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arXiv · 1701.06623

$H^\infty$-calculus for semigroup generators on BMO

Tim Ferguson·Tao Mei·Brian Simanek
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Abstract

We prove that the negative infinitesimal generator $L$ of a semigroup of positive contractions on $L^\infty$ has a bounded $H^\infty(S_\eta^0)$-calculus on the associated Poisson semigroup-BMO space for any angle $\eta>\pi/2$, provided the semigroup satisfies Bakry-Emry's $\Gamma_2 $ criterion. Our arguments only rely on the properties of the underlying semigroup and works well in the noncommutative setting. A key ingredient of our argument is a quasi monotone property for the subordinated semigroup $T_{t,\alpha}=e^{-tL^\alpha},0<\alpha<1$, that is proved in the first half of the article.

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Tim Ferguson, Tao Mei, Brian Simanek. 2017-01-23. $H^\infty$-calculus for semigroup generators on BMO. https://doi.org/10.1016/j.aim.2019.02.027

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