CONVOLUTION EQUATIONS ON THE LIE GROUP G =(−1 , 1)

The interval *G*=(−1 *,*1) turns into a Lie group under the
group operation x ◦ y : = ( x + y ) ( 1 + x y ) − 1 , x , y ∈ G . This
enables definition of the invariant measure d G ( x ) : = ( 1 − x 2 ) −
1 d x and the Fourier transformation F _{G} on the
interval *G* and, as a consequence, we can consider Fourier
convolution operators W G , a 0 : = F G − 1 a F 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 G u ( x ) = ( 1 − x
2 ) u ′ ( x ) , *x*∈ *G*. Equations are solved in the scale of
Bessel potential H p s ( G , d G ( x ) ) , 1⩽ *p*⩽∞, and
Hölder-Zygmound Z ν ( G ) , 0 *∞ spaces, adapted to the group
G. Boundedness of convolution operators (the problem of
multipliers) is discussed. The symbol a( ξ), ξ∈R,
of a convolution equation W G , a 0 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 .*