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Roots and Logarithms of Multipliers
Functional Analysis · Submitted 23 May 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that for such an $f$ and for every $t\in \mathbb{R}$, $f^t$ is also a multiplier for $H^2_n$.
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The abstract, sentence by sentence
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 $B$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $H^2_n$. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s1 | We show that for such an $f$ and for every $t\in \mathbb{R}$, $f^t$ is also a multiplier for $H^2_n$. | result 0.93 | 0.96 | 0.83 | asserted 1.00 |
| s2 | 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. | method 0.95 | 0.96 | 0.16 | asserted 1.00 |
| s3 | 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 $B$. | result 0.99 | 0.92 | 0.17 | asserted 1.00 |
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The paper
Original 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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