2. Preliminaries

A more elaborate discussion of some of the statements in this section is contained in the author’s PhD thesis [Reference Vos27].

2.1. L _p -spaces of group von Neumann algebras for noncommutative groups

Let $G$ be a locally compact group, not necessarily unimodular. We will denote the left Haar measure of such a group by $\mu \,:\!=\, \mu _G$ . Its modular function will be denoted by $\Delta$ . Recall that $\Delta \,:\, G \to ({\mathrm{\mathbb{R}}}_{\gt 0}, \times )$ is a continuous group homomorphism satisfying

\begin{equation*} \int _G f(s^{-1}) \Delta (s^{-1}) d\mu (s) = \int _G f(s) d\mu (s) = \Delta (t) \int _G f(st) d\mu (s), \qquad t \in G, f \in L_1(G). \end{equation*}

In the sequel, we will write $ds$ for the left Haar measure.

The left regular representation $\lambda$ acts on $L_2(G)$ by $(\lambda _sf)(t) = f(s^{-1}t)$ . It also defines a $*$ -representation of the $*$ -algebra $L_1(G)$ on $L_2(G)$ by the formula

\begin{equation*} \lambda (f)g = \int _G f(s) (\lambda _sg) ds = f*g. \end{equation*}

The group von Neumann algebra $\mathcal{L} G$ of $G$ is defined as

\begin{equation*} \mathcal {L} G = \{\lambda _s\,:\, s \in G\}^{\prime\prime} = \{\lambda (f)\,:\, f \in L_1(G)\}^{\prime\prime}. \end{equation*}

The group von Neumann algebra admits a canonical weight $\varphi$ , named the Plancherel weight. It is defined for $x \in \mathcal{L} G$ by

\begin{equation*} \varphi (x^*x) = \begin {cases} \|f\|_2^2 \qquad & \text {if } x = \lambda (f) \text { for some } f \in L_2(G) \\ \infty & \text {else}. \end {cases} \end{equation*}

Here, we extend $\lambda$ to all functions that define a bounded convolution operator. The Plancherel weight is tracial if and only if $G$ is unimodular. Similarly, there is a right regular representation $\rho$ defined by $(\rho _sf)(t) = \Delta ^{1/2}(s) f(ts)$ and a Plancherel weight $\psi$ on $(\mathcal{L} G)^{\prime}$ defined as $\psi (x^*x) = \|f\|^2_2$ if $x$ is given by $x = \rho (f)$ for some $f \in L_2(G)$ and $\psi (x) = \infty$ otherwise. In practice, the group $\mathcal{L} G$ is often semifinite even when $G$ is not unimodular, but it is more natural to work with the Plancherel weight. For instance, the definition of $\varphi$ gives the Plancherel identity (2.2) below.

Let $L_p(\mathcal{L} G)$ denote the Connes–Hilsum $L_p$ -space corresponding to the Plancherel weight $\psi$ on $(\mathcal{L} G)^{\prime}$ ([Reference Connes7],[Reference Hilsum11]; see also [Reference Terp25]). To keep this paper more accessible, we will not be using the Tomita–Takesaki theory or the theory of spatial derivatives, except for the last section. Instead, we will use some facts about the Connes–Hilsum $L_p$ -spaces as a black box. First, for $1 \leq p \lt \infty$ , elements of $L_p(\mathcal{L} G)$ are closed unbounded operators on $L_2(G)$ , and sums and products of such operators are densely defined and preclosed. Addition and multiplication on $L_p(\mathcal{L} G)$ are defined by taking the closures of the resulting sum resp. product, and we will use the usual addition and multiplication notations for this. For any $x \in L_p(\mathcal{L} G)$ , $1 \leq p \leq \infty$ , we have

(2.1) \begin{equation} \|\lambda _s x \lambda _t\|_{L_p(\mathcal{L} G)} = \|x\|_{L_p(\mathcal{L} G)}, \qquad s,t \in G. \end{equation}

The spatial derivative $\frac{d\varphi }{d\psi }$ is just multiplication with the modular function $\Delta$ ; see, for instance, [Reference Caspers and de la Salle3, Section 3.5] for more details. With slight abuse of notation, we will denote this operator by $\Delta$ as well. The domain of this operator is exactly the set of functions $f \in L_2(G)$ such that $\int _G \Delta ^2(s) |f(s)|^2 ds \lt \infty$ . Usually, we will only apply $\Delta$ to continuous compactly supported functions so that no technical complications can arise.

Now define $L \,:\!=\, \lambda (C_c(G) \star C_c(G))$ , where $C_c(G) \star C_c(G)$ is the linear span of elements of the form $f * g$ , $f, g \in C_c(G)$ . For $\theta \in [0,1]$ and $1 \leq p \lt \infty$ , there is an embedding $\kappa ^{\theta }_p\,:\, L \to L_p(\mathcal{L} G)$ given by

\begin{equation*} \kappa ^{\theta }_p(x) = \Delta ^{(1-\theta )/p} x \Delta ^{\theta /p}. \end{equation*}

For $p = \infty$ , we simply set ${\kappa }_p^\theta$ to be the identity. The images of the embeddings ${\kappa }^{\theta }_p$ are dense in $L_p(\mathcal{L} G)$ , which will be crucial in the rest of the paper.

We now state some other facts for later use. From the definitions of the $L_p$ -norm and the Plancherel weight, we have the following Plancherel identity:

(2.2) \begin{equation} \|\lambda (f) \Delta ^{1/2}\|_{L_2(\mathcal{L} G)} = \|f\|_2, \qquad f \in L_2(G) \cap L_1(G). \end{equation}

As a side note, this implies that the map $C_c(G) \mapsto{\kappa }_2^1(\lambda (C_c(G)))$ , $f \mapsto \lambda (f) \Delta ^{1/2}$ extends to a unitary $L_2(G) \cong L_2(\mathcal{L} G)$ . Next, a straightforward calculation yields the following commutation formulae:

(2.3) \begin{equation} \Delta ^z \lambda _s = \Delta ^z(s) \lambda _s \Delta ^z, \qquad z \in{\mathrm{\mathbb{C}}}, s \in G \end{equation}

and

(2.4) \begin{equation} \Delta ^z \lambda (f) = \lambda (\Delta ^zf) \Delta ^z, \qquad z \in{\mathrm{\mathbb{C}}}, f \in C_c(G). \end{equation}

Also, note that for $f = g * h$ , $g,h \in C_c(G)$ and $z \in{\mathrm{\mathbb{C}}}$ , we have

(2.5) \begin{equation} (\Delta ^z f)(s) = \Delta ^z(s) \int _G g(t) h(t^{-1}s) dt = \int _G \Delta ^z(t) g(t) \Delta ^z(t^{-1}s) h(t^{-1}s) dt = ((\Delta ^z g) * (\Delta ^z h))(s) \end{equation}

and therefore $\Delta ^z (C_c(G) \star C_c(G)) = C_c(G) \star C_c(G)$ . Hence, (2.4) yields that

(2.6) \begin{equation} {\kappa }^{\theta }_p(L) ={\kappa }_p^0(L), \qquad \forall \ \theta \in [0,1]. \end{equation}

2.2. Operator spaces and multiplicatively bounded maps

Let $E_1, \dots, E_n, E$ be operator spaces and $T\,:\, E_1 \times \ldots \times E_n \to E$ a linear map. For $N \geq 1$ , the multiplicative amplification $T^{(N)}\,:\, M_N(E_1) \times \ldots M_N(E_n) \to M_N(E)$ of $T$ is defined as

\begin{equation*} T^{(N)}(\alpha _1 \otimes x_1, \dots, \alpha _n \otimes x_n) = \alpha _1 \ldots \alpha _n \otimes T(x_1, \dots, x_n), \qquad \alpha _i \in M_N({\mathrm {\mathbb {C}}}), x_i \in E_i \end{equation*}

and extended linearly. The map $T$ is said to be multiplicatively bounded if

\begin{equation*} \|T\|_{mb} \,:\!=\, \sup _{N \geq 1} \|T^{(N)}\| \lt \infty . \end{equation*}

Let us now define a notion of $p$ -completely bounded maps. Set $E = L_p(\mathcal{M}, \psi )$ for some von Neumann algebra $\mathcal{M}$ and normal faithful semifinite weight $\psi$ on $\mathcal{M}^{\prime}$ . Let $x \in M_N(E)$ ; then $x$ is closable and $[x] \in L_p(M_n(\mathcal{M}), \mathrm{tr}_N \otimes \psi )$ . Here $\mathrm{tr}_N$ is the trace on $M_N({\mathrm{\mathbb{C}}})$ . Now we set $S_p^N \otimes L_p(\mathcal{M}, \psi )$ to be the space $M_n(L_p(\mathcal{M}, \psi ))$ equipped with the norm

\begin{equation*} \|x\|_{S_p^N \otimes L_p(\mathcal {M}, \psi )} \,:\!=\, \|[x]\|_{L_p(M_N(\mathcal {M}), \mathrm {tr}_N \otimes \psi )}. \end{equation*}

We say that an operator $T\,:\, L_p(\mathcal{M}, \psi ) \to L_p(\mathcal{M}, \psi )$ is $p$ -completely bounded if

\begin{equation*} \|T\|_{p-cb} \,:\!=\, \sup _{N \geq 1} \|T^{(N)}\,:\, S_p^N \otimes L_p(\mathcal {M}, \psi ) \to S_p^N \otimes L_p(\mathcal {M}, \psi )\| \lt \infty . \end{equation*}

The matrix norms satisfy

(2.7) \begin{equation} \|\alpha \otimes x\|_{S^N_p \otimes L_p(\mathcal{M}, \varphi )} = \|\alpha \|_{S^N_p} \|x\|_{L_p(\mathcal{M}, \varphi )}, \qquad \alpha \in S^N_p, x \in L_p(\mathcal{M}, \varphi ). \end{equation}

For semifinite $\mathcal{M}$ , this is an easy exercise; for arbitrary von Neumann algebras, this requires some analysis on the interplay between spatial derivatives and matrices.

We note that the norm on $S_p^N \otimes L_p(\mathcal{M}, \psi )$ does not give an operator space structure on $L_p(\mathcal{M}, \psi )$ , as it does not satisfy the axioms. However, if $\mathcal{M}$ is semifinite, then the notion of $p$ -completely bounded maps coincides with the usual notion of completely bounded maps by [Reference Pisier20].

We now define a $(p_1, \dots, p_n)$ -multiplicatively bounded norm in a similar manner. Let $1 \leq p_1, \dots, p_n, p \leq \infty$ with $p^{-1} = \sum _{i=1}^n p_i^{-1}$ . We define a map $T\,:\, L_{p_1}(\mathcal{M}) \times \ldots \times L_{p_n}(\mathcal{M}) \to L_p(\mathcal{M})$ to be $(p_1, \dots, p_n)$ -multiplicatively bounded if

\begin{equation*} \|T\|_{(p_1, \dots, p_n)-mb} \,:\!=\, \sup _{m \geq 1} \|T^{(m)}\,:\, S_{p_1}^m \otimes L_{p_1}(\mathcal {M}) \times \ldots \times S^m_{p_n} \otimes L_{p_n}(\mathcal {M}) \to S_p^m \otimes L_p(\mathcal {M})\| \lt \infty . \end{equation*}

This turns out to be the correct notion to prove our transference results. We note that this time, there does not seem to be a case for which this coincides with ‘normal’ multiplicative boundedness, as [Reference Pisier20, Lemma 1.7] does not generalise to the multilinear case. If $\mathcal{M}$ is semifinite, then the above definition does coincide with the definition of $(p_1, \dots, p_n)$ -multiplicative boundedness from [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5] and [Reference Caspers, Krishnaswamy-Usha and Vos6]. Even in the semifinite case, it is unclear if this definition corresponds to the complete boundedness of some linear map on some appropriate tensor product; see [Reference Caspers, Krishnaswamy-Usha and Vos6, Remark 2.1].

2.3. Fourier multipliers

Again let $G$ be a locally compact group. For a bounded function $\phi \,:\, G \to{\mathrm{\mathbb{C}}}$ , the associated Fourier multiplier $T_\phi \,:\, \mathcal{L} G \to \mathcal{L} G$ is given for $f \in L_1(G)$ by $\lambda (f) \mapsto \lambda (\phi f)$ , whenever this map extends weak-* continuously. This definition has been extended to $L_p(\mathcal{L} G)$ for general locally compact groups in [Reference Caspers and de la Salle3]. However, this was done only for symbols $\phi \in M_{cb}A(G)$ , that is, those symbols for which $T_\phi$ defines a completely bounded multiplier on $\mathcal{L} G$ . We give a broader definition here as preparation for the multilinear definition. Define $\overline{\mathcal{L} G}_{(\gamma )}$ to be the space of closed densely defined $\gamma$ -homogeneous operators on $L_2(G)$ . See [Reference Terp25, Section III, IV] for a definition and some properties. We use here the facts that this space contains $L_p(\mathcal{L} G)$ and that the right-hand side of (2.8) is always in $\overline{\mathcal{L} G}_{(-1/p)}$ ([Reference Terp25, III.(19) and Corollary III.34]).

For $\phi \in L_\infty (G)$ , define $T_\phi \,:\,{\kappa }_p^\theta (L) \to \overline{\mathcal{L} G}_{(-1/p)}$ by

(2.8) \begin{equation} T_\phi ({\kappa }_p^\theta (\lambda (f))) = \Delta ^{\frac{1-\theta }p} \lambda (\phi f) \Delta ^{\frac \theta p}, \qquad f \in C_c(G) \star C_c(G),\ 1 \leq p \lt \infty, \ \theta \in [0,1] \end{equation}

The map $T_\phi$ does not depend on the choice of $\theta$ ; this follows from (2.6) and some manipulations using the commutation formula (2.4). When the image of $T_\phi$ lies in $L_p(\mathcal{L} G)$ and $T_\phi$ extends continuously to a bounded map on $L_p(\mathcal{L} G)$ , we say that $\phi$ defines a $p$ -Fourier multiplier. When the extension is moreover $p$ -completely bounded on $L_p(\mathcal{L} G)$ , we say $\phi$ defines a $p$ -cb Fourier multiplier. For $\phi \in M_{cb}A(G)$ , the definition of the Fourier multiplier coincides with that of [Reference Caspers and de la Salle3]. For $p=1$ , this can be shown by using [Reference Caspers and de la Salle3, Proposition-Definition 3.5] and (2.10) below together with [Reference Caspers and de la Salle3, Theorem 6.2 (ii), (iv)]; for other $p$ , this follows by interpolation.

Let us now state some known results about when symbols define $p$ - or $p$ -cb Fourier multipliers. Let $A(G)$ be the Fourier algebra; it is defined as $A(G) = \{f * \tilde{g}\,:\, f, g \in L_2(G)\}$ where $\tilde{g}(s) = \overline{g(s^{-1})}$ . It is isomorphic to $L_1(\mathcal{L} G)$ , the predual of $\mathcal{L} G$ , through the pairing

\begin{equation*} \langle \psi, \lambda (f) \rangle = \int _G \psi (s) f(s) ds, \qquad \psi \in A(G),\ f \in L_1(G). \end{equation*}

From classical theory (see, e.g. [Reference Kaniuth and Lau16, Proposition 5.1.2]), it is known that $\phi \in M(A(G))$ if and only if $\phi$ defines a $\infty$ -Fourier multiplier. In the same way as in [Reference Caspers and de la Salle3, Proof of Definition-Proposition 3.5], one proves that in this case, $\phi$ defines a $p$ -Fourier multiplier for all $1 \leq p \leq \infty$ . Trivially, $A(G) \subseteq M(A(G))$ , and hence $A(G)$ provides us with plenty of symbols defining $p$ -Fourier multipliers; we will use this fact later on. As mentioned above, if $\phi$ is a $\infty$ -cb Fourier multiplier (i.e. $\phi \in M_{cb}A(G)$ ), then $\phi$ is also a $p$ -cb Fourier multiplier for $1 \leq p \leq \infty$ . This is proven in [Reference Caspers and de la Salle3, Definition-Proposition 3.5].

Let us now turn our attention to the multilinear case. In [Reference Todorov26], $M^{cb}_nA(G)$ was defined to be the space of all symbols $\phi \in L_\infty (G^{\times n})$ such that the map

\begin{equation*} (\lambda _{s_1}, \dots, \lambda _{s_n}) \mapsto \phi (s_1, \dots, s_n) \lambda _{s_1 \ldots s_n} \end{equation*}

extends to a multiplicatively bounded normal map $(\mathcal{L} G)^{\times n} \to \mathcal{L} G$ . In [Reference Caspers, Krishnaswamy-Usha and Vos6], multilinear Fourier multipliers on the noncommutative $L_p$ -spaces were defined for unimodular groups $G$ as follows. Let $\phi \in L_\infty (G^{\times n})$ and $1\leq p_1, \ldots, p_n, p \lt \infty$ with $p^{-1}=\sum _{i=1}^n p_i^{-1}$ . Consider the map $T_\phi \,:\, L^{\times n} \to \mathcal{L} G$ defined by

\begin{equation*} T_\phi (\lambda (f_1),\ldots, \lambda (f_n)) = \int _{G^{\times n}} \phi (t_1,\ldots, t_n) f_1(t_1)\ldots f_n(t_n) \lambda _{t_1\ldots t_n} dt_1 \ldots dt_n \end{equation*}

for $f_i \in C_c(G) \star C_c(G)$ . If this map takes values in $L_p(\mathcal{L} G)$ and extends continuously to $L_{p_1}(\mathcal{L} G) \times \ldots \times L_{p_n}(\mathcal{L} G)$ , then we say that $\phi$ defines a $(p_1, \dots, p_n)$ -Fourier multiplier. The extension is again denoted by $T_\phi$ . In case $p_i = \infty$ , we replace $L_{p_i}(\mathcal{L} G)$ by $C_\lambda ^*(G)$ in the $i$ ’th coordinate. If the extension is $(p_1, \dots, p_n$ )-multiplicatively bounded, then we say that $\phi$ defines a $(p_1, \dots, p_n)$ -mb Fourier multiplier.

