Functional calculus on real interpolation spaces for generators of $C_{0}$-groups

TL;DR

This paper investigates the functional calculus of $C_{0}$-group generators on real interpolation spaces, extending classical transference principles and establishing bounded $H^{}_{1}$-calculus results.

Contribution

It introduces interpolation versions of transference principles for bounded and unbounded groups and proves bounded $H^{}_{1}$-calculus for generators on real interpolation spaces.

Findings

Interpolation transference principles are established for bounded and unbounded groups.

Generators on Banach spaces have bounded $H^{}_{1}$-calculus on real interpolation spaces.

Additional results follow from the main calculus properties.

Abstract

We study functional calculus properties of $C_{0}$-groups on real interpolation spaces, using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded $H^{\infty}_{1}$-calculus on real interpolation spaces. Additional results are derived from this.