2 Cotlar identities and multipliers

Noncommutative integration. Throughout this text we will use liberally noncommutative integration theory and the theory of noncommutative $L_p$ -spaces. Let $\mathcal {M} \subseteq \mathcal {B}(\mathcal {H})$ be a von Neumann algebra admitting a normal semifinite and faithful tracial weight $\tau : \mathcal {M}_+ \to [0,\infty ]$ that we will henceforth just refer to as a n.s.f. trace. It is possible to construct the noncommutative $L_p$ -spaces associated to $(\mathcal {M},\tau )$ as the subset of $\tau $ -measurable operators $L_p(\mathcal {M},\tau ) \subseteq L_0(\mathcal {M},\tau )$ satisfying that

$$\begin{align*}f \in L_p(\mathcal{M},\tau) \quad \Longleftrightarrow \quad \| f \|_{p} := \tau \big( |f|^{p} \big)^{\frac{1}{p}} < \infty. \end{align*}$$

This theory, which goes back all the way to Dixmier and Segal [Reference Dixmier24, Reference Segal78], is already well understood and the interested reader can consult [Reference Terp84, Reference Pisier and Xu72, Reference Goldstein and Labuschagne32].

Let $\mathrm {G}$ be a locally compact group that we will throughout the text assume to be second countable, and let $L_2(\mathrm {G})$ be its $L_2$ -space with respect to the left Haar measure $\mu $ [Reference Folland28]. As usual, we will denote by $\lambda : \mathrm {G} \to \mathcal {U}(L_2 (\mathrm {G}))$ the left regular representation, which is the unitary representation $g \mapsto \lambda _g$ that acts by sending $\xi (h)$ to $\xi (g^{-1} h)$ . The left regular von Neumann algebra $\mathcal {L} \mathrm {G} \subseteq \mathcal {B}(L_2 (\mathrm {G}))$ of $\mathrm {G}$ is given by

$$\begin{align*}\overline{ \lambda[L_1(\mathrm{G})]^{{{\mathrm{w}}^\ast}}} \, = \, \overline{\Big\{ \lambda(\varphi) := \int_{\mathrm{G}} \varphi(g) \, \lambda_g \, d \mu(g) : \varphi \in L_1(\mathrm{G}) \Big\}^{{\mathrm{w}}^\ast}} \, \subseteq \, \mathcal{B}(L_2( \mathrm{G})). \end{align*}$$

This von Neumann algebra admits a normal, semifinite and faithful weight $\tau : \mathcal {L} \mathrm {G}_+ \to [0,\infty ]$ that satisfies the Plancherel identity, meaning that $\varphi \mapsto \lambda (\varphi )$ extends to a unitary isometry from $L_2(\mathrm {G})$ to the Gelfand-Neumark-Segal space $L_2(\mathcal {L} \mathrm {G};\tau )$ . This weight is usually referred to as the Plancherel weight [Reference Pedersen67, Chapter 7]. The weight $\tau $ is a n.s.f. trace precisely when $\mathrm {G}$ is unimodular. Thus, we will work in the natural setting of unimodular groups and refer to $\tau $ as the Plancherel trace. In this context, the Plancherel trace is given by

$$\begin{align*}\tau \left( \int_{\mathrm{G}} \varphi(g) \, \lambda_g \, d \mu(g) \right) = \varphi(e), \quad \text{ for every } \, \varphi \in C_c(\mathrm{G}) \ast C_c(\mathrm{G}). \end{align*}$$

We will denote the noncommutative $L_p$ -spaces associated to $\tau $ simply by $L_p(\mathcal {L} \mathrm {G})$ . By analogy with the classical Fourier transform, we will denote by $\widehat {f}(g)$ the value $\tau (\lambda _g^\ast f)$ , which is well-defined whenever $f \in L_1(\mathcal {L} \mathrm {G})$ . It is also worth noticing that, by Plancherel’s theorem, the map $f \mapsto \widehat {f}$ is well-defined for any $f \in L_2(\mathcal {L} \mathrm {G})$ . In fact, we have that $f \in L_2(\mathcal {L} \mathrm {G})$ can be expressed as

(2.1) $$ \begin{align} f = \lambda(\widehat{f}) := \int_{\mathrm{G}} \widehat{f}(g) \, \lambda_g \, d \mu(g), \end{align} $$

where the integral on the right-hand side converges in the $L_2$ norm.

Conditional expectations. Let $\mathcal {N} \subseteq \mathcal {M}$ be a von Neumann subalgebra of $\mathcal {M}$ , that is a $\ast $ -subalgebra that is also ultraweakly closed. If $\tau $ is a n.s.f. trace over $\mathcal {M}$ and $\tau {|}_{\mathcal {N}}$ is still semifinite, then it is easy to see that the inclusion $\iota : L_1(\mathcal {N}) \hookrightarrow L_1(\mathcal {M})$ is isometric and its dual map is a trace preserving and normal (i.e., ultraweakly continuous) conditional expectation $\mathbb {E}: \mathcal {M} \to \mathcal {N} \subseteq \mathcal {M}$ . By conditional expectation we mean a unital and completely positive map such that $\mathbb {E}{|}_{\mathcal {N}} = \mathrm {id}_{\mathcal {N}}$ . Recall that conditional expectations satisfy that $\mathbb {E} \circ \mathbb {E} = \mathbb {E}$ and that, by Tomiyama’s Theorem, see [Reference Brown and Ozawa7, Theorem 1.5.10], $\mathbb {E}$ is automatically $\mathcal {N}$ -bimodular.

Let $\mathrm {G}_0 \subseteq \mathrm {G}$ be two groups such that $\mathrm {G}_0$ is open inside $\mathrm {G}$ . Then $\mathrm {G}_0$ is unimodular if $\mathrm {G}$ is. Furthermore, the Plancherel trace $\tau _{\mathrm {G}_0}$ of $\mathcal {L} \mathrm {G}_0$ coincides with the Plancherel trace of $\mathcal {L} \mathrm {G}$ restricted to $\mathcal {L} \mathrm {G}_0$ . Therefore there is a normal and trace-preserving conditional expectation $\mathbb {E}: \mathcal {L} \mathrm {G} \to \mathcal {L} \mathrm {G}_0 \subseteq \mathcal {L} \mathrm {G}$ that is given by the Fourier multiplier associated to ${\mathbf {1}}_{\mathrm {G}_0}$ , i.e.,

$$\begin{align*}\mathbb{E}(f) \, = \, \mathbb{E} \left( \int_{\mathrm{G}} \widehat{f}(g) \, \lambda_g \, d \mu(g) \right) \, = \, \int_{\mathrm{G}} {\mathbf{1}}_{\mathrm{G}_0}(g) \, \widehat{f}(g) \, \lambda_g \, d \mu(g). \end{align*}$$

The fact that $\mathbb {E}$ is trace preserving allows us to extend $\mathbb {E}$ as a contraction to all the $L_p$ -spaces $1\leq p \leq \infty $ , $\mathbb {E}:L_p(\mathcal {L} \mathrm {G}) \to L_p(\mathcal {L} \mathrm {G}_0) \subseteq L_p(\mathcal {L} \mathrm {G})$ .

Noncommutative Cotlar identities. Most of this section up until the closed-formula characterization of the Cotlar identity in the proof of Theorem A follows closely the results obtained by Mei and Ricard and it is included here for the sake of completeness.

We are assuming that H is bounded in $L_2(\mathcal {M})$ a priori; this is not a restrictive imposition since it holds trivially in the case of Fourier multipliers. Observe as well that, although both Cotlar identities require applying H to products of functions that therefore may not be in $L_2(\mathcal {M})$ , the fact that $f \in L_2(\mathcal {M}) \cap \mathcal {M}$ and that $L_2(\mathcal {M})$ is stable by multiplications by $\mathcal {M}$ makes both equations meaningful.

We will need the following lemma. Notice that if $H: L_2(\mathcal {M}) \to L_2(\mathcal {M})$ is left $\mathcal {N}$ -modular, then its restriction to $L_2(\mathcal {N})$ composed with the expectation $\mathbb {E}:L_2(\mathcal {M}) \to L_2(\mathcal {N})$ gives a map

$$\begin{align*}\mathbb{E} \, H \, {|}_{L_2(\mathcal{N})}:L_2(\mathcal{N}) \to L_2(\mathcal{N}) \end{align*}$$

that is both bounded with norm equal to that of H and and left $\mathcal {N}$ -modular. But all left $\mathcal {N}$ -modular maps in $L_2(\mathcal {N})$ are right multiplicators by an element in $\mathcal {N}$ of norm equal to the operator norm. As such, $\mathbb {E} \, H \, {|}_{L_2(\mathcal {N}) \cap L_p(N)}$ extends to a bounded operator in $L_p(\mathcal {N})$ and

(2.2) $$ \begin{align} \big\| \mathbb{E} \, H \, {|}_{L_p(\mathcal{N})}:L_p(\mathcal{N}) \to L_p(\mathcal{N}) \big\| \leq \big\| H: L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\|. \end{align} $$

With this observation in hand, we can proceed to prove the following lemma, which is analogous to [Reference Mei and Ricard55, Proposition 3.4], but in the setting of general operators.

