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

Abstract

Abstract The interval G = ( - 1 , 1 ) {G=(-1,1)} turns into a Lie group under the group operation x ∘ y := ( x + y ) ⁢ ( 1 + x ⁢ y ) - 1 {x\circ y:=(x+y)(1+xy)^{-1}} , x , y ∈ G {x,y\in G} . This enables us to define of the invariant measure d ⁢ G ⁢ ( x ) := ( 1 - x 2 ) - 1 ⁢ d ⁢ x {dG(x):=(1-x^{2})^{-1}dx} and the Fourier transformation ℱ G {{\mathcal{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 {W^{0}_{G,a}:={\mathcal{F}}{}_{G}^{-1}a{\mathcal{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 ) {\mathfrak{D}_{G}u(x)=(1-x^{2})u^{\prime}(x)} , x ∈ G {x\kern-1.0pt\in\kern-1.0ptG} . Equations are solved in the scale of generic Bessel potential ℍ p s ⁢ ( G , d ⁢ G ⁢ ( x ) ) {\mathbb{H}^{s}_{p}(G,dG(x))} , 1 ⩽ p ⩽ ∞ {1\kern-1.0pt\leqslant\kern-1.0ptp\kern-1.0pt\leqslant\kern-1.0pt\infty} , and Hölder–Zygmund ℤ ν ⁢ ( G ) {\mathbb{Z}^{\nu}(G)} , 0 < μ , ν < ∞ {0<\mu,\nu<\infty} , spaces, adapted to the group G. The boundedness of convolution operators (the problem of multipliers) is discussed. The symbol a ⁢ ( ξ ) {a(\xi)} , ξ ∈ ℝ {\xi\in\mathbb{R}} , of a convolution equation W G , a 0 ⁢ u = f {W^{0}_{G,a}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 {G^{n}} .