Estimates of the Laplace transform on convolution Sobolev algebras

We study the range of the Laplace transform on convolution Banach algebras T(α)(tα)T(α)(tα), α>0α>0, defined by fractional derivation. We introduce Banach algebras A0(α)(C+) of holomorphic functions in the right hand half-plane which are defined using complex fractional derivation along rays leaving the origin. We prove that the range of the Laplace transform on T(α)(tα)T(α)(tα) is densely contained in A0(α)(C+). The proof makes use of the so-called Kummer functions.