Lemma 2.2. Let $\mathcal {M}$ , $\mathcal {N}$ and $\mathbb {E}$ be as above and let $H:L_2(\mathcal {M}) \to L_2(\mathcal {M})$ be a bounded and left $\mathcal {N}$ -modular map. For every $f \in L_2(\mathcal {M}) \cap \mathcal {M}$ , it holds that

(2.3) $$ \begin{align} \mathbb{E} \big[ H(f) H(f)^\ast \big] \leq \left\| H: L_2(\mathcal{M}) \to L_2(\mathcal{M}) \right\|^2 \mathbb{E} \big[ f f^\ast \big]. \end{align} $$

Furthermore, if $\mathbb {E} H = H \mathbb {E}$ , we have that for every $1 \leq p \leq \infty $

(2.4) $$ \begin{align} \big\| \mathbb{E} \big[ H \big(f \, H(f)^\ast \big) \big] \big\|_p & \leq \big\| H: L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\|^2 \, \left\| \mathbb{E} \big[ f f^\ast \big] \right\|_p \end{align} $$

(2.5) $$ \begin{align} \big\| \mathbb{E} \left[ H \big( H(f f^\ast)^\ast \big) \right] \big\|_p & \leq \big\| H: L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\|^2 \, \left\| \mathbb{E} \big[ f f^\ast \big] \right\|_p \end{align} $$

Proof. All of the points are elementary. For (2.3) first notice that every state of $\mathcal {N}$ is of the form $f \mapsto \tau ( \delta \, f)$ , where $\delta $ is a positive element of norm $1$ in the space $L_1(\mathcal {N})$ . Decomposing it as $\delta = \delta ^{\frac {1}{2}} \delta ^{\frac {1}{2}}$ gives

$$ \begin{align*} \tau \left\{ \delta \, \mathbb{E} \big[ H(f) H(f)^\ast \big] \right\} & = \tau \left\{ \delta^{\frac{1}{2}} \mathbb{E} \big[ H(f) H(f)^\ast \big] \delta^{\frac{1}{2}} \right\} \\ & = \tau \left\{ \mathbb{E} \big[ \delta^{\frac{1}{2}} H(f) H(f)^\ast \delta^{\frac{1}{2}} \big] \right\} \\ & = \tau \left\{ \mathbb{E} \big[ H(\delta^{\frac{1}{2}} f) H(\delta^{\frac{1}{2}} f)^\ast \big] \right\} \\ & = \tau \left\{ H(\delta^{\frac{1}{2}} f) H(\delta^{\frac{1}{2}} f)^\ast \right\} \\ & \leq \big\| H:L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\|^2 \, \tau \left\{ (\delta^{\frac{1}{2}} f)(\delta^{\frac{1}{2}} f)^\ast \right\} \\ & = \big\| H:L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\|^2 \, (\tau \circ \mathbb{E}) \left\{ (\delta^{\frac{1}{2}} f)(\delta^{\frac{1}{2}} f)^\ast \right\} \\ & = \big\| H:L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\|^2 \, \tau \left\{ \delta \, \mathbb{E} \big[ f f^\ast \big] \right\}. \end{align*} $$

Since this is true for every state, the operator inequality (2.3) holds.

For (2.4) we use that $\mathbb {E} H = H \mathbb {E}$ to rewrite $\mathbb {E} \left [ H \big (f \, H(f)^\ast \big ) \right ]$ as $(\mathbb {E} H \mathbb {E}) \circ \mathbb {E} \left [ \big (f \, H(f)^\ast \big ) \right ]$ . The operator norm on $\mathbb {E} H \mathbb {E}: L_p(\mathcal {M}) \to L_p(\mathcal {M})$ is bounded by that of $\mathbb {E} H {|}_{L_p(\mathcal {N})}: L_p(\mathcal {N}) \to L_p(\mathcal {N})$ , which is bounded by the norm of H by (2.2). To estimate the term $\mathbb {E} \left [ \big (f \, H(f)^\ast \big ) \right ]$ we will use the following version of Hölder’s inequality [Reference Junge41, Inequality (2.1)]

$$\begin{align*}\big\| \mathbb{E} \big[ f \, g^\ast \big] \big\|_p \leq \big\| \mathbb{E} \big[ f \, f^\ast \big]^{\frac{1}{2}} \big\|_r \, \big\| \mathbb{E} \big[ g \, g^\ast \big]^{\frac{1}{2}} \big\|_s \quad \text{ when } \quad \frac{1}{p} = \frac{1}{r} + \frac{1}{s}, \end{align*}$$

with $r = s = 2p$ and $g = H(f)$ to obtain that

$$\begin{align*}\big\| \mathbb{E} \left[ f \, H(f)^\ast \right] \big\|_p \leq \big\| \mathbb{E} \left[ f \, f^\ast \right] \big\|_p^{\frac{1}{2}} \, \big\| \mathbb{E} \left[ H(f) \, H(f)^\ast \right] \big\|_p^{\frac{1}{2}}. \end{align*}$$

Applying the inequality in (2.3) gives the result. Identity (2.5) follows immediately after using two times the fact that H and $\mathbb {E}$ commute and that $\mathbb {E} H \mathbb {E}$ has a norm in $L_p$ bounded by the norm in $L_2$ of H by (2.2).

We can now prove the following extrapolation result, which generalizes [Reference Mei and Ricard55, Theorem 3.5] to the setting of general operators.

Proposition 2.3. Let $\mathcal {N} \subseteq \mathcal {M}$ be as before and let $H:L_2(\mathcal {M}) \to L_2(\mathcal {M})$ be a left $\mathcal {N}$ -modular operator commuting with $\mathbb {E}: \mathcal {M} \to \mathcal {N}$ . If H satisfies (Cotlar_{𝔼 ⊥}) then $\forall \, 2 \leq p < \infty $

(2.6) $$ \begin{align} \big\| H: L_p(\mathcal{M}) \to L_p(\mathcal{M}) \big\| \, \lesssim \, p^\beta \, \big\| H: L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\|, \; \text{ where } \beta = \log_2(1 + \sqrt{2}). \end{align} $$

Similarly, if $H:L_2(\mathcal {M}) \to L_2(\mathcal {M})$ is a general bounded linear map that satisfies the non-relative Cotlar identity (Cotlar_nr), then the same extrapolation inequality (2.6) holds.

Proof. First, let us denote the operator norm on $L_p$ of H by $c_p := \big \| H: L_p(\mathcal {M}) \to L_p(\mathcal {M})\big \|$ . We are going to proceed by induction, assuming that $c_p < \infty $ to prove that $c_{2p} < \infty $ . Choose $f \in \mathcal {M} \cap L_2(\mathcal {M})$ with $\| f \|_{2p} \leq 1$ and notice that

(2.7) $$ \begin{align} \| H(f) \|_{2p}^2 & = \| H(f) H(f)^\ast \|_{p} \nonumber\\ & \leq \big\| \mathbb{E} \big[ H(f) H(f)^\ast \big] \big\|_{p} + \big\| \mathbb{E}^\perp \big[ H(f) H(f)^\ast \big] \big\|_{p} \nonumber \\ & \leq c_2^2 \, \big\| \mathbb{E} \big[ f f^\ast \big] \big\|_{p} + \big\| \mathbb{E}^\perp \big[ H \big(f \, H(f)^\ast \big) + H \big(f \, H(f)^\ast \big)^\ast - H \big( H(f f^\ast)^\ast \big) \big] \big\|_{p} \end{align} $$

(2.8) $$ \begin{align} & \leq c_2^2 \, \| f \|_{2p}^2 + \big\| H \big(f \, H(f)^\ast \big) \big\|_p + \big\| H \big(f \, H(f)^\ast \big)^\ast \big\|_p + \big\| H \big( H(f f^\ast)^\ast \big) \big\|_{p} \nonumber \\ & + \big\| \mathbb{E} \big[ H \big(f \, H(f)^\ast \big) \big] \big\|_{p} + \big\| \mathbb{E} \big[ H \big(f \, H(f)^\ast \big)^\ast \big] \big\|_{p} + \big\| \mathbb{E} \big[ H \big( H(f f^\ast)^\ast \big) \big] \big\|_{p} \end{align} $$

(2.9) $$ \begin{align} & \leq 2 \, c_p \, \| H(f) \|_{2p} \|f\|_{2p} + c_p^2 \| f \|_{2p} + 4 \, c_2^2 \| f \|_{2p}^2 \nonumber \\ & \leq 2 \, c_p \, \| H(f) \|_{2p} + c_p^2 + 4 \, c_2^2 \end{align} $$

We have used (Cotlar_{𝔼 ⊥}) in the second term of the sum of (2.7) and estimate (2.3) of Lemma 2.2 in the first. To pass from (2.8) to (2.9) we have used the other two identities of Lemma 2.2. Notice that (2.9) is a quadratic inequality of the form $0 \leq -t^2 + 2 c_p \, t + c_p^2 + 4 \, c_2^2$ , where $t = \| H(f)\|_{2p}$ . Since the leading term is negative, this implies a bound of $\|H(f)\|_{2p}$ in terms of $c_p$ and $c_2$ only. Taking supremum over f and using the norm density of $\mathcal {M} \cap L_{2}(\mathcal {M})$ in $L_{2p}(\mathcal {M})$ implies that $c_{2p}$ is finite. More explicitly, setting $a_p = c_p / c_2$ gives the recursive inequality

(2.10) $$ \begin{align} a_{2p}^2 \leq 2 \, a_p \, a_{2p} + a_p^2 + 4. \end{align} $$

Adding $a_{2p}^2$ to both sides in order to complete squares gives

$$\begin{align*}2 a_{2p}^2 \, \leq \, a_{2p}^2 + 2 \, a_p \, a_{2p} + a_p^2 + 4 \, = \, (a_{2p} + a_p)^2 + 4 \, \leq \, (a_{2p} + a_p + 2)^2. \end{align*}$$

After taking square roots and recursively applying the inequality above, the following is obtained

$$\begin{align*}a_{2^k} \, \leq \, C \, (1 + \sqrt{2})^{k-1}, \quad \text{ with } \quad C \leq 3 + \sqrt{2}. \end{align*}$$

This, together with Marcinkiewicz interpolation for intermediate values of p, gives the desired inequality. The non-relative case works similarly, but the $4$ extra terms that give $4 \, c_2^2$ in the right-hand side of inequality (2.9) do not appear, which only affects the absolute constant C in the final inequality.

Our source of examples for the extrapolation theorem above is taken when $\mathcal {M} = \mathcal {L} \mathrm {G}$ and $H=T_m$ is a Fourier multiplier. We will work with Cotlar identities relative to subalgebras induced by subgroups $\mathrm {G}_0 \subseteq \mathrm {G}$ . In that setting it holds that there is a normal and trace-preserving conditional expectation $\mathbb {E}: \mathcal {L} \mathrm {G} \to \mathcal {L} \mathrm {G}_0$ if and only if $\mathrm {G}_0 \subseteq \mathrm {G}$ is open. Thus, our relative Cotlar identities would be taken only with respect to open subgroups. In general, given a group $\mathrm {G}$ , when speaking about the Cotlar identity for a multiplier $T_m$ – without specifying any subgroup – we will mean, that it satisfies the non-relative Cotlar identity (Cotlar_nr) when the group $\mathrm {G}$ is continuous i.e. $\mu (\{e\}) = 0$ , while in the discrete case $\mu (\{e\}) \neq 0$ , we will instead mean that it satisfies (Cotlar_{𝔼 ⊥}) with respect to the subgroup $\{e\}$ .

We are now going to prove the equivalence between Cotlar’s identity (Cotlar_{𝔼 ⊥}) and the closed formula in Theorem A.