This definition works only for unimodular groups if $p \lt \infty$ since $L$ is not contained in $L_p(\mathcal{L} G)$ otherwise. In Section 3, we will give the definition for non-unimodular groups.

Remark 2.1. We note that a priori, the set of symbols of $(\infty, \dots, \infty )$ -mb Fourier multipliers is smaller than $M_n^{cb}A(G)$ . However, these sets are actually the same. This follows, for instance, from a combination of our results and [Reference Todorov26 , Theorem 5.5]. One does not need the complicated machinery of Section 4 however. It follows already from the proof of [ Reference Todorov26, Theorem 5.5] or from the alternative proof of the Fourier to Schur direction in [Reference Caspers, Krishnaswamy-Usha and Vos6 , Proposition 2.3], which it suffices to require that $T_\phi$ is bounded on $(C_\lambda ^*(G))^{\times n}$ .

2.4. Schatten classes and Schur multipliers

We denote by $S_p(H)$ the standard Schatten classes on the Hilbert space $H$ . If $(X, \mu )$ is some measure space, then $S_2(L_2(X))$ can be isometrically identified with the space of kernels $L_2(X \times X)$ . Through this identification, a kernel $A \in L_2(X \times X)$ corresponds to the operator $(A\xi )(s) = \int _X A(s,t) \xi (t) dt$ . This should be seen as a continuous version of matrix multiplication. We will make no distinction between an operator $A$ and its kernel. For $1 \leq p \leq 2 \leq p^{\prime} \leq \infty$ with $\frac 1p + \frac 1{p^{\prime}} = 1$ , the dual pairing between $S_p(L_2(X))$ and $S_{p^{\prime}}(L_2(X))$ is given by

(2.9) \begin{equation} \langle A, B \rangle _{p,p^{\prime}} = \int _{X^{\times 2}} A(s,t) B(t,s) dt ds, \qquad A \in S_p(L_2(X)), B \in S_2(L_2(X)). \end{equation}

This assignment is extended continuously for general $B \in S_{p^{\prime}}(L_2(X))$ . We refer to [Reference Lafforgue and de la Salle17] Section 1.2] for more details.

For $\phi \in L_\infty (X^{\times n+1})$ , the associated Schur multiplier is the multilinear map $S_2(L_2(X))\times \ldots \times S_2(L_2(X)) \to S_2(L_2(X))$ determined by

\begin{equation*} M_\phi ( A_1,\ldots, A_n)(t_0,t_n) = \int _{X^{\times n-1}} \phi (t_0,\ldots, t_n) A_1(t_0,t_1)A_2(t_1,t_2)\ldots A_n(t_{n-1},t_n) dt_1 \ldots dt_{n-1}. \end{equation*}

It follows by Cauchy–Schwarz and a straightforward calculation that $M_\phi$ does indeed take values in $S_2(L_2(X))$ (see, for instance, [Reference Caspers, Krishnaswamy-Usha and Vos6]). Now let $1\leq p,p_1,\ldots, p_n \leq \infty$ , with $p^{-1}=\sum _{i=1}^n p_i^{-1}$ . Restrict $M_\phi$ in the $i$ -th input to $S_2(L_2(X)) \cap S_{p_i}(L_2(X))$ . Assume that this restriction maps to $S_p(L_2(X))$ and has a bounded extension to $S_{p_1}(L_2(X)) \times \ldots \times S_{p_n}(L_2(X))$ . Then we say that $\phi$ defines a $(p_1, \dots, p_n)$ -Schur multiplier. Its extension is again denoted by $M_\phi$ . If $M_\phi$ is $(p_1, \dots, p_n)$ -multiplicatively bounded, then we say that $\phi$ defines a $(p_1, \dots, p_n)$ -mb Schur multiplier.

The following theorem is [Reference Caspers, Krishnaswamy-Usha and Vos6, Theorem 2.2]; see also [Reference Lafforgue and de la Salle17, Theorem 1.19] and [Reference Caspers and de la Salle3, Theorem 3.1]. It will be the starting point for the proof of Theorem4.1.

Theorem 2.2. Let $\mu$ be a Radon measure on a locally compact space $X$ and $\phi \,:\, X^{n+1} \to{\mathrm{\mathbb{C}}}$ a continuous function. Let $K\gt 0$ . The following are equivalent for $1 \leq p_1, \dots, p_n, p \leq \infty$ :

1. (i) $\phi$ defines a bounded Schur multiplier $S_{p_1}(L_2(X)) \times \ldots \times S_{p_n}(L_2(X)) \to S_p(L_2(X))$ with norm less than $K$ .

2. (ii) For every $\sigma$ -finite measurable subset $X_0$ in $X$ , $\phi$ restricts to a bounded Schur multiplier $S_{p_1}(L_2(X_0)) \times \ldots \times S_{p_n}(L_2(X_0)) \to S_p(L_2(X_0))$ with norm less than $K$ .

3. (iii) For any finite subset $F = \{s_1, \dots, s_N\} \subset X$ belonging to the support of $\mu$ , the symbol $\phi |_{F^{\times (n+1)}}$ defines a bounded Schur multiplier $S_{p_1}(\ell _2(F)) \times \ldots \times S_{p_2}(\ell _2(F)) \to S_p(\ell _2(F))$ with norm less than $K$ .

The same equivalence is true for the $(p_1, \ldots, p_n)-mb$ norms.

Now let $G$ again be a locally compact group. In general, one has $L_p(\mathcal{L} G) \cap S_p(L_2(G)) = \{0\}$ , so we cannot directly link Fourier and Schur multipliers as in the case $p = \infty$ . In Section 5, we will use the following trick from [Reference Caspers and de la Salle3] to circumvent this difficulty. Let $F \subseteq G$ be a relatively compact Borel subset of $G$ with positive measure and $P_F\,:\, L_2(G) \to L_2(F)$ , $f \mapsto 1_F f$ the orthogonal projection. Then for $x \in L_p(\mathcal{L} G)$ , one can formally define the operator $P_F x P_F$ , which lies in $S_p(L_2(G))$ . We refer to [Reference Caspers and de la Salle3, Proposition 3.3, Theorem5.1] for details.

Let $F \subseteq G$ compact. We calculate the kernel of $P_F \Delta ^a \lambda (f) \Delta ^b P_F$ : for $g \in C_c(G)$ and $s \in G$ , we have

\begin{align*} \begin{split} (P_F \Delta ^a \lambda (f) \Delta ^b P_F g)(s) &= 1_F(s) \Delta ^a(s) \int _G f(t) \Delta ^b(t^{-1} s) 1_F(t^{-1}s) g(t^{-1} s) dt \\ &= 1_F(s) \Delta ^a(s) \int _G f(st) \Delta ^b(t^{-1}) 1_F(t^{-1}) g(t^{-1}) dt \\ &= 1_F(s) \Delta ^a(s) \int _G f(st^{-1}) \Delta ^b(t) 1_F(t) \Delta (t^{-1}) g(t) dt. \end{split} \end{align*}

Hence the kernel of $P_F \Delta ^a \lambda (f) \Delta ^b P_F$ is given by

(2.10) \begin{equation} (s,t) \mapsto 1_F(s) \Delta ^a(s)f(st^{-1}) \Delta ^{b-1}(t) 1_F(t). \end{equation}

3. The definition of multilinear Fourier multipliers for non-unimodular groups

In this section, $G$ is an arbitrary locally compact group. Let $1 \leq p_1, \dots, p_n, p \leq \infty$ with $p^{-1} = \sum _{i=1}^n p_i^{-1}$ and $\phi \in L_\infty (G^{\times n})$ . In this section, we explore what a suitable definition for the Fourier multiplier $T_\phi \,:\, L_{p_1}(\mathcal{L} G) \times \ldots L_{p_n}(\mathcal{L} G) \to L_p(\mathcal{L} G)$ might be. Our first requirement is that it must coincide with the linear definition for $n=1$ ; that is, it must satisfy (2.8).

Second, we would like the definition to be compatible with interpolation arguments. More precisely, if $T_\phi$ is bounded as a map $\mathcal{L} G \times \ldots \times \mathcal{L} G \to \mathcal{L} G$ and as a map $L_{p_1}(\mathcal{L} G) \times \ldots \times L_{p_n}(\mathcal{L} G) \to L_p(\mathcal{L} G)$ , then it should also be bounded as map $L_{\frac{p_1}\nu }(\mathcal{L} G) \times \ldots \times L_{\frac{p_n}\nu }(\mathcal{L} G) \to L_{\frac{p}\nu }(\mathcal{L} G)$ for all $0 \lt \nu \lt 1$ . This means that the definition must be ‘compatible’ with the definition on $(\mathcal{L} G)^{\times n}$ , in the sense that in each input, the maps $T_\phi$ must coincide on the intersection space of some compatible couple (with respect to some $\theta$ ). This tells us what the Fourier multiplier should look like on the dense subsets ${\kappa }_{p_i}^\theta (L)$ :

Definition 3.1 (‘Wrong definition’). Let $\theta _1, \dots, \theta _n, \theta \in [0,1]$ and $x_i ={\kappa }_{p_i}^{\theta _i}(\lambda (f_i))$ for $i = 1, \dots, n$ , where $f_i \in C_c(G) * C_c(G)$ . We set

(3.1) \begin{equation} T^{\theta _1, \dots, \theta _n, \theta }_{\phi, \text{int}}(x_1, \dots, x_n) ={\kappa }^{\theta }_p(T_\phi (\lambda (f_1), \dots, \lambda (f_n))). \end{equation}

Definition 3.1 might seem reasonable at first glance; it coincides with the linear definition for $n=1$ , and it is the only option if we want interpolation results. However, there are several problems with Definition 3.1. First, the definition depends on the choice of embeddings, which is not an issue in the linear case. Second, there are several properties of multilinear Fourier multipliers on unimodular groups, which do not carry over. This includes, for instance, [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Lemma 4.3 and Lemma 4.4], which are crucial in the proof of the transference from Fourier to Schur multipliers. Moreover, if we want to prove an approximate intertwining property as in (5.1), Corollary 5.5 tells us that the definition of the Fourier multiplier has to ‘preserve products of linear multipliers’, in the sense that

\begin{equation*} T_{\phi }(x_1, \dots, x_n) = T_{\phi _1}(x_1) \ldots T_{\phi _n}(x_n) \end{equation*}

whenever $\phi (s_1, \dots, s_n) = \phi _1(s_1) \ldots \phi _n(s_n)$ . Definition 3.1 does not do this. This means that there is essentially no hope of proving the transference from Schur to Fourier multipliers either.

The above requirement on the preservation of products leads us to consider instead the following definition. Let $\theta _i \in [0,1]$ and set $a_i = \frac{1-\theta _i}{p_i}$ and $b_i = \frac{\theta _i}{p_i}$ so that ${\kappa }_{p_i}^{\theta _i}(x) = \Delta ^{a_i} x \Delta ^{b_i}$ . Now for $f_i \in C_c(G) \star C_c(G)$ , we formally define the Fourier multiplier corresponding to $\theta _1, \dots, \theta _n$ by

(3.2) \begin{equation} \begin{split} T_{\phi, (\theta _1, \dots, \theta _n)}({\kappa }^{\theta _1}_{p_1}(\lambda (f_1)),\ldots, {\kappa }^{\theta _n}_{p_n}(\lambda (f_n))) = &\int _{G^{\times n}} \phi (t_1,\ldots, t_n) f_1(t_1)\ldots f_n(t_n) \times \\ & \Delta ^{a_1} \lambda _{t_1} \Delta ^{b_1 + a_2} \lambda _{t_2} \ldots \Delta ^{b_{n-1} + a_n} \lambda _{t_n} \Delta ^{b_n} dt_1 \ldots dt_n. \end{split} \end{equation}

A priori, it is not clear how to define the integral in (3.2). After all, the integrand

\begin{equation*} H(t_1, \dots, t_n) \,:\!=\, \phi (t_1,\ldots, t_n) f_1(t_1)\ldots f_n(t_n) \Delta ^{a_1} \lambda _{t_1} \Delta ^{b_1 + a_2} \lambda _{t_2} \ldots \Delta ^{b_{n-1} + a_n} \lambda _{t_n} \Delta ^{b_n} \end{equation*}

is a function that has unbounded operators as values. However, on closer inspection, the ‘unbounded part’ of this operator doesn’t really depend on the integration variables. Indeed, using the commutation formula (2.3), we can write

\begin{align*} \begin{split} &H(t_1, \dots, t_n) \\ &\quad= \phi (t_1, \dots, t_n) f_1(t_1) \ldots f_n(t_n) \Delta ^{a_1}(t_1) \Delta ^{a_1 + a_2 + b_1}(t_2) \ldots \Delta ^{\sum _{i=1}^n a_i + \sum _{i=1}^{n-1} b_i}(t_n) \lambda _{t_1\ldots t_n} \Delta ^{1/p} \\ &\quad= \phi (t_1, \dots, t_n) (\Delta ^{\beta _1} f_1)(t_1) \ldots (\Delta ^{\beta _n} f_n)(t_n) \lambda _{t_1\ldots t_n} \cdot \Delta ^{1/p}, \qquad \beta _j = \sum _{i=1}^j a_i + \sum _{i=1}^{j-1} b_i. \\ \end{split} \end{align*}

Note here that the functions $\Delta ^{\beta _i} f_i$ are still in $C_c(G) \star C_c(G)$ by (2.5). Hence, a more rigorous way to define the Fourier multiplier is

\begin{equation*} T_{\phi, (\theta _1, \dots, \theta _n)}({\kappa }^{\theta _1}_{p_1}(\lambda (f_1)),\ldots, {\kappa }^{\theta _n}_{p_n}(\lambda (f_n))) = T_\phi (\lambda (\Delta ^{\beta _1} f_1), \ldots, \lambda (\Delta ^{\beta _n} f_n)) \Delta ^{1/p}. \end{equation*}

However, we will keep the notation from (3.2). The integral is justified through the above arguments. The latter expression also makes clear that (3.2) takes values in the space of closed densely defined $(-1/p)$ -homogeneous operators on $L_2(G)$ (see [Reference Terp25, III.(19) and Corollary III.34]). Just as in the linear case, it is not clear that (3.2) takes values in $L_p(\mathcal{L} G)$ in general; this will be part of the assumptions.

