy}{y-x}+\frac{c_{2}}{\pi}\int_{-1}^{1}\displaystyle\frac{v(y)dy}{1-xy}=g(x),\qquad
x\in(-1,1)$ (4)

and Lavrentjev-Bitsadze equation

$\displaystyle{\boldsymbol{L}}{\boldsymbol{B}}\varphi(x)=c_{0}\varphi(x)+\frac{c_{1}}{\pi}\int_{0}^{1}\left[\displaystyle\frac{1}{y-x}+\displaystyle\frac{1-2y}{x+y-2xy}\right]\varphi(y)dy=h(x),$
(5) $\displaystyle x\in G^{+}:=(0,1).$

Equations (2)-(4) and equation (5) after the variable transformation mapping
$G^{+}:=(0,1)\to G$ (see (92) below), are particular cases of the following
integro-differential equation

$\displaystyle{\boldsymbol{A}}u(x)$ $\displaystyle:=$
$\displaystyle\sum_{k=0}^{m}\left[c_{k}\mathfrak{D}^{k}_{G}u(x)+d_{k}\mathfrak{D}^{m_{k}}_{G}\int_{-1}^{1}{\mathscr{K}}_{k}\left(\frac{x-y}{1-xy}\right)(\mathfrak{D}^{n_{k}}_{G}u)(y)\frac{dy}{1-y^{2}}\right]$
(6) $\displaystyle=$ $\displaystyle w(x),\qquad x\in G=(-1,1),$

where $c_{1},\ldots,c_{m}$ and $d_{1},\ldots,d_{m}$ are complex valued
constants, $m_{0},\ldots,m_{m}$, $n_{0},\ldots,n_{m}$ are non-negative
integers,
${\mathscr{K}}_{1},\ldots,{\mathscr{K}}_{m}\in\mathbb{L}_{1}(G,dG(x))$.
${\boldsymbol{A}}=W^{0}_{G.a}$ represents a convolution operator on the Lie
group $G$ (see § 3 below) and

$\displaystyle\mathfrak{D}_{G}:=(1-x^{2})\frac{d}{dx}.$ (7)

is the natural differential operator on this group, which means that it is a
convolution operator and generates the Lie algebra on the Lie group $G$.

Equations (2)–(6) have ample of applications in Mechanics and Mathematical
physics and were investigated by many authors (see surveys in Pe06a ; SP (20),
(Du, 79, § 20), Ka (75) and the recent papers AA (21); AP (16)).

Equations (3) and (4) were solved by V. E. Petrov in Pe06a  (also see Pe06b ;
SP (20)) by using the ${\mathscr{P}}$-transformation, which he defined as the
equivalent transformation ${\mathscr{P}}=t_{*}{\mathscr{F}}x_{*}$ to the
classical Fourier transformation ${\mathscr{F}}$ on the real axes under the
transformations $t_{*}\varphi(x):=\varphi(t(x))$, $x_{*}\psi(t):=\psi(x(t))$
which are inverse to each-other. Here

$\displaystyle t(x)=\displaystyle\frac{1}{2}\ln\frac{1+x}{1-x},\qquad
x(t)=\tanh\,t,\qquad x\in G,\quad t\in\mathbb{R}.$ (8)

We have noted that the transformation ${\mathscr{P}}$ on a segment $G=(-1,1)$
considered in Pe06a  can be interpreted as the Fourier transformation on the
group $G$. Then the equations (2)–(5) are interpreted as convolutions and the
Fourier transformation (1) applied to these equations transforms them to a
simple operators of multiplication by the ”symbol” (the Fourier image of the
kernel) of the Fourier image of an unknown function. That motivated
description of multipliers and finding criteria of invertibility of
convolution operators. Also that motivated introduction of the appropriate
Bessel potential spaces, based on the differential operator $\mathfrak{D}_{G}$
to consider these equations in better space settings.

These aspects were missed in Pe06a ; Pe06b  and in the subsequent papers.
Equations (2)–(4) and some boundary value problem for the equation (5) on the
part of the unit sphere, were solved in Pe06a ; Pe06b  in the general Banach
spaceless setting, while in SP (20) equation (3) was investigated in the
Bessel potential space setting $\widetilde{\mathbb{H}}^{s}(G)$, $-1\leqslant
s\leqslant 1$, which was defined as the image of the corresponding Bessel
potential space on the axes
$\widetilde{\mathbb{H}}^{s}(G):=t_{*}\mathbb{H}^{s}(\mathbb{R})$ under the
isomorphism $t_{*}\varphi(x):=\varphi(t(x))$ (cf. (8)).

In our approach we use a natural Fourier transformation on the Lie Group
${\mathscr{F}}\hskip-2.84526pt_{G}$ (cf. § 1 below), the natural differential
operator $\mathfrak{D}_{G}$ (see (1) and (7), and the Bessel potential
operators
${\boldsymbol{\Lambda}}^{s}={\mathscr{F}}\hskip-2.84526pt_{G}^{-1}(1+|\xi|^{2})^{s/2}{\mathscr{F}}\hskip-2.84526pt_{G}$,
and define Bessel potential $\mathbb{H}^{s}_{p}(G,dG(x))$ and Sobolev
$\mathbb{W}^{m}_{p}(G,dG(x))$ spaces for arbitrary $s\in\mathbb{R}$,
$1\leqslant p\leqslant\infty$, $m=0,1,\ldots$ in a standard way. A convolution
operator
$W^{0}_{G,a}:={\mathscr{F}}\hskip-2.84526pt_{G}^{-1}a{\mathscr{F}}\hskip-2.84526pt_{G}$
on $G$ is defined and theorems on multipliers in the Bessel potential spaces
are proved. Criteria for the Fredholm property and unique solvability of
equations (2)-(6) in the Bessel potential spaces are found and explicit
formulae for their solution are indicated (see § § 4-7). Hölder-Zygmound
spaces $\mathbb{Z}^{\nu}(G)$, $0<\mu<\infty$ are defined and used to establish
a priori smoothness of solutions to equations (2)–(6).

What we expose here for the group $G$ can also be done for the multidimensinal
case $G^{n}:=G\times\cdots\times G$: Fourier transformations
${{\mathscr{F}}\hskip-2.84526pt_{G^{n}}}\\!\\!^{\pm 1}$, Bessel potential
$\mathbb{H}^{s}_{p}(G^{n},dG(x))$, Sobolev $\mathbb{W}^{m}_{p}(G^{n},dG(x))$
and Hölder-Zygmound $\mathbb{Z}^{\mu}(G^{n})$ spaces, theorems on multipliers
and solvability results for multidimensional convolution equations
$W^{0}_{G^{n},a}U:={{\mathscr{F}}\hskip-2.84526pt_{G^{n}}}\\!\\!^{-1}a{\mathscr{F}}\hskip-2.84526pt_{G^{n}}U=F$,
$U\in\mathbb{H}^{s}_{p}(G^{n},dG^{n}(x))$,
$F\in\mathbb{H}^{s-r}_{p}(G^{n},dG(x))$ and the multi-variable symbol belongs
to the multiplier class $a\in{\mathscr{M}}_{p}(G^{n})$ (see § 3). Therefore
some assertions will be formulated for multi-variable case (see, e.g.,
Proposition 5).

The paper is organized as follows. In § 1 we expose results on the Lie Group
$G=(-1,1)$-invariant Haar measure, characters, Fourier transformation and its
inverse, define Lebesgue space $\mathbb{L}_{p}(G,dG(x))$. In § 2 we define
Bessel potential $\mathbb{H}^{s}_{p}(G,dG(x))$, Sobolev
$\mathbb{W}^{m}_{p}(G,dG(x))$ and Hölder-Zygmound $\mathbb{Z}^{\mu}(G)$ spaces
($1\leqslant p\leqslant\infty$, $s\in\mathbb{R}$, $m=1,2,\ldots$, $\mu>0$,
$\gamma\in\mathbb{R}$). In § 3 convolution operators on the Lie group $G$ are
defined and a couple of theorems on their boundedness are proved. In § 4 -§ 7
we expose results on equations (2)-(5).

##  1 Lie groups $G$ and $G^{n}$

On the interval $G:=(-1,1)$ we define the following additional group operation

$\displaystyle x\circ y:=\frac{x+y}{1+xy},\qquad x,y\in G,$ (9)

which makes $G$ a Lie group. The inverse element to $x$ is $-x$ and the group
operation is the isomorphism of the Lie group $x\circ y:G\times G\rightarrow
G.$ Indeed,

$\partial_{y}\displaystyle\frac{x+y}{1+xy}=\displaystyle\frac{1+xy-x(x+y)}{(1+xy)^{2}}=\displaystyle\frac{1-x^{2}}{(1+xy)^{2}}>0;$

therefore, the function $\displaystyle\frac{x+y}{1+xy}$ is increasing with
respect of both variables $x,y\in G$ and

$\displaystyle\inf_{x,y\in
G}\displaystyle\frac{x+y}{1+xy}=\displaystyle\frac{-1-1}{1+1}=-1$
$\displaystyle\sup_{x,y\in
G}\displaystyle\frac{x+y}{1+xy}=\displaystyle\frac{1+1}{1+1}=1.$

This accomplishes the proof that the binary operation $x\circ y$ makes $G$ a
Lie group.

The invariant measure is

$dG(x):=\frac{dx}{1-x^{2}}.$

Indeed, we have

$\displaystyle dG(x\circ y)=\frac{\displaystyle
d_{x}\frac{x+y}{1+xy}}{1-\left(\displaystyle\frac{x+y}{1+xy}\right)^{2}}=\frac{1-y^{2}}{(1-x^{2})(1-y^{2})}dx=\frac{dx}{1-x^{2}}=dG(x).$
(10)

Characters of the Lie group $G$ coincides with the set of isomorpisms to the
unit circle and are given by the mappings

$x\longrightarrow
C(x,\xi):=\left(\frac{1+x}{1-x}\right)^{i\xi}\qquad\forall\,x\in G.$

Indeed, to prove this we have to check that $C(x\circ
y,\xi)=C(x,\xi)C(y,\xi),\quad\forall\,x,y\in G$. Indeed,

$\displaystyle C(x\circ y,\xi)=\left(\frac{1+(x\circ y)}{1=(x\circ
y)}\right)^{i\xi}=\left(\frac{1+\displaystyle\frac{x+y}{1+xy}}{1-\displaystyle\frac{x+y}{1+xy}}\right)^{i\xi}=\left(\frac{1+xy+x+y}{1+xy-
x-y}\right)^{i\xi}$ $\displaystyle\hskip
28.45274pt=\left(\frac{1+x}{1-x}\right)^{i\xi}\left(\frac{1+y}{1-y}\right)^{i\xi}=C(x,\xi)C(y,\xi)\qquad\forall\,x,y\in
G.$ (11)

Thus, the Fourier transformation on the space of functions on the group $G$ is
defined as follows (cf. (1)):

$\displaystyle({\mathscr{F}}\hskip-2.84526pt_{G}v)(\xi):=\int_{-1}^{1}C(y,\xi)v(y)dG(y)=\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}\frac{v(y)dy}{1-y^{2}}.$
(12)

The pull back operators, corresponding to the diffeomorphisms (8) of the Lie
groups $\mathbb{R}$ and $G$ represent isometric isomorphisms

$\displaystyle\begin{array}[]{c}(t_{*}\varphi)(x):=\varphi(t(x))=\varphi\left(\displaystyle\frac{1}{2}\ln\frac{1+x}{1-x}\right)\;:\;\mathbb{L}_{p}(\mathbb{R})\rightarrow\mathbb{L}_{p}(G,dG(x)),\\\\[8.53581pt]
(x_{*}\varphi_{0})(t):=\varphi_{0}(x(t))=\varphi_{0}(\tanh\,t):\mathbb{L}_{p}(G,dG(x))\rightarrow\mathbb{L}_{p}(\mathbb{R}),\end{array}$
(15) $\displaystyle x_{*}(t)=t^{-1}_{*}(t),\qquad
t_{*}(x)=x^{-1}_{*}(x),\qquad x\in G,\qquad t\in\mathbb{R}$

of the Lebesgue spaces $\mathbb{L}_{p}(\mathbb{R})$ and
$\mathbb{L}_{p}(G,dG(x))$, $1\leqslant p\leqslant\infty$, where the latter
weighted Lebesgue space is equipped with the norm

$\displaystyle\begin{array}[]{l}\|\varphi\big{|}\mathbb{L}_{p}(G,dG(x))\|:=\left[\displaystyle\int_{-1}^{1}|\varphi(y)|^{p}dG(y)\right]^{1/p}=\left[\displaystyle\int_{-1}^{1}|\varphi(y)|^{p}\displaystyle\frac{dy}{1-y^{2}}\right]^{1/p}\\\\[11.38109pt]
\hskip 199.16928pt\quad\mbox{for}\quad 1\leqslant p<\infty\\\\[5.69054pt]
\|\varphi\big{|}\mathbb{L}_{\infty}(G,dG(x))\|:=\|\varphi\big{|}\mathbb{L}_{\infty}(G)\|={\rm
ess}\,\sup_{x\in G}|\varphi(x)|\quad\mbox{for}\quad p=\infty.\end{array}$ (19)

The dual space to $\mathbb{L}_{p}(G,dG(x))$, with respect to the scalar
product

$\mbox{\bf{(}}\varphi,\psi\mbox{\bf{)}}_{G}:=\displaystyle\int_{-1}^{1}\varphi(y)\overline{\psi(y)}\displaystyle\frac{dy}{1-y^{2}},\quad
1<p<\infty,$

is the space $\mathbb{L}_{p^{\prime}}(G,dG(x))$,
$p^{\prime}:=\displaystyle\frac{p}{p-1}$.

