Potential operators in generalized Hölder spaces on sets in quasi-metric measure spaces without the cancellation property

We consider potential operators of order αα over sets ΩΩ in quasi-metric measure spaces and study their mapping properties from the subspace H0λ(Ω) of functions in Hölder space Hλ(Ω)Hλ(Ω) vanishing on the boundary of ΩΩ, into the space Hλ+α(Ω)Hλ+α(Ω), if λ+α<1λ+α<1. This is proved in a more general setting of generalized Hölder spaces Hω(Ω)Hω(Ω) with a given dominant ωω of modulus of continuity. Statements of such a kind are known in the Euclidean case or in the case of quasimetric measure spaces with the cancellation property. In the general case, when the cancellation property fails, our proofs are based on a special treatment of the potential of a constant function, which in general has a regularity near the boundary ∂Ω∂Ω of the type of the αα-th power of the distance to ∂Ω∂Ω. An application to the case of spatial potentials over domains in RnRn and potentials over spherical caps is given.