CONVOLUTION EQUATIONS ON THE ABELIAN GROUP $\cA(-1,1)$

The interval $j=[-1,1]$ turns into an Abelian group
$\cA(\cJ)$ under the group operation
$x+_\cJ y:=(x+y)(1+xy)^{-1},\qquad
x,y\in\cJ$. This enables definition of
the invariant measure $d_\cJ x=(1-x^2)^{-1}dx$
and the Fourier transform $\cF_\cJ$ on
the interval $\cJ$ and, as a consequence, we can
consider Fourier convolution operators
$W^0_{\cJ,\cA}:=\cF_\cJ^{-1}\cA\cF_\cJ$
on $\cJ$. This class of convolutions includes
celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also,
differential equations of arbitrary order with the natural weighted
derivative $\fD_\cJ
u(x)=-(1-x^2)u’(x)$, $t\in\cJ$.
Equations are solved in the scale of Bessel potential
$\bH^s_p(\cJ,d_\cJ
x)$, $1\leqslant
p\leqslant\infty$, and
H\”older-Zygmound
$\bZ^\nu(\cJ,(1-x^2)^\mu)$,
$0<\mu,\nu<\infty$
spaces, adapted to the group $\cA(\cJ)$.
Boundedness of convolution operators (the problem of multipliers) is
discussed. The symbol $\cA(\xi)$,
$\xi\in\bR$, of a
convolution equation
$W^0_{\cJ,\cA}u=f$ defines
solvability: the equation is uniquely solvable if and only if the symbol
$\cA$ is elliptic. The solution is written explicitely
with the help of the inverse symbol. We touch shortly the
multidimensional analogue-the Abelian group
$\cA(\cJ^n)$.