It turns out that the operator $T_{\phi, (\theta _1, \dots, \theta _n)}$ in (3.2) does not depend on the choice of $\theta _i$ ’s:

Proof. Recall that by (2.6), ${\kappa }_{p_i}^{\theta _i}(L) ={\kappa }_{p_i}^0(L)$ for $i = 1 \dots, n$ . For any such $i$ , take $a_i, b_i$ as above, that is, so that ${\kappa }_{p_i}^\theta (x) = \Delta ^{a_i} x \Delta ^{b_i}$ . Let $f_i \in C_c(G) \star C_c(G)$ and set $g_i = \Delta ^{-b_i} f_i$ . Then by (2.4), $\Delta ^{a_i} \lambda (f_i) \Delta ^{b_i} = \Delta ^{1/p_1} \lambda (g_i) \,=\!:\, x_i$ , that is, ${\kappa }_{p_i}^{\theta _i}(\lambda (f_i)) ={\kappa }_{p_i}^0(\lambda (g_i))$ . By (2.3), we find

\begin{align*} \begin{split} T_{\phi, (\theta _1, \dots, \theta _n)}(x_1, \dots, x_n) = &\int _{G^{\times n}} \phi (t_1,\ldots, t_n) f_1(t_1)\ldots f_n(t_n) \times \\ & \Delta ^{a_1} \lambda _{t_1} \Delta ^{b_1 + a_2} \lambda _{t_2} \ldots \Delta ^{b_{n-1} + a_n} \lambda _{t_n} \Delta ^{b_n} dt_1 \ldots dt_n \\ = & \int _{G^{\times n}} \phi (t_1,\ldots, t_n) \Delta ^{-b_1}(t_1) f_1(t_1) \Delta ^{-b_2}(t_2) f_2(t_2) \ldots \Delta ^{-b_n}(t_n)f_n(t_n) \times \\ & \Delta ^{1/p_1} \lambda _{t_1} \Delta ^{1/p_2} \ldots \Delta ^{1/p_n} \lambda _{t_n} dt_1 \ldots dt_n \\ = &T_{\phi, (0, \dots, 0)}(\Delta ^{1/p_1} \lambda (g_1), \dots, \Delta ^{1/p_n} \lambda (g_n)) = T_{\phi, (0, \dots, 0)}(x_1, \dots, x_n). \end{split} \end{align*}

With this issue out of the way, we can now define Fourier multipliers independent of the choice of $\theta _i$ ’s:

Definition 3.3 (‘Correct definition’). Let $1 \leq p_1, \dots, p_n, p \leq \infty$ with $p^{-1} = \sum _{i=1}^n p_i^{-1}$ . Also let $\phi \in L_\infty (G^{\times n})$ . For $i = 1, \dots, n$ , take any $a_i, b_i \in [0,1]$ such that $a_i + b_i = p_i^{-1}$ . If the map

\begin{equation*} T_\phi \,:\, {\kappa }_{p_1}^{0}(L) \times \ldots \times {\kappa }_{p_n}^0(L) \to \overline {\mathcal {L} G}_{(-1/p)} \end{equation*}

which is given for $x_i = \Delta ^{a_i} \lambda (f_i) \Delta ^{b_i}$ with $f_i \in C_c(G) \star C_c(G)$ by

(3.3) \begin{equation} \begin{split} T_\phi (x_1,\ldots, x_n) = &\int _{G^{\times n}} \phi (t_1,\ldots, t_n) f_1(t_1)\ldots f_n(t_n) \times \\ & \Delta ^{a_1} \lambda _{t_1} \Delta ^{b_1 + a_2} \lambda _{t_2} \ldots \Delta ^{b_{n-1} + a_n} \lambda _{t_n} \Delta ^{b_n} dt_1 \ldots dt_n, \end{split} \end{equation}

takes values in $L_p(\mathcal{L} G)$ and extends boundedly to $L_{p_1}(\mathcal{L} G) \times \ldots \times L_{p_n}(\mathcal{L} G)$ in the norm topology (in case $p_i = \infty$ for some $i$ , we use the space $C_\lambda ^*(G)$ instead of $L_\infty (\mathcal{L} G) = \mathcal{L} G$ in the $i$ ’th leg), then we say that $\phi$ defines a $(p_1, \dots, p_n)$ -Fourier multiplier. We denote the extension by $T_\phi$ or $T_{\phi }^{p_1, \dots, p_n}$ when we wish to emphasise the domain of the operator. This is especially useful when writing an operator norm since writing out the full domain and codomain generally makes equations too long. If $T_\phi$ is $(p_1, \dots, p_n)$ -multiplicatively bounded, then we say that $\phi$ defines a $(p_1, \dots, p_n)$ -mb Fourier multiplier.

Clearly, for $n=1$ , Definition 3.3 reduces to (2.8). It does not give the problems that Definition 3.1 does; as we saw already, it does not depend on the choice of embeddings. Moreover, the properties [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Lemma 4.3 and 4.4] do carry over, as we show in Lemmas 3.6 and 3.8. Finally, it preserves products in the following more general way: if $\phi$ is such that there exist $m \lt n$ and $\phi _1\,:\, G^{\times m} \to{\mathrm{\mathbb{C}}}$ , $\phi _2\,:\, G^{\times n-m} \to{\mathrm{\mathbb{C}}}$ such that

\begin{equation*} \phi (s_1, \dots, s_n) = \phi _1(s_1, \dots, s_m) \phi _2(s_{m+1}, \dots, s_n), \end{equation*}

then

(3.4) \begin{equation} T_{\phi }(x_1, \dots, x_n) = T_{\phi _1}(x_1, \dots, x_m) T_{\phi _2}(x_{m+1}, \dots, x_n). \end{equation}

However, we have to give up interpolation results in general. The only instances where interpolation might work is when the $L_p$ -spaces ‘in the middle’ are all equal to $C_\lambda ^*(G)$ . Indeed, in that case, we can take $\theta _1 = 0$ , $\theta _n = 1$ , $\theta = \frac{p}{p_n}$ so that the Fourier multiplier $T_\phi$ from Definition 3.3 also satisfies (3.1). We note that for $n \gt 2$ and $p_i \lt \infty$ for some $2 \leq i \leq n-1$ , (3.3) is not of the form (3.1) for any $\theta _1, \dots, \theta _n, \theta$ , and hence $T_\phi$ cannot be a compatible morphism for any ‘usual’ compatible couple structures on $(\mathcal{L} G, L_{p_i}(\mathcal{L} G))_{\theta _i}$ .

Let us now prove some properties of the multilinear Fourier multiplier for later use. Lemmas 3.6, 3.7 and 3.8 are used in the proof of Theorem4.1. Here, Lemma 3.6 generalises [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Lemma 4.3], and Lemma 3.8 generalises [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Lemma 4.4]. Since the proofs of these two lemmas require careful bookkeeping with modular functions, we will give the full details. The proof of [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Lemma 4.4] was omitted, but it is not that trivial; our argument fills that gap.

Lemma 3.6. Let $1 \leq p_j, p \leq \infty$ and fix some $1 \leq i \leq n$ . Suppose that $\phi \,:\, G^{\times n} \to{\mathrm{\mathbb{C}}}$ is bounded and measurable and set for $r,t,r^{\prime} \in G$ :

\begin{equation*} \bar {\phi }(s_1, \dots, s_n; r,t,r^{\prime}) \,:\!=\, \phi (rs_1, \dots, s_it, t^{-1} s_{i+1}, \dots, s_n r^{\prime}). \end{equation*}

Then $\phi$ defines a $(p_1, \dots, p_n)$ -Fourier multiplier (resp. $(p_1, \dots, p_n)$ -mb Fourier multiplier) iff $\bar{\phi }({\cdot}; r,t,r^{\prime})$ defines a $(p_1, \dots, p_n)$ -Fourier multiplier (resp. $(p_1, \dots, p_n)$ -mb Fourier multiplier). In that case, for $x_j \in L_{p_j}(\mathcal{L} G)$ ,

(3.5) \begin{equation} T_{\bar{\phi }({\cdot};r,t,r^{\prime})}(x_1, \dots, x_n) = \lambda _r^* T_\phi \big (\lambda _r x_1, x_2, \dots, x_i \lambda _t, \lambda _t^* x_{i+1}, \dots, x_n\lambda _{r^{\prime}}\big ) \lambda _{r^{\prime}}^*. \end{equation}

Further, we have

\begin{equation*} \|T^{p_1, \dots, p_n}_\phi \| = \|T^{p_1, \dots, p_n}_{\bar {\phi }({\cdot}; r,t,r^{\prime})}\| \end{equation*}

and $(r,t,r^{\prime}) \mapsto T_{\bar{\phi }({\cdot}; r,t,r^{\prime})}$ is strongly continuous. In the multiplicatively bounded case, we have for any $N \geq 1$

\begin{equation*} \|(T^{p_1, \dots, p_n}_\phi )^{(N)}\| = \|(T^{p_1, \dots, p_n}_{\bar {\phi }({\cdot}; r,t,r^{\prime})})^{(N)}\| \end{equation*}

as maps $S_{p_1}^N[L_{p_1}(\mathcal{L} G)] \times \ldots \times S_{p_n}^N[L_{p_n}(\mathcal{L} G)] \to S_p^N[L_p(\mathcal{L} G)]$ , and $(r,t,r^{\prime}) \mapsto T_{\bar{\phi }({\cdot}; r,t,r^{\prime})}^{(N)}$ is strongly continuous.

Proof. It is straightforward to check that for $s \in G$ , $f \in C_c(G)$ , we have $\lambda _s\lambda (f) = \lambda (\lambda _s(f)) = \lambda (f(s^{-1} {\cdot}))$ ; moreover, we have

(3.6) \begin{equation} \lambda (f)\lambda _s = \int _G f(t) \lambda _{ts} dt = \Delta (s^{-1}) \int _G f(ts^{-1}) \lambda _t dt = \Delta (s^{-1}) \lambda (f({\cdot} s^{-1})). \end{equation}

We will only make a choice for some of the embeddings and leave the rest open; this is notationally more convenient. Let $x_j = \Delta ^{a_j} \lambda (f_j) \Delta ^{b_j} \in L_{p_j}(\mathcal{L} G)$ , with $f_j \in C_c(G) \star C_c(G)$ and $a_1=b_i = a_{i+1} = b_n = 0$ (hence $b_1 = \frac 1{p_1}, a_i = \frac 1{p_i}$ , etc). We compute

\begin{align*} \begin{split} &T_{\bar{\phi }({\cdot};r,t,r^{\prime})}(x_1, \dots, x_n) \\&\quad = \int _{G^{\times n}} \bar{\phi }(s_1, \dots, s_n; r,t,r^{\prime}) f_1(s_1) \dots f_n(s_n) \lambda _{s_1} \Delta ^{b_1 + a_2} \lambda _{s_2} \dots \Delta ^{b_{n-1} + a_n}\lambda _{s_n} ds_1 \dots ds_n \\&\quad = \int _{G^{\times n}} \phi (s_1, \dots, s_n) f_1(r^{-1}s_1) \dots \Delta (t)^{-1} f_i(s_it^{-1}) f_{i+1}(ts_{i+1}) \dots \Delta (r^{\prime})^{-1} f_n(s_n (r^{\prime})^{-1}) \times \\ &\qquad \lambda _{r^{-1}s_1} \Delta ^{b_1 + a_2} \dots \Delta ^{b_{i-1} + a_i}\lambda _{s_i s_{i+1}} \Delta ^{b_{i+1}+ a_{i+2}} \dots \Delta ^{b_{n-1} + a_n} \lambda _{s_n (r^{\prime})^{-1}} ds_1 \dots ds_n \\&\quad = \lambda _r^* T_\phi \big (\widetilde{x_1}, x_2, \dots, \widetilde{x_i}, \widetilde{x_{i+1}}, \dots, \widetilde{x_n}\big ) \lambda _{r^{\prime}}^*. \end{split} \end{align*}

Here

\begin{align*} \begin{split} \widetilde{x_1} &\,:\!=\, \lambda (f_1(r^{-1}{\cdot})) \Delta ^{b_1}; \qquad \widetilde{x_i} \,:\!=\, \Delta ^{-1}(t) \Delta ^{a_i} \lambda (f_i({\cdot} t^{-1})); \\ \widetilde{x_{i+1}} &\,:\!=\, \lambda (f_{i+1}(t{\cdot})) \Delta ^{b_{i+1}}; \qquad \widetilde{x_n} \,:\!=\, \Delta ^{-1}(r^{\prime})\Delta ^{a_n} \lambda (f_n({\cdot} (r^{\prime})^{-1})). \end{split} \end{align*}

By (3.6), we can write

\begin{equation*} \widetilde {x_n} = \Delta ^{a_n} \lambda (f_n) \lambda _{r^{\prime}} = x_n \lambda _{r^{\prime}} \end{equation*}

and similarly

\begin{equation*} \widetilde {x_1} = \lambda _r x_1; \qquad \widetilde {x_i} = x_i \lambda _t; \qquad \widetilde {x_{i+1}} = \lambda _t^* x_{i+1}. \end{equation*}

Combining everything together, we conclude

\begin{equation*} T_{\bar {\phi }({\cdot};r,t,r^{\prime})}(x_1, \dots, x_n) = \lambda _r^* T_\phi \big (\lambda _r x_1, x_2, \dots, x_i \lambda _t, \lambda _t^* x_{i+1}, \dots, x_n\lambda _{r^{\prime}}\big ) \lambda _{r^{\prime}}^*. \end{equation*}

By (2.1), we have

\begin{align*} \begin{split} \|T_{\bar{\phi }({\cdot};r,t,r^{\prime})}(x_1, \dots, x_n)\|_p &= \|T_\phi \big (\lambda _r x_1, x_2, \dots, x_i \lambda _t, \lambda _t^* x_{i+1}, \dots, x_n\lambda _{r^{\prime}}\big )\|_p \\ &\leq \|T^{p_1, \dots, p_n}_{\phi }\| \|x_1\|_{p_1} \ldots \|x_n\|_{p_n}. \end{split} \end{align*}

Hence, on the dense subsets of elements $x_j$ as above, we have $\|T_{\widetilde{\phi }({\cdot} ;r,t,r^{\prime})}\| \leq \|T_{\phi }\|$ . If we set $\psi = \tilde{\phi }({\cdot}; r,t,r^{\prime})$ , then $\tilde{\psi }({\cdot}; r^{-1}, t^{-1}, (r^{\prime})^{-1}) = \phi$ . Hence, applying the above result to $\tilde{\psi }({\cdot}; r^{-1}, t^{-1}, (r^{\prime})^{-1})$ yields the reverse inequality. By density, we conclude that the first three statements of the lemma hold. By [Reference Junge and Sherman15, Lemma 2.3], the (left or right) multiplication with $\lambda _s$ , $s \in G$ is strongly continuous in $s$ . This implies the strong continuity of $(r,t,r^{\prime}) \mapsto T_{\tilde{\phi }({\cdot};r,t,r^{\prime})}$ .

Now assume $\phi$ defines a $(p_1, \dots, p_n)$ -mb Fourier multiplier and let $N \geq 1$ . Denote by $\iota _N$ the $N \times N$ -identity matrix. Then by writing out the definitions and using (3.5), we find, for $x_i \in S_{p_i}^N \otimes L_{p_i}(\mathcal{L} G)$ ,

\begin{align*} \begin{split} &T_{\bar{\phi }(\cdot ;r,t,r^{\prime})}^{(N)}(x_1, \dots, x_n) \\&\quad = (\iota _N \otimes \lambda _r^*) T_{\phi }^{(N)}((\iota _N \otimes \lambda _r) x_1, \dots, x_i(\iota _N \otimes \lambda _t), (\iota _N \otimes \lambda _t^*) x_{i+1}, \dots, x^n(\iota _N \otimes \lambda _{r^{\prime}})) (\iota _N \otimes \lambda _{r^{\prime}}^*). \end{split} \end{align*}

Hence, by a complete/matrix amplified version of the above arguments, we deduce the last two statements.

Lemma 3.7. Let $1 \leq p_j, p \leq \infty$ and fix some $1 \leq i \leq n$ . Suppose that $\phi \,:\, G^{\times n} \to{\mathrm{\mathbb{C}}}$ defines a $(p_1, \dots, p_n)$ -Fourier multiplier and $\phi _i\,:\, G \to{\mathrm{\mathbb{C}}}$ defines a $p_i$ -Fourier multiplier. Set

\begin{equation*} \bar {\phi }(s_1, \dots, s_n) = \phi (s_1, \dots, s_n) \phi _i(s_i). \end{equation*}

Then $\bar{\phi }$ defines a $(p_1, \dots, p_n)$ -Fourier multiplier and for $x_j \in L_{p_j}(\mathcal{L} G)$ ,

(3.7) \begin{equation} T_{\bar{\phi }}(x_1, \dots, x_n) = T_\phi (x_1, \dots, x_{i-1}, T_{\phi _i}(x_i), x_{i+1}, \dots, x_n). \end{equation}

In particular,

(3.8) \begin{equation} \|T_{\bar{\phi }}^{p_1, \dots, p_n}\| \leq \|T_{\phi }^{p_1, \dots, p_n}\| \|T_{\phi _i}\,:\, L_{p_i}(\mathcal{L} G) \to L_{p_i}(\mathcal{L} G)\|. \end{equation}

Proof. For $x_j \in{\kappa }_{p_j}^0(L)$ (or any other embedding), it follows directly from writing out the definitions that (3.7) holds (cf. (2.8)). By density, (3.7) holds for general $x_j \in L_{p_j}(\mathcal{L} G)$ , which implies (3.8), so $T_{\bar{\phi }}^{p_1, \dots, p_n}$ is bounded.

Lemma 3.8. Let $1 \leq p_1, \dots, p_n \leq \infty$ . Let $q_j^{-1} = \sum _{i=j}^n p_i^{-1}$ and suppose that $\phi _j\,:\, G \to{\mathrm{\mathbb{C}}}$ defines a $q_j$ -Fourier multiplier for $1 \leq j \leq n$ . Set

\begin{equation*} \bar {\phi }(s_1, \dots, s_n) = \phi _1(s_1 \ldots s_n) \phi _2(s_2 \ldots s_n) \ldots \phi _n(s_n). \end{equation*}

Then $\bar{\phi }$ defines a $(p_1, \dots p_n)$ -Fourier multiplier, and for $x_i \in L_{p_i}(\mathcal{L} G)$ , we have

(3.9) \begin{equation} T_{\bar{\phi }}(x_1, \dots, x_n) = T_{\phi _1}(x_1 T_{\phi _2}(x_2 \ldots T_{\phi _n}(x_n)\ldots )). \end{equation}

Proof. We first show (3.9) on the dense subset ${\kappa }^0_{p_i}(L) \times \ldots \times{\kappa }^0_{p_n}(L)$ . The lemma then follows from the boundedness of the $T_{\phi _i}$ together with Hölder’s inequality.

