论文标题
作用在树状结构上的组的非交换性cotlar身份
Noncommutative Cotlar identities for groups acting on tree-like structures
论文作者
论文摘要
令$ t_m $为非交换性傅立叶乘数。在最近的工作中，MEI和RICARD引入了Cotlar身份的非共同类似物，以证明某些乘数在免费组的非交换性$ L_P $ - 空格上有限。在这里，我们以$ m $的方式研究了Cotlar类型的身份，为他们提供了封闭的特征： \ [ \ big（m（g h） - m（g）\ big）\，\ big（m（g^{ - 1}） - m（h）\ big）= 0， \; \ forall g \ in \ mathrm {g} \ setMinus \ {e \}，h \ in \ mathrm {g}。 \]我们设法使用几何论点证明，如果$ x $是一棵树 - 或更一般而言，$ \ mathbb {r} $ - 树 - $ \ mathrm {g} $ acts and $ \ m $ tact and $ m $ lifts to foruct $ \ wideTilde {m} \ {x_0 \} $，然后$ m $满足Cotlar的身份，因此$ t_m $以$ l_p $为$ 1 <p <\ infty $。该结果在$ \ mathbb {r} $ - 树和傅立叶乘数上建立了组动作之间的新连接。我们表明，当动作具有全球固定点时，$ m $是微不足道的。这种机械使我们能够同时概括MEI和RICARD的自由群体变换，以及左下群中希尔伯特转换的理论，这是从阿尔维森的亚基亚尔代数来看。使用低音 - 列理论，我们构建了组中傅立叶乘数的新示例。其中包括诸如Baumslag-Solitar群体之类的新家庭。我们还表明，$ \ mathrm {psl} _2（\ mathbb {c}）$中的天然希尔伯特变换在限制到bianchi group $ \ mathrm {psl} _2 _2（\ mathbb {z} [z} [\ sqrt {\ sqrt {\ sqrt {-1}}时，就可以满足Cotlar的身份。
Let $T_m$ be a noncommutative Fourier multiplier. In recent work, Mei and Ricard introduced a noncommutative analogue of Cotlar's identity in order to prove that certain multipliers are bounded on the non-commutative $L_p$-spaces of a free group. Here, we study Cotlar type identities in full generality, giving a closed characterization for them in terms of $m$: \[ \big( m(g h) - m(g) \big) \, \big( m(g^{-1}) - m(h) \big) = 0, \; \forall g \in \mathrm{G} \setminus \{e\}, h \in \mathrm{G}. \] We manage to prove, using a geometric argument, that if $X$ is a tree -- or more generally an $\mathbb{R}$-tree -- on which $\mathrm{G}$ acts and $m$ lifts to a function $\widetilde{m}: X \to \mathbb{C}$ that is constant on the connected subsets of $X \setminus \{x_0\}$, then $m$ satisfies Cotlar's identity and thus $T_m$ is bounded in $L_p$ for $1 < p < \infty$. This result establishes a new connection between groups actions on $\mathbb{R}$-trees and Fourier multipliers. We show that $m$ is trivial when the action has global fixed points. This machinery allows us to simultaneously generalize the free group transforms of Mei and Ricard and the theory of Hilbert transforms in left-orderable groups, which follows from Arveson's subdiagonal algebras. Using Bass-Serre theory, we construct new examples of Fourier multipliers in groups. Those include new families like Baumslag-Solitar groups. We also show that a natural Hilbert transform in $\mathrm{PSL}_2(\mathbb{C})$ satisfies Cotlar's identity when restricted to the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{-1}])$.
