Abstract
Let –iA be the generator of a C 0-group \({(U(s))_{s\in \mathbb {R}}}\) on a Banach space X and ω > θ(U), the group type of U. We prove a transference principle that allows to estimate \({||{f(A)}||}\) in terms of the \({{\rm{L}}^{p}(\mathbb {R};X)}\)-Fourier multiplier norm of \({f(\cdot \pm i \omega)}\). If X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev–de Laubenfels. If X is a UMD space, one obtains a bounded \({{\rm{H}}^\infty_1}\)-calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded \({{\rm{H}}^\infty}\)-calculus on sectors.
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Acknowledgments
I am grateful to Gordon Blower (Lancaster) for an interesting discussion and for bringing to my attention an unfinished work of his and Ian Doust’s about functional calculus on Venturi regions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Haase, M. A transference principle for general groups and functional calculus on UMD spaces. Math. Ann. 345, 245–265 (2009). https://doi.org/10.1007/s00208-009-0347-3
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DOI: https://doi.org/10.1007/s00208-009-0347-3