Proof. Expanding (Cotlar_{𝔼 ⊥}) for $T_m$ gives

$$ \begin{align*} 0 \, = \, \underbrace{\mathbb{E}^\perp \Big[ T_m(f) \, T_m(f)^\ast \Big]}_{\mathrm{(I)}} - \underbrace{\mathbb{E}^\perp \Big[ T_m \big( f \, T_m(f)^\ast \big) \Big]}_{\mathrm{(II)}} - \underbrace{\mathbb{E}^\perp \Big[ T_m \big( f \, T_m(f)^\ast \big)^\ast \Big]}_{\mathrm{(III)}} + \underbrace{\mathbb{E}^\perp \Big[ T_m \big( T_m(f f^\ast)^\ast \big) \Big]}_{\mathrm{(IV)}}. \end{align*} $$

Now, elementary computations yield that

$$ \begin{align*} \mathrm{(I)} & = \int_{\mathrm{G} \setminus \mathrm{G}_0} \int_{\mathrm{G}} \widehat{f}(g h) \, \overline{\widehat{f}(h)} \, m(gh) \, \overline{m(h)} \, \lambda_g \, d\mu(h) \, d\mu(g)\\ \mathrm{(II)} & = \int_{\mathrm{G} \setminus \mathrm{G}_0} \int_{\mathrm{G}} \widehat{f}(g h) \, \overline{\widehat{f}(h)} \, m(g) \, \overline{m(h)} \, \lambda_g \, d\mu(h) \, d\mu(g)\\ \mathrm{(III)} & = \int_{\mathrm{G} \setminus \mathrm{G}_0} \int_{\mathrm{G}} \widehat{f}(g h) \, \overline{\widehat{f}(h)} \, m(gh) \, \overline{m(g^{-1})} \, \lambda_g \, d\mu(h) \, d\mu(g)\\ \mathrm{(IV)} & = \int_{\mathrm{G} \setminus \mathrm{G}_0} \int_{\mathrm{G}} \widehat{f}(g h) \, \overline{\widehat{f}(h)} \, m(g) \, \overline{m(g^{-1})} \, \lambda_g \, d\mu(h) \, d\mu(g), \end{align*} $$

where $\widehat {f}$ is given as in (2.1). This in turn implies, using the Plancherel theorem, that

$$\begin{align*}0 = \int_{\mathrm{G}} \widehat{f}(g h) \, \overline{\widehat{f}(h)} \, \big( m(gh) - m(g) \big) \, \big( \overline{m(h) - m(g^{-1})} \big) \, d\mu(h) \quad \text{ for almost every } g \in \mathrm{G} \setminus \mathrm{G}_0. \end{align*}$$

Obviously, if the factor $G_g(h) = ( m(gh) - m(g) ) \, ( \overline {m(h) - m(g^{-1})} )$ is equal to $0$ so is the above integral and therefore (Cotlar_{𝔼 ⊥}) holds. The reciprocal is immediate in the case of discrete groups. Indeed, choose any $h_0 \in \mathrm {G}$ and assume that $g \in \mathrm {G} \setminus \mathrm {G}_0$ is fixed. Pick $\widehat {f} = \delta _{h_0} + \delta _{g h_0}$ . In order to evaluate the integral, notice that

$$\begin{align*}\widehat{f}(g h) \, \overline{\widehat{f}(h)} = \delta_{\{h = h_0\}} + \delta_{\{g^2=e\}} \cdot \delta_{\{h = g h_0\}}. \end{align*}$$

The term $\delta _{h=h_0}$ in the above sum gives ${G_g(h_0)}$ in the integral. The term in which $g^2=e$ and $h = g h_0$ gives $\overline {G_g(h_0)}$ . Therefore, $\operatorname {Re}\{ G_g(h)\}=0$ for any $h \in \mathrm {G}$ . The imaginary part is similarly shown to be $0$ . In the case of a continuous group $\mathrm {G}$ it is necessary to replace $\delta _{h_0}$ with an approximation of the unit.

With all that at hand we are ready to prove Theorem A.

Remark 2.6. Let $\alpha : \mathcal {N} \to \mathcal {N}$ be a normal and trace-preserving $\ast $ -homomorphism. It is immediate that both Proposition 2.3 and Lemma 2.2 hold if we change the condition of H being left $\mathcal {N}$ -modular by that of being left $\mathcal {N}$ -modular relative to $\alpha $ , i.e.,

$$\begin{align*}H(f \, g) = \alpha(f) \, H(g), \quad \text{ for } f \in \mathcal{N}, \, g \in L_2(\mathcal{M}) \cap \mathcal{M}. \end{align*}$$

In the case of multipliers this easy observation has deep consequences. For instance, let $\chi : \mathrm {G}_0 \to \mathbb {T}$ be a (multiplicative) character. It is a straightforward consequence of Fell’s absorption principle that the map $\lambda _g \mapsto \chi (g) \, \lambda _g$ induces a normal and trace-preserving $\ast $ -automorphism $\alpha _\chi : \mathcal {L} \mathrm {G}_0 \to \mathcal {L} \mathrm {G}_0$ . Let $H = T_m$ be a Fourier multiplier on $\mathcal {L} \mathrm {G}$ . We have that it is left $\mathcal {L} \mathrm {G}_0$ -modular with respect to $\alpha _\chi $ , i.e., $H(f \, g) = \alpha _\chi (f) \, H(g)$ , for every $f \in \mathcal {N}$ and $g \in L_2(\mathcal {M}) \cap \mathcal {M}$ iff

(2.11) $$ \begin{align} m(k \, g) = \chi(k) \, m(g), \quad \text{ for every } k \in \mathrm{G}_0, \, g \in \mathrm{G}. \end{align} $$

This is specially useful when $\mathrm {G}_0$ is Abelian since, in that case, every function on $\mathrm {G}_0$ can be expressed as a limit of linear combinations of characters by the Fourier transform. This will be exploited in Theorem 5.2. It would also be used in a forthcoming paper of the third-named author.

Tightening the constant. It is known that, on the real line $\mathrm {G} = \mathbb {R}$ the operator $L_p$ -norm of the classical Hilbert transform (HT) is given by

$$\begin{align*}\big\| H: L_p(\mathbb{R}) \to L_p(\mathbb{R}) \big\| \, = \, \max \Big\{ \tan \Big( \frac{\pi}{2 \, p} \Big), \,\, \cot\Big( \frac{\pi}{2 \, p} \Big) \Big\}, \quad \text{ for } 1 < p < \infty \end{align*}$$

see [Reference Pichorides68] or [Reference Grafakos34] for a simplified proof. These constants grow asymptotically like p as $p \to \infty $ and like $1/(p-1)$ as $p \to 1^{+}$ , and those are the growth orders that we conjecture optimal in the noncommutative case as well. An interesting observation, originally made by Gokhberg and Krupnik in the classical case [Reference Gokhberg and Krupnik31] is that Cotlar’s identity on the real line gives the optimal order of growth for the constant in terms of p. Indeed, in the classical case, the fact that $H^2 = - \mathrm {id}$ yields a recurrence relation of the form

(2.12) $$ \begin{align} c_{2p}^2 \, \leq \, 2 c_p \, c_{2 p} + 1 \end{align} $$

instead of (2.10). The lack of a term depending on $c_p^2$ gives a decisively smaller bound. Solving the quadratic inequality in (2.12) gives

$$\begin{align*}c_{2p} \leq c_p + \sqrt{c_p^2 + 1} \end{align*}$$

and that results, after applying duality and interpolation, in the optimal growth order for the constant.

In the noncommutative case, the same type of argument holds for operators $H: L_2(\mathcal {M}) \to L_2(\mathcal {M})$ satisfying that $H \, H_{\mathrm {op}} = -\mathrm {id}$ , where $H_{\mathrm {op}}(f) = H(f^\ast )^\ast $ . We have the following improvement over Proposition 2.3.

Proposition 2.7. Let $\mathcal {N} \subseteq \mathcal {M}$ and $H:L_2(\mathcal {M}) \to L_2(\mathcal {M})$ be as before and assume H is left $\mathcal {N}$ -modular, commutes with $\mathbb {E}: \mathcal {M} \to \mathcal {N}$ and satisfies that $\mathbb {E}^\perp H \, H_{\mathrm {op}} = -\mathbb {E}^\perp $ . If H satisfies (Cotlar_{𝔼 ⊥}), then

$$\begin{align*}\big\| H: L_p(\mathcal{M}) \to L_p(\mathcal{M}) \big\| \, \lesssim \, p \, \big\| H:L_2(\mathcal{M}) \to L_2(\mathcal{M}) \big\| \quad \text{ for every } p \geq 2. \end{align*}$$

The same inequality holds in the non-relative case if $H \, H_{\mathrm {op}} = -\mathrm {id}$ .

Proof. The proof is immediate once it is noticed that the property $\mathbb {E}^\perp H \, H_{\mathrm {op}} = -\mathbb {E}^\perp $ implies that (Cotlar_{𝔼 ⊥}) can be rewritten as

$$\begin{align*}\mathbb{E}^\perp \big[ H(f) \, H(f)^\ast \big] \, = \, \mathbb{E}^\perp \left[ H \big( f \, H(f)^\ast \big) + H \big( f \, H(f)^\ast \big)^\ast + f f^\ast \right]. \end{align*}$$

Applying the same proof as that of Proposition 2.3 gives the recurrence $c_{2p}^2 \leq 2 \, c_{2 p} \, c_p + 1 + 3 c_2^2$ . After solving the quadratic inequality, we obtain $c_{2p} \leq c_p + \sqrt {c_p^2 + \kappa }$ , where $\kappa = 1 + 3 c_2^2$ . Iterating and applying Marcinkiewicz interpolation gives the bound. The same proof works in the non-relative case.

Observe that if $H = T_m$ is a Fourier multiplier, then $(T_m)_{\mathrm {op}} = T_{\tilde {m}}$ , for $\widetilde {m}(g) = \overline {m(g^{-1})}$ . Thus, we are asking that $m(g) \, \overline {m(g^{-1})} = -1$ for every $g \in \mathrm {G} \setminus \mathrm {G}_0$ . Similarly, since in the case of multipliers $\| \mathbb {E} H({\mathbf {1}}) \|_\infty = \| m \, {\mathbf {1}}_{\mathrm {G}_0} \|_\infty \leq \| m \|_\infty = c_2$ we can simplify the recurrence above assuming $c_2 = 1$ . In particular, we obtain the following corollary.

