The interval \({\varvec{G}}=(-1,1)\) turns into a Lie group under the group operation \(x\circ y:=(x+y)(1+xy)^{-1},\qquad x,y\in {\varvec{G}}\) . Then \({\varvec{M}}=[0,1)\) is a submonoid of \({\varvec{G}}\) (shares the same binary operation \(x\circ y\) ) and we can induce the invariant Haar measure \({\text {d}}\mu _{\varvec{M}}:=(1-x^2)^{-1}{\text {d}}x\) and the Fourier transformation \(\mathcal {F}\hspace{-2.84526pt}_{\varvec{M}}\) from \({\varvec{G}}\) to \({\varvec{M}}\) . The main object of the investigation is the Fourier convolution operator \(\varvec{W}_{{\varvec{M}},a}:=r_+\varvec{W}^0_{{\varvec{G}},a}\ell _+\) , \(\varvec{W}^0_{{\varvec{G}},a}:=\mathcal {F}\hspace{-2.84526pt}_{\varvec{G}}^{-1} a\mathcal {F}\hspace{-2.84526pt}_{\varvec{G}}\) restricted from \({\varvec{G}}\) . Theory of convolution operators \(\varvec{W}_{{\varvec{M}},a}\) on the submonoid \({\varvec{M}}\) is much more complicated, but more rich and important in applications (example of Wiener–Hopf equations on submonoid \({\varvec{M}}=[0,\infty )\) of the Lie group \({\varvec{G}}=(-\infty ,\infty )\) is a good example). Convolution equation \(\varvec{W}_{{\varvec{M}},s}\varphi =f\) in the Generic Bessel potential space setting \(f\in \mathbb {G}\mathbb {H}^{s-r}_p({\varvec{M}},{\text {d}}\mu _{\varvec{M}})\) , \(\varphi \in \mathbb {G}\mathbb {H}^s_p({\varvec{M}},{\text {d}}\mu _{\varvec{M}})\) , \(1<p<\infty \) , \(s,r\in \mathbb {R}\) , has non-trivial Fredholm index and the Fredholm property, as well as the solvability conditions for discontinuous symbol \(a(\xi )\) depend on the parameters s and p of the spaces. We expose full theory of such convolution integro-differential equations: Fredholm property and solvability criteria, index formula. Formula for solutions are available through the factorization of the symbol.
