The local geometry of
idempotent Schur multipliers

Javier Parcet, Mikael de la Salle and Eduardo Tablate

###### \pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.

A Schur multiplier is a linear map on matrices which acts on its entries by
multiplication with some function, called the symbol. We consider idempotent
Schur multipliers, whose symbols are indicator functions of smooth Euclidean
domains. Given $1<p\neq 2<\infty$, we provide a local characterization (under
some mild transversality condition) for the boundedness on Schatten
$p$-classes of Schur idempotents in terms of a lax notion of boundary
flatness. We prove in particular that all Schur idempotents are modeled on a
single fundamental example: the triangular projection. As an application, we
fully characterize the local $L_{p}$-boundedness of smooth Fourier idempotents
on connected Lie groups. They are all modeled on one of three fundamental
examples: the classical Hilbert transform, and two new examples of Hilbert
transforms that we call affine and projective. Our results in this paper are
vast noncommutative generalizations of Fefferman’s celebrated ball multiplier
theorem. They confirm the intuition that Schur multipliers share profound
similarities with Euclidean Fourier multipliers —even in the lack of a Fourier
transform connection— and complete, for Lie groups, a longstanding search of
Fourier $L_{p}$-idempotents.

Abstract

## Introduction

Schur multipliers are linear maps on matrix algebras with a great impact on
geometric group theory, operator algebras, and functional analysis. Their
definition is rather simple on discrete spaces $S_{M}(A)=(M(j,k)A_{jk})_{jk}$.
It easily extends to nonatomic $\sigma$-finite measure spaces $(\Omega,\mu)$,
by restricting to operators $A$ in $L_{2}(\Omega,\mu)$ admitting a kernel
representation over $\Omega\times\Omega$. Their role in geometric group theory
and operator algebras was first analyzed by Haagerup. His pioneering work on
free groups [19] and the research thereafter on semisimple lattices [4, 13]
encoded deep geometric properties via approximation properties with Schur
multipliers. Other interesting links can be found in [1, 35, 37, 39, 40, 42].

In 2011, stronger rigidity properties of high rank lattices were discovered by
studying $L_{p}$-approximations [25, 26]. In first place, there are no
$L_{p}$-approximations by means of Fourier or Schur multipliers over
$\mathrm{SL}_{n}(\mathbf{R})$ for $p>2+\alpha_{n}$, with $\alpha_{n}\to 0$ as
$n\to\infty$. Secondly, it turns out that this unprecedented pathology leads
to a strong form of nonamenability which is potentially useful to distinguish
the group von Neumann algebras of $\mathrm{PSL}_{n}(\mathbf{Z})$ for different
values of $n\geq 3$, the most iconic form of Connes’ rigidity conjecture. This
has strongly motivated our recent work [10, 33] with several forms of the
Hörmander-Mikhlin theorem. Nevertheless, there is still much to learn about
less regular multipliers. A key point in [26] was a careful analysis of Schur
multipliers over the $n$-sphere for symbols of the form
$M_{\varphi}(x,y)=\varphi(\langle x,y\rangle)$. More precisely, the
boundedness of $S_{M_{\varphi}}$ on the Schatten class $S_{p}$ for
$p>2+\frac{2}{n-1}$ imposes Hölder regularity conditions on $\varphi$. This
article grew from the analysis of the _spherical Hilbert transform_

$H_{\mathbf{S}}:A\mapsto\Big{(}\mathrm{sgn}\langle x,y\rangle
A_{xy}\Big{)}_{xy}.$

Is it $S_{p}$-bounded for some $\frac{2n}{n+1}<p\neq 2<\frac{2n}{n-1}$? It is
worth noting the analogy with the ball multiplier problem, which was only
known to be unbounded for $p$ outside this range before Fefferman’s celebrated
contribution [15]. Our main result completely solves this problem:
$H_{\mathbf{S}}$ is $S_{p}$-unbounded unless $n=1$ or $p=2$. In fact we
characterize $S_{p}$-boundedness for a much larger class of idempotents.

Let $M,N$ be two differentiable manifolds with the Lebesgue measure coming
from any Riemmanian structure on them. Consider a $\mathcal{C}^{1}$-domain
$\Sigma\subset M\times N$ so that its boundary $\partial\Sigma$ is a smooth
hypersurface, which is locally represented by level sets of some real-valued
$\mathcal{C}^{1}$-functions with nonvanishing gradients. We say that $\Sigma$
is transverse at a point $(x,y)$ in its boundary when the tangent space of
$\partial\Sigma$ at $(x,y)$ maps surjectively on each factor $T_{x}M$ and
$T_{y}N$. In that case, both sections
$\partial\Sigma_{x}=\big{\\{}y^{\prime}\in
N\mid(x,y^{\prime})\in\partial\Sigma\big{\\}}$ and
$\partial\Sigma^{y}=\big{\\{}x^{\prime}\in
M\mid(x^{\prime},y)\in\partial\Sigma\big{\\}}$ become codimension $1$
manifolds on a neighbourhood of $y$ and $x$ respectively.

######  Theorem A .

_Let $p\in(1,\infty)\setminus\\{2\\}$ and consider a $\mathcal{C}^{1}$-domain
$\Sigma\subset M\times N$. Then the following statements are equivalent for
any transverse point $(x_{0},y_{0})\in\partial\Sigma\hskip-2.0pt:$_

  1. (1)

$S_{p}$-boundedness. _The idempotent Schur multiplier $S_{\Sigma}$ whose
symbol equals $1$ on $\Sigma$ and $0$ elsewhere is bounded on
$S_{p}(L_{2}(U),L_{2}(V))$ for some pair of neighbourhoods $U,V$ of
$x_{0},y_{0}$ in $M,N$._

  2. (2)

Zero-curvature condition. _There are neighbourhoods $U,V$ of $x_{0},y_{0}$ in
$M,N$ such that the tangent spaces $T_{y}(\partial\Sigma_{x_{1}})$ and
$T_{y}(\partial\Sigma_{x_{2}})$ coincide for any pair of points
$(x_{1},y),(x_{2},y)\in\partial\Sigma\cap(U\times V)$._

  3. (3)