Corollary 2.8. Let $\mathrm {G}$ be a group and let $m: \mathrm {G} \to \mathbb {C}$ be a function satisfying (Cotla ^r) relative to a closed subgroup $\mathrm {G}_0 \subseteq \mathrm {G}$ , and such that m is left $\mathrm {G}_0$ -invariant and $m(g)\overline {m(g^{-1})} = -1$ , for every $g \in \mathrm {G} \setminus \mathrm {G}_0$ . Then

$$\begin{align*}\big\| T_m: L_p(\mathcal{L} \mathrm{G}) \to L_p(\mathcal{L} \mathrm{G}) \big\|_{\mathrm{cb}} \,\, \lesssim \,\, \Big( \frac{p^2}{p-1} \Big) \, \| m \|_\infty. \end{align*}$$

Both Proposition 2.7 and Corollary 2.8 are still true if one changes the value of $m(g) \, \overline {m(g^{-1})}$ from $-1$ to any other constant independent of g.

Non-Fourier multiplier examples. All of the machinery developed or reviewed in this Section is formulated in a way that works beyond the case of Fourier multipliers. As an illustration, another family of examples comes from Schur multipliers. Indeed, let $\mathcal {D} \subset \mathcal {B}(\ell _2 \mathbb {Z})$ be the finite span of the matrix units $e_{j \, k}$ . A Schur multiplier is a linear map $S_m: \mathcal {D} \subseteq \mathcal {B}(\ell _2 \mathbb {Z}) \to \mathcal {B}(\ell _2 \mathbb {Z})$ defined by sending the matrix unit $e_{j \, k}$ to $m(j,k) \, e_{j \, k}$ for some function $m:\mathbb {Z} \times \mathbb {Z} \to \mathbb {C}$ – called the symbol of $S_m$ . Schur multipliers are $\ell _\infty (\mathbb {Z})$ -bimodular, with $\ell _\infty (\mathbb {Z})$ sitting inside $\mathcal {B}(\ell _2 \mathbb {Z})$ as the diagonal subalgebra. As such, if we take in Theorem 2.3 $\mathcal {M}$ to be $\mathcal {B}(\ell _2 \mathbb {Z})$ and $\mathcal {N}$ to be $\ell _\infty (\mathbb {Z})$ , we have that any bounded symbol m satisfying (Cotlar_{𝔼 ⊥}) extends boundedly to all Schatten classes $S_p$ with $1 < p < \infty $ . The same computations of Theorem 2.4 can be performed in this setting, yielding that

(2.13) $$ \begin{align} S_m \text{ satisfies } (Cotlar_{\mathbb{E}^\perp}) & \text{ relative to } \ell_\infty \nonumber\\ & \Longleftrightarrow \big( m(j,k) - m(j,\ell) \big) \, \big( m(\ell,j) - m(\ell,k) \big) \, {\mathbf{1}}_{\{j \neq \ell\}} = 0, \; \forall j, k, \ell \in \mathbb{Z}. \end{align} $$

This makes verifying the Cotlar identity for Schur multipliers straightforward. For instance, the triangular truncation $H = S_m$ is the Schur multiplier with symbol $m(j,k) = \mathrm {sgn}(j - k)$ . It is trivial to verify that it satisfies identity (2.13). Indeed, if $m(j,k) \neq m(j,\ell )$ it is because $j < \ell $ and $j \geq k$ or vice–versa. If the first case holds, then $k \leq j < \ell $ , which implies that $m(\ell ,j) = m(\ell ,k)$ . The other case is checked similarly. Since $H \, H^{\mathrm {op}} = - \mathrm {id}$ , we recover the sharp bound of the operator-norm on $S_p$ as $p \to 1^{+}, \infty $ . This is not a novel result, since the triangular truncation is a very well understood object whose $S_p$ -boundedness can be obtained from an array of techniques ranging from subdiagonal algebras [Reference Randrianantoanina74] to Fourier-Schur transference for amenable groups [Reference Neuwirth and Ricard61, Reference Caspers and de la Salle11] and smooth Schur multiplier techniques [Reference Conde-Alonso, González-Pérez, Parcet and Tablate14]. Nevertheless, it signals the possibility of exploiting Cotlar identities in settings beyond Fourier multipliers.

The convex hull of Cotlar-type multipliers. A natural question is which class of multipliers m can be shown to be bounded in $L_p(\mathcal {L} \mathrm {G})$ by being represented as a convex combination of multipliers satisfying (Cotlar_{𝔼 ⊥}) or natural modifications of them. To that end, notice that if m is an $L_p$ -bounded multiplier, then so is $g \mapsto m(h^{-1} \theta (g) r)$ , where $h, r \in \mathrm {G}$ , $\theta \in \mathrm {Aut}(\mathrm {G})$ and their norms coincide. Let us denote the group of transformations of $\mathrm {G}$ given by $g \mapsto h^{-1} \theta (g) r$ by the affine transformations $\mathrm {Aff}(\mathrm {G})$ of $\mathrm {G}$ . Observe that, if we define $\mathrm {G}^\Delta = (\mathrm {G} \oplus \mathrm {G}) / \Delta $ , where $\Delta $ is the subgroup of diagonal central elements, $\Delta = \{ (a,a) : a \in \mathcal {Z}(\mathrm {G}) \} \subseteq \mathrm {G} \oplus \mathrm {G}$ , then there is a faithful representation that sends $(h, r)$ to $g \mapsto h^{-1} \, g \, r$ . A trivial computation gives that

$$\begin{align*}\mathrm{Aff}(\mathrm{G}) \cong \mathrm{G}^\Delta \rtimes \mathrm{Aut}(\mathrm{G}), \end{align*}$$

with the natural action. It is clear that, if $\mu \in M(\mathrm {Aut}(\mathrm {G}))$ is a finite signed measure and $m: \mathrm {Aff}(\mathrm {G}) \times \mathrm {G} \to \mathbb {C}$ is a bounded map such that $g \mapsto m(\alpha , g)$ satisfies (Cotlar_{𝔼 ⊥}) for every $\alpha $ , then

$$\begin{align*}m(g) \, = \, \int_{\mathrm{Aff}(\mathrm{G})} m(\alpha, \alpha(g)) \, d \mu(\alpha), \end{align*}$$

is clearly bounded in $L_p(\mathcal {L} \mathrm {G})$ for every $1 < p < \infty $ . We could add more flexibility to this technique by allowing the map $g \mapsto m(\alpha , g)$ to be the product of k terms satisfying (Cotlar_{𝔼 ⊥}). Let us call this class $\mathrm {coCot}^k(\mathrm {G})$ . We leave mostly unexplored the following natural problem:

This remains as an underexplored approach to prove the boundedness of Fourier multipliers over groups without recourse to noncommutative analogues of singular integral theory. In fact, observe that in the classical case of $\mathbb {R}$ any function of bounded variation lies in the convex hull of (translations of) the classical Hilbert transform and therefore $m \in \mathrm {BV}(\mathbb {R}) \implies m \in \mathrm {coCot}^1(\mathbb {R})$ , see [Reference Duoandikoetxea25, Corollary 3.8]. In higher dimensions the behaviour is even richer. For instance, let $m:\mathbb {R}^2 \to \mathbb {C}$ be a homogeneous function, i.e., a function satisfying that $m(\lambda \, \xi ) = m(\xi )$ for every $\lambda> 0$ . Clearly, m depends only on its angular component $m{|}_{\mathbb {T}}$ . We have that

$$\begin{align*}m{|}_{\mathbb{T}} \in \mathrm{BV}(\mathbb{T}) \quad \implies \quad m \in \mathrm{coCot}^2(\mathbb{R}^2), \end{align*}$$

where $\mathrm {BV}(\mathbb {T})$ is the space of functions of bounded variation on the torus. To see that, just notice that the indicator function ${\mathbf {1}}_{\Sigma _\theta }$ of the sector $\Sigma _\theta \subseteq \mathbb {R}^2$ of all vectors whose polar angle $\omega $ lies in $[0,\theta )$ can be expressed as the product of the characteristic functions of two half-planes, each of which satisfies a Cotlar identity. Homogeneous functions of angular bounded variation can be expressed as convex combinations of different sector indicators.

Generalizations. It worth noticing that the identity (Cotlar_{𝔼 ⊥}) can be generalized in a natural way by changing the equality by an operator inequality

(Cotlar_𝔼^⊥≤) $$\begin{align} \mathbb{E}^\perp \big[ H(f) \, H(f)^\ast \big] \, \leq \, \mathbb{E}^\perp \left[ H \big( f \, H(f)^\ast \big) + H \big( f \, H(f)^\ast \big)^\ast - H \big( H(f f^\ast)^\ast \big) \right] + \Lambda \, \mathbb{E} [f \, f^\ast], \end{align} $$

for every $f \in \mathcal {M} \cap L_2(\mathcal {M})$ and some constant $\Lambda \geq 0$ . It is clear that this inequality implies the same bound (2.6) of Proposition 2.3 for left $\mathcal {N}$ -modular operators, just with a constant growing like $\mathrm {O}(1 + \sqrt {\Lambda })$ . Similarly, there exists a closed-formula characterization of Fourier multipliers $T_m$ with left $\mathrm {G}_0$ -invariant symbol m satisfying (Cotlar_{𝔼 ⊥}^≤). To formulate such characterization let us define

$$\begin{align*}\Omega_m(g,h) = \big( m(g) - m(g h^{-1}) \big) \, \overline{\big( m(h) - m(h g^{-1}) \big)} \, {\mathbf{1}}_{\mathrm{G} \setminus \mathrm{G}_0}(g h^{-1}) + \Lambda \, {\mathbf{1}}_{\mathrm{G}_0}(g h^{-1}) \end{align*}$$

and notice that (Cotlar_{𝔼 ⊥}^≤) is actually equivalent to

$$\begin{align*}0 \, \leq \, \int_{\mathrm{G}} \int_{\mathrm{G}} \widehat{f}(g h) \, \overline{\widehat{f}(h)} \, \Omega_m(g h, h) \, \lambda_g \, d\mu(h) \, d\mu(g). \end{align*}$$

The inequality above is equivalent to the (integral) positivity of the operator-valued kernel $\Omega _m(g,h) \, \lambda _{gh^{-1}}$ . Since multiplication by $\lambda _{g^{-1}} \cdot \lambda _h$ preserves positivity, this is equivalent to requiring that $\Omega _m$ be an integrally positive kernel, i.e.,

(2.14) $$ \begin{align} 0 \, \leq \, \int_{\mathrm{G}} \int_{\mathrm{G}} \widehat{f}(g) \, \overline{\widehat{f}(h)} \, \Omega_m(g, h) \, d\mu(h) \, d\mu(g), \;\;\;\; \text{ for every } \widehat{f} \in L_1(\mathrm{G}) \cap L_\infty(\mathrm{G}). \end{align} $$

When $\mathrm {G}$ is discrete, this condition implies that $\Omega _m$ factors as

$$\begin{align*}\Omega_m(g,h) = \big\langle \xi(h), \xi(g) \big\rangle, \end{align*}$$

where $\xi : \mathrm {G} \to \mathcal {H}$ is a bounded function with values in a Hilbert space $\mathcal {H}$ , see [Reference Bekka, de la Harpe and Valette3, Theorem C.1.4]. The reason why we need to add the term $\Lambda \, \mathbb {E}[f \, f^\ast ]$ to the Cotlar identity when $\mathrm {G}_0$ is open is that, without it, we will be forcing the kernel $\Omega _m(g,h)$ to be positive definite while also vanishing on the diagonal $g = h$ , which would imply that it is identically zero. In the non-relative case, this extra term can be omitted. This more general Cotlar identity leads to the following problem, which we leave unexplored.