Using the transformation of the variables (8) and taking into account the
connection between differentials

$dt=\displaystyle\frac{1}{2}d\left[\ln\frac{1+x}{1-x}\right]=\displaystyle\frac{(1-x)+(1+x)}{2\displaystyle\frac{1+x}{1-x}(1-x)^{2}}dx=\displaystyle\frac{dx}{1-x^{2}}$

we easily prove the following equality between the norms (isometrical
isomorphism of the spaces)

$\displaystyle\|\varphi_{0}\big{|}\mathbb{L}_{p}(G,dG(x))\|:=\left[\int_{-1}^{1}|\varphi_{0}(x)|^{p}\,dG(x)\right]^{1/p}$
(20)
$\displaystyle=\left[\int_{-1}^{1}\left|\varphi\left(\displaystyle\frac{1}{2}\ln\frac{1+x}{1-x}\right)\right|^{p}\,\displaystyle\frac{dx}{1-x^{2}}\right]^{1/p}=\left[\int_{-\infty}^{\infty}|\varphi(t)|^{p}\,dt\right]^{1/p}=\|\varphi\big{|}\mathbb{L}_{p}(\mathbb{R})\|,$

where we used the substitution
$t=\displaystyle\frac{1}{2}\ln\displaystyle\frac{1+x}{1-x}$,
$dt=\displaystyle\frac{dx}{1-x^{2}}$ and

$\varphi\in\mathbb{L}_{p}(\mathbb{R}),\qquad\varphi_{0}(x):=t_{*}\varphi(x)=\varphi\left(\displaystyle\frac{1}{2}\ln\frac{1+x}{1-x}\right),\quad\varphi_{0}\in\mathbb{L}_{p}(G,dG(x)).$

The Fourier transform ${\mathscr{F}}\hskip-2.84526pt_{G}\varphi_{0}(\xi)$ of a
function $\varphi_{0}(x)$ on the Lie group $G$ and the Fourier transform
${\mathscr{F}}_{\mathbb{R}}\varphi(\xi)$ of a function $\varphi(t)$ on the Lie
group $\mathbb{R}$ are related by the formulae

$\displaystyle\hskip-8.53581pt\begin{array}[]{l}({\mathscr{F}}\hskip-2.84526pt_{G}\varphi_{0})(\xi)={\boldsymbol{D}}_{2}{\mathscr{F}}_{\mathbb{R}}(x_{*}\varphi_{0})(\xi)={\mathscr{F}}_{\mathbb{R}}(x_{*}\varphi_{0})(2\xi),\qquad\xi\in\mathbb{R},\\\\[8.53581pt]
({{\mathscr{F}}}_{G}^{-1}\psi)(x)=(t_{*}{\mathscr{F}}^{-1}_{\mathbb{R}}{\boldsymbol{D}}_{1/2}\psi)(x)=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\infty}^{\infty}\left(\frac{1+x}{1-x}\right)^{-i\xi}\hskip-14.22636pt\psi(\xi/2)d\xi,\quad
x\in G,\\\\[8.53581pt] \hskip 142.26378pt\varphi_{0}\in
C^{\infty}_{0}(G),\qquad\psi\in C^{\infty}_{0}(\mathbb{R}),\end{array}$ (24)

where ${\boldsymbol{D}}_{\lambda}\psi(\xi):=\psi(\lambda\xi)$ is the dilation
operator and $x_{*}$ is defined in (15). $C^{\infty}_{0}(\mathbb{R})$ and
$C^{\infty}_{0}(G)$ denote the space of $C^{\infty}-$functions with compact
supports on the sets $G=(-1,1)$ and $\mathbb{R}$.

The isomorphism of spaces (15) and (20) via transformation (8) was noted in
Pe06a  (also see SP (20)) and applied to the definition of the transformation
(12), which was called there ${\mathscr{P}}$-transformation.

The following Parseval’s equality is valid:

$\displaystyle\mbox{\bf{(}}{\mathscr{F}}\hskip-2.84526pt_{G}\varphi,{\mathscr{F}}\hskip-2.84526pt_{G}\psi\mbox{\bf{)}}_{\mathbb{R}}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\hskip-8.53581pt{\mathscr{F}}\hskip-2.84526pt_{G}\varphi(\xi)\overline{{\mathscr{F}}\hskip-2.84526pt_{G}\psi(\xi)}d\xi=\int_{-\infty}^{\infty}{\mathscr{F}}\hskip-2.84526pt_{G}\varphi(\xi)\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{-i\xi}\frac{\overline{\psi(y)}dy}{1-y^{2}}d\xi$
(25) $\displaystyle=$
$\displaystyle\pi\int_{-1}^{1}\overline{\psi(y)}\frac{1}{\pi}\int_{-\infty}^{\infty}\left(\frac{1+y}{1-y}\right)^{-i\xi}{\mathscr{F}}\hskip-2.84526pt_{G}\varphi(\xi)d\xi\frac{dy}{1-y^{2}}$
$\displaystyle=$
$\displaystyle\pi\int_{-1}^{1}\overline{\psi(y)}{\mathscr{F}}\hskip-2.84526pt_{G}^{-1}{\mathscr{F}}\hskip-2.84526pt_{G}\varphi(y)dG(y)=\pi\int_{-1}^{1}\varphi(y)\overline{\psi(y)}dG(y)$
$\displaystyle=$ $\displaystyle\pi\mbox{\bf{(}}\varphi,\psi\mbox{\bf{)}}_{G}.$

From (25) follows that the mappings

$\displaystyle\begin{array}[]{l}\displaystyle\frac{1}{\sqrt{\pi}}{\mathscr{F}}\hskip-2.84526pt_{G}\;:\;\mathbb{L}_{2}(G,dG(x))\to\mathbb{L}_{2}(\mathbb{R}),\\\\[8.53581pt]
\sqrt{\pi}{{\mathscr{F}}}_{G}^{-1}\;:\;\mathbb{L}_{2}(\mathbb{R})\to\mathbb{L}_{2}(G,dG(x))\end{array}$
(28)

are isometric isomorphisms of the spaces.

Note, that the direct product $G^{n}:=G\times\ldots\times G$ of $n$ copies of
intervals $G:=(-1,1)$ endowed with the group operation

$\displaystyle x\circ y:=\left(x_{1}\circ y_{1},\ldots,x_{n}\circ
y_{n}\right),$ $\displaystyle
x=(x_{1},\ldots,x_{n})^{\top},y=(y_{1},\ldots,y_{n})^{\top}\in G^{n},$

makes $G^{n}$ a Lie group. The inverse element to $x$ is $-x$ and the
invariant measure is

$dG(x):=dG(y_{1})\cdots
dG(y_{n})=\frac{dx_{1}}{1-x^{2}_{1}}\cdots\frac{dx_{n}}{1-x^{2}_{n}},\quad
x=(x_{1},\ldots,x_{n})\in G^{n}.$

The Fourier transformation on the space of functions on the group $G^{n}$ is:

$\displaystyle({\mathscr{F}}\hskip-2.84526pt_{G}v)(\xi):=\int_{G^{n}}\left(\frac{1+y}{1-y}\right)^{i\xi}v(y)dG(y):=\int_{G^{n}}\prod_{k=1}^{n}\left(\frac{1+y_{k}}{1-y_{k}}\right)^{i\xi_{k}}v(y)dG(y),$
$\displaystyle\xi\in\mathbb{R}^{n}.$

Further we will formulate and prove assertions for the group $G$, although all
of them from §§ 2-3, except the second part of Theorem 3.2, are valid for the
group $G^{n}$.

##  2 Function spaces on $G$

Let $\mathbb{S}(\mathbb{R})$ denote the Schwartz space of fast decaying
$C^{\infty}(\mathbb{R})-$functions, which eliminate at $\pm\infty$ together
with the derivatives $[(1+t^{2})]^{m}(d^{k}\varphi)(t)/dt^{k}$ for arbitrary
integers $m,k=0,1,\ldots$. $\mathbb{S}(G)$ denotes the image of
$\mathbb{S}(\mathbb{R})$ under the transformation (cf. (15))

$\displaystyle t_{*}\;:\;\mathbb{S}(\mathbb{R})\to\mathbb{S}(G)$ (29)

and consists of $C^{\infty}(G)-$functions $\varphi(x)$ which eliminate at $\pm
1$ together with derivatives $[\ln(1-x^{2})]^{m}(\mathfrak{D}^{k}\varphi)(x)$
for arbitrary integers $m,k=0,1,\ldots$. The notations
$\mathbb{S}^{\prime}(\mathbb{R})$ and $\mathbb{S}^{\prime}(G)$ are used for
the dual Schwartz space of distributions. It is well known that the spaces
$\mathbb{S}(\mathbb{R})$ and $\mathbb{S}^{\prime}(\mathbb{R})$ are invariant
under the Fourier transformations ${\mathscr{F}}_{\mathbb{R}}^{\pm 1}$ and,
therefore, due to (24), (29), the Fourier transformations
${\mathscr{F}}\hskip-2.84526pt_{G}^{\pm 1}$ map the spaces as follows:

$\displaystyle\begin{array}[]{rcl}{\mathscr{F}}\hskip-2.84526pt_{G}&:&\mathbb{S}(G)\to\mathbb{S}(\mathbb{R}),\\\\[2.84526pt]
&:&\mathbb{S}^{\prime}(G)\to\mathbb{S}^{\prime}(\mathbb{R}).\end{array}\hskip
28.45274pt\begin{array}[]{rcl}{\mathscr{F}}\hskip-2.84526pt_{G}^{-1}&:&\mathbb{S}(\mathbb{R})\to\mathbb{S}(G),\\\\[2.84526pt]
&:&\mathbb{S}^{\prime}(\mathbb{R})\to\mathbb{S}^{\prime}(G).\end{array}$ (34)

Let $W^{0}_{a}$ denote the Fourier convolution operator on the real axes

$W^{0}_{a}\varphi:={\mathscr{F}}^{-1}_{\mathbb{R}}a{\mathscr{F}}_{\mathbb{R}}\varphi,\qquad\varphi\in\mathbb{S}(\mathbb{R})$

and $a(\xi)$ is called its symbol (cf. Du (79)).

The notation $\mathbb{H}^{m}_{p}(\mathbb{R})=\mathbb{W}_{p}^{m}(\mathbb{R})$
with $1\leqslant p\leqslant\infty,\;m\in\mathbb{N}_{0}$ refers to the Sobolev
space of which represents the closure of the Schwartz space
$\mathbb{S}(\mathbb{R})$ under the norm

$\displaystyle\|f\,|\,\mathbb{W}_{p}^{m}(\mathbb{R})\|\,:=\,\left(\sum\limits_{k=0}^{m}\|\partial^{k}_{t}f\,|\,\mathbb{L}_{p}(\mathbb{R})\|^{p}\right)^{1/p}$

for $1\leqslant p<\infty$, with the usual ${\rm ess}\,\sup$-norm modification
for $p=\infty$:

$\|f\,|\,\mathbb{W}_{\infty}^{m}(\mathbb{R})\|\,:=\,\sum\limits_{k=0}^{m}{\rm
ess}\sup\limits_{t\in\mathbb{R}}|\partial^{k}_{t}f(t)|\,.$

The notation $\mathbb{H}^{s}_{p}(\mathbb{R})$ with
$s\in\mathbb{R},\;1\leqslant p\leqslant\infty$ refers to the Bessel potential
space on the Lie group $\mathbb{R}$, which represents the closure of the
Schwartz space $\mathbb{S}(\mathbb{R})$ under the norm

$\displaystyle\|f\,|\,\mathbb{H}_{p}^{s}(\mathbb{R})\|\,:=\,\|{\mathscr{F}}^{-1}_{\mathbb{R}}\langle\xi\rangle^{s}{\mathscr{F}}_{\mathbb{R}}f\,|\,\mathbb{L}_{p}(\mathbb{R})\|=\,\|W^{0}_{\langle\cdot\rangle^{s}}f\,|\,\mathbb{L}_{p}(\mathbb{R})\|<+\infty,$
$\displaystyle\langle\xi\rangle^{s}:=(1+|\xi|^{2})^{s/2}$

for $1\leqslant p\leqslant\infty$. We will apply the standard convention and
write $\mathbb{H}^{s}(G)$ for the Hilbert space $\mathbb{H}^{s}_{2}(G)$,
dropping the subscript index $p=2$. Due to the classical Parseval’s equality
for the axes $\mathbb{R}$ the formula

$\displaystyle\|f\,|\,\mathbb{H}^{s}(\mathbb{R})\|_{0}:=\left(\int_{\mathbb{R}}\left|(1+\xi^{2})^{s/2}{\mathscr{F}}_{\mathbb{R}}f(\xi)\right|^{2}\,d\xi\right)^{1/2}$
(35)

provides an equivalent norm in the Hilbert space $\mathbb{H}^{s}(\mathbb{R})$.

If $\langle\xi\rangle^{-r}a(\xi)$ has bounded variation on $\mathbb{R}$, than
$W^{0}_{a}\;:\;\mathbb{H}^{s}_{p}(\mathbb{R})\to\mathbb{H}^{s-r}_{p}(\mathbb{R})$
extends to a bounded operator for arbitrary $s,r\in\mathbb{R}$, $1<p<\infty$
(see, e.g., Du (79)).