We make the slightly stronger claim that for any $\phi _2, \dots, \phi _n$ as in the assumptions and any $x_i \in{\kappa }^0_{p_i}(L)$ , there exists a compactly supported function $g\,:\, G \to{\mathrm{\mathbb{C}}}$ such that for all $\phi _1$ as in the assumptions,

(3.10) \begin{equation} T_{\bar{\phi }}(x_1, \dots, x_n) = \Delta ^{\frac 1{q_1}} \lambda (\phi _1g) = T_{\phi _1}(x_1 T_{\phi _2}(x_2 \ldots T_{\phi _n}(x_n)\ldots )). \end{equation}

We will prove (3.10) with induction on $n$ . We will need this intermediate step in order to expand the outer Fourier multiplier in the right-hand side of (3.9).

The case $n=1$ follows directly from (2.8). Now assume that (3.10) holds for any choice of $\phi _1, \dots, \phi _{n-1}$ as above and $x_1, \dots, x_{n-1}$ with $x_i \in{\kappa }^0_{p^{\prime}_i}(L)$ . Fix functions $\phi _1, \dots, \phi _n$ as in the assumptions and $x_1, \dots, x_n$ so that $x_i = \Delta ^{\frac 1{p_i}} \lambda (f_i)$ for $f_i \in C_c(G) \star C_c(G)$ . Take $g$ compactly supported such that for any $\psi \,:\, G \to{\mathrm{\mathbb{C}}}$ defining a $q_2$ -Fourier multiplier,

(3.11) \begin{equation} T_{\psi }(x_2 T_{\phi _3}(x_3 \ldots T_{\phi _n}(x_n)\ldots )) = \Delta ^{\frac 1{q_2}} \lambda (\psi g) = T_{\bar{\psi }}(x_2, \dots, x_n) \end{equation}

where $\bar{\psi }(s_2, \dots, s_n) = \psi (s_2 \ldots s_n) \phi _3(s_3\ldots s_n) \ldots \phi _n(s_n)$ . We calculate

\begin{align*} \begin{split} T_{\phi _1}(x_1 T_{\phi _2}(x_2 \ldots T_{\phi _n}(x_n)\ldots )) &\stackrel{(3.11)}= T_{\phi _1}(\Delta ^{\frac 1{p_1}} \lambda (f_1) \Delta ^{\frac 1{q_2}} \lambda (\phi _2 g)) \\ &\stackrel{(2.4)}= T_{\phi _1}(\Delta ^{\frac 1{q_1}} \lambda ((\Delta ^{-\frac 1{q_2}} f_1)*(\phi _2g))) \\ &\stackrel{(2.8)}= \Delta ^{\frac 1{q_1}} \lambda (\phi _1 ((\Delta ^{-\frac 1{q_2}} f_1)*(\phi _2g))). \end{split} \end{align*}

This shows the second equality from (3.10). Continuing the previous equation,

\begin{align*} \begin{split} T_{\phi _1}(x_1 T_{\phi _2}(x_2 \ldots T_{\phi _n}(x_n)\ldots )) &= \int _G \phi _1(t) \left (\int _G (\Delta ^{-\frac 1{q_2}} f_1)(s_1) (\phi _2g)(s_1^{-1} t) ds_1\right ) \Delta ^{\frac 1{q_1}} \lambda _t dt \\ &= \int _G \int _G \phi _1(s_1t) (\Delta ^{-\frac 1{q_2}} f_1)(s_1) (\phi _2g)(t) \Delta ^{\frac 1{q_1}} \lambda _{s_1t} dt ds_1 \\ &\stackrel{(2.3)}= \int _G f_1(s_1) \Delta ^{\frac 1{p_1}} \lambda _{s_1} \int _G \phi _1(s_1t) \phi _2(t) g(t) \Delta ^{\frac 1{q_2}} \lambda _t dt ds_1 \\ &= \int _G f_1(s_1) \Delta ^{\frac 1{p_1}} \lambda _{s_1} \Delta ^{\frac 1{q_2}} \lambda (\phi _1(s_1 {\cdot}) \phi _2 g) ds_1. \\ \end{split} \end{align*}

Applying (3.11) again but now with $\phi _1(s_1 {\cdot}) \phi _2$ in place of $\psi$ , we get

\begin{align*} \begin{split} &T_{\phi _1}(x_1 T_{\phi _2}(x_2 \ldots T_{\phi _n}(x_n)\ldots )) \\ &\quad = \int _G f_1(s_1) \Delta ^{\frac 1{p_1}}\lambda _{s_1} \int _{G^{\times n-1}} \phi _1(s_1s_2 \ldots s_n) \phi _2(s_2 \ldots s_n) \phi _3(s_3 \ldots s_n) \ldots \phi _n(s_n) \times \\ & \qquad \qquad \qquad \qquad \qquad \qquad f_2(s_2) \ldots f_n(s_n) \Delta ^{\frac 1{p_2}} \lambda _{s_2} \ldots \Delta ^{\frac 1{p_n}} \lambda _{s_n} ds_2 \ldots ds_n ds_1 \\ &\quad= T_{\bar{\phi }}(x_1, \dots, x_n). \end{split} \end{align*}

Finally, we calculate a convenient form for the kernel of a corner of the Fourier multiplier for use in Theorem 5.1.

Lemma 3.9. Let $F \subseteq G$ compact and $x_i = \Delta ^{a_i} \lambda (f_i) \Delta ^{b_i} \in L_{p_i}(\mathcal{L} G)$ as above for $f_i \in C_c(G) \star C_c(G)$ . Then the kernel of $P_F T_\phi (x_1, \dots, x_n) P_F$ is given by

\begin{align*} \begin{split} (t_0,t_n) \mapsto 1_F(t_0) 1_F(t_n) \int _{G^{\times n-1}} & \phi (t_0t_1^{-1},\ldots, t_{n-1}t_n^{-1}) f_1(t_0t_1^{-1})\ldots f_n(t_{n-1}t_n^{-1}) \times \\ & \Delta ^{a_1}(t_0) \Delta ^{b_1 + a_2}(t_1) \ldots \Delta ^{b_n}(t_n) \Delta ((t_1 \ldots t_n)^{-1}) dt_1\ldots dt_{n-1}. \end{split} \end{align*}

Proof. Let $g \in C_c(G) \star C_c(G)$ and $t_0 \in G$ . Then the function $P_F T_\phi ^{\theta _1, \dots, \theta _n}(x_1, \dots, x_n) P_F g$ is given by

\begin{align*} \begin{split} t_0 &\mapsto 1_F(t_0) \int _{G^{\times n}} \phi (t_1,\ldots, t_n) f_1(t_1)\ldots f_n(t_n) \times \\ & \qquad (\Delta ^{a_1} \lambda _{t_1} \Delta ^{b_1 + a_2} \lambda _{t_2} \ldots \Delta ^{b_{n-1} + a_n} \lambda _{t_n} \Delta ^{b_n} P_F g)(t_0) dt_1 \ldots dt_n \\ &= 1_F(t_0) \int _{G^{\times n}} \phi (t_1,\ldots, t_n) f_1(t_1)\ldots f_n(t_n) \Delta ^{a_1}(t_0) \Delta ^{b_1 + a_2}(t_1^{-1}t_0) \times \ldots \\ & \qquad \Delta ^{b_{n-1} + a_n}(t_{n-1}^{-1} \dots t_1^{-1} t_0) (\Delta ^{b_n}1_F g)(t_n^{-1} \dots t_1^{-1} t_0) dt_1 \ldots dt_n \\ &= 1_F(t_0) \int _{G^{\times n}} \phi (t_0t_1, t_2, \ldots, t_n) f_1(t_0t_1) f_2(t_2) \ldots f_n(t_n) \Delta ^{a_1}(t_0) \Delta ^{b_1 + a_2}(t_1^{-1}) \times \\ & \qquad \Delta ^{b_2 + a_3}(t_2^{-1} t_1^{-1} ) \ldots \Delta ^{b_{n-1} + a_n}(t_{n-1}^{-1} \dots t_1^{-1}) (\Delta ^{b_n}1_F g)(t_n^{-1} \dots t_1^{-1}) dt_1 \ldots dt_n \\ &= 1_F(t_0) \int _{G^{\times n}} \phi (t_0 t_1^{-1}, t_2,\ldots, t_n) f_1(t_0 t_1^{-1}) f_2(t_2)\ldots f_n(t_n) \Delta ^{a_1}(t_0) \Delta ^{b_1 + a_2}(t_1) \times \\ & \qquad \Delta ^{b_2 + a_3}(t_2^{-1} t_1) \ldots \Delta ^{b_{n-1} + a_n}(t_{n-1}^{-1} \dots t_2^{-1} t_1) (\Delta ^{b_n}1_F g)(t_n^{-1} \dots t_2^{-1} t_1) \Delta (t_1^{-1}) dt_1 \ldots dt_n \\ &= \ldots \\ &= 1_F(t_0) \int _{G^{\times n}} \phi (t_0 t_1^{-1},\ldots, t_{n-1} t_n^{-1}) f_1(t_0 t_1^{-1})\ldots f_n(t_{n-1} t_n^{-1}) \Delta ^{a_1}(t_0) \Delta ^{b_1 + a_2}(t_1) \times \ldots \\ & \qquad \Delta ^{b_{n-1} + a_n}(t_{n-1}) (\Delta ^{b_n}1_F g)(t_n) \Delta ((t_1 \ldots t_n)^{-1}) dt_1 \ldots dt_n. \\ \end{split} \end{align*}

It follows that the kernel has the required form.

4. Fourier to Schur transference

Let $G$ be a locally compact first countable group. In this section, we prove the transference from Fourier to Schur multipliers for such groups. An important ingredient will be the following ‘split’ coordinate-wise convolution: fix functions $\varphi _k \in C_c(G) \star C_c(G) \subseteq A(G)$ such that $\|\varphi _k\|_1 = 1$ and the supports of $\varphi _k$ form a decreasing neighbourhood basis of $\{e\}$ . In other words, $(\varphi _k)$ is an approximate unit for the Banach $*$ -algebra $L_1(G)$ . Note that we use the first countability of $G$ here. Now, given a bounded function $\phi \,:\, G^{\times n} \to{\mathrm{\mathbb{C}}}$ and some fixed $1 \leq i \leq n$ , we define

\begin{equation*} \phi _{t_1, \dots, t_n}(s_1, \dots, s_n) = \phi (t_1^{-1} s_1 t_2, \ldots, t_{i-1}^{-1} s_{i-1} t_i, t_i^{-1} s_i, s_{i+1} t_{i+1}^{-1}, t_{i+1} s_{i+2} t_{i+2}^{-1}, \dots, t_{n-1} s_n t_n^{-1}) \end{equation*}

and

(4.1) \begin{equation} \begin{split} &\phi _k(s_1, \dots, s_n) \,:\!=\, \int _{G^{\times n}} \phi _{t_1, \dots, t_n}(s_1, \dots, s_n) \left ( \prod _{j=1}^n \varphi _k(t_j)\right ) dt_1 \ldots dt_n \\ &\quad = \int _{G^{\times n}} \phi _{t_1, \dots, t_n}(e, \dots, e) \left ( \prod _{j=1}^i \varphi _k(s_j\ldots s_i t_j) \right ) \left ( \prod _{j=i+1}^n \varphi _k(t_js_{i+1} \ldots s_j) \Delta (s_{i+1} \ldots s_j) \right ) dt_1 \ldots dt_n. \end{split} \end{equation}

The last term of (4.1) in combination with Lemma 3.8 will allow us to reduce the problem to the linear case. The necessity of the ‘split’ between indices $i$ and $i+1$ will be explained later.

Theorem 4.1. Let $G$ be a locally compact first countable group, and let $1 \leq p \leq \infty$ , $1\lt p_1, \ldots, p_{n-1} \leq \infty$ be such that $p^{-1} = \sum _{i=1}^n p_i^{-1}$ . Let $\phi \in C_b(G^{\times n})$ and define $\widetilde{\phi } \in C_b(G^{\times n + 1})$ by

\begin{equation*} \widetilde {\phi }(s_0, \ldots, s_n) = \phi (s_0 s_1^{-1}, s_1 s_2^{-1}, \ldots, s_{n-1} s_n^{-1}), \qquad s_i \in G. \end{equation*}

If $\phi$ is defines a $(p_1, \ldots, p_n)$ -mb Fourier multiplier $T_\phi$ of $G$ , then $\widetilde{\phi }$ defines a $(p_1, \ldots, p_n)$ -mb Schur multiplier $M_{\widetilde{\phi }}$ of $G$ . Moreover,

\begin{align*} \begin{split} & \Vert M_{\widetilde{\phi }}\,:\, S_{p_1}(L_2(G)) \times \ldots \times S_{p_n}(L_2(G)) \rightarrow S_{p}(L_2(G)) \Vert _{(p_1,\ldots, p_n)-mb}\\ & \qquad \leq \Vert T_{\phi }\,:\, L_{p_1}(\mathcal{L}G ) \times \ldots \times L_{p_n}(\mathcal{L}G ) \rightarrow L_{p}(\mathcal{L}G ) \Vert _{(p_1,\ldots, p_n)-mb}. \end{split} \end{align*}

Proof. Let $F \subseteq G$ finite with $|F| = N$ . By Theorem 2.2, it suffices to show that the norm of

\begin{equation*} M_{\widetilde {\phi }}\,:\, S_{p_1}(\ell _2(F)) \times \ldots S_{p_n}(\ell _2(F)) \to S_p(\ell _2(F)) \end{equation*}

and its matrix amplifications are bounded.

For $s \in F$ , let $p_s \in B(\ell _2(F))$ be the orthogonal projection onto the span of $\delta _s$ . Define the unitary $U = \sum _{s \in F} p_s \otimes \lambda _s \in B(\ell _2(F)) \otimes \mathcal{L} G$ . In the case $p = \infty$ , the Fourier to Schur transference is proven through the transference identity

\begin{equation*} T_\phi ^{(N)}(U(a_1 \otimes 1)U^*, \ldots, U(a_n \otimes 1)U^*) = U (M_{\widetilde {\phi }}(a_1, \dots, a_n) \otimes 1)U^*, \qquad a_i \in B(\ell _2(F)). \end{equation*}

The idea is to do something similar in the case $p \lt \infty$ . However, the unit does not embed in $L_p(\mathcal{L} G)$ , so we need to use some approximation of the unit instead. We construct this as follows: let $\mathcal{V} = (V_i)_{i \in{\operatorname{\mathbb{N}}}}$ be a decreasing symmetric neighbourhood basis of the identity (this is possible because $G$ is first countable). For $V \in \mathcal{V}$ , we define the operator

\begin{equation*} k_V = \|1_V \Delta ^{-1/4}\|_2^{-1} \lambda (1_V \Delta ^{-1/4}) \Delta ^{1/2} \in L_2(\mathcal {L} G) \end{equation*}

which is proven to be self-adjoint in [Reference Caspers, Parcet, Perrin and Ricard8, Section 8.3]. Let $k_V = u_V h_V$ be its polar decomposition. Then we have $h_V^{2/p} \in L_p(\mathcal{L} G)$ , and by (2.2), $\|h_V^{2/p}\|_p = 1$ . Now for any $V \in \mathcal{V}$ , we have, by (2.7),

(4.2) \begin{equation} \|M_{\widetilde{\phi }}(a_1, \dots, a_n)\|_{S_p^N} = \|M_{\widetilde{\phi }}(a_1, \dots, a_n) \otimes h_V^{\frac 2p}\|_{S_p^N \otimes L_p(\mathcal{L} G)}, \qquad a_i \in B(\ell _2(F)). \end{equation}

Next, fix an $i \in \{1, \dots, n\}$ such that $\bar{p}_1 \,:\!=\, \left (\sum _{l=1}^i p_l^{-1}\right )^{-1} \gt 1$ and $\bar{p}_2 \,:\!=\, \left (\sum _{l=i+1}^n p_l^{-1}\right )^{-1} \gt 1$ . This is possible by our assumption that $p_1, \dots, p_n \gt 1$ . We now define the functions $\phi _{t_1, \dots, t_n}$ and $\phi _k$ as in (4.1) for the chosen $i$ .

The condition $\bar{p}_1, \bar{p}_2 \gt 1$ is necessary for the use of Proposition 4.3 at the end of the proof of Lemma 4.2; this also explains why we need the ‘split’ in the pointwise convolutions. If $p\gt 1$ , then one can take $i=n$ in which case the proof of Lemma 4.2 simplifies somewhat. Note that in [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5] and [Reference Caspers, Krishnaswamy-Usha and Vos6], the convolutions were defined for $i = n-1$ . In the latter paper, this creates a problem in case $p_n = \infty, p = 1$ ; this problem is resolved by splitting instead at some $i$ chosen as above.