3 Groups acting on $\mathbb {R}$ -trees

Let X be a Hausdorff topological space. An arc $\gamma $ on X is a subset of X that is the image of an injective continuous function mapping $[0, 1]$ onto $\gamma $ . The space X is said to be uniquely arcwise connected, or UAC, if any two points in X are joined by a unique arc. We say that a group $\mathrm {G}$ acts on a UAC space X if $\mathrm {G}$ acts by homeomorphisms on X.

If, in addition, the UAC space X is metrisable and there is a metric $d: X \times X \to \mathbb {C}$ such that the unique arc joining two points is isometric to a closed interval of the real line, then $(X, d)$ is called an $\mathbb {R}$ -tree. We will say that a group $\mathrm {G}$ acts on an $\mathbb {R}$ -tree X if it acts on it by isometries.

Observe that this definition is topological in nature since the underlying space is required to be arcwise connected. An alternative route to $\mathbb {R}$ -trees can be taken by defining them as hyperbolic spaces with $\delta = 0$ , i.e., every triangle is a tripod. These two definitions, although equivalent in spirit, are slightly different. Namely, a tree seen as a discrete set with the edge metric is a $0$ -hyperbolic metric space but not an $\mathbb {R}$ -tree in our definition. This is not a problem since trees can still be seen as a subclass of $\mathbb {R}$ -trees by treating them as simplicial trees, i.e. the one-dimensional simplicial complexes obtained from the incidence information of the tree. We also have that, given an $\mathbb {R}$ -tree X, if the set of points whose complement has three or more connected components is discrete in X, then X is a simplicial tree. The first definition of $\mathbb {R}$ -trees was given by Tits [Reference Tits, Bass, P.J. and J.85], then Morgan and Shalen [Reference Morgan and Shalen58], following earlier results of Alperin and Moss [Reference Alperin and Moss1], drew attention to the theory of $\mathbb {R}$ -trees by showing how to compactify a generalization of Teichmuller space for a finitely generated group using $\mathbb {R}$ -trees. We refer the reader to [Reference Bestvina, Daverman and Sher5] for more on $\mathbb {R}$ -trees.

We define the following two models for a group acting on a UAC space. Let $\mathrm {G} \curvearrowright X$ be a topological action and $x_0 \in X$ a selected point. We will say that a bounded measurable function $\varphi : X\setminus \{x_0\} \to \mathbb {C}$ is constant along arcs iff $\varphi (x) = \varphi (y)$ if there is an arc connecting x and y inside $X \setminus \{x_0\}$ . This is equivalent to decomposing $X \setminus \{x_0\}$ as a union of arcwise connected subsets and imposing the function to be constant over those subsets.

Model 1. Let $\mathrm {G} \curvearrowright X$ be a topological action on a UAC space, $x_0 \in X$ a point and $\widetilde {m}: X \to \mathbb {C}$ a bounded measurable function such that

1. (i) $\widetilde {m}$ , restricted to $X \setminus \{x_0\}$ , is constant along arcs.

2. (ii) $\widetilde {m}$ is invariant under the action $\mathrm {St}_{x_0} \curvearrowright X \setminus \{x_0\}$ .

Then we define the multiplier $m: \mathrm {G} \to \mathbb {C}$ as

$$\begin{align*}m(g) = \widetilde{m}(g \cdot x_0) \end{align*}$$

and fix $(\mathrm {G}_0, \mathrm {G})$ as $(\mathrm {St}_{x_0}, \mathrm {G})$ .

This definition has the drawback that the invariance under $\mathrm {St}_{x_0}$ of m can make the symbol constant outside $\mathrm {St}_{x_0}$ in some cases. We introduce the following, more involved, model.

Model 2. Let us fix two distinct constants $C_1$ , $C_2 \in \mathbb {C}$ . Let similarly $\mathrm {G} \curvearrowright X$ be a topological action on a UAC space and $x_0 \in X$ a point. Choose $X_0 \subseteq X \setminus \{x_0\}$ to be an arcwise connected subset. We define $m: \mathrm {G} \to \mathbb {C}$ to be

$$ \begin{align*} m(g) \, = \, \begin{cases} 0 & \text{when } \, g \in \mathrm{St}_{x_0} \, \text{ and } \, g \cdot X_0 = X_0\\ C_1 & \text{when } \, g \in \mathrm{St}_{x_0} \, \text{ and } \, g \cdot X_0 \neq X_0 \\ C_1 & \text{when } \, g \not\in \mathrm{St}_{x_0} \, \text{ and } \, g \cdot x_0 \not\in X_0 \\ C_2 & \text{when } \, g \cdot x_0 \in X_0. \end{cases} \end{align*} $$

We will also fix $\mathrm {G}_0$ to be $\mathrm {St}_{x_0} \cap \{ g \in \mathrm {G} : g \cdot X_0 = X_0 \}$ .

Observe that Model 2 is a natural modification of Model 1 for the function $\widetilde {m}: X \to \mathbb {C}$ given by $\widetilde {m} = C_1 {\mathbf {1}}_{X\setminus (\{ x_0\}\cup X_0)} + C_2 {\mathbf {1}}_{X_0}$ . The main difference is that we extend the value of $C_1$ to a portion of the stabilizer.

Proof. The statement in point (i) has already been proved in the introduction. Thus, we concentrate on point (ii). The fact that m is left invariant under the action of $\mathrm {G}_0 = \{ g \in \mathrm {G} : g \cdot x_0 = x_0 \} \cap \{ g \in \mathrm {G} : g \cdot X_0 = X_0 \}$ is immediate. Now, all we have to do is to verify (Cotla ^r). To that end, let us divide the group as a disjoint union $\mathrm {G} = \mathrm {G}_0 \cup \mathrm {G}_1 \cup \mathrm {G}_2 \cup \mathrm {G}_3$ , where

$$ \begin{align*} \mathrm{G}_0 & = \{ g \in \mathrm{G} : g \cdot x_0 = x_0 \} \cap \{ g \in \mathrm{G} : g \cdot X_0 = X_0 \} \\ \mathrm{G}_1 & = \{ g \in \mathrm{G} : g \cdot x_0 = x_0 \} \cap \{ g \in \mathrm{G} : g \cdot X_0 \neq X_0 \} \\ \mathrm{G}_2 & = \{ g \in \mathrm{G} : g \cdot x_0 \neq x_0 \} \cap \{ g \in \mathrm{G} : g \cdot x_0 \not\in X_0 \} \\ \mathrm{G}_3 & = \{ g \in \mathrm{G} : g \cdot x_0 \neq x_0 \} \cap \{ g \in \mathrm{G} : g \cdot x_0 \in X_0 \}. \end{align*} $$

Observe also that, since m is both left and right $\mathrm {G}_0$ -invariant, it is enough to verify (Cotla ^r) for $g, h \in \mathrm {G} \setminus \mathrm {G}_0$ . Assume that $m(g^{-1}) \neq m(h)$ , otherwise we are done, our aim is to show that $m(g h) = m(g)$ . We will proceed by cases. First, assume that $g \in \mathrm {G}_1$ . This is equivalent to $g^{-1} \in \mathrm {G}_1$ and therefore $h \in \mathrm {G}_3$ . But then, $g \, h \cdot x_0 \not \in X_0$ and $g \, h \cdot x_0 \neq x_0$ . Therefore $g \, h \in \mathrm {G}_2$ and we get $m(g h) = m(g)$ . In the case of $g \in \mathrm {G}_2$ we have the whole range of possibilities and h can belong to $\mathrm {G}_1$ , $\mathrm {G}_2$ or $\mathrm {G}_3$ . In the case of $g \in \mathrm {G}_2$ and $h \in \mathrm {G}_1$ we have that $g h \in \mathrm {G}_2$ . Indeed, $g h \cdot x_0 = g \cdot x_0 \not \in X_0$ and it is immediate that $g h$ is not in the stabilizer of $x_0$ . For the second case of $g \in \mathrm {G}_2$ and $h \in \mathrm {G}_2$ the condition $m(g^{-1}) \neq m(h)$ implies that $g^{-1} \cdot x_0$ and $h \cdot x_0$ live in distinct arcwise connected subsets of $X \setminus \{x_0\}$ . But then there is a unique path connecting both points that passes through the root. Applying g to the whole arc gives that $g \cdot x_0$ and $g h \cdot x_0$ lie in the same arcwise connected subset and do not stabilize $x_0$ , see Figure 1. Therefore $m(g h) = m(g)$ . The third case is given by $g \in \mathrm {G}_2$ and $h \in \mathrm {G}_3$ . Observe that if $g \in \mathrm {G}_2$ , then $g^{-1}$ can only be inside $\mathrm {G}_2$ or $\mathrm {G}_3$ . But the case $g^{-1} \in \mathrm {G}_3$ can be easily discarded since it will contradict the assumption $m(g^{-1}) \neq m(h)$ . But if $g^{-1} \in \mathrm {G}_2$ and $h \in \mathrm {G}_3$ , then $g^{-1} \cdot x_0$ and $h \cdot x_0$ live in distinct arcwise connected subsets and we can proceed like in the previous case. It remains to check the case of $g \in \mathrm {G}_3$ . We have that h can be in either $\mathrm {G}_1$ , $\mathrm {G}_2$ or $\mathrm {G}_3$ . In the first case we deduce that $g h \in \mathrm {G}_3$ . For the second one we have that if $g \in \mathrm {G}_3$ and $h \in \mathrm {G}_2$ , then we can assume that $g^{-1} \in \mathrm {G}_3$ , the only other choice being $g^{-1} \in \mathrm {G}_2$ which will contradict the assumption $m(g^{-1}) \neq m(h)$ . But this implies that $g^{-1} \cdot x_0$ and $h \cdot x_0$ live in different arcwise connected subsets of $X \setminus \{x_0\}$ and we can apply the argument in Figure 1. Lastly, if $g \in \mathrm {G}_3$ and $h \in \mathrm {G}_3$ we obtain similarly that $g^{-1}$ can only lie in $\mathrm {G}_2$ . The same path argument applies. Applying Theorem B, we get the $L_p$ -boundedness of $T_m$ .