It is well known that the derivative
$\partial_{t}\varphi(t)=\varphi^{\prime}(t)=(W^{0}_{-i\xi}\varphi)(t)$ on the
real axes $\mathbb{R}$ is a convolution operator and its symbol is $-i\xi$:

$\displaystyle({\mathscr{F}}_{\mathbb{R}}\partial_{t}\varphi)(\xi)=-i\xi({\mathscr{F}}_{\mathbb{R}}\varphi)(\xi),\qquad\xi\in\mathbb{R}.$
(36)

Let $0<\alpha<1,\;\;\;1\leqslant p\leqslant\infty{}.$ Then the space
$\mathbb{Z}_{p}^{\alpha}(\mathbb{R})$ consists of functions

$\mathbb{Z}_{p}^{\alpha}(\mathbb{R})=\left\\{\varphi\in\mathbb{L}_{p}(\mathbb{R})\;\;:\;\;\|\varphi\|_{\mathbb{Z}^{\alpha{}}}=\sup_{x,t\in{\boldsymbol{R}}}\frac{\mid\varphi(x)-\varphi(t)\mid}{\mid
x-t\mid^{\alpha}}<\infty\right\\}$

and is endowed with the norm

$\|\varphi\|_{\mathbb{Z}_{p}^{\alpha{}}}=\|\varphi\|_{\mathbb{L}_{p}}+\|\varphi\|_{\mathbb{Z}^{\alpha{}}}.$
(37)

To extend the definition of the space $\mathbb{Z}_{p}^{\alpha}(\mathbb{R})$ to
the case $\alpha\geqslant 1$ the Poisson integral is involved (cf. (St, 70, §
III.2)

$\displaystyle
P_{y}\varphi(x)=\int_{\mathbb{R}}P_{y}(x-t)\varphi(t)dt=W^{0}_{a_{y}}f(x),\hskip
28.45274ptP_{y}(x)=\frac{1}{\pi}\frac{y}{x^{2}+y^{2}},$ $\displaystyle
a_{y}(\xi)=\exp(-\mid\xi\mid y),\hskip
14.22636pty>0,\;\;\;x,\xi\in\mathbb{R}.$

$P_{y}\varphi(x)$ approximates $\varphi(x)$ (cf. (St, 70, § III.2))

$\lim_{y\rangle 0}\|P_{y}\varphi-\varphi\|_{p}=0,\hskip
28.45274pt\varphi\in\mathbb{L}_{p}(\mathbb{R}),\qquad 1<p<\infty{}.$

###### Lemma 1

For $0<\alpha<1$, $1\leqslant p\leqslant\infty$ the expressions

$\displaystyle\|\varphi\|^{(0)}_{\mathbb{Z}^{\alpha{}}_{p}}=\|\varphi\|_{p}+\sup_{y>0}y^{1-\alpha{}}\|D_{y}P_{y}\varphi\|_{\infty{}},\quad
D_{x}:=\displaystyle\frac{d}{dy}$
$\displaystyle\|\varphi\|_{\mathbb{Z}^{\alpha}_{p}}=\|\varphi\|_{p}+\sup_{y>0}y^{1-\alpha{}}\|D_{x}P_{y}\varphi\|_{\infty{}},\hskip
28.45274ptk=1,2,...n.$

define equivalent norms in the space $\mathbb{Z}_{p}^{\alpha}(\mathbb{R})$.

The proof of the formulated Lemma for the case $p=\infty$ is exposed in (St,
70, § V.4), while for $1<p<\infty$ in DS (93). $\Box$

The foregoing lemma leads to the following definition of the space
$\mathbb{Z}_{p}^{\alpha}(\mathbb{R})$ for $0<\alpha\leqslant\infty$,
$1\leqslant p\leqslant\infty$:

$\displaystyle\mathbb{Z}_{p}^{\alpha}(\mathbb{R})$ $\displaystyle=$
$\displaystyle\Big{\\{}\varphi\in\mathbb{L}_{p}(\mathbb{R})\;:\;\|\varphi\|_{\mathbb{Z}_{p}^{\alpha}}=\|\varphi\|_{p}$
(38) $\displaystyle+$
$\displaystyle\sup_{y>0}y^{k-\alpha{}}\|D_{y}^{k}W^{0}_{a_{y}}\varphi\|_{\infty{}}<\infty,\qquad
k=[\alpha{}]+1\Big{\\}},$

where $[\alpha]$ denotes the integer part of $\alpha.$

Moreover, the definition (38) can be applied for the definition of the Hölder-
Zygmound space $\mathbb{Z}_{p}^{\alpha}(G)$ on the Lie group:

$\displaystyle\mathbb{Z}_{p}^{\alpha}(G)$ $\displaystyle=$
$\displaystyle\Big{\\{}\psi\in\mathbb{L}_{p}(G,dG(x))\;:\;\|\psi\|_{\mathbb{Z}_{p}^{\alpha}}=\|\varphi\|_{p}$
$\displaystyle+$
$\displaystyle\sup_{y>0}y^{k-\alpha{}}\|\mathfrak{D}_{y}^{k}W^{0}_{G,a_{y}}\varphi\|_{\infty{}}<\infty,\qquad
k=[\alpha{}]+1\Big{\\}}$

as the image of the space $\mathbb{Z}_{p}^{\alpha}(\mathbb{R})$ under the
transformation $t_{*}$ (cf. (15)):
$\mathbb{Z}^{\alpha}_{p}(G):=t_{*}\mathbb{Z}^{\alpha}_{p}(\mathbb{R}):=\left\\{\varphi_{0}=t_{*}\varphi\;:\;\varphi\in\mathbb{Z}^{\alpha}_{p}(\mathbb{R})\right\\}$.

Further properties of the space $\mathbb{Z}^{\alpha}_{p}(\mathbb{R})$ (valid
also for $\mathbb{Z}^{\alpha}_{p}(G)$) one can found in DS (93); St (70).

In the next Lemma 2 we have collected some formulae on
${\mathscr{F}}\hskip-2.84526pt_{G}$-transformations. Most of them can also be
found in Pe06a .

###### Lemma 2

The following holds:

$\displaystyle\partial_{t}\varphi(t)=\partial_{t}(x_{*}\varphi_{0})(t)$
$\displaystyle=$ $\displaystyle
x_{*}\mathfrak{D}_{G}\varphi_{0}(t),\quad\varphi_{0}=t_{*}\varphi\in\mathbb{C}_{0}^{\infty}(G),\quad
t\in\mathbb{R},$ (39a)
$\displaystyle\partial_{t}=W^{0}_{-i\xi},\qquad\mathfrak{D}_{G}$
$\displaystyle=$ $\displaystyle(1-x^{2})\partial_{x}=W^{0}_{G,-2i\xi},$ (39c)
$\displaystyle\hskip-93.89409pt({\mathscr{F}}\hskip-2.84526pt_{G}f)(\xi)=-\frac{1}{2i\xi}\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}f^{\prime}(y)dy,\quad
f\in\mathbb{C}_{0}^{\infty}(G),\quad\xi\in\mathbb{R},$
$\displaystyle({\mathscr{F}}\hskip-2.84526pt_{G}\mathfrak{D}_{G}\varphi_{0})(\xi)$
$\displaystyle=$
$\displaystyle-2i\xi({\mathscr{F}}\hskip-2.84526pt_{G}\varphi_{0})(\xi),\qquad\xi\in\mathbb{R},$
(39d)
$\displaystyle\left({\mathscr{F}}\hskip-2.84526pt_{G}\frac{1}{y}\right)(\xi)$
$\displaystyle=$ $\displaystyle\pi i\coth(\pi\xi),\qquad\xi\in\mathbb{R},$
(39e)
$\displaystyle\left({\mathscr{F}}\hskip-2.84526pt_{G}\sqrt{1-y^{2}}\right)(\xi)$
$\displaystyle=$
$\displaystyle\frac{\pi}{\cosh(\pi\xi)},\qquad\xi\in\mathbb{R},$ (39f)
$\displaystyle\left({\mathscr{F}}\hskip-2.84526pt_{G}\frac{\sqrt{1-y^{2}}}{y}\right)(\xi)$
$\displaystyle=$ $\displaystyle\pi i\tanh(\pi\xi),\qquad\xi\in\mathbb{R},$
(39g) $\displaystyle({\mathscr{F}}\hskip-2.84526pt_{G}y)(\xi)$
$\displaystyle=$ $\displaystyle\frac{\pi
i}{\sinh(\pi\xi)},\qquad\xi\in\mathbb{R},$ (39h)
$\displaystyle({\mathscr{F}}\hskip-2.84526pt_{G}(1-y^{2}))(\xi)$
$\displaystyle=$
$\displaystyle\frac{2\pi\xi}{\sinh\pi\xi},\qquad\xi\in\mathbb{R}.$ (39i) Note,
that the integrals in (39e) and in (39g) are understood in the Cauchy mean
value sense.

Proof: We have:

$\displaystyle
t_{*}\partial_{t}x_{*}\varphi_{0}(x)=(t_{*}\partial_{t}\varphi_{0}(\tanh
t))(x)=(t_{*}(\partial_{x}\varphi)_{0}(\tanh t)\partial_{t}\tanh t)(x)$
$\displaystyle\hskip
42.67912pt=\partial_{x}\varphi_{0}(x)t_{*}\left(\displaystyle\frac{1}{\cosh^{2}t}\right)(x)=\partial_{x}\varphi_{0}(x)(t_{*}(1-\tanh^{2}t))(x)$
$\displaystyle\hskip
42.67912pt=\partial_{x}\varphi_{0}(x)(1-x^{2})=(1-x^{2})\partial_{x}\varphi_{0}(x)=\mathfrak{D}_{G}\varphi_{0}(x),\quad
x\in G.$

Formulae (39a) and (39c) are consequences of the proved formula.

Formulae (39c) follows from (39d). Indeed,

$\displaystyle-\frac{1}{2i\xi}\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}f^{\prime}(y)dy$
$\displaystyle=$
$\displaystyle-\frac{1}{2i\xi}\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}(\mathfrak{D}_{G}f)(y)\frac{dy}{1-y^{2}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2i\xi}({\mathscr{F}}\hskip-2.84526pt_{G}\mathfrak{D}_{G}f)(\xi)=({\mathscr{F}}\hskip-2.84526pt_{G}f)(\xi).$

o prove (39e) we use equality (24), formula (cf. (GR, 07, 3.511.5))

$\int_{0}^{\infty}\frac{\sinh(at)\cosh(bt)\,dt}{\sinh\,t}=\frac{\pi}{2}\frac{\sin(a\pi)}{\cos(a\pi)+\cos(b\pi)},\qquad\qquad{\rm
Re}(a+b)<1$ (40)

and proceed as follows:

$\displaystyle\left({\mathscr{F}}_{G}\frac{1}{y}\right)(\xi)$ $\displaystyle=$
$\displaystyle\left({\mathscr{F}}_{\mathbb{R}}t_{*}^{-1}\frac{1}{y}\right)(2\xi)=\int_{-\infty}^{\infty}e^{i2\xi
t}\coth\,t\,dt$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\frac{\sinh(2\xi
ti)\cosh\,t\,dt}{\sinh\,t}=-\pi\frac{\sin(2\pi\xi i)}{1-\cos(2\pi\xi i)}$
$\displaystyle=$ $\displaystyle-\pi
i\frac{\sinh(2\pi\xi)}{1-\cosh(2\pi\xi)}=\pi
i\frac{\sinh(\pi\xi)\cosh(\pi\xi)}{\sinh^{2}(\pi\xi)}$ $\displaystyle=$
$\displaystyle\pi i\coth(\pi\xi),\qquad\xi\in\mathbb{R}.$

To prove (39f) we use equality (24), formula (cf. (GR, 07, 3.511.4))

$\int_{0}^{\infty}\frac{\cosh(at)\,dt}{\cosh(bt)}=\frac{\pi}{2b}\sec\frac{a\pi}{2b},\qquad{\rm
Re}\,b>{\rm Re}\,a$

and proceed as follows:

$\displaystyle\left({\mathscr{F}}_{G}\sqrt{1-y^{2}}\right)(\xi)$
$\displaystyle=$
$\displaystyle\left({\mathscr{F}}_{\mathbb{R}}t_{*}^{-1}\sqrt{1-y^{2}}\right)(2\xi)=\int_{-\infty}^{\infty}e^{2t\xi
i}\sqrt{1-\tanh^{2}t}dt$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\frac{\cosh(2\xi ti)dt}{\cosh\,t}=\frac{\pi}{\cos(\pi\xi
i)}=\frac{\pi}{\cosh(\pi\xi)},\qquad\xi\in\mathbb{R}.$

To prove (39g) we use equality (24), formula (40) and proceed as follows:

$\displaystyle\left({\mathscr{F}}_{G}\frac{\sqrt{1-y^{2}}}{y}\right)(\xi)$
$\displaystyle=$
$\displaystyle\left({\mathscr{F}}_{\mathbb{R}}t_{*}^{-1}\frac{\sqrt{1-y^{2}}}{y}\right)(2\xi)=\int_{-\infty}^{\infty}e^{2t\xi
i}\coth\,t\sqrt{1-\tanh^{2}t}dt$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\frac{\sinh(2\xi ti)\,dt}{\sinh\,t}=\pi\frac{\sin(2\pi\xi
i)}{\cos(2\pi\xi i)+1}$ $\displaystyle=$ $\displaystyle\pi
i\frac{\sinh(2\pi\xi)}{\cosh(2\pi\xi)+1}=\pi
i\frac{\sinh(\pi\xi)\cosh(\pi\xi)}{\cosh^{2}(\pi\xi)}$ $\displaystyle=$
$\displaystyle\pi i\tanh(\pi\xi),\qquad\xi\in\mathbb{R}.$

To prove (39h) we apply, consequtively, formula (24), replace the variable
under the integral $y=-\tanh\,t$, apply formula (cf. (GR, 07, 3.511.7))

$\displaystyle\int_{0}^{\infty}\frac{\sinh(at)\sinh(bt)dt}{\cosh(ct)}=\frac{\pi}{c}\frac{\sin\displaystyle\frac{a\pi}{2c}\sin\displaystyle\frac{b\pi}{2c}}{\cos\displaystyle\frac{a\pi}{c}+\cos\displaystyle\frac{b\pi}{c}}\qquad{\rm
Re}\,a+{\rm Re}\,b<{\rm Re}\,c$

and proceed as follows:

$\displaystyle\left({\mathscr{F}}_{G}y\right)(\xi)$ $\displaystyle=$
$\displaystyle\left({\mathscr{F}}_{\mathbb{R}}(x_{*}y)\right)(2\xi)=\int_{-\infty}^{\infty}e^{i2\xi
t}\tanh\,t\,dt$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\frac{\sinh(2\xi
ti)\sinh\,t\,dt}{\cosh\,t}=2\pi\frac{\sin(\pi\xi
i)\sin\displaystyle\frac{\pi}{2}}{\cos(2\pi\xi i)+\cos\pi}$ $\displaystyle=$
$\displaystyle 2\pi i\frac{\sinh(\pi\xi)}{\cosh(2\pi\xi)-1}=\frac{\pi
i}{\sinh(\pi\xi)},\qquad\xi\in\mathbb{R}.$

To prove the last formula (39i) we change the variable under the integral
$y=\tanh\,t$

$\displaystyle({\mathscr{F}}_{G}(1-y^{2}))(\xi)$ $\displaystyle=$
$\displaystyle\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}(1-y^{2})\frac{dy}{1-y^{2}}=\int_{-\infty}^{\infty}e^{2t\xi
i}(1-\tanh^{2}t)dt$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}e^{2t\xi
i}\frac{dt}{\cosh^{2}t}=2\int_{0}^{\infty}\frac{\cos(2\xi t)}{\cosh^{2}t}dt$

and by applying formula (cf. (GR, 07, 3.982.1))

$\displaystyle\int_{0}^{\infty}\frac{\cos(at)dt}{\cosh^{2}\beta
t}=\frac{a\pi}{2\beta^{2}\sinh\displaystyle\frac{a\pi}{2\beta}},\qquad{\rm
Re}\,\beta>0,\quad a>0.$

we immediately derive (39i).