Triangular truncation representation.  _There are neighbourhoods $U,V$ of the
points $x_{0},y_{0}$ in $M,N$ and $\mathcal{C}^{1}$-functions
$f_{1}:U\to\mathbf{R}$ and $f_{2}:V\to\mathbf{R}$, such that the domain
$\Sigma\cap(U\times V)=\big{\\{}(x,y)\in U\times
V:f_{1}(x)>f_{2}(y)\big{\\}}$._

Theorem A characterizes the local behavior of $S_{p}$-bounded idempotent Schur
multipliers at transverse points. This includes spherical Hilbert transforms
and vastly amplifies Fefferman’s ball multiplier theorem [15]. When
$M=N=\mathbf{R}^{n}$ and
$\mathbf{n}(x,y)=(\mathbf{n}_{1}(x,y),\mathbf{n}_{2}(x,y))$ is a normal vector
to $\partial\Sigma$ at $(x,y)$, transversality means that both $n$-dimensional
components $\mathbf{n}_{1},\mathbf{n}_{2}$ are nonzero. The zero-curvature
condition means that $\mathbf{n}_{2}(x_{1},y)$ and $\mathbf{n}_{2}(x_{2},y)$
are parallel—equivalent forms in terms of $\mathbf{n}_{1}(x,y_{1})$ and
$\mathbf{n}_{1}(x,y_{2})$ instead, or simpler formulations for
$\mathcal{C}^{2}$-domains will be also discussed after the proof of Theorem A.
Fefferman’s Fourier analytic framework corresponds to sets of the form

$\Sigma=\Big{\\{}(x,y):x-y\in\Omega\Big{\\}}$

for an Euclidean domain $\Omega$, which turn out to be fully transverse and
partly explains why transversality did not appear so far. It is also quite
remarkable that Theorem A holds beyond Euclidean spaces, since Schur
multipliers on general manifolds lack to admit a Fourier transform connection.
It has come as a surprise that even after substantial and necessary
differences with Fefferman’s formulation, a similar phenomenon applies in such
a general context. In a different direction, a global nonlocal
characterization of $S_{p}$-bounded idempotent Schur multipliers also follows
for relatively compact fully transverse domains $\Sigma$. Let us note that
this class of domains has no representatives in the Fourier analytic setting,
since Toeplitz-like domains $\Sigma=\big{\\{}(x,y):x-y\in\Omega\big{\\}}$ are
not relatively compact.

$\bullet$$y$$x_{1}$$x_{2}$$\partial\Sigma_{x_{1}}$$\partial\Sigma_{x_{2}}$$y\in\partial\Sigma_{x_{1}}\hskip-2.0pt\cap\partial\Sigma_{x_{2}}$$T_{y}\partial\Sigma_{x_{1}}\neq
T_{y}\partial\Sigma_{x_{2}}$

Figure 1. Failure of (2) for spherical Hilbert transforms
$H_{{\mathbf{S},\delta}}$

Here $H_{{\mathbf{S},\delta}}=S_{\Sigma}$ with
$\Sigma=\big{\\{}(x,y)\in\mathbf{S}^{n}\times\mathbf{S}^{n}:\langle
x,y\rangle>\delta\big{\\}}$ for $n=2$.

Theorem A has profound consequences for Fourier multipliers on Lie group von
Neumann algebras. Smooth Fourier multipliers on group algebras were
intensively investigated over the last decade [5, 11, 17, 22, 23, 30, 33]. The
nonsmooth theory concerns a longstanding search to classify idempotent Fourier
$L_{p}$-multipliers, but the geometric behavior of their symbols is very
sensitive to the underlying group. Bożejko and Fendler [3] studied an analog
of Fefferman’s ball multiplier theorem in the free group for $|1/p-1/2|>1/6$,
though the general case $p\neq 2$ stands open since then. More recently, Mei
and Ricard found a large class of free Hilbert transforms in their remarkable
work [29]. The search for Hilbert transforms on general groups also includes
crossed products and groups acting on tree-like structures [18, 34].

In this paper, we shall give a complete characterization of the local boundary
behavior for completely bounded idempotent Fourier multipliers on arbitrary
Lie groups. Our result is more easily stated for simply connected groups. We
refer to Section 2 for the statement of the result on general Lie groups
(Theorem 2.3) and for the precise definitions of local Fourier multipliers.

######  Theorem B .

_Let $p\in(1,\infty)\setminus\\{2\\}$. Let $\mathrm{G}$ be a simply connected
Lie group, $\Omega\subset\mathrm{G}$ a $\mathcal{C}^{1}$-domain and
$g_{0}\in\partial\Omega$ a point in the boundary of $\Omega$. The following
are equivalent_:

  1. (1)

_$\chi_{\Omega}$ defines locally at $g_{0}$ a completely bounded Fourier
$L_{p}$-multiplier._

  2. (2)

_There is a smooth action $\mathrm{G}\to\mathrm{Diff}(\mathbf{R})$ by
diffeomorphisms on the real line, such that $\Omega$ coincides on a
neighbourhood of $g_{0}$ with $\\{g\in\mathrm{G}\mid g\cdot 0>g_{0}\cdot
0\\}$._

Lie himself classified Lie groups admitting (local) actions by diffeomorphisms
on the real line [28]. This classification into three types (translation,
affine, and projective) gives rise to the following three fundamental examples
of a group $\mathrm{G}$ with a smooth domain $\Omega$:

  * i)

The real line $\mathrm{G}_{1}=\mathbf{R}$ with $\Omega_{1}=(0,\infty)$.

  * ii)

The affine group
$\mathrm{G}_{2}=\operatorname{Aff}_{+}(\mathbf{R})$\star\star$\star$Affine
increasing bijections $x\mapsto ax+b$ for $a\in\mathbf{R}_{+}^{*}$ and
$b\in\mathbf{R}$, isomorphic to $\mathbf{R}\rtimes\mathbf{R}_{+}^{*}$. and
$\Omega_{2}=\\{ax+b:b>0\\}$.