Observe that in the case of $\mathrm {G} = \mathbb {R}$ , the Hilbert transform is also of weak type $(1,1)$ , i.e., $H:L_1(\mathbb {R}) \to L_{1,\infty }(\mathbb {R})$ and bounded between $L_\infty (\mathbb {R})$ and the space of bounded mean oscillation functions $\mathrm {BMO}(\mathbb {R})$ . Both endpoint spaces give – by either complex or real interpolation with $L_2$ – the optimal order for the operator $L_p$ norm of H. The following problem remains open.

This problem presents at least three challenges. The first difficulty comes from the fact that weak type $(1,1)$ bounds are difficult to obtain for noncommutative singular integral type operators. They are known in some semicommutative examples [Reference Parcet62, Reference Cadilhac, Conde-Alonso and Parcet8] but open in the case of Quantum Euclidean spaces [Reference González-Pérez, Junge and Parcet33] and in most group settings besides left-orderable groups [Reference Randrianantoanina74]. The second challenge is that the specific endpoint space $\mathrm {X}_\infty $ to be used for m in Model 1 or Model 2 has to be defined in terms of the geometry of $\mathrm {G} \curvearrowright X$ and it cannot just be the usual noncommutative $\mathrm {BMO}$ space, see [Reference Junge and Mei43, Reference Mei53, Reference Mei54]. Indeed, there is a natural unital and completely positive semigroup $S_t: \mathcal {L} \mathbb {F}_2 \to \mathcal {L} \mathbb {F}_2$ in the free group algebra given by $S_t(\lambda _g) = e^{-t|g|}$ , see [Reference Haagerup36]. This semigroup allows to construct a natural and interpolating semigroup $\mathrm {BMO}$ space $\mathrm {BMO}(\mathcal {L} \mathbb {F}_2)$ , see [Reference Junge, Mei and Parcet44]. But it is known that the multipliers (MR) are unbounded from $L_\infty (\mathcal {L} \mathbb {F}_2)$ to $\mathrm {BMO}(\mathcal {L} \mathbb {F}_2)$ , see [Reference Mei, Ricard and Xu56, Appendix A]. The third difficulty comes from the fact that the classical technique, employed by Kolmogorov [Reference Kolmogorov50], of comparing a singular integral operator with a maximal one is delicate in this context since some of the operators obtained are not positivity preserving, which makes interpolating maximal functions an open problem [Reference Junge and Xu47]. The technique of Kolmogorov has been used in the noncommutative case in [Reference Hong, Lai and Xu40]. We will also mention that some tentative progress in the direction of Problem 3.3 has been made. For instance, Gała̧zka and Osękowski [Reference Gała̧zka and Osȩkowski30] have lowered the operator $L_p$ -norm of the Fourier multiplier $m: \mathbb {F}_2 \to \mathbb {C}$ that depends on the starting letter from $\mathrm {O}(p^\beta )$ to $\mathrm {O}(p \log p)$ as $p \to \infty $ .

Now, we are going to study Models 1 and 2 in the context of $\mathbb {R}$ -trees. Observe that, since any $\mathbb {R}$ -tree action is an action of the underlying UAC topological space, Proposition 3.1 above works for $\mathbb {R}$ -trees. Furthermore, in the case of group actions on $\mathbb {R}$ -trees we have the following result connecting the existence of global fixed points with the form of the Fourier multiplier $T_m$ . We are going to say that the multipliers m coming from Models 1 and 2 are trivial iff they are constant for any $g \in \mathrm {G} \setminus \mathrm {G}_0$ .

Proof. We will prove first (i) for m as in Model 1. Assume that the action has a global fixed point $x_1 \in X$ . If $x_1$ coincides with the root $x_0$ then $m = 0$ and there is nothing to prove. Therefore we assume $x_0 \neq x_1$ . Similarly, we can assume without loss of generality that $\mathrm {St}_{x_0}\neq \mathrm {G}$ , since otherwise $m = 0$ . Pick $g \in \mathrm {G}$ and assume that $g \cdot x_0$ lies in an arcwise connected subset of $X \setminus \{x_0\}$ different from that of $x_1$ . Then, there is a unique path joining $x_1$ and $g \cdot x_0$ that passes through the root $x_0$ , see Figure 2. But since $x_1$ is fixed by any $g\in \mathrm {G}$ , after applying g to the path we obtain a larger path, which contradicts the fact that the action is isometric.

Figure 2

The action of g over the path connecting the global fixed point and $g \cdot x_0$ .

Figure 2 Long description

The diagram consists of two panels separated by a central arrow labeled g pointing from left to right.

In the left panel, a curved black line connects three points. The top point is x sub 1, located within a pink circular region. The middle point is x sub 0. The bottom point is g dot x sub 0, located within a light blue circular region.

In the right panel, the transformation is shown. The path starting from x sub 1 (now in a larger pink region) passes through x sub 0 and reaches g dot x sub 0 (now in a larger blue region). A new red segment continues from g dot x sub 0 to a final point labeled g squared dot x sub 0. A red X mark is placed below this final red segment.

This implies that $g^2 \cdot x_0 = g \cdot x_0$ and so $g \cdot x_0 = x_0$ , which contradicts the assumptions. Therefore, $g \cdot x_0$ belongs to the same connected subset of $x_1$ for every g that do not stabilize the root $x_0$ , that is, $m(g)$ is constant for any $g\in \mathrm {G}\setminus \mathrm {St}_{x_0}$ .

For the case of m as in Model 2, let $x_1 \in X$ be a global fixed point. We can again consider without loss of generality that $x_0 \neq x_1$ and that $\mathrm {St}_{x_0}\neq \mathrm {G}$ . By those assumptions, there exists $g_0 \in \mathrm {G}$ with $g_0 \cdot x_0 \in X \setminus \{x_0\}$ . We have two possibilities for $g_0$ . If $g_0 \cdot x_0 \neq x_0$ and $g_0 \cdot x_0 \not \in X_0$ , then $m(g_0) = C_1$ , while if $g_0 \cdot x_0 \in X_0$ then $m(g_0) = C_2$ . If we are in the first case $g_0 \cdot x_0 \not \in X_0$ , then the multiplier m will be trivial unless there exists a $g_1 \in \mathrm {G}$ such that $g_1 \cdot x_0 \in X_0$ . Let us obtain a contradiction. First, we claim that the global fixed point $x_1 \not \in X_0$ . Assume $x_1 \in X_0$ . Then, there is a path joining $x_1$ and $g_0 \cdot x_0$ that passes through the root $x_0$ , see Figure 3. But applying $g_0$ to the path gives that $g_0 \cdot x_0 = x_0$ which is a contradiction. Therefore $x_1 \not \in X_0$ , but since $g_1 \cdot x_0 \in X_0$ , we can build a path from $x_1$ to $g_1 \cdot x_0$ that passes through the root $x_0$ and, repeating the same argument as before, obtain that $g_1 \cdot x_0 = x_0$ , which is a contradiction. For the second case $g_0 \cdot x_0 \in X_0$ . First, we notice that $x_1$ must live in $X_0$ , if not, proceeding as before we will get that $g_0 \cdot x_0 = x_0$ , which is a contradiction. In order for m to be nontrivial there should be a $g_1 \in \mathrm {G}$ such that either $g_1 \cdot x_0 = x_0$ and $g_1 \cdot X_0 \neq X_0$ or $g_1 \cdot x_0 \neq x_0$ and $g_1 \cdot x_0 \not \in X_0$ . For the first case, let us choose a point $x_2 \in X_0$ . Then, $g_1 \cdot x_2 \not \in X_0$ and similarly $g_1 \cdot x_2 \neq x_0$ . Now, construct a path joining $x_1$ with $g_1 \cdot x_2$ . Since $x_1 \in X_0$ but $g_1 \cdot x_2$ belongs to a different connected subset, the arc joining them passes through the root $x_0$ . Let us apply $g_1^{-1}$ to the whole arc. Since we have that $g_1^{-1} \cdot x_0 = x_0$ , we obtain an arc that joins $x_1$ and $x_2$ and passes through the root $x_0$ . But this is a contradiction with the fact that $x_1$ and $x_2$ both belong to $X_0$ , see Figure 4, since X is a UAC space the arc joining two elements in the same connected subset of $X \setminus \{x_0\}$ cannot pass trough the root $x_0$ . The remaining case is when $g_1 \cdot x_0 \neq x_0$ and $g_1 \cdot x_0 \not \in X_0$ . Then, the arc joining $x_1 \in X_0$ and $g_1 \cdot x_1$ passes through $x_0$ . Applying $g_1$ gives a contradiction with the fact that the action is isometric and that $g_1 \cdot x_0 \neq x_0$ . For the other point, if $\mathrm {G}$ is finitely generated, the converse statement is also true and it is a corollary of [Reference Bowditch6, Lemma 5.2]. This lemma tells us that when a finitely generated group $\mathrm {G}$ acts non-trivially on an $\mathbb {R}$ -tree X, there exist a point $y_0$ and elements $g, h \in \mathrm {G}$ such that $y_0$ lies in the arc between $g \cdot y_0$ and $h \cdot y_0$ and $y_0$ , $g \cdot y_0$ and $h \cdot y_0$ are all different. Therefore, if $\mathrm {G}$ does not have property (F $\mathbb {R}$ ), we can choose $x_0$ as $y_0$ and $X_0$ as the maximally connected subset of $g \cdot y_0$ . Then, there is an $h \in \mathrm {G}$ such that $h \cdot x_0 \neq x_0$ and $h \cdot x_0 \not \in X_0$ . Thus, sending g and h to different values $C_1$ and $C_2$ gives a nontrivial multiplier within Model 2.

Figure 3

The action of $g_0$ over the path connecting $x_1$ and $g_0 \cdot x_0 \not \in X_0$ .