Formulae (39a)-(39i) are proved. $\Box$

Note that according to (39a) and (39c) the counterpart of the derivative
$\partial_{t}$ on the Lie group $\mathbb{R}$ is the derivative
$\mathfrak{D}_{G}:=(1-x^{2})\partial_{x}$ on the Lie group $G$ and both of
them are convolution operators.

The notation $\mathbb{W}^{m}_{p}(G,dG(x))$ with $1\leqslant
p\leqslant\infty,\;m\in\mathbb{N}_{0}$ refers to the Sobolev space of
functions which represents the closure of the Schwartz space $\mathbb{S}(G)$
under the norm

$\|f\,|\,\mathbb{W}_{p}^{m}(G,dG(x))\|\,:=\,\left(\sum\limits_{k=0}^{m}\|\mathfrak{D}^{k}_{G}f\,|\,\mathbb{L}_{p}(G,dG(x))\|^{p}\right)^{1/p}$

for $1\leqslant p<\infty$, with the usual ${\rm ess}\,\sup$-norm modification
for $p=\infty$:

$\|f\,|\,\mathbb{W}_{\infty}^{m}(G)\|\,:=\,\sum\limits_{k=0}^{m}{\rm
ess}\,\sup\limits_{x\in G}|\mathfrak{D}^{k}_{G}f(x)|\,.$

The functions from $\mathbb{W}^{m}_{p}(G,dG(x))\subset\mathbb{L}_{p}(G,dG(x))$
have distributional derivatives $\mathfrak{D}_{G}^{k}\varphi$ of order
$k=1,\ldots,m$, which belong to the Lebesgue space $\mathbb{L}_{p}(G,dG(x))$.

The notation $\mathbb{H}^{s}_{p}(G))=\mathbb{H}^{s}_{p}(G,dG(x))$ with
$s\in\mathbb{R},\;1\leqslant p\leqslant\infty$ refers to the Bessel potential
space on the Lie group $\mathbb{R}$, where the norm is defined as follows

$\displaystyle\|f\,|\,\mathbb{H}_{p}^{s}(G,dG(x))\|\,:=\,\|{{\mathscr{F}}}_{G}^{-1}\langle\xi\rangle^{s}{\mathscr{F}}\hskip-2.84526pt_{G}f\,|\,\mathbb{L}_{p}(G,dG(x))\|$
$\displaystyle=\,\|W^{0}_{\langle\cdot\rangle^{s}}f\,|\,\mathbb{L}_{p}(G,dG(x))\|<+\infty,\qquad\langle\xi\rangle^{s}:=(1+|\xi|^{2})^{s/2}$
(41)

for $1\leqslant p\leqslant\infty$. We will apply the standard convention and
write $\mathbb{H}^{s}(G,dG(x))$ for the Hilbert space
$\mathbb{H}^{s}_{2}(G,dG(x))$, by dropping the subscript index $p=2$. Due to
the isomorphisms (28) the following

$\displaystyle\|f\,|\,\mathbb{H}^{s}(G,dG(x))\|_{0}:=\left(\int_{-1}^{1}\left|(1+\xi^{2})^{s/2}{\mathscr{F}}\hskip-2.84526pt_{G}\,f(\xi)\right|^{2}\,d\xi\right)^{1/2}$

defines an equivalent norm on $\mathbb{H}^{s}(\mathbb{R})$.

Let ${\mathscr{L}}(\mathfrak{B}_{1},\mathfrak{B}_{2})$ denote the space of all
linear bounded operators mapping the Banach spaces
$\mathfrak{B}_{1}\longrightarrow\mathfrak{B}_{2}$.

Based on formulae (36) and (39d) the following is proved (cf. Du (79); Tr (95)
for the case $\mathbb{H}_{p}^{m}(\mathbb{R})=\mathbb{W}_{p}^{m}(\mathbb{R})$).

###### Proposition 1

For arbitrary $1<p<\infty$ and an integer $m\in\mathbb{N}_{0}$ the the Bessel
potential space $\mathbb{H}^{m}_{p}(G,dG(x))$ and the Sobolev space
$\mathbb{W}^{m}_{p}(G,dG(x))$ have equivalent norms and are topologically
isomorphic.

Due to the isomorphisms

$\displaystyle\begin{array}[]{rcl}t_{*}&:&\mathbb{H}_{p}^{s}(\mathbb{R})\longrightarrow\mathbb{H}^{s}_{p}(G,dG(x)),\qquad
1\leqslant p\leqslant\infty,\quad s\in\mathbb{R},\\\\[8.53581pt]
t_{*}&:&\mathbb{Z}^{\mu}_{p}(\mathbb{R})\longrightarrow\mathbb{Z}^{\mu}_{p}(G),\qquad
0<\mu<\infty,\quad 1\leqslant p\leqslant\infty,\end{array}$ (44)

we can justify the following propositions, valid for the Bessel potential and
Hölder-Zygmound spaces on the Euclidean space $\mathbb{R}$.

###### Proposition 2

(see (Tr, 95, § 2.4.2)) Let $s_{0},s_{1},r_{0},r_{1}\in\mathbb{R},\quad
1\leqslant p_{0},p_{1},q_{0},q_{1}<\infty,$ $0<\ t<1$ and

$\begin{array}[]{c}\displaystyle\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}},\quad\displaystyle\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}},\quad
s=(1-\theta)s_{0}+\theta s_{1}\,,\quad r=(1-\theta)r_{0}+\theta
r_{1}\,.\end{array}$

If
$A\in{\mathscr{L}}_{j}:={\mathscr{L}}(\mathbb{H}^{s_{j}}_{p_{j}}(G,dG(x)),\mathbb{H}^{r_{j}}_{q_{j}}(G,dG(x)))$,
$j=0,1$, then $A$ is bounded between the interpolated spaces
$A\in{\mathscr{L}}:={\mathscr{L}}(\mathbb{H}^{s}_{p}(G,dG(x)),\mathbb{H}^{r}_{q}(G,dG(x)))$
and the norm is estimated as follows

$\|A\big{|}{\mathscr{L}}\|\leqslant\|A\big{|}{\mathscr{L}}_{0}\|^{1-\theta}\|A\big{|}{\mathscr{L}}_{1}\|^{\theta}.$

###### Proposition 3

(see (Tr, 95, § 2.7.2)) Let $0<\mu_{0},\mu_{1},\nu_{0},\nu_{1}<\infty$,
$0<\gamma,\theta<1$ and

$\mu=(1-\theta)\mu_{0}+\theta\mu_{1}\,,\quad\nu=(1-\theta)\nu_{0}+\theta\nu_{1}\,.$

If
$A\in{\mathscr{L}}_{j}:={\mathscr{L}}(\mathbb{Z}^{\mu_{j}}_{p}(G),\mathbb{Z}^{\nu_{j}}_{p}(G)$,
$j=0,1$, then $A$ is bounded between the interpolated spaces
$A\in{\mathscr{L}}:={\mathscr{L}}(\mathbb{Z}^{\mu}_{p}(G),\mathbb{Z}^{\nu}_{p}(G)$
and the norm is estimated as follows

$\|A\big{|}{\mathscr{L}}\|\leqslant\|A\big{|}{\mathscr{L}}_{0}\|^{1-\theta}\|A\big{|}{\mathscr{L}}_{1}\|^{\theta}.$

######  Proposition 4 (see Tr (95); DS (93))

Let $s,r\in\mathbb{R}$, $1\leqslant p\leqslant\infty$, $0<\mu<\infty$ and
$\langle\xi\rangle^{r}:=(1+|\xi|^{2})^{1/2}$. The Bessel potential operator

$\displaystyle\begin{array}[]{rcl}{\boldsymbol{\Lambda}}^{r}:=W^{0}_{G,\langle\cdot\rangle^{r}}&:&\mathbb{H}_{p}^{s}(G,dG(x))\rightarrow\mathbb{H}_{p}^{s-r}(G),\\\\[5.69054pt]
&:&\mathbb{Z}_{p}^{\mu}(G)\rightarrow\mathbb{Z}_{p}^{\mu-r}(G).\end{array}$
(47)

is an isometric isomorphism for the firs pair of spaces and is an isomorphism
for the second pair of spaces.

##  3 Convolutions and multipliers on $G$

Now we consider a convolution on the Lie group $G=(-1,1)$

$\displaystyle({\boldsymbol{K}}_{0}\varphi)(x)=(k_{0}*_{G}\varphi_{0})(x):=\int_{-1}^{1}k_{0}(x\circ(-y))\varphi_{0}(y)dG(y)$
$\displaystyle=\int_{-1}^{1}k_{0}\left(\frac{x-y}{1-xy}\right)\varphi_{0}(y)\frac{dy}{1-y^{2}}.$

Convolution (3) is equivalent with the corresponding convolution on the Lie
group $\mathbb{R}$ (the Fourier convolution)

$\displaystyle({\boldsymbol{K}}\varphi)(t)=(k*_{\mathbb{R}}\varphi)(t)=\int_{-\infty}^{\infty}k(t-\tau)\varphi(\tau)\,d\tau$
(48)

and the equivalence is established with the help of isomorphisms $t_{*}$ and
$x_{*}$ (cf. (15). Indeed,

$\displaystyle t_{*}{\boldsymbol{K}}(x_{*}\varphi_{0})(x)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}k\left(\displaystyle\frac{1}{2}\ln\frac{1+x}{1-x}-t\right)(x_{*}\varphi_{0})(t)dt$
(49) $\displaystyle=$
$\displaystyle\int_{-1}^{1}k\left(\displaystyle\frac{1}{2}\ln\frac{1+x}{1-x}-\displaystyle\frac{1}{2}\ln\frac{1+y}{1-y}\right)\varphi_{0}(y)dG(y)$
$\displaystyle=$
$\displaystyle\int_{-1}^{1}k\left(\displaystyle\frac{1}{2}\ln\displaystyle\frac{1+\displaystyle\frac{x-y}{1-xy}}{1-\displaystyle\frac{x-y}{1-xy}}\right)\varphi_{0}(y)dG(y)$
$\displaystyle=$
$\displaystyle\int_{-1}^{1}k_{0}\left(\displaystyle\frac{x-y}{1-xy}\right)\,\varphi_{0}(y)dG(y)=({\boldsymbol{K}}_{0}\varphi_{0})(x),\quad
x\in G,$

where we have inserted $t=\displaystyle\frac{1}{2}\ln\frac{1+y}{1-y}$. Vice
versa, we start from the convolution (3):

$\displaystyle
x_{*}{\boldsymbol{K}}_{0}(t_{*}\varphi)(t)=\int_{-1}^{1}k_{0}\left(\frac{\tanh
t-y}{1-y\tanh t}\right)(t_{*}\varphi)(y)dG(y)$ $\displaystyle\hskip
56.9055pt=\int_{-\infty}^{\infty}k_{0}\left(\displaystyle\frac{\tanh
t-\tanh\tau}{1-\tanh\tau\tanh t}\right)\varphi(\tau)d\tau$
$\displaystyle\hskip
56.9055pt=\int_{-\infty}^{\infty}k_{0}\left(\tanh(t-\tau)\right)\varphi(\tau)d\tau=({\boldsymbol{K}}\varphi)(t),\quad
t\in\mathbb{R},$ (50)

where $t_{*},x_{*}=t^{-1}_{*}$ are defined in (15) and we have inserted
$y=\tanh\tau$. Hence, the kernels of convolution operators $k_{0}(x)$ in (3)
and $k(t)$ in (48) are connected as follows:

$\displaystyle k_{0}(x):=(t_{*}k)(x),\quad k(t)=(x_{*}k_{0})(t),\quad x\in
G,\quad t\in\mathbb{R}.$ (51)

The connection (49) and (3) between the convolution operators (3) and (48) was
noted in Pe06a  and in SP (20) and used to investigate Prandtl equation,
Tricomi equation (see Pe06a ; Pe06b ) and Prandtl equation in the Bessel
potential spaces $\widetilde{\mathbb{H}}^{s}(G,dG(x))$, $0\leqslant s\leqslant
1$ (see SP (20)). These equations are addressed in §§ 4-5 below.

