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arXiv · 2405.14401

Roots and Logarithms of Multipliers

Jingbo Xia·Congquan Yan·Danjun Zhao·Jingming Zhu
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Abstract

By now it is a well-known fact that if $f$ is a multiplier for the Drury-Arveson space $H^2_n$, and if there is a $c>0$ such that $|f(z)|\geq c$ for every $z\in B$, then the reciprocal function 1/f is also a multiplier for $H^2_n$. We show that for such an $f$ and for every $t\in \mathbb{R}$, $f^t$ is also a multiplier for $H^2_n$. We do so by deriving a differentiation formula for $R^m(f^th)$.Moreover, by this formula the same result holds for spaces $H_{m,s}$ of the Besov-Dirichlet type. The same technique also gives us the result that for a non-vanishing multiplier $f$ of $H^2_n$, $log f$ is a multiplier of $H^2_n$ if and only if log $f$ is bounded on $B$.

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Jingbo Xia, Congquan Yan, Danjun Zhao, Jingming Zhu. 2024-05-23. Roots and Logarithms of Multipliers. https://arxiv.org/abs/2405.14401

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