Figure 3 Long description

A mathematical diagram with two main sections separated by a curved arrow labeled g sub 0 pointing from left to right.

Left section: A curved black path connects three points. The top point x sub 1 is inside a pink circular region labeled X sub 0. The middle point is x sub 0. The bottom point is g sub 0 dot x sub 0, which sits inside a light blue circular region.

Right section: The same path from the left is repeated, starting at x sub 1 in a pink region X sub 0 and passing through x sub 0 and g sub 0 dot x sub 0. However, a new red segment extends from g sub 0 dot x sub 0 to a final point g sub 0 squared dot x sub 0. A large red X is placed below this final segment, indicating an invalid or restricted path extension.

Figure 4

The action of $g_1^{-1}$ over the path connecting $x_1$ and $g_1 \cdot x_2$ .

Many groups we are familiar with admit actions on $\mathbb {R}$ -trees. For example, every finitely generated hyperbolic group. Indeed, a finitely generated group is hyperbolic if and only if every asymptotic cone of the group is an $\mathbb {R}$ -tree [Reference Gromov35]. Then, the action of a hyperbolic group on its Cayley graph induces an action on the asymptotic cone of the Cayley graph. Moreover, any surface group having Euler characteristic less than $-1$ acts freely on an $\mathbb {R}$ -tree [Reference Morgan and Shalen59].

On the other hand, there are many examples of groups for which any action on an $\mathbb {R}$ -tree has a global fixed point. When this happens the group $\mathrm {G}$ is said to have property (F $\mathbb {R}$ ), see [Reference Bestvina, Daverman and Sher5, Reference Shalen80]. The following corollary of Proposition 3.4 characterizes groups with property (F $\mathbb {R}$ ).

One natural question, that we leave open, is whether there are functions $m: \mathrm {G} \to \mathbb {C}$ satisfying (Cotlar) for $\mathrm {G}_0 = \{e\}$ and such that they do not lift to a function $\widetilde {m}$ on a UAC space like in Model 1 or 2 on which $\mathrm {G}$ acts. One way to prove the non-existence of such lift $\widetilde {m}$ would be a procedure to assemble from the group $\mathrm {G}$ and the function m a $\mathrm {G}$ -space X on which $\mathrm {G}$ acts naturally and a lift $\widetilde {m}: X \to \mathbb {C}$ satisfying hypothesis like those of Models 1 and 2. So far, this reverse construction has escaped us.

4 Left orderable groups

Recall that a total or linear order $\preceq $ is an order relation such that, given any two points x, y then it holds that $x \preceq y$ or $y \preceq x$ . A left-orderable group is a group admitting a left-invariant total order, i.e., $(\mathrm {G}, \preceq )$ satisfies that $g \preceq h$ if and only if $k g \preceq k h$ . Recall that, as we defined in the introduction, every left-orderable group has a sign function $\mathrm {sgn}: \mathrm {G} \to \mathbb {C}$ that assigns $+1$ or $-1$ depending on whether $e \prec g$ or $g \prec e$ . We will prove that $H = T_{\mathrm {sgn}}$ is bounded.

Apart from the direct proof above, it is interesting to notice that these Hilbert transform type multipliers on left-orderable groups can be put into the framework of Model 1 and Theorem B. To do that we need to use the well-known characterization of countable left-orderable groups as order-preserving groups of homeomorphisms of the real line $\mathbb {R}$ , which is a UAC space. The first reference we found for this is [Reference Holland39].

Observe as well that, given a homeomorphism of the real line $f: \mathbb {R} \to \mathbb {R}$ it is either orientation preserving or orientation reversing and that the composition of two orientation reversing maps is orientation preserving. Therefore, there is a group homomorphism $\mathrm {Homeo}(\mathbb {R}) \to \mathbb {Z}_2 \cong \{1,-1\}$ that associate each homeomorphism of $\mathbb {R}$ with $+1$ or $-1$ depending on whether the homeomorphism is orientation preserving or reversing. As a consequence each discrete group $\mathrm {G}$ acting on $\mathbb {R}$ has an index $2$ subgroup that is left-orderable. So, when the UAC space X in Model 1 is equal to $\mathbb {R}$ , the group $\mathrm {G}$ is left-orderable and have an index $2$ subgroup in which the multiplier m essentially coincides with the sign function.

The left invariant order of a left-orderable group is by no means unique. Thus, the sign functions associated to different orders may give different $L_p$ -bounded Fourier multiplier. Similarly, left-orderable groups can have actions on UAC spaces other than $\mathbb {R}$ which will yield different multipliers still within our Model 1. An example of this will be that of Baumslag-Solitar groups $\mathrm {BS}(1,n)$ , which both admit nontrivial actions on their Bass-Serre trees and are left-orderable.

Left orderable groups and subdiagonal algebras. Finally, let us mention that the Hilbert transform $H=T_{\mathrm {sgn}}: L_p(\mathcal {L} \mathrm {G}) \to L_p(\mathcal {L} \mathrm {G})$ of a left-orderable group can be considered as a particular case of the Hilbert transforms associated with subdiagonal algebras which have been studied in [Reference Randrianantoanina74], see also [Reference Pisier and Xu72, Section 8]. Such algebras were introduced by Arveson [Reference Arveson2]. Indeed, let $(\mathcal {M}, \tau )$ be a von Neumann algebra with a n.s.f. trace and let $\mathbb {E}: \mathcal {M} \to \mathcal {D} \subseteq \mathcal {M}$ be a $\tau $ -preserving conditional expectation onto a von Neumann subalgebra $\mathcal {D} \subseteq \mathcal {M}$ . A finite subdiagonal algebra $\mathcal {H}^\infty (\mathcal {M}) \subseteq \mathcal {M}$ with respect to $\mathbb {E}$ is a weak- $\ast $ closed, non self-adjoint algebra such that

1. (i) $\mathbb {E}(f \, g) = \mathbb {E}(f) \, \mathbb {E}(g)$ ,

2. (ii) $\displaystyle \big \{f + g^\ast : f, g \in \mathcal {H}^\infty (\mathcal {M}) \big \} = \mathcal {M}$ and

3. (iii) $\mathcal {H}^\infty (\mathcal {M}) \cap (\mathcal {H}^\infty (\mathcal {M}) )^\ast = \mathcal {D}$ .

The notation comes from the fact that, for the algebra $L_\infty (\mathbb {T})$ and the integral as expectation, the Hardy space $\mathcal {H}^\infty (\mathbb {T})$ gives a finite subdiagonal algebra. Other examples are the $n \times n$ upper triangular matrices, which are a finite subdiagonal algebra with respect to the diagonal subalgebra $\ell _\infty ^n \subseteq M_n(\mathbb {C})$ , and more generally the nest algebras [Reference Davidson20] of totally ordered families of projections. In this setting any $f \in \mathcal {M}$ admits a unique decomposition as

$$\begin{align*}f = g + \delta + h^\ast, \quad \text{ with } g, h \in {\mathcal{H}_0^\infty(\mathcal{M})}, \,\, \delta \in \mathcal{D}, \end{align*}$$

where $\mathcal {H}_0^\infty (\mathcal {M}) = \{f\in \mathcal {H}^\infty (\mathcal {M}): \mathbb {E}(f)=0\}$ . The Hilbert transform associated to the finite subdiagonal algebra $\mathcal {H}^\infty (\mathcal {M})$ is thus $H(f) = -i g + i h^\ast $ . For these family of operators Randrianantoanina [Reference Randrianantoanina74] proved their weak type $(1,1)$ , which after interpolation gives the optimal constant in terms of p. In the case of a discrete left-orderable group $\mathcal {L} \mathrm {G}$ with the subalgebra $\mathbb {C} {\mathbf {1}} \subseteq \mathcal {L} \mathrm {G}$ and the conditional expectation given by $f \mapsto \tau (f) {\mathbf {1}}$ , we have that

$$\begin{align*}\mathcal{H}^\infty_0 (\mathcal{L} \mathrm{G}) \, = \, \bigg\{ f = \sum_{e \prec g} \widehat{f}(g) \, \lambda_g : f \in \mathcal{L} \mathrm{G} \bigg\} \end{align*}$$

is a finite subdiagonal algebra whose Hilbert transform coincides with the multiplier $-i T_{\mathrm {sgn}}$ in Theorem C. Thus, by Corollary 2.8, Cotlar identities can be used to recover previously known results with optimal constant.

5 Multipliers from Bass-Serre theory

The theory of discrete groups acting on trees (without edge inversions) is very well understood due to the work of Serre. The interested reader can find more on this topic in [Reference Serre79]. Here we are going to briefly explain how it is possible to use this theory to build examples of multipliers satisfying Cotlar’s identity and to understand previously known examples like free group multipliers from [Reference Mei and Ricard55, Reference Mei, Ricard and Xu56] under a geometric lens.

Given a graph of groups $\mathbb {X}$ , which we will assume is connected, its fundamental group $\pi _1(\mathbb {X})$ can be defined in two ways, with the second one being more handy when constructing the Bass-Serre tree of $\mathbb {X}$ .

• (D.i) The first construction, denoted by $\pi _1(\mathbb {X}, x_0)$ and briefly explained in the introduction, consists of the subgroup of $F(\mathbb {X})$ given by words of type c, where c is a closed path in X that starts and ends at the base point $x_0 \in \mathrm {Vert}(X)$ .

• (D.ii) For the second construction fix a spanning tree $T \subseteq X$ , that is a tree containing all vertices of X. The fundamental group $\pi _1(\mathbb {X};T)$ can be constructed as

  $$\begin{align*}\pi_1(\mathbb{X}; T) \, = \, F(\mathbb{X}) / \langle y : y \in \mathrm{Edge}(T) \rangle, \end{align*}$$
  where $\langle y : y \in \mathrm {Edge}(T) \rangle $ is the normal subgroup generated by all the edges y in the spanning tree.

The first definition is independent of the choice of $x_0$ , while the second is independent of the choice of spanning tree T. Both are isomorphic, see [Reference Serre79], and we will usually denote the resulting group just by $\pi _1(\mathbb {X})$ . Recall from the introduction that a word g of type c is in normal form if, informally, it can not be shortened applying (1.3). More formally

• • Given a word $g = |c,r|$ of type c, it is in normal form if either $m = 0$ and $r_0 \neq e$ or, in the case in which $m \geq 1$ , it holds that for every $1 \leq j \leq m - 1$ , $r_j \not \in \mathrm {H}_{y_j}^{y_j}$ when $y_{j+1} = {\overline {y}_j}$ .