###### Theorem 3.1

Let $1\leqslant p\leqslant\infty$, $s\in\mathbb{R}$, $\mu>0$. The convolution
operator ${\bf K_{0}}$ on the Lie Group $G$ (cf. (3)) is bounded in the
Lebesgue and space

$\displaystyle\begin{array}[]{rcl}{\bf
K_{0}}&:&\mathbb{L}_{p}(G,dG(x))\to\mathbb{L}_{p}(G,dG(x))\\\\[5.69054pt]
&:&\mathbb{Z}_{p}^{\mu}(G)\rightarrow\mathbb{Z}_{p}^{\mu}(G).\end{array}$ (54)

provided its kernel $k_{0}(x)$ satisfies the condition
$k_{0}\in\mathbb{L}_{1}(G,dG(x))$, i.e.,

$\|k_{0}\big{|}\mathbb{L}_{1}(G,dG(x)\|=\int_{-1}^{1}|k_{0}(y)|dG(y)<\infty.$

and

$\displaystyle\|{\boldsymbol{K}}_{0}\varphi_{0}\big{|}\mathbb{L}_{p}(G,dG(x))\|\leqslant\|k_{0}\big{|}\mathbb{L}_{1}(G,dG(x))\|\|\varphi_{0}\big{|}\mathbb{L}_{p}(G,dG(x))\|.$
(55)

Proof: For the proof in case of Hölder-Zygmound space
$\mathbb{Z}_{p}^{\mu}(G)$ we quote (DS, 93, Theorem 4.12).

Fourier convolution operator ${\bf K}$ on the Lie Group $\mathbb{R}$ (cf.
(48)) is bounded in the space $\mathbb{L}_{p}(\mathbb{R})$ for a kernel
$k\in\mathbb{L}_{1}(\mathbb{R})$. Then
${\boldsymbol{K}}_{0}=W^{0}_{G,{\mathscr{K}}}$,
${\mathscr{K}}:=({\mathscr{F}}k_{0})(\xi)$, $\xi\in\mathbb{R}$ it is bounded
in $\mathbb{L}_{p}(G,dG(x))$ (cf. (51), (24)). Due to Proposition 7 proved
below.

The direct proof for the space $\mathbb{L}_{p}(G,dG(x))$ goes as follows:

$\displaystyle\|{\boldsymbol{K}}_{0}\varphi_{0}\big{|}\mathbb{L}_{p}(G,dG(x))\|=\left[\int_{-1}^{1}\left|\int_{-1}^{1}k_{0}\left(\frac{x-y}{1-xy}\right)\frac{\varphi_{0}(y)}{1-y^{2}}dy\right|^{p}\frac{dx}{1-x^{2}}\right]^{1/p}$
$\displaystyle\hskip
42.67912pt\leqslant\Bigg{[}\int_{-1}^{1}\left\\{\int_{-1}^{1}\left[\left|k_{0}\left(\frac{x-y}{1-xy}\right)\right|^{1/p}\frac{|\varphi_{0}(y)|}{(1-y^{2})^{1/p}}\right]\right.$
$\displaystyle\hskip
71.13188pt\left.\left[\left|k_{0}\left(\frac{x-y}{1-xy}\right)\right|^{1/p^{\prime}}\frac{1}{(1-y^{2})^{1/p^{\prime}}}\right]dy\right\\}^{p}\frac{dx}{1-x^{2}}\Bigg{]}^{1/p}$
$\displaystyle\hskip
42.67912pt\leqslant\Bigg{[}\int_{-1}^{1}\int_{-1}^{1}\left|k_{0}\left(\frac{x-y}{1-xy}\right)\right|\frac{|\varphi_{0}(y)|^{p}dydx}{(1-x^{2})(1-y^{2})}$
$\displaystyle\hskip
71.13188pt\left\\{\int_{-1}^{1}\left|k_{0}\left(\frac{x-y}{1-xy}\right)\right|\frac{dy}{1-y^{2}}\right\\}^{p/p^{\prime}}\Bigg{]}^{1/p}$
$\displaystyle\hskip
42.67912pt=\Bigg{[}\int_{-1}^{1}\frac{|\varphi_{0}(y)|^{p}dy}{1-y^{2}}\int_{-1}^{1}\left|k_{0}\left(\frac{x-y}{1-xy}\right)\right|\frac{dx}{1-x^{2}}$
$\displaystyle\hskip
71.13188pt\left\\{\int_{-1}^{1}\left|k_{0}\left(\frac{x-y}{1-xy}\right)\right|\frac{dy}{1-y^{2}}\right\\}^{p/p^{\prime}}\Bigg{]}^{1/p}.$

In the integrals we change the variables, using the invariance of the Haar
measure with respect to the group operation (cf. (10)):

$\displaystyle\mbox{if}\quad
t=\displaystyle\frac{x-y}{1-xy},\quad\mbox{then}\quad dG(t)=dG(x)=-dG(y).$

and obtain

$\displaystyle\|{\boldsymbol{K}}_{0}\varphi_{0}\big{|}\mathbb{L}_{p}(G,dG(x))\|$
$\displaystyle\hskip
42.67912pt\leqslant\left[\int_{-1}^{1}\frac{|\varphi_{0}(y)|^{p}dy}{1-y^{2}}\int_{-1}^{1}\frac{|k_{0}(t)|dt}{1-t^{2}}\left\\{-\int_{1}^{-1}\frac{|k_{0}(t)|dt}{1-t^{2}}\right\\}^{p/p^{\prime}}\right]^{1/p}$
$\displaystyle\hskip
42.67912pt=\|k_{0}\big{|}\mathbb{L}_{1}(G,dG(x)\|\|\varphi_{0}\big{|}\mathbb{L}_{p}(G,dG(x)\|$

and (55) is proved. $\Box$

Note, that

$\displaystyle\mathscr{F_{\mathbb{R}}}[k\ast_{\mathbb{R}}\varphi](\xi)={\mathscr{K}}(\xi)\Phi(\xi),\quad{\mathscr{K}}(\xi)=\mathscr{F_{\mathbb{R}}}k(\xi),\quad\Phi(\xi)=\mathscr{F_{\mathbb{R}}}\varphi(\xi),$
(56)
$\displaystyle{\mathscr{F}}\hskip-2.84526pt_{G}[k_{0}\ast_{G}\varphi_{0}](\xi)={\mathscr{K}}_{0}(\xi)\Phi_{0}(\xi),\quad{\mathscr{K}}_{0}(\xi)={\mathscr{F}}\hskip-2.84526pt_{G}k_{0}(\xi),$
$\displaystyle\Phi_{0}(\xi)={\mathscr{F}}\hskip-2.84526pt_{G}\varphi_{0}(\xi),\quad\xi\in\mathbb{R}$
(57)

and ${\mathscr{K}}_{0}(\xi)$, ${\mathscr{K}}(\xi)$ are called the symbols of
the corresponding convolution operators ${\bf K_{0}}$ in (3), ${\bf K}$ in
(48). These symbols differ by constant dilation of the argument
${\mathscr{K}}_{0}(\xi)={\mathscr{K}}(2\xi)$. Indeed, from (24) and (56)
follows:

$\displaystyle{\mathscr{K}}(2\xi)=(\mathscr{F_{\mathbb{R}}}k)(2\xi)={\mathscr{F}}_{\mathbb{R}}(x_{*}k_{0})(2\xi)=\int_{-\infty}^{\infty}e^{2t\xi
i}k_{0}(\tanh\,t)dt$
$\displaystyle=\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}k_{0}(y)dG(y)={\mathscr{K}}_{0}(\xi),\qquad\xi\in\mathbb{R}.$
(58)

It is well known and follows from (56) that the Fourier convolution operator
${\bf K}$ in (48) is written in the form (see, e.g., Du (79))

$\displaystyle({\boldsymbol{K}}\varphi)(t)=(k\ast_{\mathbb{R}}\varphi)(t)={\mathscr{F}}^{-1}_{\mathbb{R}}{\mathscr{K}}{\mathscr{F}}_{\mathbb{R}}\varphi(t)=:W^{0}_{\mathscr{K}}\varphi(t),\quad
t\in\mathbb{R}.$ (59)

Similarly, due to (3), the convolution operator ${\bf K_{0}}$ in (3) is
written in the form

$\displaystyle({\boldsymbol{K}}_{0}\varphi_{0})(x)=(k_{0}\ast_{G}\varphi_{0})(x)={{\mathscr{F}}}_{G}^{-1}{\mathscr{K}}_{0}{\mathscr{F}}\hskip-2.84526pt_{G}\varphi_{0}(x)=:W^{0}_{G,{\mathscr{K}}_{0}}\varphi_{0}(x),$
(60) $\displaystyle x\in G$

Let $\mathfrak{M}_{p}(\mathbb{R})=\mathfrak{M}(\mathbb{L}_{p}(\mathbb{R}))$,
$1\leqslant p\leqslant\infty$, denote the algebra of Fourier
$\mathbb{L}_{p}(\mathbb{R})$-multipliers, consisting of functions for which
the Fourier convolution operator $W^{0}_{a}$,
$a\in\mathfrak{M}_{p}(\mathbb{R})$ is bounded in the Lebesgue space
$W^{0}_{a}\;:\;\mathbb{L}_{p}(\mathbb{R})\to\mathbb{L}_{p}(\mathbb{R})$ (cf.
Hr (60); Du (79)).

Let
$\mathbb{W}(\mathbb{R}):=\\{a(\xi)\;:\;a(\xi)=c+{\mathscr{F}}k(\xi)=c+{\mathscr{F}}\hskip-2.84526pt_{G}k_{0}\;:\;k\in\mathbb{L}_{1},\quad
k_{0}\in\mathbb{L}_{1}(G,dG(x))\\}$ denote the Wiener class of functions. Norm
in $\mathbb{W}(\mathbb{R})$ is introduced as follows:

$\|a\|:=|c_{0}|+\|k\big{|}\mathbb{L}_{1}(\mathbb{R})\|=|c_{0}|+\|k_{0}\big{|}\mathbb{L}_{1}(G,dG(x))\|.$

###### Theorem 3.2

Let $1\leqslant p\leqslant\infty$. The convolution operator $W^{0}_{G,a}$ on
the Lie Group $G$ (cf. (3) and (60)) is bounded in the Lebesgue space

$\displaystyle
W^{0}_{G,a}\;:\;\mathbb{L}_{p}(G,dG(x))\to\mathbb{L}_{p}(G,dG(x))$ (61)

if and only if the symbol $a(\xi)$ is a Fourier
$\mathbb{L}_{p}(\mathbb{R})$-multiplier, $a\in\mathfrak{M}_{p}(\mathbb{R})$.

In particular, $W^{0}_{G,a}$ is bounded in (61) for all symbols $a(\xi)$ with
a bounded variation $a\in\mathbb{V}_{1}(\mathbb{R})$ and from the Wiener class
$\mathbb{W}(\mathbb{R})$.

Proof: The proof of the first part follows from the isomorphism properties
(20) and the equivalence of convolution operators

$\displaystyle W^{0}_{G,a_{0}}=t_{*}W^{0}_{a}x_{*},\qquad a_{0}(\xi)=a(2\xi),$
(62)

which, in its turn, follows from (3), (59) and (60).

The inclusion $\mathbb{V}_{1}(\mathbb{R})\subset\mathfrak{M}_{p}(\mathbb{R})$
was proved by S. B. Stechkin in St (50) (also see (Du, 79, Theorem 2.11) for a
proof). The inclusion
$\mathbb{W}(\mathbb{R})\subset\mathfrak{M}_{p}(\mathbb{R})$ for the Wiener
class of functions follows from Theorem 3.1 and the first part of the present
theorem. $\Box$

Let us expose here the celebrated Mikhlin-Hörmander-Lizorkin theorem on
multipliers, which we formulate for the multi-variable case.

######  Proposition 5 (Mikhlin-Hörmander-Lizorkin)

. Let $1<p<\infty$, $s,r\in\mathbb{R}$. If a function $a(\xi)$ satisfies the
estimates

$|\xi|^{\alpha}|\partial^{\alpha}_{\xi}a(\xi)|\leqslant
M_{\alpha}\langle\xi\rangle^{r}<\infty,\qquad\xi\in\mathbb{R}^{n}$ (63)

for all $\alpha\in\mathbb{N}^{n}_{0}$, then
$a\in\mathfrak{M}^{r}_{p}(\mathbb{R}^{n})$ and

$\displaystyle\|W_{G^{n},a}^{0}u\big{|}\mathbb{H}^{s-r}_{p}(\mathbb{R}^{n})\|\leqslant
C_{p}(a)\|u\big{|}\mathbb{H}^{s}_{p}(\mathbb{R}^{n})\|$ (64)

for some constant $C_{p}(a)<\infty$.

Proof: Due to the multi-variable analogues of Proposition 7 and Theorem 3.4,
the multiplier class $\mathfrak{M}_{p}(\mathbb{R}^{n})$ is the same for the
spaces $\mathbb{L}_{p}(G^{n},dG^{n}(x))$ and
$\mathbb{H}^{s}_{p}(G^{n},dG^{n}(x))$ and
$\mathfrak{M}^{r}_{p}(\mathbb{R}^{n})=(1+|\xi|^{2})^{r/2}\mathfrak{M}_{p}(\mathbb{R}^{n})$.
Therefore, the proof is reduced to the case of miltipliers on the Lebesgue
space. Then the formulated assertion follows from the classical Mikhlin-
Hörmander-Lizorkin theorem (cf. (Hr, 83, v. I, Theorem 7.9.5)). $\Box$

###### Proposition 6

. Let $1<p<\infty$, $\mu>r>0$. If a function $a(\xi)$ satisfies the condition
of the foregoing Proposition 5, then the convolution operator $W^{0}_{G,a}$ is
bounded in the Hölder-Zygmound space
$\mathbb{Z}_{p}^{\mu}(G)\to\mathbb{Z}_{p}^{\mu-r}(G)$.

Proof: For the proof we quote (DS, 93, Theorem 4.8). $\Box$

###### Theorem 3.3

Let $1\leqslant p\leqslant\infty$, $\mu>0$. If the function
$a\in\mathbb{V}_{1}(\mathbb{R})$ has the bounded variation
$a\in\mathbb{V}_{1}(\mathbb{R})$, the convolution operator $W^{0}_{G,a}$ is
bounded in the Hölder-Zygmound space

$W^{0}_{G,a}\;:\;\mathbb{Z}^{\mu}_{p}(G)\to\mathbb{Z}^{\mu}_{p}(G).$

Proof: The convolution operator $W^{0}_{G,-\operatorname{sign}}$ is bounded in
$\mathbb{Z}^{\mu}_{p}(G)$ due to Proposition 6, since
$\operatorname{sign}\,\xi$ and its derivative
$\xi\partial_{\xi}\operatorname{sign}\,\xi=\xi\delta(\xi)=0$ are bounded
functions.