Let $a_1, \ldots, a_n \in B(\ell _2(F))$ . By continuity of $\phi$ , we have that $\phi _k \to \phi$ pointwise. Indeed, for $\varepsilon \gt 0$ , we can take $K$ such that for $t_1, \dots, t_n \in \mathrm{supp} \varphi _K$ , $|\phi _{t_1, \dots, t_n}(s_1, \dots, s_n) - \phi (s_1, \dots, s_n)| \lt \varepsilon$ . Then for $k \gt K$ , we get $|\phi _k(s_1, \dots, s_n) - \phi (s_1, \dots, s_n)| \lt \varepsilon$ . Since we are working in finite dimensions, this implies

\begin{equation*} M_{\widetilde {\phi _k}}(a_1, \dots, a_n) \to M_{\widetilde {\phi }}(a_1, \dots, a_n) \end{equation*}

in $S^N_p$ . Together with (4.2), we find

\begin{align*} \begin{split} \|M_{\widetilde{\phi }}(a_1, \dots, a_n)\|_{S_p^N} &= \lim _k \limsup _{V \in \mathcal{V}} \|M_{\widetilde{\phi _k}}(a_1, \dots, a_n) \otimes h_V^{\frac 2p}\|_{S_p^N \otimes L_p(\mathcal{L} G)} \\ &= \lim _k \limsup _{V \in \mathcal{V}} \|U(M_{\widetilde{\phi _k}}(a_1, \dots, a_n) \otimes h_V^{\frac 2p})U^*\|_{S_p^N \otimes L_p(\mathcal{L} G)} \\ &\leq \limsup _k \limsup _{V \in \mathcal{V}} \|T_{\phi _k}^{(N)}(U(a_1 \otimes h_V^{\frac 2{p_1}})U^*, \ldots, U(a_n \otimes h_V^{\frac 2{p_n}})U^*)\|_{S_p^N \otimes L_p(\mathcal{L} G)} \\ &\quad + \limsup _k \limsup _{V \in \mathcal{V}} \|T_{\phi _k}^{(N)}(U(a_1 \otimes h_V^{\frac 2{p_1}})U^*, \ldots, U(a_n \otimes h_V^{\frac 2{p_n}})U^*) \\ &\qquad \qquad \qquad \qquad \qquad - U(M_{\widetilde{\phi _k}}(a_1, \dots, a_n) \otimes h_V^{\frac 2p})U^*\|_{S_p^N \otimes L_p(\mathcal{L} G)} \\ &\,:\!=\, A + B. \end{split} \end{align*}

First, we have

\begin{align*} \begin{split} A &\leq \limsup _k \limsup _{V \in \mathcal{V}} \|T_{\phi _k}^{(N)} \| \|a_1 \otimes h_V^{\frac 2{p_1}}\|_{S_{p_1}^N \otimes L_{p_1}(\mathcal{L} G)} \ldots \|a_n \otimes h_V^{\frac 2{p_n}}\|_{S_{p_n}^N \otimes L_{p_n}(\mathcal{L} G)} \\ &= \limsup _k \|T_{\phi _k}^{(N)}\| \|a_1\|_{S_{p_1}^N} \ldots \|a_n\|_{S_{p_n}^N}. \end{split} \end{align*}

By repeated use of Lemma 3.6 (in particular, we can use Fubini because of the strong continuity property) we find

\begin{equation*} \|T_{\phi _k}^{(N)}\| \leq \int _{G^{\times n}} \|T_\phi ^{(N)}\| \left ( \prod _{i=1}^n |\varphi _k(t_j)| \right ) dt_1 \ldots dt_n = \|T_\phi ^{(N)}\| \|\varphi _k\|_1^n = \|T_\phi ^{(N)}\| \leq \|T_\phi \|_{(p_1, \dots, p_n)-mb}. \end{equation*}

and hence

\begin{equation*} A \leq \|T_{\phi }\|_{(p_1, \dots, p_n)-mb} \|a_1\|_{S_{p_1}^N} \ldots \|a_n\|_{S_{p_n}^N}. \end{equation*}

It remains to show that $B = 0$ . By the triangle inequality, it suffices to show this for $a_i = E_{r_{i-1}, r_i}$ , $r_0, \dots, r_n \in F$ (for other combinations of matrix units, the term below becomes 0). In that case we get, by applying Lemma 3.6 in the second equality:

\begin{align*} \begin{split} & T_{\phi _k}^{(N)}(U(E_{r_0, r_1} \otimes h_V^{\frac 2{p_1}})U^*, \ldots, U(E_{r_{n-1},r_n} \otimes h_V^{\frac 2{p_n}})U^*) - U(M_{\widetilde{\phi _k}}(E_{r_0, r_1}, \dots, E_{r_{n-1},r_n}) \otimes h_V^{\frac 2p})U^*\\ &\quad= E_{r_0, r_n} \otimes \left ( T_{\phi _k}(\lambda _{r_0}h_V^{\frac 2{p_1}} \lambda _{r_1}^*, \ldots, \lambda _{r_{n-1}}h_V^{\frac 2{p_n}} \lambda _{r_n}^*) - \phi _k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) \lambda _{r_0} h_V^{\frac 2p}\lambda _{r_n}^* \right ) \\ &\quad = E_{r_0, r_n} \otimes \lambda _{r_0} \left ( T_{\phi _k(r_0 {\cdot} r_1^{-1}, \ldots, r_{n-1} {\cdot} r_n^{-1})}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - \phi _k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) h_V^{\frac 2p} \right ) \lambda _{r_n}^*. \end{split} \end{align*}

Hence,

(4.3) \begin{equation} B = \limsup _k \limsup _{V \in \mathcal{V}} \left \| T_{\phi _k(r_0{\cdot} r_1^{-1}, \ldots, r_{n-1} {\cdot} r_n^{-1})}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - \phi _k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) h_V^{\frac 2p} \right \|_{L_p(\mathcal{L} G)}. \end{equation}

The limit over $k$ exists and is 0; we postpone the proof to Lemma 4.2 below.

For the multiplicatively bounded estimate, we prove using similar methods that for $K \geq 1$ and $a_1, \dots, a_n \in M_K(B(\ell _2(F)))$ ,

\begin{equation*} \|M_{\widetilde {\phi }}^{(K)}(a_1, \dots, a_n)\|_{S_p^{KN}} \leq \|T_\phi ^{(KN)}\| \|a_1\|_{S_p^{KN}} \ldots \|a_n\|_{S_p^{KN}}. \end{equation*}

Here, we use $1_{M_K} \otimes U$ in place of $U$ . Moreover, by the triangle inequality, it suffices to prove the estimate for $B$ for $a_i = E_{j_{i-1},j_i} \otimes E_{r_{i-1},r_i}$ , with $1 \leq j_i \leq K$ and $r_i \in F$ ; the expression for $B$ then reduces to (4.3) again.

The following Lemma is similar to [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Lemma 4.6]. In our case, we have $x_j = 1$ , which allows us to avoid the SAIN condition used in that paper; on the other hand, we work with translated functions, and our result works for non-unimodular groups. Already in [Reference Caspers, Krishnaswamy-Usha and Vos6, Theorem 3.1], it was explained how to adapt the proof of [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Lemma 4.6] for the translated functions. However, this paper only considered unimodular groups. Here, we spell out the full proof for the convenience of the reader.

Lemma 4.2. In the proof of Theorem 4.1 , we have that

\begin{equation*} \lim _k \limsup _{V \in \mathcal {V}} \left \| T_{\phi _k(r_0 {\cdot} r_1^{-1}, \ldots, r_{n-1} {\cdot} r_n^{-1})}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - \phi _k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) h_V^{\frac 2p} \right \|_{L_p(\mathcal {L} G)} = 0. \end{equation*}

The main idea is to reduce the problem to the linear case using (4.1) and apply the following result for linear Fourier multipliers:

Proposition 4.3. Let $\mathcal{V}$ be a symmetric neighbourhood basis of the identity of $G$ . Let $2 \leq q \lt p \leq \infty$ or $1 \leq p \lt q \leq 2$ . Assume $\psi \in C_b(G)$ defines a Fourier multiplier on $L_p(\mathcal{L} G)$ . Then we have

\begin{equation*} \lim _{V \in \mathcal {V}} \|T_{\psi }(h_V^{2/q}) - \psi (1) h_V^{2/q}\|_{L_q(\mathcal {L} G)} \to 0. \end{equation*}

The proof of Proposition 4.3 is essentially a matter of combining results and remarks from [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Proposition 3.9] and [Reference Caspers, Parcet, Perrin and Ricard8, Claim B and Section 8] and applying Haagerup reduction to [Reference Conde-Alonso, Parcet and Ricard9, Lemma 3.1] to generalise that estimate to general von Neumann algebras. We give more details in Section 6.

Proof of Lemma 4.2. The idea is to use a dominated convergence argument in the last expression of (4.1). However, the functions $\phi _k$ need not be integrable. We work around this by multiplying with compactly supported functions that are close to 1 around $e$ so that as $V \in \mathcal{V}$ decreases to $\{e\}$ , we are just ‘multiplying by 1’ in the limit. Define a function $\zeta \in C_c(G) \cap A(G)$ with $\zeta (e) = 1$ , which is positive definite and (therefore) satisfies $\|T_{\zeta }\,:\, L_p(\mathcal{L} G) \to L_p(\mathcal{L} G)\| \leq 1$ for all $1 \leq p \leq \infty$ . Next let

\begin{equation*} \zeta _j(s) = \zeta (r_{j-1}^{-1} s r_j), \qquad 1 \leq j \leq n,\ s \in G. \end{equation*}

We define a product function as follows:

\begin{equation*} (\phi (\zeta _1, \dots, \zeta _n))(s_1, \dots, s_n) = \phi (s_1, \dots, s_n) \zeta _1(s_1) \ldots \zeta _n(s_n). \end{equation*}

Then

\begin{align*} &\| T_{\phi _k(r_0 {\cdot} r_1^{-1}, \ldots, r_{n-1} {\cdot} r_n^{-1})}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - \phi _k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) h_V^{\frac 2p} \|_{L_p(\mathcal{L} G)} \\ &\quad\leq \| (\phi (\zeta _1, \dots, \zeta _n))_k (r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) h_V^{\frac 2p}- \phi _k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) h_V^{\frac 2p} \|_{L_p(\mathcal{L} G)} \\ &\qquad + \| T_{(\phi (\zeta _1, \dots, \zeta _n))_k(r_0 {\cdot} r_1^{-1}, \ldots, r_{n-1} {\cdot} r_n^{-1})}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - (\phi (\zeta _1, \dots, \zeta _n))_k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) h_V^{\frac 2p} \|_{L_p(\mathcal{L} G)} \\ &\qquad + \|T_{\phi _k(r_0 {\cdot} r_1^{-1}, \ldots, r_{n-1} {\cdot} r_n^{-1})}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - T_{(\phi (\zeta _1, \dots, \zeta _n))_k(r_0 {\cdot} r_1^{-1}, \ldots, r_{n-1} {\cdot} r_n^{-1})}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) \|_{L_p(\mathcal{L} G)} \\&\quad \,=\!:\,\ A_{k,V} + B_{k,V} + C_{k,V}. \end{align*}

Here, $\phi ((\zeta _1, \dots, \zeta _n))_k$ is defined again by (4.1) for the same $i$ . We will estimate these terms separately. We start by showing that $\lim _k \limsup _{V \in \mathcal{V}} A_{k,V}$ and $\lim _k \limsup _{V \in \mathcal{V}} C_{k,V}$ are 0, essentially reducing the problem to the integrable functions $\phi (\zeta _1, \dots, \zeta _n)$ . We then apply the idea mentioned above to show that $\lim _{V \in \mathcal{V}} B_{k,V} = 0$ for any $k$ .

First, since $\psi _k \to \psi$ pointwise for any $\psi \in C_b(G)^{\times n}$ , we have

\begin{align*} \begin{split} \limsup _{V \in \mathcal{V}} A_{k,V} &= |(\phi (\zeta _1, \dots, \zeta _n))_k (r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1}) - \phi _k(r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1})|\\ &\to |\phi (r_0r_1^{-1}, \ldots, r_{n-1}r_n^{-1})(1 - \zeta _1(r_0r_1^{-1}) \ldots \zeta _n(r_{n-1} r_n^{-1}))| = 0. \end{split} \end{align*}

Next, we estimate the limit in $k$ of $\limsup _{V \in \mathcal{V}} C_{k,V}$ . Set $u_j = t_j^{-1} r_{j-1}$ , $v_j = t_j r_j$ and

\begin{equation*}C_V(t_1, \dots, t_n) = \|T_{\eta }(h_V^{\frac 2{p_1}}, \dots, h_V^{\frac 2{p_n}})\|_{L_p(\mathcal {L} G)},\end{equation*}

where

\begin{equation*} \eta = (\phi - \phi (\zeta _1, \dots, \zeta _n) )(u_1 \cdot u_2^{-1}, \ldots, u_{i-1} \cdot u_i^{-1}, u_i \cdot r_i^{-1}, r_i \cdot v_{i+1}^{-1}, v_{i+1} \cdot v_{i+2}^{-1}, \dots, v_{n-1} \cdot v_n^{-1}). \end{equation*}

Thanks to the strong continuity statement of Lemma 3.6, we can use Fubini to deduce

(4.4) \begin{equation} C_{k,V} \leq \int _{G^{\times n}} C_V(t_1, \dots, t_n) \left (\prod _{i=1}^n |\varphi _k(t_j)|\right ) dt_1 \ldots dt_n. \end{equation}

Now set

\begin{align*} \begin{split} &y_{j,V} = \lambda _{u_j}h_V^{\frac 2{p_j}} \lambda _{u_{j+1}}^*\ \text{for } 1 \leq j \leq i-1, \qquad y_{i,V} = \lambda _{u_i} h_V^{\frac 2{p_i}} \lambda _{r_i}^*, \qquad \quad y_{i+1,V} = \lambda _{r_i} h_V^{\frac 2{p_{i+1}}} \lambda _{v_{i+1}}^*, \\ &y_{j,V} = \lambda _{v_{j-1}}h_V^{\frac 2{p_j}} \lambda _{v_j}^* \quad \text{for } i+2 \leq j \leq n. \end{split} \end{align*}

Denote $\iota _q$ for the identity operator on $L_q(\mathcal{L} G)$ . The symbol $1$ is used both for the constant 1-function and the number 1. Then we get the following estimate, where we apply Lemma 3.6 and (2.1) in the first line and Lemma 3.7 and the assumption that $T_\zeta$ is a contraction in the third line:

(4.5) \begin{equation} \begin{split} C_V(t_1, \dots, t_n) &= \|T_{(\phi - \phi (\zeta _1, \dots, \zeta _n))}(y_{1,V}, \dots, y_{n,V})\|_{L_p(\mathcal{L} G)}\\ &\leq \sum _{j=1}^n \|T_{\phi (1, \dots, 1, (\zeta _j - 1), \zeta _{j+1}, \dots, \zeta _n)}(y_{1,V}, \dots, y_{n,V}) \|_{L_p(\mathcal{L} G)} \\ &\leq \|T_{\phi }\,:\, L_{p_1} \times \ldots \times L_{p_n} \to L_p\| \sum _{j=1}^n\left ( \|(T_{\zeta _j} - \iota _{p_j})(y_{j,V})\|_{L_{p_j}(\mathcal{L} G)} \prod _{i \neq j} \|y_{i,V}\|_{p_i}\right ). \end{split} \end{equation}

By (2.1), we have $\|y_{j,V}\|_{p_j} = 1$ . Further, by applying again Lemma 3.6 and Proposition 4.3,

\begin{equation*} \|(T_{\zeta _j} - \iota _{p_j})(y_{j,V})\|_{L_{p_j}(\mathcal{L} G)} = \|T_{\zeta _j(u_j{\cdot} u_{j+1}^{-1}) - 1}(h_V^{\frac 2{p_j}})\|_{L_{p_j}(\mathcal{L} G)} \to |\zeta _j(u_j u_{j+1}^{-1}) - 1| \end{equation*}

for $1 \leq j \leq i-1$ . Filling in the definition of $\zeta _j$ ,

\begin{equation*} |\zeta _j(u_j u_{j+1}^{-1}) - 1| = |\zeta (r_{j-1}^{-1} t_j^{-1} r_{j-1} r_j^{-1} t_{j+1} r_j) - 1| \end{equation*}

and this equals 0 when evaluated at $t_j, t_{j+1} = e$ . Similarly, we find for $i \leq j \leq n$ that $\lim _{V \in \mathcal{V}} \|(T_{\zeta _j} - \iota )(y_{j,V})\|_{L_{p_j}(\mathcal{L} G)}$ exists and equals 0 when evaluated at the identity in the corresponding $t_1, \dots, t_n$ . Moreover, all these values are bounded by 2. Going back to (4.4), let us write $M\,:\!=\,\|T_{\phi }\,:\, L_{p_1} \times \ldots \times L_{p_n} \to L_p\|$ . We find

(4.6) \begin{equation} \begin{split} C_{k,V} \leq & \int _{G^{\times n}} M \left (\sum _{j=1}^n \|(T_{\zeta _j} - \iota )(y_{j,V})\|_{L_{p_j}(\mathcal{L} G)} \right ) \left (\prod _{j=1}^n |\varphi _k(t_j)| \right ) dt_1 \ldots dt_n. \end{split} \end{equation}

The integrand of (4.6) is bounded by the integrable function $2M\prod _{i=1}^n |\varphi _k(t_j)|$ . Hence, by Lebesgue’s dominated convergence theorem, the right-hand side of (4.6) converges in $V$ . We find that

\begin{align*} \begin{split} &\limsup _{V \in \mathcal{V}} C_{k,V} \leq M \int _{G^{\times n}} \left ( \sum _{j=1}^n \lim _{V \in \mathcal{V}} \|(T_{\zeta _j} - \iota )(y_{j,V})\|_{L_{p_j}(\mathcal{L} G)} \right )\left (\prod _{i=1}^n |\varphi _k(t_j)|\right ) dt_1 \dots dt_n. \end{split} \end{align*}

This quantity goes to 0 in $k$ . This concludes the proof for $C_{k,V}$ .