  * iii)

The universal covering group
$\mathrm{G}_{3}=\widetilde{\mathrm{PSL}}_{2}(\mathbf{R})${\dagger}{\dagger}${\dagger}$The
action
$\alpha:\widetilde{\mathrm{PSL}}_{2}(\mathbf{R})\curvearrowright\mathbf{R}$ is
obtained by lifting the standard action of $\mathrm{PSL}_{2}(\mathbf{R})$ on
the projective line to the universal covers. If $p:\mathbf{R}\to
P^{1}(\mathbf{R})$ denotes the universal cover, then the universal cover of
$\mathrm{SL}_{2}(\mathbf{R})$ is identified with the group of homeomorphisms
$g:\mathbf{R}\to\mathbf{R}$ for which there is
$A\in\mathrm{PSL}_{2}(\mathbf{R})$ such that $p\circ g=A\cdot p$. with
$\Omega_{3}=\\{g:\alpha_{g}(0)>0\\}$.

Lie’s classification implies the following interesting consequence.

######  Corollary B1 .

_Conditions_ (1) _and_ (2) _in Theorem_ B _are equivalent to_ :

  * (3)

_There is $j\in\\{1,2,3\\}$ and a smooth surjective homomorphism
$f\colon\mathrm{G}\to\mathrm{G}_{j}$ such that the domain $\Omega$ coincides
on a neighbourhood of $g_{0}$ with $g_{0}f^{-1}(\Omega_{j})$._

_Therefore, there are three fundamental models for Hilbert transforms on Lie
groups._

The above examples define (not only local, but) global completely bounded
$L_{p}$ Fourier multipliers for $1<p<\infty$. Example i) is the classical
Hilbert transform and Example iii) follows from recent Cotlar identities for
unimodular groups [18]. The nonunimodular Example ii) will be properly
justified in Example 2.8.

Theorem B and Corollary B1 give very satisfactory descriptions of completely
bounded Fourier idempotents in arbitrary Lie groups. It is certainly
surprising that these multipliers are modeled out of exactly three fundamental
examples, the classical Hilbert transform and its affine and projective
variants. It also shows that every $\mathcal{C}^{1}$-idempotent is
automatically $\mathcal{C}^{\infty}$. This rigidity property collides head-on
with the much more flexible scenario of Theorem A. In the following result, we
further describe Fourier $L_{p}$-idempotents for large classes of Lie groups.

######  Corollary B2 .

_Let $p\in(1,\infty)\setminus\\{2\\}$ and let $\mathrm{G}$ be a Lie group_:

  * i)

_If $\mathrm{G}$ is simply connected and nilpotent, every cb-$L_{p}$-bounded
smooth Fourier idempotent is locally of the form $H\circ\varphi$, for the
Hilbert transform $H$ and some continuous homomorphism
$\varphi:\mathrm{G}\to\mathbf{R}$._

  * ii)

_If $\mathrm{G}$ is a simple Lie group which is not locally isomorphic to
$\mathrm{SL}_{2}(\mathbf{R})$, then $\mathrm{G}$ does not carry any smooth
Fourier idempotent which is locally completely $L_{p}$-bounded on its group
von Neumann algebra._

  * iii)

_If $\mathrm{G}$ is locally isomorphic to $\mathrm{SL}_{2}(\mathbf{R})$, then
$\mathrm{G}$ carries a unique local Fourier idempotent which is completely
$L_{p}$-bounded on its group algebra $($up to left/right translations$)$ given
by $g\mapsto\frac{1}{2}\big{(}1+\mathrm{sgn}\,\mathrm{Tr}(ge_{12})\big{)}$._

As an illustration for stratified Lie groups, the homomorphism $\varphi$
corresponds on the Lie algebra level with the projection onto any
1-dimensional subspace of the first stratum. The second statement above
spotlights the singular nature of harmonic analysis over simple Lie groups. It
also yields an alternative way to answer our motivating question: the
spherical Hilbert transform $H_{\mathbf{S}}$ is not $L_{p}$-bounded for any
$p\neq 2$. Finally, as we shall justify, the third statement gives a
straightforward solution (in the negative) to Problem A in [18]. We refer to
[36, 38] for the operator space background necessary for this paper.

The plan of the paper is as follows. Section 1 is devoted to idempotent Schur
multipliers. It contains the proof of Theorem A and several discussions,
including our analysis of the spherical Hilbert transform. Section 2 is
devoted to Fourier multipliers. It contains the proof of Theorem B and its
corollaries. The proof relies on a result of independent interest on the local
transference between Fourier and Schur multipliers for arbitrary locally
compact groups, Theorem 2.1.

##  1\. Idempotent Schur multipliers

In this section we give a complete proof of Theorem A. We begin by recalling
some particularly flexible changes of variables for Schur symbols, which
preserve the $S_{p}$-norm of the corresponding Schur multipliers on nonatomic
spaces. Then, we prove the implications (1) $\Rightarrow$ (2) $\Rightarrow$
(3) $\Rightarrow$ (1) in Theorem A separately. We shall finish with some
comments and applications to spherical Hilbert transforms.

###  1.1. Schur multipliers

Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite measure spaces. Given $1\leq
p<\infty$, let $S_{p}(L_{2}(X),L_{2}(Y))$ be the space all of bounded linear
operators $T:L_{2}(X)\to L_{2}(Y)$ with $\operatorname{Tr}|T|^{p}<\infty$,
which is a Banach space for the norm below

$\|T\|_{S_{p}}=\big{(}\operatorname{Tr}\hskip
1.0pt|T|^{p}\big{)}^{\frac{1}{p}}.$

When $p=2$, the Schatten class $S_{2}(L_{2}(X),L_{2}(Y))$ is the space of
Hilbert-Schmidt operators $L_{2}(X)\to L_{2}(Y)$. It coincides with
$L_{2}(X\times Y)$, regarding any $L_{2}$-function $(x,y)\mapsto K(x,y)$ as
the kernel of the corresponding Hilbert-Schmidt operator

$T_{K}f(y)=\int_{X}K(x,y)f(x)d\mu(x).$

Given $m\in L_{\infty}(X\times Y)$, the Schur $S_{p}$-multiplier with symbol
$m$ is defined (when it exists) as the unique bounded linear map $S_{m}$ on
$S_{p}(L_{2}(X),L_{2}(Y))$ which assigns $T_{K}=(K(x,y))_{x\in X,y\in Y}\in
S_{2}\cap S_{p}$ to $(m(x,y)K(x,y))_{x\in X,y\in Y}=S_{m}(T_{K})$. We shall
write $\left\|m\right\|_{MS_{p}}$ for its norm, with the convention
$\left\|m\right\|_{MS_{p}}=\infty$ if $S_{m}$ does not exist.