Further the boundedness of convolution operator $W^{0}_{G,a}$,
$a\in\mathbb{V}_{1}(\mathbb{R})$ (Stechkin’s theorem) is proved as in the
original paper St (50) (the proof of Stechkin, modified by Matsaev, is exposed
also in (Du, 79, Theorem 2.11)). $\Box$

###### Remark 1

Let $M(\mathbb{Z}^{\mu}_{p}(G))$ denote the space of multipliers in the
Hölder-Zygmound space $\mathbb{Z}^{\mu}_{p}(G)$, namely, $a\in
M(\mathbb{Z}^{\mu}_{p}(G))$ if and only if $W^{0}_{G,a}$ is bounded in
$\mathbb{Z}^{\mu}_{p}(G)$.

Summarizing the exposed results we note that $M(\mathbb{Z}^{\mu}_{p}(G))$
contains the Wiener class $W(\mathbb{R})$, functions of bounded variation
$V_{1}\mathbb{R})$ and functions which satisfy the Mikhlin-Hörmander
conditions $a(\xi)$ and $\xi\partial_{\xi}a(\xi)$ are uniformly bounded on
$\mathbb{R}$.

###### Lemma 3

Let

$\displaystyle
a(\xi)=c_{0}-ic_{1}\tanh(h\xi)+\frac{c_{2}}{\cosh(h\xi)},\qquad\xi\in\mathbb{R},$
(65)

where $c_{0},c_{1},c_{2}\in\mathbb{C}$, $h\in\mathbb{R}$, are constants.

If $a(\xi)$ is elliptic, i.e., $\inf_{\xi\in\mathbb{R}}|a(|\xi)|>0$, the
inverse $a^{-1}(\xi)$ has a similar representation:

$\displaystyle\hskip
71.13188pta^{-1}(\xi)=d_{0}-d_{1}\tanh(h\xi)+a_{-1}(\xi),$
$\displaystyle\hskip 0.0pta_{-1}\in C^{\infty}(\mathbb{R}),\quad
a_{-1}(\xi)={\mathscr{O}}\left(\cosh^{-1}(h\xi)\right)={\mathscr{O}}\left(e^{-|h\xi|}\right),$
$\displaystyle d_{0}=\frac{(c_{0}+ic_{1})^{-1}+(c_{0}-ic_{1})^{-1}}{2},\qquad
d_{1}=\frac{(c_{0}+ic_{1})^{-1}-(c_{0}-ic_{1})^{-1}}{2}.$

In particular, if $h\not=0$ (3) implies $a_{-1}\in\mathbb{S}(\mathbb{R})$.

Proof: It is easy to check, that

$\displaystyle\left[c_{0}-ic_{1}\tanh(h\xi)\right]\left[d_{0}-d_{1}\tanh(h\xi)\right]=c_{0}d_{0}+ic_{1}d_{1}\tanh^{2}(h\xi)$
$\displaystyle\hskip
19.91692pt-[c_{0}d_{1}+ic_{1}d_{0}]\tanh(h\xi)=c_{0}d_{0}+ic_{1}d_{1}\tanh^{2}(h\xi)$
$\displaystyle\hskip
19.91692pt=c_{0}d_{0}+ic_{1}d_{1}-ic_{1}d_{1}[1-\tanh^{2}(h\xi)]=1-\frac{ic_{1}d_{1}}{\cosh^{2}(h\xi)},\quad\xi\in\mathbb{R}.$

therefore,

$1\equiv
a(\xi)a^{-1}(\xi)=1-\frac{ic_{1}d_{1}}{\cosh^{2}(h\xi)}+[c_{0}-ic_{1}\tanh(h\xi)]a_{-1}(\xi)+\frac{c_{2}a^{-1}(\xi)}{\cosh(h\xi)}$

Then,

$\displaystyle
a_{-1}(\xi)=[c_{0}-ic_{1}\tanh(h\xi)]^{-1}\left[\frac{ic_{1}d_{1}}{\cosh^{2}(h\xi)}-\frac{c_{2}a^{-1}(\xi)}{\cosh(h\xi)}\right]$
$\displaystyle={\mathscr{O}}\left(\cosh^{-1}(h\xi)\right)={\mathscr{O}}\left(e^{-|h\xi|}\right),\quad|\xi|\to\infty,$
(67)

since, due to the ellipticity of $a(\xi)$,

$\displaystyle\inf_{|\xi|>R}\left|c_{0}-ic_{1}\tanh(h\xi)\right|\not=0$ (68)

for $R>0$ sufficiently large.

From (3) and (68) follows the claimed asymptotic in (65). $\Box$

###### Remark 2

Cauchy singular integral operator

$\displaystyle{\boldsymbol{S}}_{G}u(x):=\frac{1}{\pi
i}\int_{-1}^{1}\frac{u(y)dy}{y-x}Let.=W^{0}_{\sigma({\boldsymbol{S}}_{G})}u(x)-{\boldsymbol{F}}(u),\qquad
x\in G$ (69)

is represented as a difference of a convolution operator on the Lie group
modulo and one dimensional operator (functional)

${\boldsymbol{S}}_{G}u=W^{0}_{\sigma({\boldsymbol{S}}_{G})}u(x)-{\boldsymbol{F}}(u),\qquad{\boldsymbol{F}}(u)=\langle
F,u\rangle:=\displaystyle\frac{1}{\pi i}\displaystyle\int_{-1}^{1}yu(y)dG(y).$

Both operators in this difference are unbounded in the space
$\mathbb{L}_{p}(G,dG(x))$, $1<p<\infty$.

Indeed,

$\displaystyle{\boldsymbol{S}}_{G}u(x)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\pi
i}\int_{-1}^{1}\displaystyle\frac{1-y^{2}}{y-x}u(y)dG(y)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{\pi
i}\int_{-1}^{1}\displaystyle\frac{1-xy}{y-x}u(y)dG(y)-\displaystyle\frac{1}{\pi
i}\int_{-1}^{1}yu(y)dG(y)$ $\displaystyle=$ $\displaystyle\left(\frac{1}{\pi
iy}\ast_{G}u\right)(x)-{\boldsymbol{F}}(u)=W^{0}_{\sigma({\boldsymbol{S}}_{G})}u(x)-{\boldsymbol{F}}(u),$
$\displaystyle\sigma({\boldsymbol{S}}_{G})(\xi)$ $\displaystyle=$
$\displaystyle{\mathscr{F}}\hskip-2.84526pt_{G}\left(\frac{1}{\pi
iy}\right)(\xi)=\coth\,\pi\xi,\qquad\xi\in\mathbb{R}.$

and in the last line we used formula (39e).

That ${\boldsymbol{F}}$ is unbounded is easily checked.

As for the convolution operator, the symbol
$\sigma({\boldsymbol{S}}_{G})(\xi)$ is unbounded and can not belong to any
multiplier class
$\sigma({\boldsymbol{S}}_{G})\not\in\mathfrak{M}_{p}(\mathbb{R})$,
$\forall\,p\in[1,\infty]$. $\Box$

Obviously, $\mathfrak{M}_{p}(\mathbb{R})$ is a Banach algebra, endowed with
the norm
$\|a\big{|}\mathfrak{M}_{p}(\mathbb{R})\|:=\|W^{0}_{G,a}\big{|}\mathbb{L}_{p}(G,dG(x))\|$,
because

$\displaystyle W^{0}_{G,a}W^{0}_{G,b}=W^{0}_{G,ab}\qquad\forall
a,b\in\mathfrak{M}_{p}(\mathbb{R}).$ (70)

######  Proposition 7 (see Hr (60); Du (79))

Let $1<p<\infty$ and $s\in\mathbb{R}$. The multiplier class of the space
$\mathbb{H}_{p}^{s}(G,dG(x))$ is independent of the parameter $s\in\mathbb{R}$

$\mathfrak{M}(\mathbb{H}_{p}^{s}(G,dG(x)))=\mathfrak{M}(\mathbb{L}_{p}(G,dG(x)))=\mathfrak{M}_{p}(\mathbb{L}_{p}(\mathbb{R}))=\mathfrak{M}_{p}(\mathbb{R})\,.$

The norm of a convolution operator $W^{0}_{G,a}$,
$a\in\mathfrak{M}_{p}(\mathbb{R})$ in the Bessel potential space
$\mathbb{H}^{s}_{p}(G,dG(x))$ on the Lie group is independent of $s$:

$\|W^{0}_{G,a}|{\mathscr{L}}(\mathbb{H}_{p}^{s}(G,dG(x)))\|=\|W^{0}_{G,a}|{\mathscr{L}}(\mathbb{L}_{p}(G,dG(x)))\|\qquad\forall
s\in\mathbb{R}.$

Let $\mathfrak{M}^{r}_{p}(\mathbb{R})$, $1<p<\infty$, $r\in\mathbb{R}$, denote
the class of functions

$\mathfrak{M}^{r}_{p}(\mathbb{R}):=\left\\{\langle\xi\rangle^{r}a(\xi)\;:\;a\in\mathfrak{M}_{p}(\mathbb{R})\right\\},\qquad\langle\xi\rangle^{r}=(1+|\xi|^{2})^{r/2}$
(71)

and use $\mathfrak{M}_{p}(\mathbb{R})$ for $\mathfrak{M}^{0}_{p}(\mathbb{R})$.

###### Theorem 3.4

Let $1<p<\infty$, $s,r\in\mathbb{R}$. The convolution operator

$W^{0}_{G,a}\>:\>\mathbb{H}_{p}^{s}(G,dG(x))\rightarrow\mathbb{H}_{p}^{s-r}(G,dG(x))$
(72)

is bounded if and only if $a\in\mathfrak{M}^{r}_{p}(\mathbb{R})$.

The convolution operator $W^{0}_{G,a}$ in (72) is Fredholm if only the symbol
is elliptic

$\displaystyle\inf_{\xi\in\mathbb{R}}\left|\langle\xi\rangle^{-r}a(\xi)\right|>0.$
(73)

If the symbol $a_{-r}(\xi):=\langle\xi\rangle^{-r}a(\xi)$ has a bounded
variation $a_{-r}\in\mathbb{V}_{1}(\mathbb{R})$ or belongs to the Wiener
class, $a_{-r}\in\mathbb{W}(\mathbb{R})$, the ellipticity of the symbol (73)
is sufficient for $W^{0}_{G,a}$ to be invertible in the setting (72) and the
inverse operator is $W^{0}_{G,a^{-1}}$.

Proof: Due to the property (70) and Proposition 7, the operator $W^{0}_{G,a}$
in (72) is equivalently lifted to the operator

$\displaystyle{\boldsymbol{\Lambda}}^{s-r}W^{0}_{G,a}{\boldsymbol{\Lambda}}^{-s}=W^{0}_{G,\langle\xi\rangle^{-r}a}\;:\;\mathbb{L}_{p}(G,dG(x))\rightarrow\mathbb{L}_{p}(G,dG(x)).$
(74)

The equivalent lifting (74) shows that the operator $W^{0}_{G,a}$ in (72) and
the operator $W^{0}_{G,\langle\xi\rangle^{-r}a}$ in (74) are simultaneously
bounded or not, are simultaneously Fredholm or not, are simultaneously
invertible or not. Therefore, we only need to prove the assertion for $r=0$.

Let us note that $W^{0}_{G,a}$ is translation invariant

$\displaystyle\tau^{0}_{y}W^{0}_{G,a}u(x)=W^{0}_{G,a}\tau^{0}_{y}u(x),\quad\tau^{0}_{y}u(x):=u\left(\frac{x+y}{1+xy}\right),\quad
x,y\in G.$ (75)

This property can be checked directly, but it also follows from the
equivalence of operators $W^{0}_{G,a}$ and $W^{0}_{a}$ (cf. (62)) since
$W^{0}_{a}$ is translation invariant

$\tau_{h}W^{0}_{a}\varphi(t)=W^{0}_{a}\tau_{h}\varphi(t),\qquad\tau_{h}\varphi(t):=\varphi(t+h),\qquad
t,h\in\mathbb{R}.$

If we assume that $W^{0}_{G,a}\in{\mathscr{L}}(\mathbb{L}_{p}(G,dG(x)))$ is
Fredholm, it is invertible. This follows since $W^{0}_{G,a}$ is translation
invariant (75) and is proved as for the Fourier convolution operator
$W^{0}_{a}$ in Du (79, 84) (I. Simonenko proved this property first in Si (65)
for a singular integral operator).

If $W^{0}_{G,a}\in{\mathscr{L}}(\mathbb{L}_{p}(G,dG(x)))$ is invertible, the
dual (adjoint) operator
$W^{0}_{G,\overline{a}}\in{\mathscr{L}}(\mathbb{L}_{p^{\prime}}(G,dG(x)))$,
$p^{\prime}=p/(p-1)$, is also invertible. Then is invertible
$W^{0}_{G,a}=JW^{0}_{G,\overline{a}}J$, where $Ju(x)=\overline{u(x)}$ is the
complex conjugation. Then, by interpolation, the inverse is bounded in the
space ${\mathscr{L}}(\mathbb{L}_{2}(G,dG(x)))$ and, therefore,
$W^{0}_{G,a}\in{\mathscr{L}}(\mathbb{L}_{2}(G,dG(x)))$ is invertible. Since
the Fourier transforms
${\mathscr{F}}^{0}_{G}=\displaystyle\frac{1}{\sqrt{\pi}}{\mathscr{F}}\hskip-2.84526pt_{G}$
and
$({\mathscr{F}}^{0}_{G})^{-1}:=\sqrt{\pi}{\mathscr{F}}\hskip-2.84526pt_{G}^{-1}$
map the corresponding spaces isometrically (maintaining the norms; see (28)),
the operator

$({\mathscr{F}}^{0}_{G}W^{0}_{G,a}({\mathscr{F}}^{0}_{G})^{-1}U)(\xi)=a(\xi)U(\xi),\qquad
U\in\mathbb{L}_{2}(\mathbb{R})$

is also invertible in the space $\mathbb{L}_{p}(\mathbb{R})$. But the latter
represents a multiplication operator by the symbol $a(\xi)$ in the space
$\mathbb{L}_{2}(\mathbb{R})$. The invertibility of the multiplication operator
by a function $aI$ implies the ellipticity of the function $a(\xi)$.