Finally, we prove that $\lim _{V \in \mathcal{V}} B_{k,V} = 0$ for any $k$ . We fix a $k$ for the remainder of the proof. Recall that since $\varphi _k \in A(G)$ , $T_{\varphi _k}$ is bounded on $L_q(\mathcal{L} G)$ for any $1 \leq q \leq \infty$ . Moreover, since $\varphi _k \in C_c(G) \star C_c(G)$ , we also have $\varphi _k \Delta \in C_c(G) \star C_c(G) \subseteq A(G)$ (cf. the calculation before (2.6)), hence $T_{\varphi _k \Delta }$ is also bounded on $L_q(\mathcal{L} G)$ for any $q$ .

We may assume, by scaling $\varphi _k$ if necessary, that $T_{\varphi _k}\,:\, L_q(\mathcal{L} G) \to L_q(\mathcal{L} G)$ and $T_{\varphi _k \Delta }\,:\, L_q(\mathcal{L} G) \to L_q(\mathcal{L} G)$ are contractions for any Hölder combination $q$ of $p_1, \dots, p_n$ . Of course, this means that $\|\varphi _k\|_1$ need no longer be $1$ from now on. Set

\begin{align*} \begin{split} \psi _k(s_1, \dots, s_n; t_1, \dots, t_n) &\,:\!=\, \left ( \prod _{j=1}^i \varphi _k(r_{j-1} s_j\ldots s_{i} r_{i}^{-1} t_j)\right )\\ & \quad \times \left ( \prod _{j=i+1}^n \varphi _k(t_j r_{i} s_{i+1} \ldots s_j r_j^{-1}) \Delta (r_{i} s_{i+1} \ldots s_j r_j^{-1})\right ) \\ &\,=\!:\, \psi _k^1(s_1, \dots, s_i; t_1, \dots, t_i) \psi _k^2(s_{i+1}, \dots, s_n; t_{i+1}, \dots, t_n). \end{split} \end{align*}

By using the last term of (4.1) and Fubini, we get

(4.7) \begin{equation} \begin{split} B_{k,V} &\leq \int _{G^{\times n}} |(\phi (\zeta _1, \dots, \zeta _n))_{t_1, \dots, t_n}(e, \dots, e)| \\ & \qquad \times \| T_{\psi _k({\cdot}; t_1, \dots, t_n)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - \psi _k(1, \ldots, 1; t_1, \dots t_n) h_V^{\frac 2p} \|_{L_p(\mathcal{L} G)} dt_1 \ldots dt_n.\\ \end{split} \end{equation}

Note that $|\varphi _k| \leq 1$ by the assumed contractivity of $T_{\varphi _k}$ . Indeed, for $s \in G$ , apply $T_{\varphi _k}$ to $\lambda _s$ to deduce that $|\varphi _k(s)| \leq 1$ . Hence, $|\psi _k(1, \dots, 1; t_1, \dots, t_n)| \leq \prod _{j=i+1}^n \Delta (r_ir_j^{-1})$ . Moreover, from the expression (4.9) below, we see that $\|T_{\psi _k({\cdot}; t_1, \dots, t_n)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}})\|_{L_p(\mathcal{L} G)} \leq \Delta ((t_{i+1}, \dots, t_n)^{-1})$ . Since $\phi (\zeta _1, \dots, \zeta _n)$ is compactly supported, the integrand of (4.7) is dominated by an integrable function. Hence by the Lebesgue dominated convergence theorem, it suffices to show that the term

(4.8) \begin{equation} \| T_{\psi _k({\cdot}; t_1, \dots, t_n)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - \psi _k(1, \ldots, 1; t_1, \dots t_n) h_V^{\frac 2p} \|_{L_p(\mathcal{L} G)} \end{equation}

goes to 0 in $V$ for any choice of $t_1, \dots, t_n \in G$ .

Fix $t_1, \dots, t_n \in G$ . For $1 \leq j \leq i$ , set $q_j^{-1} = \sum _{l=j}^{i} p_l^{-1}$ (so $q_1 = \bar{p}_1$ ) and $T_j = T_{\varphi _k(r_{j-1} {\cdot} r_{i}^{-1} t_j)}$ . By Lemma 3.6, $T_j$ is a contraction on $L_{q_j}(\mathcal{L} G)$ . For $i+1 \leq j \leq n$ , set $q_j^{-1} = \sum _{l=j}^n p_l^{-1}$ (so $q_{i+1} = \bar{p}_2$ ) and $T_j = T_{\varphi _k(t_j r_i{\cdot} r_j^{-1}) \Delta (r_i {\cdot} r_j^{-1})}$ . We can estimate the norm of $T_j: L_{q_j}(\mathcal{L} G) \to L_{q_j}(\mathcal{L} G)$ by using again Lemma 3.6:

\begin{equation*} \|T_j\| = \Delta (t_j^{-1}) \|T_{\varphi _k(t_j r_i{\cdot} r_j^{-1}) \Delta (t_j r_i{\cdot} r_j^{-1})}\| = \Delta (t_j^{-1}) \|T_{\varphi _k \Delta }\| \leq \Delta (t_j^{-1}). \end{equation*}

Now, by Lemma 3.8, we have

\begin{align*} \begin{split} T_{\psi _k^1(\cdot ; t_1, \dots, t_i)}(h_V^{\frac 2{p_1}}, \dots, h_V^{\frac 2{p_i}}) &= T_1(h_V^{\frac 2{p_1}} T_2(h_V^{\frac 2{p_2}} \dots T_i(h_V^{\frac 2{p_i}}) \dots )), \\ T_{\psi _k^2(\cdot ; t_{i+1}, \dots, t_n)}(h_V^{\frac 2{p_{i+1}}}, \dots, h_V^{\frac 2{p_n}}) &= T_{i+1}(h_V^{\frac 2{p_{i+1}}} T_{i+2}(h_V^{\frac 2{p_{i+2}}} \dots T_n(h_V^{\frac 2{p_n}}) \dots )). \end{split} \end{align*}

Clearly, $T_{\psi _k^1(\cdot ; t_1, \dots, t_i)}$ is contractive as a map on $L_{p_1}(\mathcal{L} G) \times \ldots \times L_{p_i}(\mathcal{L} G)$ . Let $x_j \in L_{p_j}(\mathcal{L} G)$ with $\|x_j\|_{L_{p_j}(\mathcal{L} G)} \leq 1$ ; then, from (3.4),

(4.9) \begin{equation} \begin{split} \|T_{\psi _k({\cdot} ; t_1, \dots, t_n)}(x_1, \dots, x_n)\|_{L_p(\mathcal{L} G)} &\leq \|T_{\psi ^2_k({\cdot} ; t_1, \dots, t_n)}(x_{i+1}, \dots, x_n)\|_{L_{\bar{p}_2}(\mathcal{L} G)} \\ &\leq \Delta ((t_{i+1} \ldots t_n)^{-1}). \end{split} \end{equation}

This validates the use of the dominated convergence theorem above. Now we go back to estimating (4.8). Using subsequently the triangle inequality and Hölder’s inequality (with again [3.4]), we find

\begin{align*} \begin{split} &\| T_{\psi _k({\cdot}; t_1, \dots, t_n)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - \psi _k(1, \ldots, 1; t_1, \dots t_n) h_V^{\frac 2p} \|_{L_p(\mathcal{L} G)} \\ &\quad\leq \|T_{\psi ^1_k({\cdot}; t_1, \dots, t_i)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_i}}) \cdot \psi _k^2(1, \dots, 1; t_{i+1}, \dots, t_n) h_V^{\frac 2{\bar{p}_2}} - \psi _k(1, \ldots, 1; t_1, \dots t_n) h_V^{\frac 2p} \|_{L_p(\mathcal{L} G)} \\ &\qquad + \| T_{\psi _k({\cdot}; t_1, \dots, t_n)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_n}}) - T_{\psi ^1_k({\cdot}; t_1, \dots, t_i)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_i}}) \cdot \psi _k^2(1, \dots, 1; t_{i+1}, \dots, t_n) h_V^{\frac 2{\bar{p}_2}}\|_{L_p(\mathcal{L} G)}\\ &\quad\leq \left (\prod _{j=i+1}^n \Delta (r_ir_j^{-1})\right ) \|T_{\psi ^1_k({\cdot}; t_1, \dots, t_i)}(h_V^{\frac 2{p_1}}, \ldots, h_V^{\frac 2{p_i}}) - \psi _k^1(1, \dots, 1; t_1, \dots, t_i) h_V^{\frac 2{\bar{p}_1}}\|_{L_{\bar{p}_1}(\mathcal{L} G)} \\ &\qquad + \|T_{\psi _k^2(\cdot ; t_{i+1}, \dots, t_n)}(h_V^{\frac 2{p_{i+1}}}, \dots, h_V^{\frac 2{p_n}}) - \psi _k^2(1, \dots, 1; t_{i+1}, \dots, t_n) h_V^{\frac 2{\bar{p}_2}} \|_{L_{\bar{p}_2}(\mathcal{L} G)} \\ &\quad \,=\!:\, B_{k,V}^1 + B_{k,V}^2. \end{split} \end{align*}

We show only that $\lim _{V \in \mathcal{V}} B_{k,V}^2 = 0$ ; the equality $\lim _{V \in \mathcal{V}} B_{k,V}^1 = 0$ follows similarly and is in fact slightly easier since the $T_j$ are contractions for $j \leq i$ . Now set, for $i \leq j \leq n$ ,

\begin{equation*} R_{j,V} \,:\!=\, \left (\prod _{l=j+1}^{n} \varphi _k(t_l r_i r_l^{-1}) \right ) T_{i+1}(h_V^{\frac 2{p_{i+1}}} \ldots T_j( h_V^{\frac 2{q_j}}) \ldots ). \end{equation*}

Here $R_{i,V} = \prod _{l=i+1}^{n-1} \varphi _k(t_l r_i r_l^{-1}) h_V^{\frac 2{q_1}}$ . Then

\begin{equation*} B_{k,V}^2 \leq \sum _{j=i+1}^n \|R_{j,V} - R_{j-1, V}\|_{L_{\bar {p}_2}(\mathcal {L} G)}. \end{equation*}

Recall that $|\varphi _k| \leq 1$ . Hence

\begin{align*} \begin{split} &\|R_{j,V} - R_{j-1,V}\|_{L_{\bar{p}_2}(\mathcal{L} G)} \\ &\quad= \left (\prod _{l=j+1}^n |\varphi _k(t_l r_i r_l^{-1})|\right ) \|T_{i+1}(h_V^{\frac 2{p_{i+1}}} \ldots T_{j-1}(h_V^{\frac 2{p_{j-1}}}(T_j(h_V^{\frac 2{q_j}}) - \varphi _k(t_j r_i r_j^{-1}) h_V^{\frac 2{q_j}})) \ldots )\|_{L_{\bar{p}_2}(\mathcal{L} G)} \\ &\quad\leq \Delta ((t_{i+1} \ldots t_n)^{-1}) \|T_j(h_V^{\frac 2{q_j}}) - \varphi _k(t_j r_i r_j^{-1}) h_V^{\frac 2{q_j}}\|_{L_{q_j}(\mathcal{L} G)}. \end{split} \end{align*}

We know that $q_j \gt \bar{p}_2 \gt 1$ for any $i+1 \leq j \leq n$ . Additionally, $T_j$ is bounded on $\mathcal{L} G$ and $L_1(\mathcal{L} G)$ . By Proposition 4.3, the above terms converge to 0 in $V$ . Hence, $\lim _{V \in \mathcal{V}} B_{k,V}^2 = 0$ . This finishes the proof.

5. Schur to Fourier transference for amenable groups

In this section, we extend [Reference Caspers, Krishnaswamy-Usha and Vos6, Proof of Theorem 4.1], that is, the transference from Schur multipliers to Fourier multipliers for amenable groups, to the non-unimodular setting. The proof is essentially the same but with extra technicalities due to the modular function. We also fixed a small mistake in the proof, as will be mentioned at the relevant spot.

Recall [Reference Paterson19, Theorem 4.10] that $G$ is amenable iff it satisfies the following Følner condition: for any $\varepsilon \gt 0$ and any compact set $K\subseteq G$ , there exists a compact set $F \subseteq G$ with nonzero measure such that $\frac{\mu ((sF \setminus F) \cup (F \setminus sF))}{\mu (F)} \lt \varepsilon$ for all $s\in K$ . This allows us to construct a net $F_{(\varepsilon, K)}$ of such Følner sets using the ordering $( \varepsilon _1, K_1) \leq (\varepsilon _2,K_2)$ if $\varepsilon _1 \geq \varepsilon _2, K_1 \subseteq K_2$ .

Theorem 5.1. Let $G$ be a locally compact, amenable group and let $1\leq p,p^{\prime},p_1,\ldots, p_n \leq \infty$ be such that $\frac 1p=\sum _{i=1}^n \frac 1{p_i} = 1 - \frac 1{p^{\prime}}$ . Let $\phi \in L_\infty (G^{\times n})$ and define $\widetilde{\phi } \in L_\infty (G^{\times n + 1})$ by

\begin{equation*} \widetilde {\phi }(s_0, \ldots, s_n) = \phi (s_0 s_1^{-1}, s_1 s_2^{-1}, \ldots, s_{n-1} s_n^{-1}), \qquad s_i \in G. \end{equation*}

Assume that $\widetilde{\phi }$ defines a $(p_1,\ldots, p_n)$ -Schur multiplier of $G$ . Then there is a net $I$ , and there are complete contractions $i_{q,\alpha }\,:\, L_q(\mathcal{L} G) \to S_q(L_2(G))$ , $\alpha \in I$ , such that for all $f_i,f \in C_c(G)\star C_c(G)$ ,

(5.1) \begin{equation} \left |\langle i_{p,\alpha }(T_\phi (x_1,\ldots, x_n)),i_{p^{\prime},\alpha }(y) \rangle - \langle M_{\widetilde{\phi }}( i_{p_1,\alpha }(x_1),\ldots, i_{p_n,\alpha }(x_n) ), i_{p^{\prime},\alpha }(y) \rangle \right | \stackrel{\alpha }\to 0, \end{equation}

where $x_i = \Delta ^{a_i}\lambda (f_i)\Delta ^{b_i} \in L^{p_i}(\mathcal{L} G),\ y = \Delta ^a \lambda (f) \Delta ^b \in L^{p^{\prime}}(\mathcal{L} G)$ (i.e. $a_i + b_i = \frac 1{p_i}$ ). In a similar way, the matrix amplifications of $i_{q,\alpha }$ approximately intertwine the multiplicative amplifications of the Fourier and Schur multipliers.

Proof. Let $F_\alpha, \alpha \in I$ be a Følner net for $G$ , where $I$ is the index set consisting of pairs $(\varepsilon, K)$ for $\varepsilon \gt 0$ , $K \subseteq G$ compact and the ordering as described above.

Let $P_\alpha =P_{F_\alpha }$ be the projection of $L_2(G)$ onto $L_2(F_\alpha )$ . Consider the maps $i_{p,\alpha }\,:\, L_p(\mathcal{L} G)\to S_p(L_2(G))$ defined by $i_{p,\alpha }(x) = \mu (F_\alpha )^{-1/p} P_{\alpha }x P_{\alpha }$ . They are contractions by [Reference Caspers and de la Salle3, Theorem5.1]. By replacing $G$ by $G \times SU(2)$ , one proves that they are in fact complete contractions (see also the last paragraph of [Reference Caspers and de la Salle3, Proof of Theorem 5.2]).