• • Let $|c,r|$ and $|c,\mu |$ be two words of type c with $r = (r_0, \, r_1, \, \dots , r_m)$ and $\mu = (\mu _0, \, \mu _1, \, \dots , \mu _m)$ . They are said to be equivalent if $\mu _0 = r_0 a_1^{\bar y_1}$ and $\mu _j=(a_j^{y_j})^{-1} r_j a_{j+1}^{\bar y_{j+1}}$ , where $a_j \in \mathrm {H}_{y_j}$ and $a_j^{y_j}$ and $a_j^{\bar {y}_j}$ are the corresponding images in $\mathrm {H}_{y_j}^{y_j}$ and $\mathrm {H}_{\bar {y_j}}^{\bar {y_j}}$ .

We have that two words of type c in normal form are equivalent if and only if they represent the same element in $\pi _1(\mathbb {X}, x_0)$ . Observe also that any $g \in \pi _1 (\mathbb {X}, x_0)$ admits a unique normal form up to equivalence.

The fundamental group of a graph of groups unifies several constructions.

• • Homotopy groups of $\mathbf {1}$ -dimensional simplicial complexes. Any graph X is naturally a graph of groups by putting the trivial group in each vertex and edge. In this case $\pi _1(\mathbb {X})$ is naturally isomorphic to the (usual) fundamental group $\pi _1(X)$ of the associated $1$ -dimensional simplicial complex, see [Reference Rotman77, Chapter 11].

• • Amalgamated free products. Let X be a connected graph with two vertices $x_1$ and $x_2$ associated to groups $\mathrm {G}_1$ and $\mathrm {G}_2$ and connected by a single edge y to which a common subgroup $\mathrm {H}$ of $\mathrm {G}_1$ and $\mathrm {G}_2$ is attached. Then, the fundamental group of this graph of groups is the amalgamated free product $\pi _1(\mathbb {X}) = \mathrm {G}_1 \ast _{\mathrm {H}} \mathrm {G}_2$ , see Figure 5, and when $\mathrm {H} = \{e\}$ the usual free product is obtained.

  Figure 5

  Amalgamated free product.

  

• • HNN extensions. Let $\mathrm {G}$ be a group and $\mathrm {H}_1$ and $\mathrm {H}_2$ two isomorphic subgroups $\mathrm {H}_1, \, \mathrm {H}_2 \subseteq \mathrm {G}$ . Fix an isomorphism $\theta : \mathrm {H}_1 \to \mathrm {H}_2$ . The HNN extension of $\mathrm {G}$ relative to $\theta $ , see [Reference Rotman77], is the smallest extension of $\mathrm {G}$ such that $\theta $ is implemented by an inner automorphism, i.e. $\theta (h) = t \, h \, t^{-1}$ , for some t and every $h \in \mathrm {H}_1$ . In the case of a group given by the presentation $\mathrm {G} = \langle S \, | \, R \rangle $ , its HNN extension is

  (5.1) $$ \begin{align} \big\langle S, \, t \, {\big|} \, R, \; t \, h \, t^{-1} = \theta(h) \, \big\rangle, \end{align} $$
  where h runs through $\mathrm {H}_1$ . This construction can be obtained as the fundamental group of a graph of groups with a single vertex with group $\mathrm {G}$ and an edge y whose associated group is $\mathrm {H}_1$ and the two inclusions are given by $\alpha _{y}(h) = h$ and $\alpha _{\bar {y}}(h) = \theta (h)$ , see Figure 6.
  Figure 6

  HNN extension.

  

Let $\mathbb {X}$ be a graph of groups whose underlying graph we will denote by X and let $\pi = \pi _1(\mathbb {X})$ be its fundamental group. We will recall the construction of the Bass-Serre tree $\widetilde {X}$ of $\mathbb {X}$ whose vertices are

$$\begin{align*}\mathrm{Vert}(\widetilde{X}) = \bigsqcup_{x \in \mathrm{Vert}(X)} \pi / \mathrm{G}_{x}, \end{align*}$$

while its edges are given by

$$\begin{align*}\mathrm{Edge}_+(\widetilde{X}) = \bigsqcup_{y \in \mathrm{Edge}_+(X) } \pi / \mathrm{H}_{\bar y}^{\bar y} \;\;\;\;\; \text{and} \;\;\;\;\; \mathrm{Edge}_-(\widetilde{X}) = \bigsqcup_{y \in \mathrm{Edge}_-(X) } \pi/\mathrm{H}_{\bar y}^{\bar y}, \end{align*}$$

where $\mathrm {Edge}_+(X)$ is a fixed orientation of the edges of X. In the definition above we are including only the edges of a fixed orientation of $\widetilde {X}$ induced by the orientation of X. We can define the endpoint of each edge as follows

(5.2) $$ \begin{align} \begin{array}{lll} \mathrm{o}(g \, \mathrm{H}_{\bar y}^{\bar y}) = g \mathrm{G}_{\mathrm{o}(y)}, & \mathrm{t}(g \mathrm{H}_{\bar y}^{\bar y}) = g g_y \mathrm{G}_{\mathrm{t}(y)} & \text{ for } \, y \in \mathrm{Edge}_+(X), \,\, g \in \pi, \\ \mathrm{o}( g \mathrm{H}_{\bar y}^{\bar y}) = g g_y^{-1}\mathrm{G}_{\mathrm{o}(y)}, & \mathrm{t}(g \mathrm{H}_{\bar y}^{\bar y})= g \mathrm{G}_{\mathrm{t}(y)} & \text{ for } y \in \mathrm{Edge}_-(X), \,\, g \in \pi, \end{array} \end{align} $$

where $g_{y}$ is the image of y under the canonical projection from $F(\mathbb {X})$ to $\pi _1(\mathbb {X};T)$ . The group $\pi $ acts on $\widetilde {X}$ by left multiplication. The Bass-Serre tree construction also allows us to produce actions on trees for the fundamental group of any graph of groups. We will exploit this to obtain examples of Fourier multipliers satisfying Cotlar’s identity.

We are ready to prove the main theorem of this section.

Observe that, by the definition of the Bass-Serre tree of $\mathbb {X}$ in equation (5.2), the set of edges in $\widetilde {X}$ connected to the vertex $\widetilde {x}_0 = e \cdot \mathrm {G}_{x_0} \in \mathrm {Vert}(\widetilde {X})$ is given, up to orientation, by the disjoint union of $\mathrm {G}_{x_0} / \mathrm {H}_{\bar {y}}^{\bar {y}}$ , where, again, y runs over every edge in X starting with $x_0$ . This already establishes that the functions depending only on the initial segment and the functions that are constant on the connected components of $\widetilde {X} \setminus \{\widetilde x_0\}$ are in a natural and bijective correspondence. The following proposition, whose proof is immediate, asserts that m falls into Model 1. This provides an alternative proof of Theorem D via Proposition 3.1.(i).

In the case in which $\mathrm {G}_{x_0}$ is Abelian, the more restrictive condition that $m:\pi \to \mathbb {C}$ depends on the initial edge can be removed and a multiplier theorem still holds for symbols depending on the starting segment. Let m be a symbol depending on the starting segment and $\widetilde {m}: W_{x_0} \to \mathbb {C}$ be its lift to the space of initial segments defined in (1.5). We will denote by $\widetilde {m}_y$ the restriction of $\widetilde {m}$ to $\mathrm {G}_{x_0}/\mathrm {H}_{\bar {y}}^{\bar {y}}$ . We have the following result.

Proof. The proof is almost immediate. For (i) first notice that $m_{y,\chi }$ depends on the starting segment and that, given $a \in \mathrm {G}_{x_0}$ , $\sigma _{y,\chi }$ satisfies that $\sigma _{y,\chi }(a \, g ) = \chi (a) \, \sigma _{y,\chi }(g)$ . Thus, since it satisfies (Cotla ^r) relative to $\mathrm {G}_{x_0}$ , by Remark 2.6 we have that it is a bounded multiplier in $L_p$ for $1<p<\infty $ satisfying the bound (1.1) in Theorem A. The fact that $\widehat {\mathrm {G}}_y$ is compact implies that $\rho _y$ is bounded with the same bound as $T_{\sigma _{y,\chi }}$ . For the lower bound, notice that, since the Haar measure of $\widehat {\mathrm {G}}_y$ is a probability measure, ${\mathbf {1}}_{\widehat {\mathrm {G}}_y}$ is of norm one on $L_{p'}(\widehat {\mathrm {G}}_y)$ . But evaluating against it gives

$$\begin{align*}\big\| \rho_y(f) \big\|_{L_p(\widehat{\mathrm{G}}_y;L_p(\mathcal{L} \pi))} \geq \bigg\| \int_{\widehat{\mathrm{G}}_y} T_{m_{y,\chi}}(f) \, d \widehat{\mu}(\chi) \bigg\|_{L_p(\mathcal{L} \pi)} = \| f \|_p. \end{align*}$$

This shows that $\rho _y$ is a quasi-isometry, which gives (i). For (ii), just notice that if m is a symbol depending on the starting segment and that is $0$ outside $\mathrm {G}_y$ , then the following diagram commutes

and together with point (i), this yields the result. In the general case of m depending on the starting segment, we can decompose it as a sum of symbols $\widetilde {m}{|}_{\mathrm {G}_y}$ and build a direct sum map

$$\begin{align*}\rho: L_p(\mathcal{L} \pi) \longrightarrow \bigoplus_{\mathrm{o}(y) = x_0}{} L_p(\widehat{\mathrm{G}}_y;L_p(\mathcal{L} \pi)) \end{align*}$$

by taking $\rho _y$ in each of the components.

Now, we have all the tools required to illustrate our theory with the examples.

Examples: Amalgamated free products. This first example is already known, but our proof gives a concise geometric interpretation of it. Let $\{\mathrm {G}_i: i \geq 1 \}$ be a family of groups and A a common subgroup. Let us denote the injective homomorphisms by $\alpha _i : A \rightarrow \mathrm {G}_i$ , for $i \geq 1$ . Now, consider the graph of groups $\mathbb {X}$ in Figure 7, whose underlying tree X connects the point $x_0$ , with $\mathrm {G}_{x_0} = A$ , with the points $x_i$ , $i \geq 1$ , whose groups are $\mathrm {G}_i$ . Meanwhile, the groups on the edges $y_i$ are all isomorphic to A with the embedding into $\mathrm {G}_{x_0} = A$ being the identity map and the embedding into $\mathrm {G}_i$ being $\alpha _i$ .