If
$a_{-r}\in\mathbb{V}_{1}(\mathbb{R})\cap\mathbb{W}(\mathbb{R})\subset\mathfrak{M}_{p}(\mathbb{R})$
is elliptic, then
$a^{-1}_{-r}\in\mathbb{V}_{1}(\mathbb{R})\cap\mathbb{W}(\mathbb{R})\subset\mathfrak{M}_{p}(\mathbb{R})$
and, due to the property (74) $W^{0}_{G,a^{-1}}$ is the inverse operator to
$W^{0}_{G,a}$:

$W^{0}_{G,a^{-1}}W^{0}_{G,a}=W^{0}_{G,a}W^{0}_{G,a^{-1}}=W^{0}_{G,1}=I.\hskip
128.0374pt\mbox{$\Box$\vskip 6.0pt plus 2.0pt minus 2.0pt\par}$

###### Remark 3

For $p=2$ the ellipticity condition (73) is necessary and sufficient for the
convolution operator

$\displaystyle
W^{0}_{G,a}\>:\>\mathbb{H}^{s}(G,dG(x))\rightarrow\mathbb{H}^{s-r}(G,dG(x))$
(76)

with a symbol $a\in\mathfrak{M}_{p}(\mathbb{R})$ to be invertible, because
$\mathfrak{M}_{2}(\mathbb{R})=\mathbb{L}_{\infty}(\mathbb{R})$.

But for $p\not=2$ there exist multipliers $a\in\mathfrak{M}_{p}(\mathbb{R})$,
which are elliptic $|a(\xi)|\geqslant 1$ and even continuous, but
$a^{-1}\not\in\mathfrak{M}_{p}(\mathbb{R})$ (see (Ig, 69, Theorem 6)).

##  4 Convolution integro-differential equation

The symbol of the integro-differential equation (6) is

$\displaystyle{\mathscr{A}}(\xi):=\sum_{k=0}^{m}\left[c_{k}(-2i\xi)^{k}+d_{k}(-2i\xi)^{m_{k}+n_{k}}({\mathscr{F}}\hskip-2.84526pt_{G}{\mathscr{K}}_{k})(\xi)\right],\qquad\xi\in\mathbb{R},$
(77)
$\displaystyle({\mathscr{F}}\hskip-2.84526pt_{G}{\mathscr{K}}_{k})(\xi):=\int_{-1}^{1}\left(\frac{1+y}{1-y}\right)^{i\xi}\frac{{\mathscr{K}}_{k}(y)dy}{1-y^{2}},\quad
k=1,\ldots,m.$

###### Theorem 4.1

Let $1<p<\infty$, $s\in\mathbb{R}$, $m_{k}+n_{k}\leqslant m$,
$k=1,2,\ldots,m$. The operator ${\boldsymbol{A}}$ in (6) in the setting

$\displaystyle{\boldsymbol{A}}\;:\;\mathbb{H}^{s}_{p}(G,dG(x))\to\mathbb{H}^{s-m}_{p}(G,dG(x))$
(78)

is Fredholm if and only if its symbol is elliptic:

$\displaystyle\inf_{\xi\in\mathbb{R}}\left|\frac{{\mathscr{A}}(\xi)}{(1+\xi^{2})^{m/2}}\right|>0.$
(79)

If the ellipticity condition (79) holds, the inverse operator is the following
convolution (pseudodifferential) operator
${\boldsymbol{A}}^{-1}=W^{0}_{G,{\mathscr{A}}^{-1}}$.

Proof: Due to Theorem 3.4 the operator ${\boldsymbol{A}}$ in (6) is bounded in
the setting (78). Due to formulae (3) and (39c), (39d) the following equality
holds

$({\mathscr{F}}\hskip-2.84526pt_{G}{\boldsymbol{A}}u)(\xi)={\mathscr{A}}(\xi)({\mathscr{F}}\hskip-2.84526pt_{G}u)(\xi),\qquad\mbox{\rm
or}\quad{\boldsymbol{A}}={{\mathscr{F}}}_{G}^{-1}{\mathscr{A}}{\mathscr{F}}\hskip-2.84526pt_{G}=W^{0}_{G,{\mathscr{A}}}.$

It is clear, that $(1+\xi^{2})^{-m/2}{\mathscr{A}}(\xi)$ belongs to the Wiener
algebra $W(\mathbb{R})$ and, due to the Wiener’s theorem, the inverse symbol
$(1+\xi^{2})^{m/2}{\mathscr{A}}^{-1}(\xi)$ also belongs to the Wiener’s
algebra $W(\mathbb{R})$, provided the symbol is elliptic. But then the inverse
symbol is an $\mathbb{L}_{p}$-multiplier for all $1<p<\infty$ (see Theorem
3.2).

Therefore, due to Theorem 3.4, ${\boldsymbol{A}}=W^{0}_{G,{\mathscr{A}}}$ is
invertible if and only if the ellipticity condition (79) holds and the inverse
operator is ${\boldsymbol{A}}^{-1}=W^{0}_{G,{\mathscr{A}}^{-1}}$. $\Box$

##  5 Prandtl equation

###### Theorem 5.1

Let $1<p<\infty$, $s\in\mathbb{R}$ and
$f_{0}\in\mathbb{H}^{s-1}_{p}(G,dG(x))$, where $f_{0}(x):=(1-x^{2})f(x)$. The
Prandtl Equation (3) has a unique solution $u\in\mathbb{H}^{s}_{p}(G,dG(x))$
if and only if its symbol is elliptic:

$\displaystyle\inf_{\xi\in\mathbb{R}}\left|\frac{{\mathscr{P}}(\xi)}{\sqrt{1+|\xi|^{2}}}\right|>0,\qquad{\mathscr{P}}(\xi):=c_{0}+2c_{1}\xi\coth(\pi\xi).$
(80)

If the ellipticity condition (80) holds, the solution is

$\displaystyle u(x)=(W^{0}_{G,{\mathscr{P}}^{-1}}f_{0})(x),\qquad
f_{0}(x)=(1-y^{2})f(x).$ (81)

If the condition (80) fails, equation (3) is not even Fredholm in the
following setting $f_{0}\in\mathbb{H}^{s-1}_{p}(G,dG(x))$ and
$u\in\mathbb{H}^{s}_{p}(G,dG(x))$.

In particular, if $f\in\mathbb{H}^{r}_{p}(G,dG(x))$, $r>1/p-1=-1/p^{\prime}$,
$p^{\prime}=p/(p-1)$, then the solution, if it exists, belongs to Hölder-
Zygmound space $u\in\mathbb{Z}^{\mu}_{p}(G)$ for $\mu<r+1/p^{\prime}$.

Proof: Let us multiply equation (3) by $1-x^{2}$ and rewrite it in the
following equivalent form:

$\displaystyle{\boldsymbol{P}}_{0}u(x)=c_{0}u(x)+(1-x^{2})\frac{c_{1}}{\pi}\int_{-1}^{1}\displaystyle\frac{u^{\prime}(y)dy}{y-x}=f_{0}(x),$
(82)

where $f_{0}(x)=(1-x^{2})f(x)$, $f_{0}\in\mathbb{H}^{s-1}(G,dG)$.

By using the equality

$\frac{1-x^{2}}{y-x}=\frac{1-xy}{y-x}+x,\qquad x,y\in G$

and the property of a solution $u(-1)=u(+1)=0$ we give equation (82) the
following form:

$\displaystyle\hskip-19.91692pt{\boldsymbol{P}}_{0}u(x)=c_{0}u(x)+\displaystyle\frac{c_{1}}{\pi}(1-x^{2})\int_{-1}^{1}\displaystyle\frac{u^{\prime}(y)dy}{y-x}=c_{0}u(x)+\displaystyle\frac{c_{1}}{\pi}\int_{-1}^{1}u^{\prime}(y)\displaystyle\frac{1-xy}{y-x}dy$
$\displaystyle+\displaystyle\frac{c_{1}x}{\pi}\int_{-1}^{1}u^{\prime}(y)dy=c_{0}u(x)+\displaystyle\frac{c_{1}}{\pi}\int_{-1}^{1}(1-y^{2})u^{\prime}(y)\displaystyle\frac{1-xy}{y-x}\displaystyle\frac{dy}{1-y^{2}}$
$\displaystyle=c_{0}u(x)+\frac{c_{1}}{\pi}\left[\mathfrak{D}_{G}u\ast_{G}\displaystyle\frac{1}{y}\right](x)=f_{0}(x),\qquad
x\in G,$ (83)

where $(v\ast_{G}w)(x)$ is the convolution (see (3)) and $\mathfrak{D}_{G}$ is
the derivative (see (7)) on the Lie group $G$.

By applying the Fourier transformation ${\mathscr{F}}\hskip-2.84526pt_{G}$
(see (12)) to the equation (5) and taking into account formulae (3), (39d),
(39e), we find the following:

$\displaystyle\hskip-19.91692ptc_{0}U(\xi)+\displaystyle\frac{c_{1}}{\pi}{\mathscr{F}}\hskip-2.84526pt_{G}(\mathfrak{D}_{G}u)(\xi){\mathscr{F}}\hskip-2.84526pt_{G}\left(\displaystyle\frac{1}{y}\right)(\xi)={\mathscr{P}}(\xi)U(\xi)=F_{0}(\xi),$
$\displaystyle{\mathscr{P}}(\xi):=c_{0}+2c_{1}\xi\coth(\pi\xi),\qquad
U(\xi):=({\mathscr{F}}\hskip-2.84526pt_{G}u)(\xi),\quad\xi\in\mathbb{R}.$ (84)

Solvability (Fredholmness) of the equation (3) under conditions of Theorem 5.1
means invertibility (Fredholmness) of the operator

$\displaystyle{\boldsymbol{P}}_{0}=W^{0}_{G,{\mathscr{P}}}\;:\;\mathbb{H}^{s}_{p}(G,dG(x))\to\mathbb{H}^{s-1}_{p}(G,dG(x)).$
(85)

It is clear, that $(1+\xi^{2})^{-1/2}{\mathscr{P}}(\xi)$ has bounded variation
(belongs to $\mathbb{V}_{1}(\mathbb{R})$) and the inverse symbol
$(1+\xi^{2})^{m/2}{\mathscr{A}}^{-1}(\xi)$ also has bounded variation,
provided the symbol is elliptic. But then the inverse symbol is an
$\mathbb{L}_{p}$-multiplier for all $1<p<\infty$ (see Theorem 3.2).

From Theorem 3.4 follows that the operator
${\boldsymbol{P}}_{0}=W^{0}_{G,{\mathscr{P}}}$ in (85) is invertible if and
only if the ellipticity condition (80) holds and is not Fredholm if the
ellipticity condition fails. The solution to equation (82) (and to equation
(3)) is represented by formula (81).

The assertion about a priori smoothness of a solution to equation (3) follows
from the inclusion $u\in\mathbb{H}^{r+1}_{p}(G,dG(x))$ and the Sobolev’s
embedding theorem (cf. Tr (95) for details):

$\displaystyle\mathbb{H}^{r+1}_{p}(G,dG(x))\subset\mathbb{Z}_{p}^{\mu}(G)\qquad{\rm
for}\quad 0<\mu<r+1-\frac{1}{p}=r+1/p^{\prime}.$ (86)

The proof is completed. $\Box$

##  6 Tricomi equation

###### Theorem 6.1

Let $1<p<\infty$, $s\in\mathbb{R}$, $g_{0}\in\mathbb{H}^{s}_{p}(G,dG(x))$,
where $g_{0}(x):=(1-x^{2})^{1/2}g(x)$. Tricomi Equation (4) has a unique
solution $v(x)$, such that
$v_{0}:=(1-x^{2})^{1/2}v\in\mathbb{H}^{s}_{p}(G,dG(x))$, if and only if its
symbol is elliptic:

$\displaystyle\inf_{\xi\in\mathbb{R}}|{\mathscr{T}}(\xi)|>0,\qquad{\mathscr{T}}(\xi):=c_{0}-ic_{1}\tanh(\pi\xi)+\frac{c_{2}}{\cosh(\pi\xi)},\quad\xi\in\mathbb{R}.$
(87)

If the ellipticity condition (87) holds, the solution is represented as
follows

$\displaystyle v(x)$ $\displaystyle=$ $\displaystyle
d_{0}g(x)-\frac{d_{1}}{\pi}\int_{-1}^{1}\sqrt{\frac{1-y^{2}}{1-x^{2}}}\displaystyle\frac{g(y)dy}{y-x}$
(88)
$\displaystyle+\int_{-1}^{1}\sqrt{\frac{1-y^{2}}{1-x^{2}}}k_{\boldsymbol{T}}\left(\frac{x-y}{1-xy}\right)g(y)dy,\qquad
x\in G,$

where

$d_{0}=\displaystyle\frac{(c_{0}+ic_{1})^{-1}+(c_{0}-ic_{1})^{-1}}{2},\qquad
d_{1}=\displaystyle\frac{(c_{0}+ic_{1})^{-1}-(c_{0}-ic_{1})^{-1}}{2}$

and $k_{\boldsymbol{T}}\in\mathbb{S}(G)$ is the inverse Fourier transform of
the inverse symbol:

$\displaystyle
k_{\boldsymbol{T}}(x):=({{\mathscr{F}}}_{G}^{-1}{\mathscr{T}}_{-1})(x)=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\infty}^{\infty}\left(\frac{1+x}{1-x}\right)^{-i\xi}{\mathscr{T}}_{-1}(\xi)d\xi,\quad
x\in G,$
$\displaystyle{\mathscr{T}}_{-1}(\xi)={\mathscr{T}}^{-1}(\xi)-d_{0}-d_{1}\tanh(\pi\xi),\quad{\mathscr{T}}_{-1}\in\mathbb{S}(\mathbb{R}).$

If the condition (87) fails, equation (4) is not Fredholm in the space setting
$v_{0},g_{0}\in\mathbb{H}^{s}_{p}(G,G(x))$, where $v_{0},g_{0}$ are defined
above.

