On measuring unboundedness of the $H^\infty$-calculus for generators of analytic semigroups

TL;DR

This paper studies the unboundedness of the $H^$-calculus for generators of analytic semigroups, providing bounds on the operator norm growth near zero and exploring conditions for boundedness.

Contribution

It establishes precise bounds on the $H^$-calculus unboundedness, generalizing previous results and analyzing the impact of square function estimates.

Findings

Bound $b(ps)$ grows as $|ps|$ approaches zero, with order $|ps| ext{log}$ for general cases.

Bound $b(ps)$ remains bounded ($O(1)$) for bounded calculi.

Square function estimates lead to a bound of order $ ext{log}^{1/2}|ps|$.

Abstract

We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$ in general, whereas $b(\varepsilon)=\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield $b(\varepsilon)=\mathcal{O}(\sqrt{|\log\varepsilon|})$.