The following general fact will be crucial in our proof of Theorem A. It
evidences a much greater flexibility of Schur multipliers compared to Fourier
multipliers. The proof follows from [26], we include the argument below.

######  Lemma 1.1.

Let $(X,\mu),(X^{\prime},\mu^{\prime}),(Y,\nu),(Y^{\prime},\nu^{\prime})$ be
atomless $\sigma$-finite measure spaces and $f\colon X\to X^{\prime}$ and
$g\colon Y\to Y^{\prime}$ be measurable maps. Assume the pushforward measures
$f_{*}\mu$ and $g_{*}\nu$ are absolutely continuous with respect to the
measures $\mu^{\prime}$ and $\nu^{\prime}$ respectively. Then, for every $m\in
L_{\infty}(X^{\prime}\times Y^{\prime})$

$\displaystyle\left\|m\circ(f\times
g)\right\|_{MS_{p}(L_{2}(X,\mu),L_{2}(Y,\nu))}$
$\displaystyle=\left\|m\right\|_{MS_{p}(L_{2}(X^{\prime},f_{*}\mu),L_{2}(Y^{\prime},g_{*}\nu))}$
$\displaystyle\leq\left\|m\right\|_{MS_{p}(L_{2}(X^{\prime},\mu^{\prime}),L_{2}(Y^{\prime},\nu^{\prime}))}.$

###### Proof.

The last inequality follows directly from [26, Lemma 1.9] and the absolute
continuity assumption. To prove the first identity and lighten the notation,
let us assume for simplicity that $(X,\mu)=(Y,\nu)$,
$(X^{\prime},\mu^{\prime})=(Y^{\prime},\nu^{\prime})$ and $f=g$. Let
$\mathcal{B},\mathcal{B}^{\prime}$ be the underlying $\sigma$-algebras, and
consider $\mathcal{A}:=f^{-1}(\mathcal{B}^{\prime})$. Then $f$ allows to
identify $L_{2}(X^{\prime},\mathcal{B}^{\prime},f_{*}\mu)$ with
$L_{2}(X,\mathcal{A},\mu)$. In particular we have

$\left\|m\circ(f\times
f)\right\|_{MS_{p}(L_{2}(X,\mathcal{A},\mu))}=\left\|m\right\|_{MS_{p}(L_{2}(X^{\prime},\mathcal{B}^{\prime},f_{*}\mu))},$

and similarly for the cb-norm. On the other hand, [26, Lemma 1.13] implies
that the cb norms of $m\circ(f\times f)$ on $S_{p}(L_{2}(\mathcal{A},\mu))$
and $S_{p}(L_{2}(\mathcal{B},\mu))$ coincide, so we deduce

$\left\|m\circ(f\times
f)\right\|_{cbMS_{p}(L_{2}(X,\mathcal{B},\mu))}=\left\|m\right\|_{cbMS_{p}(L_{2}(X^{\prime},\mathcal{B}^{\prime},f_{*}\mu))}.$

Then [26, Theorem 1.18] allows us to conclude. Indeed, our assumptions that
$\mu$ and $\mu^{\prime}$ have no atoms imply that both cb-norms are equal to
their norms. ∎

###  1.2. Proof of Theorem A: Boundedness implies zero-curvature

In this paragraph we prove (1) $\Rightarrow$ (2) from the statement of Theorem
A. The key new idea we introduce is an amplification of Meyer’s lemma [15,
Lemma 1], which directly connects Schur idempotents with Meyer’s classical
condition, to which Fefferman’s construction may be applied. Taking charts, we
can and will assume that $M$ and $N$ are open subsets of $\mathbf{R}^{m}$ and
$\mathbf{R}^{n}$ respectively. We shall further assume for simplicity that
$m=n$, our argument applies as well when $m\neq n$. Given
$z\in\partial\Sigma$, let
$\mathbf{n}(z)=(\mathbf{n}_{1}(z),\mathbf{n}_{2}(z))\in\mathbf{R}^{n}\oplus\mathbf{R}^{n}$
be a normal to $\partial\Sigma$ at $z$ pointing towards $\Sigma$.

######  Lemma 1.2.

Consider a $\mathcal{C}^{1}$-domain
$\Sigma\subset\mathbf{R}^{n}\times\mathbf{R}^{n}$ and open sets
$U,V\subset\mathbf{R}^{n}$ such that $\partial\Sigma$ intersects $U\times V$.
Let $S_{\Sigma}$ be the idempotent Schur multiplier whose symbol is the
characteristic function of $\Sigma$ and assume that it is bounded on
$S_{p}(L_{2}(U),L_{2}(V))$ with norm $C$. Let $x_{1},x_{2},\ldots,x_{N}\in U$
and $y\in V$ such that $z_{j}=(x_{j},y)$ is a transverse point in the boundary
$\partial\Sigma$ for every $j=1,2,\ldots,N$. Define
$u_{j}=\mathbf{n}_{2}(z_{j})$ and consider functions
$f_{1},f_{2},\ldots,f_{N}\in L_{p}(\mathbf{R}^{n})$. Then, we have

$\Big{\|}\Big{(}\sum_{j=1}^{N}\big{|}H_{u_{j}}(f_{j})\big{|}^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}(\mathbf{R}^{n})}\leq
C\Big{\|}\Big{(}\sum_{j=1}^{N}|f_{j}|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}(\mathbf{R}^{n})}.$

Here we write $H_{u}$ for the $u$-directional Hilbert transform
$\widehat{H_{u}f}(\xi)=\chi_{\langle\xi,u\rangle>0}\widehat{f}(\xi)$.

The implication (1)$\Rightarrow$(2) in Theorem A follows from Lemma 1.2.
Indeed, by the transversality assumption, the map
$z\mapsto\mathbf{n}_{2}(z)/\|\mathbf{n}_{2}(z)\|$ is continuous on a
neighbourhood of the transverse point $(x_{0},y_{0})$ in Theorem A. Moreover,
for $y$ close to $y_{0}$ we have that $\partial\Sigma^{y}$ is locally a
manifold, so is connected. Therefore, if (2) was not true, there would exist
$y$ close to $y_{0}$ such that the subset of the sphere
$X=\\{\mathbf{n}_{2}(x^{\prime},y)/\|\mathbf{n}_{2}(x^{\prime},y)\|:x^{\prime}\in\partial\Sigma^{y}\cap
U\\}$ contains a connected subset not reduced to a point. According to (1) and
Lemma 1.2, this would imply that the square function inequality there holds
uniformly in $L_{p}(\mathbf{R}^{n})$ for any finite set in a continuum of
directions in the $(n-1)$-sphere. However, Fefferman’s main result in his
proof of the ball multiplier theorem [15] claims that such a uniform
inequality cannot hold. In fact, Fefferman stated it for $n=2$ but the result
in arbitrary dimension follows from the $2$-dimensional case by K. de Leeuw’s
restriction theorem [27]. In particular, the zero-curvature condition (2) must
hold.

###### Proof of Lemma _1.2_..

The proof relies on the following two claims:

  * (A)

Let $(x,y)$ be a transverse point in the boundary of $\partial\Sigma$ and let
$T\in\textrm{GL}_{n}(\mathbf{R})$ be such that
$T^{*}\mathbf{n}_{1}(x,y)=-\mathbf{n}_{2}(x,y)$. Then, the following identity
holds for almost every $\xi,\eta\in\mathbf{R}^{n}$

$\hskip 20.0pt\lim_{\varepsilon\to 0^{+}}\chi_{\Sigma}\big{(}x+\varepsilon
T\xi,y+\varepsilon\eta\big{)}=\frac{1}{2}\big{(}1+\mathrm{sgn}\big{\langle}\mathbf{n}_{2}(x,y),\eta-\xi\big{\rangle}\big{)}.$

  * (B)

Let $u_{j}$ be as in the statement. Then the Schur multiplier

$\hskip
17.0pt\big{(}(\xi,j),\eta\big{)}\in\big{(}\mathbf{R}^{n}\times\\{1,\dots,N\\}\big{)}\times\mathbf{R}^{n}\mapsto\frac{1}{2}\big{(}1+\mathrm{sgn}\langle\eta-\xi,u_{j}\rangle\big{)}$

is bounded on
$S_{p}(L_{2}(\mathbf{R}^{n}),L_{2}(\mathbf{R}^{n}\times\\{1,2,\ldots,N\\}))$
with norm $\leq\left\|\chi_{\Sigma}\right\|_{MS_{p}}$.

Assuming the validity of the above claims, we may now conclude the proof using
standard transference ideas that go back at least to the work of Bożejko and
Fendler [2]. We know from [7, Theorem 5.2] that there is an ultrafilter
$\mathcal{U}$ on $\mathbf{N}$ and a completely isometric map

$j_{p}\colon
L_{p}(\mathbf{R}^{n})\to\prod_{\mathcal{U}}S_{p}(L_{2}(\mathbf{R}^{n}))$

that intertwines Fourier and Schur multipliers. This gives
$j_{p}(H_{u}(f))=S_{m_{u}}(j_{p}(f))$ for every $u\in\mathbf{R}^{n}$ and $f\in
L_{p}(\mathbf{R}^{n})$. Here
$m_{u}(\xi,\eta)=\frac{1}{2}(1+\mathrm{sgn}\langle\eta-\xi,u\rangle)$. As a
consequence we have for every $f_{1},f_{2},\ldots,f_{N}\in
L_{p}(\mathbf{R}^{n})$

$\displaystyle\Big{\|}\Big{(}\sum_{j=1}^{N}|H_{u_{j}}(f_{j})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}(\mathbf{R}^{n})}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\Big{\|}\sum_{j=1}^{N}H_{u_{j}}(f_{j})\otimes
e_{j,1}\Big{\|}_{L_{p}(\mathbf{R}^{n};S_{p})}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\Big{\|}\sum_{j=1}^{N}j_{p}(H_{u_{j}}(f_{j}))\otimes
e_{j,1}\Big{\|}_{S_{p}}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\Big{\|}\sum_{j=1}^{N}S_{m_{u_{j}}}(j_{p}(f_{j}))\otimes
e_{j,1}\Big{\|}_{S_{p}}.$

According to claim (B) we deduce

$\displaystyle\Big{\|}\Big{(}\sum_{j=1}^{N}|H_{u_{j}}(f_{j})|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}(\mathbf{R}^{n})}\\!\\!\\!$
$\displaystyle\leq$
$\displaystyle\\!\\!\\!\|\chi_{\Sigma}\|_{MS_{p}}\Big{\|}\sum_{j=1}^{N}j_{p}(f_{j})\otimes
e_{j,1}\Big{\|}_{S_{p}}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\|\chi_{\Sigma}\|_{MS_{p}}\Big{\|}\Big{(}\sum_{j=1}^{N}|f_{j}|^{2}\Big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}(\mathbf{R}^{n})}.$

Thus, the assertion is a consequence of claim (B), for which we need to
justify claim (A) first. To do so, we can assume that
$\Sigma=f^{-1}(0,\infty)$ for a $\mathcal{C}^{1}$-submersion $f\colon U\times
V\to\mathbf{R}$. Then $\nabla f(x,y)=(\nabla_{x}f(x,y),\nabla_{y}f(x,y))$ is a
normal vector to the boundary $\partial\Sigma=f^{-1}(0)$ at every
$(x,y)\in\partial\Sigma$. Thus, replacing $f$ by a multiple, we can assume
that its gradient is $(\mathbf{n}_{1}(x,y),\mathbf{n}_{2}(x,y))$. Then, the
Taylor expansion of $f$ gives

$\displaystyle f\big{(}x+\varepsilon T\xi,y+\varepsilon\eta\big{)}$
$\displaystyle=$
$\displaystyle\varepsilon\big{\langle}\mathbf{n}_{1}(x,y),T\xi\big{\rangle}+\varepsilon\big{\langle}\mathbf{n}_{2}(x,y),\eta\big{\rangle}+o(\varepsilon)$
$\displaystyle=$
$\displaystyle\varepsilon\big{\langle}\mathbf{n}_{2}(x,y),\eta-\xi\big{\rangle}+o(\varepsilon).$

Therefore, if $\eta-\xi$ is not orthogonal to $\mathbf{n}_{2}(x,y)$ (a
condition that holds for almost every $\xi$ and $\eta$), we have
$\chi_{\Sigma}(x+\varepsilon
T\xi,y+\varepsilon\eta)=\frac{1}{2}(1+\mathrm{sgn}\langle\mathbf{n}_{2}(x,y),\eta-\xi\rangle)$
for every $\varepsilon>0$ small enough. This proves claim (A).

To prove claim (B) we apply (A). More precisely, let
$T_{j}\in\textrm{GL}_{n}(\mathbf{R})$ be such that
$T_{j}^{*}\mathbf{n}_{1}(x_{j},y)=-\mathbf{n}_{2}(x_{j},y)=-u_{j}$ for every
$j=1,2,\ldots,N$. The existence of these maps is clear, because by the
transversality assumption both $\mathbf{n}_{1}(x_{j},y)$ and
$\mathbf{n}_{2}(x_{j},y)$ are nonzero vectors in $\mathbf{R}^{n}$ and
$\textrm{GL}_{n}(\mathbf{R})$ acts transitively on them. By Lemma 1.1 the
Schur multiplier with symbol

$m_{\varepsilon}\big{(}(\xi,j),\eta\big{)}=\chi_{\Sigma}\big{(}x_{j}+\varepsilon
T_{j}\xi,y+\varepsilon\eta\big{)}$

is bounded with norm $\leq\left\|\chi_{\Sigma}\right\|_{MS_{p}}$ for every
$\varepsilon>0$. Taking $\varepsilon\to 0^{+}$, we obtain that the almost
everywhere limit of $m_{\varepsilon}$ is $S_{p}$-bounded with norm
$\leq\left\|\chi_{\Sigma}\right\|_{MS_{p}}$. However this limit is
$(1+\mathrm{sgn}\langle\eta-\xi,u_{j}\rangle)/2$ from claim (A). This proves
claim (B). ∎

######  Remark 1.3.

_Taking_

$\Sigma=\big{\\{}(x,y):x-y\in\Omega\big{\\}}$

_for some smooth domain $\Omega$, Lemma 1.2 reduces to the classical Meyer’s
lemma._

###  1.3. Proof of Theorem A: Zero-curvature implies triangular truncations

The implication (2)$\Rightarrow$(3) is a general geometric statement
concerning transverse hypersurfaces in manifolds of product type. Note that
both conditions are invariant by diffeomorphisms of product type, that is of
the form $(x,y)\mapsto(\phi(x),\psi(y))$. It will be useful to have a
description of a local normal form, analogous to the fact that hypersurfaces
are locally the diffeomorphic image of a hyperplane.

######  Lemma 1.4.

Let $\Sigma\subset\mathbf{R}^{n}\times\mathbf{R}^{n}$ be a
$\mathcal{C}^{1}$-domain and $(x_{0},y_{0})$ be a transverse point of its
boundary. Then, there are diffeomorphisms $\phi$ and $\psi$ from
neighbourhoods $U$ and $V$ of $x_{0}$ and $y_{0}$ into $\mathbf{R}^{n}$
satisfying $\phi(x_{0})=\psi(y_{0})=0$ and

$\Sigma\cap(U\times
V)=(\phi\times\psi)^{-1}\Big{\\{}\big{(}(s,\tilde{x}),y\big{)}:s>g(\tilde{x},y)\Big{\\}}$

for some $\mathcal{C}^{1}$-function
$g\colon\mathbf{R}^{n-1}\times\mathbf{R}^{n}\to\mathbf{R}$ satisfying
$g(0,y)=\langle y,e_{1}\rangle$ for every $y$.

###### Proof.

Fix $z_{0}=(x_{0},y_{0})$, by the transversality assumption that
$T_{z_{0}}\partial\Sigma$ surjects onto the second coordinate,
$T_{z_{0}}\partial\Sigma\cap(\mathbf{R}^{n}\oplus 0)\neq\mathbf{R}^{n}\oplus
0$. Equivalently, we have $\mathbf{n}_{1}(z_{0})\neq 0$. In particular, we may
construct a local diffeomorphism $\phi$ on the first variable, satisfying that
$\phi(x_{0})=0$ and $(e_{1},0)\notin T_{z_{0}}\partial\Sigma$. Then, according
to the implicit function theorem, there is a $\mathcal{C}^{1}$-function
$h\colon\mathbf{R}^{n-1}\times\mathbf{R}^{n}\to\mathbf{R}$ such that

$\phi\times\mathrm{id}(\Sigma)=\Big{\\{}\big{(}(s,\tilde{x}),y\big{)}\mid
s>h(\tilde{x},y)\Big{\\}}$

on a neighbourhood of $(0,y_{0})$. Moreover, the function $h(0,\cdot)$
vanishes at $y_{0}$ and by the second half of the transversality assumption,
it has nonzero differential at $y_{0}$. By the implicit function theorem
again, there is a diffeomorphism $\psi$ from a neighbourhood of $y_{0}$ into
$\mathbf{R}^{n}$ vanishing at $y_{0}$ and such that
$h(0,y)=\langle\psi(y),e_{1}\rangle$ for every $y$ close enough to $y_{0}$.
Taking $g(\tilde{x},y)=h(\tilde{x},\psi^{-1}(y))$, we get

$\phi\times\psi(\Sigma)=\Big{\\{}\big{(}(s,\tilde{x}),y\big{)}:s>g(\tilde{x},y)\Big{\\}}$

in a neighbourhood $\phi(U)\times\psi(V)$ of $(0,0)$ as expected. Also
$g(0,y)=\langle y,e_{1}\rangle$. ∎

With this in hand, we can prove the implication (2)$\Rightarrow$(3). Both
conditions are unchanged if we replace $(z_{0},\Sigma)$ by
$(\phi\times\psi(z_{0}),\phi\times\psi(\Sigma))$ for any pair $(\phi,\psi)$ of
local diffeomorphisms. By Lemma 1.4, we may assume without loss of generality
that $\Sigma=\\{(x,y):s>g(\tilde{x},y)\\}$ locally around $z_{0}=(0,0)$ for
$g\colon\mathbf{R}^{n-1}\times\mathbf{R}^{n}\to\mathbf{R}$ a
$\mathcal{C}^{1}$-function satisfying $g(0,y)=\langle y,e_{1}\rangle$. Then,
for every point $(x,y)\in\partial\Sigma$ where $x=(s,\tilde{x})$, we have
$T_{y}\partial\Sigma_{x}=\ker(d_{y}g(\tilde{x},y))$. Consider
$\tilde{U}\subset U$ and $\tilde{V}\subset V$ convex neighbourhoods of $0$
such that $(g(\tilde{x},y),\tilde{x})\in U$ for every
$(x,y)\in\tilde{U}\times\tilde{V}$. Then, (2) applied to
$x_{1}=(g(\tilde{x},y),\tilde{x})$ and $x_{2}=(g(0,y),0)$ yield

$\ker d_{y}g(\tilde{x},y)=\ker
d_{y}g(0,y)=\operatorname{span}\\{e_{2},e_{3},\ldots,e_{n}\\}\quad\mbox{for
every $(x,y)\in\tilde{U}\times\tilde{V}$.}$

In particular, $\partial_{y_{j}}g(\tilde{x},y)=0$ for every $j\geq 2$, so
$g(\tilde{x},y)=w(\tilde{x},\langle y,e_{1}\rangle)$ (since $\tilde{V}$ is
convex) for a $\mathcal{C}^{1}$-function
$w:\mathbf{R}^{n-1}\times\mathbf{R}\to\mathbf{R}$ satisfying $w(0,t)=t$ for
all $t$. By the implicit function theorem, we obtain that locally
$\\{(s,\tilde{x},t):s>w(\tilde{x},t)\\}$ is of the form
$\\{(s,\tilde{x},t):t>u(s,\tilde{x})\\}$ for a $\mathcal{C}^{1}$-function
$u\colon\mathbf{R}^{n}\to\mathbf{R}$. This proves (3).

###  1.4. Proof of Theorem A: Transference on triangular truncations

The implication (3) $\Rightarrow$ (1) in Theorem A is immediate from the
boundedness of the triangular projection on Schatten $p$-classes for
$1<p<\infty$ and the transference Lemma 1.1 above. This completes the proof of
Theorem A. $\square$

###  1.5. Relatively compact domains.

Using a partition of unity argument, it is not difficult to prove that Theorem
A holds globally for relatively compact fully transverse domains $\Sigma$.
More precisely, let $p\in(1,\infty)\setminus\\{2\\}$ and consider a relatively
compact domain $\Sigma$ in $M\times N$ which is transverse at every point of
$\partial\Sigma$. Then $S_{\Sigma}$ is an $S_{p}$-bounded multiplier if and
only if any of the equivalent conditions (2) and (3) in the statement of
Theorem A holds at every point of the boundary.

######  Remark 1.5.

_The fact that $\Sigma$ is relatively compact is crucial in the preceding
argument. For instance, (2) holds trivially at every boundary point for every
fully transverse $\mathcal{C}^{1}$-domain of $\mathbf{R}\times\mathbf{R}$. But
there are examples of such domains —which are Toeplitz, arising from Fourier
symbols— that do not define an $S_{p}$ multiplier for any $p\neq 2$. An
explicit construction is given in [6, Appendix A]._

At this point, it is interesting to observe the difference here between
Fourier and Schur idempotents. We know from Fefferman’s theorem [15] that
there are no Fourier $L_{p}$-idempotents associated to smooth compact domains.
However, there are plenty such Schur idempotents: necessarily nonToeplitz,
since Toeplitz symbols give rise to Fourier idempotents. A funny instance is
precisely given by other forms of ball multipliers
$\Sigma_{R}=\\{(x,y)\hskip-1.0pt\in\hskip-1.0pt\mathbf{R}^{n}\times\mathbf{R}^{n}\hskip-2.0pt:|x|^{2}+|y|^{2}<R^{2}\\}$,
which are clearly $S_{p}$-bounded and have been recently used by Chuah-Liu-Mei
in their recent paper [9, Example 4.4]. Theorem A proves in addition that the
spheres $\partial\Sigma_{R}$ satisfy the zero-curvature condition (2). More
intriguing examples are the spherical Hilbert transforms defined in the
Introduction as

$H_{\mathbf{S}}:A\mapsto\Big{(}\mathrm{sgn}\langle x,y\rangle
A_{xy}\Big{)}_{x,y\in\mathbf{S}^{n}}.$

More generally, we also define $H_{\mathbf{S},\delta}(A)=(\chi_{\langle
x,y\rangle>\delta}A_{xy})$ for any $\delta\in(-1,1)$. The case $\delta=0$
corresponds to $\frac{1}{2}(1+\mathrm{sgn}\langle x,y\rangle)$ which is
formally equivalent to $H_{\mathbf{S}}$ above.

######  Corollary 1.6.

Let us fix $1<p\neq 2<\infty$. Then, the $n$-dimensional spherical Hilbert
transforms $H_{\mathbf{S},\delta}$ are all $S_{p}$-bounded for $n=1$ and
$S_{p}$-unbounded for $n\geq 2$.

###### Proof.

Spherical Hilbert transforms arise from relatively compact domains whose
boundary is fully transverse. In particular, we may apply Theorem A. Next, in
dimension $1$ the assertion follows since the zero-curvature condition (2) is
trivially satisfied. Alternatively, the symbol can be expressed as a
triangular truncation in terms of the polar coordinates of $x$ and $y$. When
$n\geq 2$, it is easily checked that the tangent spaces at
$\partial\Sigma_{x_{1}}$ and $\partial\Sigma_{x_{2}}$ differ at their
intersection points. This was illustrated for $n=2$ in Figure 1. Theorem A
implies the assertion. ∎

######  Remark 1.7.

_Alternatively, Corollary 1.6 also follows as a special case of Corollary B2.
Indeed, Lemma 1.1 implies that $H_{\mathbf{S},\delta}$ has the same norm as
the Schur multiplier on $\mathrm{SO}(n+1)\times\mathrm{SO}(n+1)$ with symbol
$(g,h)\mapsto\mathrm{sgn}((g^{-1}h)_{1,1})$, which by [7] coincides with the
cb-norm of the Fourier multipler with symbol $g\mapsto\mathrm{sgn}(g_{1,1})$.
But for $n\geq 2$ $\mathrm{SO}(n+1)$ is a simple Lie group not locally
isomorphic to $\mathrm{SL}_{2}(\mathbf{R})$, so it does not carry _any_
idempotent multiplier._