In particular, if $g_{0}\in\mathbb{H}^{s}_{p}(G,dG(x))$, $s>1/p$, then the
solution, if it exists, belongs to weighted Hölder-Zygmound space:
$v(x)=\displaystyle\frac{v_{0}(x)}{\sqrt{1-x^{2}}}$, where
$v_{0}\in\mathbb{Z}^{\mu}_{p}(G)$ for $\mu<s-1/p$.

Proof: Let us multiply equation (4) by $\sqrt{1-x^{2}}$, use the notation of
functions $v_{0}$ and $g_{0}$ introduced above, and rewrite (4) in the
following equivalent form:

$\displaystyle{\boldsymbol{T}}_{0}v_{0}(x)=c_{0}v_{0}(x)+\frac{c_{1}}{\pi}\int_{-1}^{1}\frac{\sqrt{(1-x^{2})(1-y^{2})}}{y-x}v_{0}(y)dG(y)$
$\displaystyle+\frac{c_{2}}{\pi}\int_{-1}^{1}\frac{\sqrt{(1-x^{2})(1-y^{2})}}{1-xy}v_{0}(y)dG(y)$
$\displaystyle=c_{0}v_{0}(x)-\frac{c_{1}}{\pi}\int_{-1}^{1}\frac{1-xy}{x-y}\sqrt{1-\left(\frac{x-y}{1-xy}\right)^{2}}v_{0}(y)dG(y)$
$\displaystyle+\frac{c_{2}}{\pi}\int_{-1}^{1}\sqrt{1-\left(\frac{x-y}{1-xy}\right)^{2}}v_{0}(y)dG(y)$
$\displaystyle=c_{0}v_{0}(x)-\frac{c_{1}}{\pi}(y^{-1}\sqrt{1-y^{2}}\ast_{G}v_{0})(x)+\frac{c_{2}}{\pi}(\sqrt{1-y^{2}}\ast_{G}v_{0})(x)=g_{0}(x),$
(89)

where $(v\ast_{G}w)(x)$ is the convolution on the Lie group $G$ (see (12)).

By applying the Fourier transformation ${\mathscr{F}}\hskip-2.84526pt_{G}$
(see (12)) to the equation (6) and taking into account formulae (3), (39f),
(39g), we find the following:

$\displaystyle{\mathscr{T}}(\xi)V_{0}(\xi)$ $\displaystyle=$
$\displaystyle\left[c_{0}-\displaystyle\frac{c_{1}}{\pi}\left({\mathscr{F}}\hskip-2.84526pt_{G}\frac{\sqrt{1-y^{2}}}{y}\right)(\xi)+\displaystyle\frac{c_{2}}{\pi}{\mathscr{F}}\hskip-2.84526pt_{G}\left(\sqrt{1-y^{2}}\right)(\xi)\right]V_{0}(\xi)$
$\displaystyle=$ $\displaystyle
G_{0}(\xi),\qquad{\mathscr{T}}(\xi):=c_{0}-ic_{1}\tanh(\pi\xi)+\frac{c_{2}}{\cosh(\pi\xi)},$
$\displaystyle V_{0}(\xi):=({\mathscr{F}}\hskip-2.84526pt_{G}v_{0})(\xi),\quad
G_{0}(\xi):=({\mathscr{F}}\hskip-2.84526pt_{G}g_{0})(\xi),\quad\xi\in\mathbb{R}.$

The solvability (Fredholmness) of equation (4) under conditions of Theorem 6.1
means the invertibility (Fredholmness) of the operator

$\displaystyle{\boldsymbol{T}}_{0}=W^{0}_{G,{\mathscr{T}}}\;:\;\mathbb{H}^{s}_{p}(G,dG(x))\to\mathbb{H}^{s}_{p}(G,dG(x)).$
(91)

From Theorem 3.4 follows that the operator
${\boldsymbol{T}}_{0}=W^{0}_{G,{\mathscr{T}}}$ in (91) is invertible if and
only if the ellipticity condition (87) holds. The solution to equation (6) is
$v_{0}(x)={\boldsymbol{T}}^{-1}_{0}g_{0}(x)=W^{0}_{G,{\mathscr{T}}^{-1}}g_{0}(x)$
and, therefore, the solution to equation (4) is

$v(x)=(1-x^{2})^{-1/2}(W^{0}_{G,{\mathscr{T}}^{-1}}(1-y^{2})^{1/2}g)(x).$

From Lemma 3 follows, that

$\displaystyle{\mathscr{T}}^{-1}(\xi)=d_{0}-d_{1}\tanh(\pi\xi)+{\mathscr{T}}_{-1}(\xi),\qquad
k_{\boldsymbol{T}}(\xi)=({{\mathscr{F}}}_{G}^{-1}{\mathscr{T}}_{-1})(\xi),$
$\displaystyle{\mathscr{T}}_{-1}\in\mathbb{S}(\mathbb{R})\quad\mbox{\rm and,
therefore,}\quad k_{\boldsymbol{T}}\in\mathbb{S}(G)$

and formula (88) is proved.

The concluding assertion about a priori smoothness of a solution to equation
(4) follows from the Sobolev’s embedding theorem (86). $\Box$

##  7 Lavrentjev-Bitsadze equation

###### Theorem 7.1

Let $1<p<\infty$, $s\in\mathbb{R}$. The Lavrentjev-Bitsadze Equation (5) has a
unique solution $\varphi$ in the setting

$\displaystyle{\boldsymbol{V}}h,{\boldsymbol{V}}\varphi\in\mathbb{H}^{s}_{p}(G,dG(x)),\quad{\boldsymbol{V}}\psi(x):=(1-x^{2})\psi\left(\frac{1+x}{2}\right),\quad
x\in G,$ (92)

if and only if its symbol is elliptic:

$\displaystyle\inf_{\xi\in\mathbb{R}}|{\mathscr{L}}{\mathscr{B}}(\xi)|>0,\qquad{\mathscr{L}}{\mathscr{B}}(\xi):=c_{0}-ic_{1}\tanh\frac{\pi\xi}{2},\quad\xi\in\mathbb{R}.$
(93)

If the ellipticity condition (93) holds, the solution is

$\displaystyle\varphi(x)$ $\displaystyle=$ $\displaystyle
d_{0}h(x)-\frac{d_{1}}{\pi
i}\int_{0}^{1}\left[\displaystyle\frac{1}{y-x}+\displaystyle\frac{1-2y}{x+y-2xy}\right]\displaystyle\frac{h(y)dy}{y-x}$
(94)
$\displaystyle+2\int_{0}^{1}k_{{\boldsymbol{L}}{\boldsymbol{B}}}\left(\frac{x-y}{x+y-2xy}\right)h(y)dy,\qquad
x\in G^{+},$

where

$d_{0}=\displaystyle\frac{1}{2}\left[{(c_{0}+ic_{1})^{-1}+(c_{0}-ic_{1})^{-1}}\right],\qquad
d_{1}=\displaystyle\frac{1}{2}\left[{(c_{0}+ic_{1})^{-1}-(c_{0}-ic_{1})^{-1}}\right]$

and $k_{{\boldsymbol{L}}{\boldsymbol{B}}}\in\mathbb{S}(G)$ is the inverse
Fourier transform of the modified symbol:

$\displaystyle
k_{{\boldsymbol{L}}{\boldsymbol{B}}}(x):=({{\mathscr{F}}}_{G}^{-1}{\mathscr{L}}{\mathscr{B}}_{-1})(x)=\displaystyle\frac{1}{\pi}\displaystyle\int_{-\infty}^{\infty}\left(\frac{1-x}{1+x}\right)^{-i\xi}{\mathscr{L}}{\mathscr{B}}_{-1}(\xi)d\xi,\quad
x\in G^{+},$
$\displaystyle{\mathscr{L}}{\mathscr{B}}_{-1}(\xi)={\mathscr{L}}{\mathscr{B}}^{-1}(\xi)-d_{0}-d_{1}\tanh\frac{\pi\xi}{2},$
$\displaystyle{\mathscr{L}}{\mathscr{B}}_{-1}\in
C^{\infty}(\mathbb{R}),\quad{\mathscr{L}}{\mathscr{B}}_{-1}(\pm\infty)=0,\quad{\mathscr{L}}{\mathscr{B}}^{\prime}_{-1}\in\mathbb{S}(\mathbb{R}).$

If the condition (93) fails, equation (5) is not Fredholm in the setting (92).

In particular, if ${\boldsymbol{V}}h\in\mathbb{H}^{s}_{p}(G,dG(x))$, $s>1/p$
and the symbol is elliptic, the solution belongs to Hölder-Zygmound space
$\mathbb{Z}^{\mu}(G)$ for $\mu<s-1/p$.

Proof: In the equation (5) we change variables, introduce new unknown
functions

$\displaystyle\begin{array}[]{r}t=\displaystyle\frac{1+x}{2},\quad\tau=\displaystyle\frac{1+y}{2},\quad
w(x)={\boldsymbol{V}}\varphi(x),\quad h_{0}(x)={\boldsymbol{V}}h(x),\quad
x,y\in G,\end{array}$ (96)

multiply both sides by $1-x^{2}$ and obtain:

$\displaystyle\hskip-28.45274pt{\boldsymbol{L}}{\boldsymbol{B}}_{0}w(x)=c_{0}w(x)+\frac{c_{1}}{\pi}\int_{-1}^{1}\frac{1-x^{2}}{1-y^{2}}\left[\frac{1}{y-x}-\frac{y}{1-xy}\right]w(y)dy$
$\displaystyle=c_{0}w(x)+\frac{c_{1}}{\pi}\int_{-1}^{1}\frac{(1-x^{2})(1-y^{2})}{(y-x)(1-xy)}\frac{w(y)dy}{1-y^{2}}$
(97)
$\displaystyle=c_{0}w(x)+\frac{c_{1}}{\pi}\int_{-1}^{1}\left[\frac{x-y}{1-xy}-\frac{1-xy}{x-y}\right]\frac{w(y)dy}{1-y^{2}}=h_{0}(x),\quad
x\in G,$

because

$\displaystyle\frac{(1-x^{2})(1-y^{2})}{(y-x)(1-xy)}=\frac{x-y}{1-xy}-\frac{1-xy}{x-y}.$
(98)

Thus, ${\boldsymbol{L}}{\boldsymbol{B}}_{0}$ is a $G$-convolution operator
with the kernel $t-\displaystyle\frac{1}{t}$:

$\displaystyle{\boldsymbol{L}}{\boldsymbol{B}}_{0}w(x)=c_{0}w(x)+\frac{c_{1}}{\pi}\left(t-\frac{1}{t}\right)*_{G}w(x),\qquad
x\in G.$ (99)

The symbol of the operator is (cf. (39e) and (39h)):

$\displaystyle{\mathscr{L}}{\mathscr{B}}(\xi)$ $\displaystyle=$ $\displaystyle
c_{0}+\frac{c_{1}}{\pi}({\mathscr{F}}\hskip-2.84526pt_{G}t)(\xi)-\frac{c_{1}}{\pi}\left({\mathscr{F}}\hskip-2.84526pt_{G}\frac{1}{t}\right)(\xi)$
(100) $\displaystyle=$ $\displaystyle
c_{0}+ic_{i}\left[\frac{1}{\sinh(\pi\xi)}-\coth(\pi\xi)\right]$
$\displaystyle=$ $\displaystyle
c_{0}+\frac{ic_{1}(1-\cosh(\pi\xi))}{\sinh(\pi\xi)}=c_{0}-ic_{1}\tanh\frac{\pi\xi}{2},\qquad\xi\in\mathbb{R}.$

From Theorem 3.4 follows that the operator
${\boldsymbol{L}}{\boldsymbol{B}}_{0}=W^{0}_{G,{\mathscr{T}}}$ in (7) is
invertible if and only if the ellipticity condition (93) holds. The solution
to equation (94) is
$w(x)={\boldsymbol{L}}{\boldsymbol{B}}^{-1}_{0}h_{0}(x)=W^{0}_{G,{\mathscr{L}}{\mathscr{B}}^{-1}}{\boldsymbol{V}}h(x)$
and, therefore, solution to equation (5) is

$\varphi(x)={\boldsymbol{V}}^{-1}w(x):=\frac{w(1-2x)}{x(1-x)}=(W^{0}_{G,{\mathscr{L}}{\mathscr{B}}^{-1}}{\boldsymbol{V}}h)(x),$

Since ${\mathscr{L}}{\mathscr{B}}(\pm\infty)=c_{0}\pm ic_{1}$, from Lemma 3 we
conclude

$\displaystyle{\mathscr{L}}{\mathscr{B}}^{-1}(\xi)=d_{0}-d_{1}\tanh\frac{\pi\xi}{2}+{\mathscr{L}}{\mathscr{B}}_{-1}(\xi),\qquad{\mathscr{L}}{\mathscr{B}}_{-1}\in\mathbb{S}(\mathbb{R}),$
$\displaystyle
k_{{\boldsymbol{L}}{\boldsymbol{B}}}(x)=({\mathscr{F}}\hskip-2.84526pt_{G}^{-1}{\mathscr{L}}{\mathscr{B}}_{-1})(x),\quad
x\in G,\qquad k_{{\boldsymbol{L}}{\boldsymbol{B}}}\in\mathbb{S}(G)$

and formula (94) is proved, from which follows formula (94) by the change of
the integration variable.

The concluding assertion about a priori smoothness of a solution to equation
(5) follows from the Sobolev’s embedding theorem (86). $\Box$

###### Acknowledgements.

The investigation is supported by the grant of the Shota Rustaveli Georgian
National Science Foundation FR-19-676

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