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

Abstract

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 us to define 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 a G - 1 ℱ 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 ∈ G . Equations are solved in the scale of generic Bessel potential ℍ p s ⁢ ( G , d ⁢ G ⁢ ( x ) ) , 1 ⩽ p ⩽ ∞ , and Hölder–Zygmund ℤ ν ⁢ ( G ) , 0 < μ , ν < ∞ , spaces, adapted to the group G. The 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 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 .