Boundary values of holomorphic semigroups and fractional integration

Abstract

The concept of boundary values of holomorphic semigroups in a general Banach space is studied. As an application, we consider the Riemann-Liouville semigroup of integration operator in the little Hölder spaces ${\mathrm{lip}}_0^{\alpha }[0,1],0<\alpha <1$ and prove that it admits a strongly continuous boundary group, which is the group of fractional integration of purely imaginary order. The corresponding result for the $L^p$-spaces ($1<p<\infty$) has been known for some time, the case $p=2$ dating back to the monograph by Hille and Phillips. In the context of $L^p$ spaces, we establish the existence of the boundary group of the Hadamard fractional integration operators using semigroup methods. In the general framework, using a suitable spectral decomposition,we give a partial treatment of the inverse problem, namely: Which $C_0$-groups are boundary values of some holomorphic semigroup of angle $\pi /2$?