# Boundary values of holomorphic semigroups and fractional integration

## TL;DR

This paper investigates boundary values of holomorphic semigroups in Banach spaces, focusing on fractional integration operators in H"older and L^p spaces, and explores which groups are boundary values of such semigroups.

## Contribution

It extends the theory of boundary values of holomorphic semigroups to fractional integration operators in various function spaces and characterizes boundary groups via spectral decomposition.

## Key findings

- The Riemann-Liouville semigroup admits a strongly continuous boundary group in little H"older spaces.
- Boundary groups of Hadamard fractional integration operators are established in L^p spaces using semigroup methods.
- Partial characterization of C_0-groups as boundary values of holomorphic semigroups of angle π/2.

## 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 H\"older spaces $\rm{lip}_0^\alpha[0,\, 1] , \, 0<\alpha<1$ 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 $\pi/2$?