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

The interval G = (- 1, 1) turns into a Lie group under the group operation x circle y := ( x + y)( 1 + xy)(-1), x, y is an element of G. This enables us to define of the invariant measure dG(x) := (1 - x(2))(-1)dx and the Fourier transformation F-G on the interval G and, as a consequence, we can consider Fourier convolution operatorsW(G,a)(0) := F(G)(-1)aF(G) on G. This class of convolutions includes the celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the generic differential operator D(G)u(x) = (1 - x(2))u'(x), x is an element of G. Equations are solved in the scale of generic Bessel potential H-p(s)(G, dG(x)), 1 <= p <= infinity, and Holder-Zygmund Z(nu)(G), 0 < mu, nu < infinity, spaces, adapted to the group G. The boundedness of convolution operators (the problem of multipliers) is discussed. The symbol a(xi), xi is an element of R, of a convolution equation W-G,a(0) u = f defines solvability as follows: 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. Also, we shortly touch upon the multidimensional analogue - the Lie group G(n).