Unbounded functional calculus for bounded groups with applications

Abstract

In this paper, we develop the unbounded extension of the Hille–Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Levy–Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by fractional powers of group generators, where the power is not an odd integer, and a shifted abstract Grunwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines.