Convolution equations on the Lie group (-1,1)

The interval $j=[-1,1]$ turns into an Abelian group $\text{cA}(\text{cJ})$ under the group operation $x{+}_{\text{cJ}}y:=(x+y)(1+xy{)}^{-1},x,y\in \text{cJ}$. This enables definition of the invariant measure $d_{\text{cJ}}x=(1-x^2{)}^{-1}dx$ and the Fourier transform ${\text{cF}}_{\text{cJ}}$ on the interval $\text{cJ}$ and, as a consequence, we can consider Fourier convolution operators $W_{\text{cJ},\text{cA}}^0:={\text{cF}}_{\text{cJ}}^{-1}\text{cA}{\text{cF}}_{\text{cJ}}$ on $\text{cJ}$. This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative ${\text{fD}}_{\text{cJ}}u(x)=-(1-x^2)u^{\mathrm{\prime }}(x)$, $t\in \text{cJ}$. Equations are solved in the scale of Bessel potential ${\text{bH}}_p^s(\text{cJ},d_{\text{cJ}}x)$, $1⩽p⩽\mathrm{\infty }$, and Hölder-Zygmound ${\text{bZ}}^{\nu }(\text{cJ},(1-x^2{)}^{\mu })$, $0<\mu ,\nu <\mathrm{\infty }$ spaces, adapted to the group $\text{cA}(\text{cJ})$. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol $\text{cA}(\xi )$, $\xi \in \text{bR}$, of a convolution equation $W_{\text{cJ},\text{cA}}^{0}u=f$ defines solvability: the equation is uniquely solvable if and only if the symbol $\text{cA}$ is elliptic. The solution is written explicitely with the help of the inverse symbol. We touch shortly the multidimensional analogue-the Abelian group $\text{cA}({\text{cJ}}^n)$.