Now fix $\alpha$ . From (2.10), we deduce

\begin{align*} M_{\widetilde{\phi }} (i_{p_1, \alpha }(x_1),\ldots, i_{p_n, \alpha }(x_n)) (t_0,t_{n}) \\ = \frac{1}{\mu (F_\alpha )^{1/p}} 1_{F_\alpha }(t_0)1_{F_\alpha }(t_n) \int _{F_\alpha ^{\times n-1}} &\phi (t_0t_1^{-1},\ldots, t_{n-1}t_n^{-1}) f_1(t_0t_1^{-1})\ldots f_n(t_{n-1}t_n^{-1}) \times \\ & \Delta ^{a_1}(t_0) \Delta ^{b_1 + a_2}(t_1) \ldots \Delta ^{b_n}(t_n) \Delta ((t_1 \ldots t_n)^{-1}) dt_1 \ldots dt_{n-1}. \end{align*}

From Lemma 3.9, we have a similar expression for the kernel of $i_{p,\alpha }(T_\phi (x_1, \dots, x_n))$ :

\begin{align*} \begin{split} (t_0,t_n) \mapsto \frac 1{\mu (F_{\alpha })^{1/p}} 1_{F_{\alpha }}(t_0) 1_{F_{\alpha }}(t_n) & \int _{G^{\times n-1}} \phi (t_0t_1^{-1},\ldots, t_{n-1}t_n^{-1}) f_1(t_0t_1^{-1})\ldots f_n(t_{n-1}t_n^{-1}) \times \\ & \Delta ^{a_1}(t_0) \Delta ^{b_1 + a_2}(t_1) \ldots \Delta ^{b_n}(t_n) \Delta ((t_1 \ldots t_n)^{-1}) dt_1\ldots dt_{n-1}. \end{split} \end{align*}

Now we need to take the pairing of these kernels with $i_{p^{\prime},\alpha }(y)$ and calculate their difference. To that end, we define the following function $\Phi$ :

\begin{align*} \begin{split} \Phi (t_0,\ldots, t_n) = &\phi (t_0t_1^{-1},\ldots, t_{n-1}t_n^{-1}) f_1(t_0t_1^{-1})\ldots f_n(t_{n-1}t_n^{-1}) f(t_n t_0^{-1}) \times \\ & \Delta ^{a_1+b}(t_0) \Delta ^{b_1 + a_2}(t_1) \ldots \Delta ^{b_n+a}(t_n) \Delta ((t_0t_1 \ldots t_n)^{-1}), \end{split} \end{align*}

and the function $\Psi _{\alpha }$ :

\begin{equation*} \Psi _\alpha (t_0,\ldots, t_n) = 1_{F_\alpha }(t_0) 1_{F_\alpha } (t_n) - 1_{F_\alpha ^{\times n+1}}(t_0,\ldots, t_n) = 1_{F_\alpha \times (F_{\alpha }^{\times n-1})^c \times F_{\alpha }}(t_0, \dots, t_n). \end{equation*}

Note that in [Reference Caspers, Krishnaswamy-Usha and Vos6], the indicator function was mistakenly taken over $F_\alpha \times (F_\alpha ^c)^{\times n-1} \times F_{\alpha }$ instead. This correction leads to an extra term $n$ in the choice of the lower bound of $\alpha$ at the end. Also note that a priori, it is not clear that $T_\phi (x_1, \dots, x_n)$ lies in $L_p(\mathcal{L} G)$ , and hence it is not clear that $i_{p,\alpha }(T_\phi (x_1, \dots, x_n))$ lies in $S_p(L_2(G))$ . However, both $i_{p,\alpha }(T_\phi (x_1, \dots, x_n))$ and $i_{p^{\prime},\alpha }(y)$ are given by integration against a kernel in $L_2(G \times G)$ , so the pairing (2.9) is still well-defined as a pairing in $S_2(L_2(G))$ instead. Now we have

(5.2) \begin{equation} \begin{split} &\vert \langle i_{p,\alpha }(T_\phi (x_1,\ldots, x_n)),i_{p^{\prime},\alpha }(y) \rangle - \langle M_{\widetilde{\phi }}( i_{p_1,\alpha }(x_1),\ldots, i_{p_n,\alpha }(x_n) ), i_{p^{\prime},\alpha }(y) \rangle \vert \\ &\quad= \left \vert \frac{1}{\mu (F_\alpha )} \int _{G^{\times n+1}} \Phi (t_0,\ldots, t_n) \Psi _\alpha (t_0,\ldots, t_n) dt_0 \ldots dt_n \right \vert \\ \end{split} \end{equation}

Let $K \subseteq G$ be some compact set such that $\mathrm{supp}(f_j),\mathrm{supp}(f)\subseteq K$ and $e \in K$ . Let $t_0, \dots, t_n$ be such that both $\Phi (t_0, \dots, t_n)$ and $\Psi _\alpha (t_0, \dots, t_n)$ are nonzero. Since $\Psi _\alpha (t_0, \dots, t_n)$ is nonzero, we must have $t_0, t_n \in F_\alpha$ and $t_i \notin F_\alpha$ for some $i \in \{1, \dots, n-1\}$ . Since $\Phi (t_0, \dots, t_n)$ is nonzero, there are $k_1,\ldots, k_n \in K$ such that $t_{n-1} = k_n t_n,\ t_{n-2} = k_{n-1}k_n t_n,\ \ldots, \ t_0 = k_1 \ldots k_n t_n$ . Hence we find

(5.3) \begin{equation} \begin{split} t_n &\in F_\alpha \cap (k_1\ldots k_n)^{-1}F_\alpha \setminus \left ( (k_2\ldots k_n)^{-1}F_\alpha \cap \ldots \cap k_n^{-1}F_\alpha \right ) \\ &\subseteq F_\alpha \setminus \left ( (k_2\ldots k_n)^{-1}F_\alpha \cap \ldots \cap k_n^{-1}F_\alpha \right ) \\ &= (F_\alpha \setminus (k_2\ldots k_n)^{-1}F_\alpha ) \cup \ldots \cup (F_\alpha \setminus k_n^{-1} F_\alpha ) \end{split} \end{equation}

We want to apply a change of variables in (5.2). Let us first look at a simple case: assume $g \in L_1(G \times G)$ is such that $g(s,t) \neq 0$ only when $st^{-1} \in K$ . Then

\begin{align*} \begin{split} \int _{G^{\times 2}} g(s,t) ds dt &= \int _{G^{\times 2}} 1_K(s t^{-1}) g(s, t) ds dt = \int _{G^{\times 2}} 1_K(s) g(s t, t) \Delta (t) ds dt \\ &= \int _G \int _K g(k_1 t, t) \Delta (t) dk_1 dt \end{split} \end{align*}

where we renamed the variable $s$ in the last line. Applying the above formula twice for a function $g \in L_1(G^{\times 3})$ such that $g(r,s,t) \neq 0$ only when $rs^{-1} \in K$ , $st^{-1} \in K$ , we get

\begin{align*} \begin{split} \int _{G^{\times 3}} g(r,s,t) dr ds dt &= \int _{G^{\times 2}} \int _K g(k_1 s, s, t) \Delta (s) dk_1 ds dt \\ &= \int _G \int _{K^{\times 2}} g(k_1k_2 t, k_2t, t) \Delta (k_2t) \Delta (t) dk_1 dk_2 dt. \end{split} \end{align*}

Carrying on like this, we obtain

(5.4) \begin{equation} \begin{split} &\left \vert \frac{1}{\mu (F_\alpha )} \int _{G^{\times n+1}} \Phi (t_0,\ldots, t_n) \Psi _\alpha (t_0,\ldots, t_n) dt_0 \ldots dt_n \right \vert \\ &\quad= \bigg \vert \frac{1}{\mu (F_\alpha )} \int _{K^{\times n}} \int _{G} \Phi (k_1\ldots k_n t_n, \ldots, k_n t_n, t_n) \Psi _\alpha (k_1\ldots k_nt_n,\ldots, k_nt_n, t_n) \times \\ & \qquad \Delta (k_2 \dots k_n t_n) \dots \Delta (k_nt_n) \Delta (t_n) dt_n dk_1 \ldots dk_n \bigg \vert . \\ \end{split} \end{equation}

Note that $a + b + \sum _{i=1}^n a_i + b_i = 1$ , hence

(5.5) \begin{equation} \begin{split} &|\Phi (k_1\ldots k_n t_n, \ldots, k_n t_n, t_n)|\Delta (k_2 \dots k_n t_n) \dots \Delta (k_nt_n) \Delta (t_n) \\ &\quad\leq \|\phi f_1 \dots f_n f\|_{\infty } \Delta ^{a_1+b}(k_1\ldots k_n t_n) \Delta ^{b_1+a_2+1}(k_2 \ldots k_n t_n) \ldots \Delta ^{b_{n-1} + a_n+1}(k_nt_n) \times \\ & \qquad \Delta ^{b_n + a+1}(t_n)\Delta (k_1^{-1} k_2^{-2} \dots k_n^{-n} t_n^{-n-1}) \\ &\quad= \|\phi f_1 \dots f_n f\|_{\infty } \Delta ^{a_1+b-1}(k_1) \Delta ^{a_1+a_2+b_1+b-1}(k_2) \ldots \Delta ^{1-b_n-a-1}(k_n) \\ &\quad\leq \|\phi f_1 \dots f_n f\|_{\infty } C_{K,n} \,=\!:\, M. \end{split} \end{equation}

Here the constant $C_{K,n}$ can be chosen to be dependent only on $K$ and $n$ (and $G$ ).

Applying (5.2), (5.4), (5.5) and (5.3) consecutively, we get

(5.6) \begin{equation} \begin{split} &\vert \langle i_{p,\alpha }(T_\phi (x_1,\ldots, x_n)),i_{p^{\prime},\alpha }(y) \rangle _{p,p^{\prime}} - \langle M_{\widetilde{\phi }}( i_{p_1,\alpha }(x_1),\ldots, i_{p_n,\alpha }(x_n) ), i_{p^{\prime},\alpha }(y) \rangle \vert \\ &\quad= \bigg \vert \frac{1}{\mu (F_\alpha )} \int _{K^{\times n}} \int _{G} \Phi (k_1\ldots k_n t_n, \ldots, k_n t_n, t_n) \Psi _\alpha (k_1\ldots k_nt_n,\ldots, k_nt_n, t_n) \times \\ & \qquad \Delta (k_2 \dots k_n t_n) \dots \Delta (k_nt_n) \Delta (t_n) dt_n dk_1 \ldots dk_n \bigg \vert \\ &\quad\leq \frac{M}{\mu (F_\alpha )} \int _{K^{\times n}} \int _G \Psi _\alpha (k_1\ldots k_nt_n,\ldots, k_nt_n, t_n) dt_n dk_1 \ldots dk_n \\ &\quad= \frac{M}{\mu (F_\alpha )} \int _{K^{\times n}} \mu \left (F_\alpha \cap (k_1\ldots k_n)^{-1}F_\alpha \setminus \left ( (k_2\ldots k_n)^{-1}F_\alpha \cap \ldots \cap k_n^{-1}F_\alpha \right )\right ) dk_1 \ldots dk_n\\ &\quad\leq \frac{M}{\mu (F_\alpha )} \int _{K^{\times n}} \sum _{i=2}^n \mu (F_\alpha \setminus (k_i \ldots k_n)^{-1}F_\alpha ) dk_1 \ldots dk_n\\ & \quad\leq M (n-1) \mu (K)^n \sup _{k \in K^{1-n}} \frac{\mu (F_{\alpha } \setminus kF_{\alpha })}{\mu (F_\alpha )}. \end{split} \end{equation}

Using the ordering described earlier, if the index $\alpha \geq (\varepsilon \times \left (Mn \mu _G(K)^n\right )^{-1}, K^{1-n})$ , then the Følner condition implies that (5.6) is less than $\varepsilon$ , and hence the limit (5.1) holds.

From (5.1), it follows from writing out the definitions that the matrix amplifications of $i_{p,\alpha }$ also approximately intertwine the multiplicative amplifications of the Fourier and Schur multipliers; that is, for $\beta _i \in S_{p_i}^N, \beta \in S_{p^{\prime}}^{N}$ , we have

(5.7) \begin{equation} \begin{split} &\Big |\langle{\mathrm{id}} \otimes i_{p,\alpha }(T^{(N)}_\phi (\beta _1\otimes x_1,\ldots, \beta _n \otimes x_n)),{\mathrm{id}}\otimes i_{p^{\prime},\alpha }(\beta \otimes y) \rangle _{p,p^{\prime}} \\ &\quad - \langle M^{(N)}_{\widetilde{\phi }}({\mathrm{id}}\otimes i_{p_1,\alpha }(\beta _1 \otimes x_1),\ldots, {\mathrm{id}}\otimes i_{p_n,\alpha }(\beta _n \otimes x_n) ), {\mathrm{id}}\otimes i_{p^{\prime},\alpha }(\beta \otimes y) \rangle \Big | \to 0 \end{split} \end{equation}

Corollary 5.2. Let $G$ be an amenable locally compact group and $1 \leq p, p_1, \dots, p_n \leq \infty$ be such that $\frac 1p=\sum _{i=1}^n \frac 1{p_i}$ . Let $\phi \in L_\infty (G^{\times n})$ . If $\widetilde{\phi }$ is the symbol of a $(p_1, \dots, p_n)$ -bounded (resp. multiplicatively bounded) Schur multiplier then $\phi$ is the symbol of a $(p_1, \dots, p_n)$ -bounded (resp. multiplicatively bounded) Fourier multiplier. Moreover,

\begin{equation*} \|T_\phi \|_{(p_1, \dots, p_n)} \leq \|M_{\widetilde {\phi }}\|_{(p_1, \dots, p_n)}, \qquad \|T_\phi \|_{(p_1, \dots, p_n)-mb} \leq \|M_{\widetilde {\phi }}\|_{(p_1, \dots, p_n)-mb}. \end{equation*}

Proof. Let $x_i$ be as in the hypotheses of Theorem5.1, and let $i_{p,\alpha }$ be as in the proof of Theorem5.1. Let $p^{\prime}$ be the Holder conjugate of $p$ . In [Reference Caspers and de la Salle3, Theorem 5.2], it is proven that

(5.8) \begin{equation} \langle i_{p,\alpha }(x), i_{p^{\prime}, \alpha }(y) \rangle _{p,p^{\prime}}\to \langle x, y \rangle _{p,p^{\prime}}, \qquad x \in L_p(\mathcal{L} G),\ y \in L_{p^{\prime}}(\mathcal{L} G). \end{equation}

Note that this inequality also holds and in fact is explicitly proven for $p = \infty$ ; by symmetry, it also holds for $p=1$ . We remark that this result also uses the Følner condition.

Let $\varepsilon \gt 0$ . Then we can find $y = \Delta ^{a} \lambda (f) \Delta ^b$ , $f \in C_c(G) \star C_c(G)$ such that $\|y\|_{L_{p^{\prime}}(\mathcal{L} G)} \leq 1$ and

\begin{equation*} \|T_{\phi }(x_1, \dots, x_n)\|_{L_p(\mathcal {L} G)} \leq |\langle T_{\phi }(x_1, \dots, x_n), y \rangle | + \varepsilon . \end{equation*}

Next, by (5.8) and Theorem5.1, we can find $\alpha \in I$ such that the following two inequalities hold:

\begin{equation*} |\langle T_{\phi }(x_1, \dots, x_n), y \rangle - \langle i_{p,\alpha }(T_{\phi }(x_1, \dots, x_n)), i_{p^{\prime}, \alpha }(y) \rangle | \lt \varepsilon \end{equation*}

and

\begin{equation*} \vert \langle i_{p,\alpha }(T_\phi (x_1,\ldots, x_n)),i_{p^{\prime},\alpha }(y) \rangle _{p,p^{\prime}} - \langle M_{\widetilde {\phi }}( i_{p_1,\alpha }(x_1),\ldots, i_{p_n,\alpha }(x_n) ), i_{p^{\prime},\alpha }(y) \rangle _{p,p^{\prime}} \vert \lt \varepsilon . \end{equation*}

By combining these inequalities, we find

\begin{align*} \begin{split} \|T_{\phi }(x_1, \dots, x_n)\|_{L_p(\mathcal{L} G)} &\leq |\langle M_{\widetilde{\phi }}( i_{p_1,\alpha }(x_1),\ldots, i_{p_n,\alpha }(x_n) ), i_{p^{\prime},\alpha }(y) \rangle _{p,p^{\prime}}| + 3\varepsilon \\ &\leq \|M_{\widetilde{\phi }}\|_{(p_1, \dots, p_n)} \prod _{i=1}^n \|x_i\|_{L_{p_i}(\mathcal{L} G)} + 3\varepsilon \end{split} \end{align*}

The elements $x_i$ as chosen above are norm dense in $L_{p_i}(\mathcal{L} G)$ (resp. $C^*_\lambda (G)$ when $p_i = \infty$ ); hence, we get the required bound. The multiplicative bound follows similarly.

We can now extend the result of [Reference Caspers, Krishnaswamy-Usha and Vos6, Corollary 4.3] to non-unimodular groups. It is also a multiplicatively bounded, non-unimodular version of [Reference Caspers, Janssens, Krishnaswamy-Usha and Miaskiwskyi5, Theorem 4.5]. Moreover, we no longer need the SAIN condition, and the subgroup no longer needs to be discrete.

Corollary 5.4. Let $G$ be a locally compact, first countable group, and let $1 \leq p \leq \infty$ and $1 \lt p_1, \dots, p_n \leq \infty$ with $p^{-1} = \sum _{i=1}^n p_i^{-1}$ . Let $\phi \in C_b(G^{\times n})$ , which defines a $(p_1, \dots, p_n)$ -mb Fourier multiplier, and let $H \leq G$ be an amenable subgroup. Then

\begin{equation*} \|T_{\phi |_{H^{\times n}}}\|_{(p_1, \dots, p_n)-mb} \leq \|T_{\phi }\|_{(p_1, \dots, p_n)-mb} \end{equation*}

Proof. The associated inequality for Schur multipliers follows from Theorem2.2. Now Corollary 5.2 (using amenability of $H$ ) and Theorem4.1 yield the result.

