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arXiv · 1904.03960

Boundary values of holomorphic semigroups and fractional integration

Omar EL-Mennaoui·Valentin Keyantuo·Ahmed Sani
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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 $\rm{lip}_0^α[0,\, 1] , \, 0<α<1$ 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 $π/2$?

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Omar EL-Mennaoui, Valentin Keyantuo, Ahmed Sani. 2019-04-08. Boundary values of holomorphic semigroups and fractional integration. https://arxiv.org/abs/1904.03960

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