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In this paper we define Bessel potentials in Ahlfors regular spaces using aCoifman type approximation of the identity, and show they improve regularityfor Lipschitz, Besov and Sobolev-type functions. We prove density and embeddingresults for the Sobolev potential spaces defined by them. Finally, viafractional derivatives, we find that for small orders, these Bessel potentialsare inversible, and show a way to characterize potential spaces, using singularintegrals techniques, such as the $T1$ theorem. Moreover, this characterizationallows us to prove these spaces in fact coincide with the classical potentialSobolev spaces in the Euclidean case.

Bessel potentials in Ahlfors regular metric spaces