In the next corollary, we prove a necessary condition for a ‘Fourier multiplier’ to satisfy (5.1) for the embeddings $i_{p,\alpha }$ defined above. This was used in the discussion in Section 3.

Corollary 5.5. Fix $n \gt 1$ , $1 \leq p_1, \dots, p_n, p, p^{\prime} \leq \infty$ such that $\frac 1p = \sum _{i=1}^n \frac 1{p_i} = 1 - \frac 1{p^{\prime}}$ , and let $\theta _1, \dots, \theta _n, \theta, \theta ^{\prime} \in [0,1]$ . Let $i_{p,\alpha }$ be as in the proof of Theorem 5.1 . Assume that for each $\phi \in L_\infty (G^{\times n})$ , we have a map $S_\phi \,:\,{\kappa }^{\theta _1}_{p_i}(L) \times \ldots \times{\kappa }^{\theta _n}_{p_n}(L) \to{\kappa }^{\theta }_p(L)$ satisfying

(5.9) \begin{equation} \left |\langle i_{p,\alpha }(S_\phi (x_1,\ldots, x_n)),i_{p^{\prime},\alpha }(y) \rangle _{p,p^{\prime}} - \langle M_{\widetilde{\phi }}( i_{p_1,\alpha }(x_1),\ldots, i_{p_n,\alpha }(x_n) ), i_{p^{\prime},\alpha }(y) \rangle _{p,p^{\prime}} \right | \stackrel{\alpha }\to 0. \end{equation}

for $x_i \in{\kappa }_{p_i}^{\theta _i}(L)$ , $y \in{\kappa }_{p^{\prime}}^{\theta ^{\prime}}(L)$ . Now let $\phi (s_1, \dots, s_n) = \phi _1(s_1)\ldots \phi _n(s_n)$ for some functions $\phi _1, \dots, \phi _n \in L^\infty (G)$ . Then $S_\phi$ must satisfy

\begin{equation*} S_{\phi }(x_1, \dots, x_n) = T_{\phi _1}(x_1) \ldots T_{\phi _n}(x_n) \end{equation*}

for $x_i \in{\kappa }^{\theta _i}_{p_i}(L)$ , $i = 1, \dots, n$ .

Proof. Fix some $y \in{\kappa }^{\theta ^{\prime}}_{p^{\prime}}(L)$ . By density, it suffices to show that

\begin{equation*} \langle S_\phi (x_1, \dots, x_n), y\rangle = \langle T_{\phi _1}(x_1) \ldots T_{\phi _n}(x_n), y\rangle . \end{equation*}

By (5.8), it suffices to show

\begin{equation*} \lim _{\alpha \in I} |\langle i_{p,\alpha }(S_\phi (x_1, \dots, x_n) - T_{\phi _1}(x_1) \ldots T_{\phi _n}(x_n)), i_{p^{\prime},\alpha }(y)\rangle | = 0. \end{equation*}

By running the proof of Theorem5.1 with the constant 1 function in place of $\phi$ and $T_{\phi _i}(x_i)$ in place of $x_i$ , we find that

\begin{equation*} \lim _{\alpha \in I} |\langle i_{p,\alpha }(T_{\phi _1}(x_1) \ldots T_{\phi _n}(x_n)) - i_{p_1, \alpha }(T_{\phi _1}(x_1)) \ldots i_{p_n, \alpha }(T_{\phi _n}(x_n)), i_{p^{\prime},\alpha }(y) \rangle | = 0. \end{equation*}

Since multiplication with $\phi _i$ only maps $C_c(G) \star C_c(G)$ to $C_c(G)$ , we no longer need to have that $T_{\phi _i}(x_i) \in{\kappa }_{p_i}^{\theta _i}(L)$ , so we cannot apply Theorem5.1 directly. But since we have $\phi = 1$ , this does not give any technical complications in the proof.

Using the kernel representations, it is straightforward to show that

\begin{align*} \begin{split} i_{p_1, \alpha }(T_{\phi _1}(x_1)) \ldots i_{p_n, \alpha }(T_{\phi _n}(x_n)) &= M_{\widetilde{\phi _1}}(i_{p_1, \alpha }(x_1)) \ldots M_{\widetilde{\phi _n}}(i_{p_n,\alpha }(x_n)) \\ &= M_{\widetilde{\phi }}(i_{p_1, \alpha }(x_1), \dots, i_{p_n,\alpha }(x_n)). \end{split} \end{align*}

By combining the above observations with (5.9), we get the required result.

6. Linear intertwining result

In this section, we sketch the proof of Proposition 4.3. The main ingredient to be added to already existing results is the extension of [Reference Conde-Alonso, Parcet and Ricard9, Lemma 3.1] to general von Neumann algebras via Haagerup reduction. The Haagerup reduction method is described by Theorem6.1, proved for $\sigma$ -finite von Neumann algebras in [Reference Haagerup, Junge and Xu12] and extended to the weight case in [Reference Caspers, Parcet, Perrin and Ricard8, Section 8]. We will assume that the reader is familiar with Tomita–Takesaki theory, conditional expectations and such. We refer to [Reference Takesaki24] for the background.

Denote by $\sigma ^\varphi$ the modular automorphism group of a normal faithful semifinite (nfs) weight $\varphi$ . Recall that the centraliser $\mathcal{N}_\varphi$ of a nfs weight $\varphi$ on a von Neumann algebra $\mathcal{M}$ is given by

\begin{equation*} \mathcal {N}_\varphi = \{x \in \mathcal {M}\,:\, \sigma _t^\varphi (x) = x\ \forall t \in {\mathrm {\mathbb {R}}}\}. \end{equation*}

We denote by $D_\varphi$ the spatial derivative with respect to $\varphi$ (and some weight on the commutant, whose choice is unimportant). Assume that $T\,:\, \mathcal{M} \to \mathcal{M}$ is unital completely positive (ucp) and satisfies $\varphi \circ T \leq \varphi$ . Then by [Reference Haagerup, Junge and Xu12, Section 5], $T$ ‘extends’ to a map $T^{(p)}$ on $L_p(\mathcal{M})$ , in the sense that $T^{(p)}(D_\varphi ^{1/2p} x D_\varphi ^{1/2p}) = D_\varphi ^{1/2p} T(x) D_\varphi ^{1/2p}$ for $x \in{\mathfrak{m}}_{\widehat \varphi }$ . If $T$ satisfies $\sigma _s^\varphi \circ T = T \circ \sigma _s^\varphi$ , then we moreover have $T^{(p)}(D_\varphi ^{\theta /p} x D_\varphi ^{(1-\theta )/p}) = D_\varphi ^{\theta /p} T(x) D_\varphi ^{(1-\theta )/p}$ for any $0 \leq \theta \leq 1$ and $x \in{\mathfrak{m}}_{\widehat \varphi }$ .

In particular, the conditional expectations ${\mathcal{E}},{\mathcal{E}}_n$ ‘extend’ to maps ${\mathcal{E}}^{(p)},{\mathcal{E}}_n^{(p)}$ from $L_p({\mathcal{R}}, \widehat{\varphi })$ to $L_p(\mathcal{M}, \varphi )$ resp. $L_p({\mathcal{R}}_n, \widehat{\varphi })$ . The following statement is [Reference Caspers, Parcet, Perrin and Ricard8, Lemma 8.3]:

(6.1) \begin{equation} \lim _{n \to \infty }\|{\mathcal{E}}_n^{(p)}(x) - x\|_p = 0, \qquad 1 \leq p \lt \infty, \ x \in L_p({\mathcal{R}}, \hat{\varphi }). \end{equation}

We need a few more facts; we refer to [Reference Caspers, Parcet, Perrin and Ricard8, Section 8.2] for the details. First, there is an isometric isomorphism ${\kappa }_p\,:\, L_p({\mathcal{R}}_n, \widehat{\varphi }) \to L_p({\mathcal{R}}_n, \varphi _n)$ given by ${\kappa }_p(D_{\widehat{\varphi }}^{1/2p}xD_{\widehat \varphi }^{1/2p}) = e^{a_n/2p} x e^{a_n/2p}$ for $x \in{\mathfrak{m}}_{\widehat \varphi }$ . Next, assume that $T\,:\, \mathcal{M} \to \mathcal{M}$ is ucp and preserves $\varphi$ and $\sigma _s^\varphi$ . Then by [Reference Haagerup, Junge and Xu12, Section 4], there exists an extension $\widehat{T}\,:\,{\mathcal{R}} \to{\mathcal{R}}$ , which is also ucp and preserves $\widehat{\varphi }$ and $\sigma _s^{\widehat \varphi }$ . Hence $\widehat{T}$ itself also ‘extends’ to the various noncommutative $L_p$ -spaces. Moreover, the following diagram commutes:

Note that since $\varphi _n$ is tracial on ${\mathcal{R}}_n$ , the $\widehat{T}$ in the rightmost upwards arrow is actually an extension of the operator $\widehat{T}$ on ${\mathcal{R}}_n$ , so we do not need to use the notation $\widehat{T}^{(p)}$ here.

Finally, for $1 \leq p,q \lt \infty$ , we define the Mazur maps $M_{p,q}\,:\, L_p(\mathcal{M}) \to L_q(\mathcal{M})$ by $x \mapsto u |x|^{p/q}$ , where $x = u|x|$ is the polar decomposition of $x$ . The Mazur maps satisfy ${\kappa }_q \circ M_{p,q} = M_{p,q} \circ{\kappa }_p$ ; see, for instance, [Reference Ricard23, end of Section 3]. We are now ready to state and prove the generalisation of [Reference Conde-Alonso, Parcet and Ricard9, Lemma 3.1] for general von Neumann algebras. This result was already shown for $2 \lt p \lt \infty$ in [Reference Caspers, Parcet, Perrin and Ricard8, Section 8], but we will prove the result for all $1 \lt p \lt \infty$ at once since this does not take any extra effort.

Lemma 6.2. Let $(\mathcal{M}, \varphi )$ be a von Neumann algebra equipped with nfs weight. Let $T\,:\, \mathcal{M} \to \mathcal{M}$ be a unital completely positive map satisfying $\varphi \circ T = \varphi$ and $T \circ \sigma _s^\varphi = \sigma _s^\varphi \circ T$ for all $s \in{\mathrm{\mathbb{R}}}$ . Then there exists a universal constant $C\gt 0$ such that for any $x \in L_2(\mathcal{M})$ and $1 \lt p \lt \infty$ :

\begin{equation*} \|T^{(p)}(M_{2,p}(x)) - M_{2,p}(x)\|_p \leq C \|T^{(2)}(x) - x\|_2^\theta \|x\|_2^{1-\theta }, \end{equation*}

where $\theta = \frac 14 \min \{\frac p2, \frac 2p\}$ .

Proof. The proof runs via Haagerup reduction, using the estimates for the semifinite case from [Reference Caspers, Parcet, Perrin and Ricard8, Claim B] for $p \gt 2$ and [Reference Conde-Alonso, Parcet and Ricard9, Lemma 3.1] for $p \lt 2$ . Note that the latter was stated only for finite von Neumann algebras, but the same proof works for the semifinite case as well.

Set $y = M_{2,p}(x)$ . Since $T = \widehat{T}$ on $\mathcal{M}$ and $L_p(\mathcal{M}, \varphi ) \hookrightarrow L_p({\mathcal{R}}, \widehat \varphi )$ canonically and isometrically, we have $T^{(p)}(y) = \widehat{T}^{(p)}(y)$ and

\begin{equation*} \|T^{(p)}(y) - y\|_{L_p(\mathcal {M}, \varphi )} = \|\widehat {T}^{(p)}(y) - y\|_{L_p({\mathcal {R}}, \widehat \varphi )}.\end{equation*}

Now fix $n \geq 1$ . Then

\begin{align*} \begin{split} \|{\mathcal{E}}_n^{(p)}(\widehat{T}^{(p)}(y)) -{\mathcal{E}}_n^{(p)}(y)\|_{L_p({\mathcal{R}}_n, \widehat \varphi )} &= \|{\kappa }_p\left ({\mathcal{E}}_n^{(p)}(\widehat{T}^{(p)}(y)) -{\mathcal{E}}_n^{(p)}(y)\right )\|_{L_p({\mathcal{R}}_n, \varphi _n)} \\ &= \|\widehat{T}({\kappa }_p({\mathcal{E}}_n^{(p)}(y))) -{\kappa }_p({\mathcal{E}}_n^{(p)}(y))\|_{L_p({\mathcal{R}}_n, \varphi _n)}. \end{split} \end{align*}

Now we can apply the result for the semifinite case on ${\kappa }_p({\mathcal{E}}_n^{(p)}(y))$ to obtain

\begin{align*} \begin{split} &\|{\mathcal{E}}_n^{(p)}(\widehat{T}^{(p)}(y)) -{\mathcal{E}}_n^{(p)}(y)\|_{L_p({\mathcal{R}}_n, \widehat \varphi )} \\ &\quad \leq \ C \|\hat{T}(M_{p,2}({\kappa }_p({\mathcal{E}}_n^{(p)}(y)))) - M_{p,2}({\kappa }_p({\mathcal{E}}_n^{(p)}(y)))\|_{L_2({\mathcal{R}}_n, \varphi _n)}^\theta \cdot \|M_{p,2}({\kappa }_p({\mathcal{E}}_n^{(p)}(y)))\|_{L_2({\mathcal{R}}_n, \varphi _n)}^{1-\theta } \\ &\quad= C \|{\kappa }_2(\hat{T}^{(2)}(M_{p,2}({\mathcal{E}}_n^{(p)}(y)))) -{\kappa }_2(M_{p,2}({\mathcal{E}}_n^{(p)}(y)))\|_{L_2({\mathcal{R}}_n, \varphi _n)}^\theta \cdot \|{\kappa }_2(M_{p,2}({\mathcal{E}}_n^{(p)}(y)))\|_{L_2({\mathcal{R}}_n, \varphi _n)}^{1-\theta } \\ &\quad= C \|\hat{T}^{(2)}(M_{p,2}({\mathcal{E}}_n^{(p)}(y))) - M_{p,2}({\mathcal{E}}_n^{(p)}(y))\|_{L_2({\mathcal{R}}_n, \widehat \varphi )}^\theta \cdot \|M_{p,2}({\mathcal{E}}_n^{(p)}(y))\|_{L_2({\mathcal{R}}_n, \widehat \varphi )}^{1-\theta } \\ &\quad\,=\!:\, C A_n^\theta B_n^{1-\theta }.\\ \end{split} \end{align*}

By the triangle inequality, the main result from [Reference Ricard23] and (6.1), we find

\begin{align*} \begin{split} B_n &\leq \|M_{p,2}({\mathcal{E}}_n^{(p)}(y)) - M_{p,2}(y)\|_{L_2({\mathcal{R}}_n, \widehat \varphi )} + \|M_{p,2}(y)\|_{L_2({\mathcal{R}}_n, \widehat \varphi )} \\ &\leq C_{x,p} \|{\mathcal{E}}_n^{(p)}(y) - y\|_{L_p({\mathcal{R}}_n, \widehat \varphi )}^{\min \{\frac p2, 1\}} + \|x\|_{L_2({\mathcal{R}}_n, \widehat \varphi )} \to \|x\|_{L_2(\mathcal{M}, \varphi )} \end{split} \end{align*}

for some constant $C_{x,p}$ independent of $n$ . Similarly, we find

\begin{equation*} A_n \leq C_{x,p}\|\widehat {T}^{(2)} - 1_{{\mathcal {R}}}\| \|{\mathcal {E}}_n^{(p)}(y) - y\|_{L_p({\mathcal {R}}_n, \widehat \varphi )}^{\min \{\frac p2, 1\}} + \|\widehat {T}^{(2)}(x) - x\|_{L_2({\mathcal {R}}_n, \widehat \varphi )} \to \|T^{(2)}(x) - x\|_{L_2(\mathcal {M}, \varphi )}. \end{equation*}

Hence, taking limits and applying again (6.1), we conclude

\begin{equation*} \|T^{(p)}(y) - y\|_p = \lim _{n \to \infty } \|{\mathcal {E}}_n^{(p)}(\widehat {T}^{(p)}(y)) - {\mathcal {E}}_n^{(p)}(y)\|_{L_p({\mathcal {R}}_n, \widehat \varphi )} \leq C \|T^{(2)}(x) - x\|_2^\theta \|x\|_2^{1-\theta }.\end{equation*}

Proof of Proposition 4.3. We indicate only the changes to [Reference Caspers, Parcet, Perrin and Ricard8, Proof of Claim B]. The statement we have to prove is precisely [Reference Caspers, Parcet, Perrin and Ricard8, Equation (9)], but without the $u_j$ (this is just a different choice based on convenience). The $T_\zeta$ constructed in [Reference Caspers, Parcet, Perrin and Ricard8, Proof of Claim B] is a $\varphi$ -preserving ucp map that commutes with the modular automorphism group; one can see this from (2.4). Hence, we can apply Lemma 6.2 on $T_\zeta$ and $h_V$ to show [Reference Caspers, Parcet, Perrin and Ricard8, Equation (10)] (but without the $u_j$ ). Then, setting $z_j = h_V^{2/q}$ , the rest of the proof is the same.