Generic Bessel Potential Spaces on Lie Groups

The purpose of the present work is to define suitable (generic) Bessel Potential Spaces (GBPS) for convolution integro-differential equations (CIDE) on a Lie group G, which is homomorphic to the Euclidean space \(\mathbb {R}^n\) . Is proved that ellipticity of the symbol is necessary condition for the Fredholm property of all CIDE in GBPS, but not sufficient. Ellipticity is sufficient condition for unique solvability of CIDE for \(p=2\) and, for \(p\neq 2\) , only for CIDE with symbols from the subalgebra of \(\mathbb {L}_p\) -multipliers.