######  Remark 1.8.

_We may also consider the symbols
$\Sigma_{\delta}=\\{(x,y)\in\mathbf{R}^{n}:\langle x,y\rangle>\delta\\}$ in
the full Euclidean space for $n\geq 2$. In this case, Theorem A gives
$S_{p}$-unboundedness for $(n,\delta)\neq(2,0)$. By [26, Theorem 1.18] and
since $S_{\Sigma_{0}}=H_{\mathbf{S},0}\otimes\mathrm{id}_{\mathbf{R}_{+}}$, it
turns out that $S_{p}$-boundedness for $(n,\delta)=(2,0)$ follows from
Corollary 1.6._

###  1.6. Curvature on smoother domains.

Our curvature condition (2) admits an alternative formulation under additional
regularity. Let $\Sigma$ be a $\mathcal{C}^{2}$-domain. Then
$\Sigma\cap(U\times V)=\big{\\{}(x,y):F(x,y)>0\big{\\}}$ for some
$\mathcal{C}^{2}$-function $F:M\times N\to\mathbf{R}$ and small enough
neighbourhoods $U,V$. Our curvature condition holds if and only if we have

$\Big{\langle}d_{x}\hskip-1.0ptd_{y}F(x,y),u\otimes
v\Big{\rangle}:=u^{\mathrm{t}}\cdot\Big{(}\partial_{x_{j}}\partial_{y_{k}}F(x,y)\Big{)}_{j,k}\cdot
v=0$

for $(u,v)\in\ker d_{x}F(x,y)\times\ker d_{y}F(x,y)$ at every
$(x,y)\in\partial\Sigma\cap(U\times V)$. The argument is quite simple. By
fixing boundary points $(x,y)$ and vectors $(u,v)$ as specified above, let
$\gamma:[0,1]\to\partial\Sigma^{y}\cap U$ be a curve with $\gamma(0)=x$ and
$\gamma^{\prime}(0)=u$, and set $h(s)=d_{y}F(\gamma(s),y)$. The curvature
condition (2) means that $h(s)=\alpha(s)h(0)$ for some nonvanishing function
$\alpha:[0,1]\to\mathbf{R}$. In particular, we get

$\Big{\langle}d_{x}\hskip-1.0ptd_{y}F(x,y),u\otimes
v\Big{\rangle}=\big{\langle}h^{\prime}(0),v\big{\rangle}=\alpha^{\prime}(0)\big{\langle}h(0),v\big{\rangle}=0.$

Reciprocally, assume that the $\mathcal{C}^{2}$-curvature condition above
holds. Consider a curve $\gamma\colon[0,1]\to\partial\Sigma^{y}\cap U$ and
define $h$ as above. Since we have $\gamma^{\prime}(s)\in\ker
d_{x}F(\gamma(s),y)$ and $h^{\prime}(s)=\gamma^{\prime}(s)^{\mathrm{t}}\cdot
d_{x}d_{y}F(\gamma(s),y)$ by construction, it turns out that $\langle
h^{\prime}(s),v\rangle$ equals $\langle
d_{x}d_{y}F(\gamma(s),y),\gamma^{\prime}(s)\otimes v\rangle$ for any $v\in\ker
d_{y}F(\gamma(s),y)$. Applying the $\mathcal{C}^{2}$-curvature condition, this
implies that $h^{\prime}(s)$ is parallel to $h(s)$ for every $s$, which leads
to the ODE

$\left.\begin{array}[]{rcl}h^{\prime}(s)&=&\lambda(s)h(s)\\\
h(0)&=&d_{y}F(x,y)\end{array}\right\\}\Rightarrow
h(s)=\exp\Big{(}\int_{0}^{s}\lambda(t)dt\Big{)}h(0)=\alpha(s)h(0)$

for a nonvanishing $\alpha:[0,1]\to\mathbf{R}$. This implies condition (2) in
Theorem A.

######  Remark 1.9.

_In this form, ( 2) is invariant under exchanging $x$ and $y$, which is clear
a posteriori without the $\mathcal{C}^{2}$ assumption, since both (1) and (3)
are. On the other hand, condition (2) in Theorem A seems new, while its
$\mathcal{C}^{2}$-form above is quite similar to the _rotational curvature_
$\det[d_{x}d_{y}F(x,y)]$ defined by Stein in [43, XI.3.1]_.

###  1.7. On the transversality condition.

The transversality assumption has been essential in our proofs of
(1)$\Rightarrow$(2)$\Rightarrow$(3) in Theorem A, but it is not clear that it
is really needed for the statement. Indeed, conditions (1) and (3) make sense
without it, and (2) is already meaningful if one only assumes that
$\mathbf{n}_{2}(x_{0},y_{0})\neq 0$, and we do have an example where the
equivalence fails. It is likely that such examples can be found, but probably
not for domains with analytic boundary. We leave these questions as open
problems. In the degenerate case where $\mathbf{n}_{1}$ is identically $0$, or
equivalently when $\Sigma$ is locally of the form
$\Sigma=\\{(x,y):y\in\Omega\\}$, all conditions in Theorem A hold. The
$S_{p}$-boundedness is in that case even true for $1\leq p\leq\infty$ because
the Schur multiplier whose symbol is the indicator function of $\Sigma$ is
just the right-multiplication by the orthogonal projection on $L_{2}(\Omega)$.

##  2\. Idempotent Fourier multipliers on Lie groups

Let $\mathrm{G}$ be a Lie group, that we equip with a left Haar measure. As to
every locally compact group, we can associate to it:

  * •

Its von Neumann algebra $\mathcal{L}\mathrm{G}$.

  * •

The noncommutative $L_{p}$ spaces $L_{p}(\mathcal{L}\mathrm{G})$ for $1\leq
p<\infty$.

  * •

The Fourier $L_{p}$-multipliers $T_{m}$ with symbol
$m\colon\mathrm{G}\to\mathbf{C}$.

The group von Neumann algebra $\mathcal{L}\mathrm{G}$ is the weak-$*$ closure
in $B(L_{2}(\mathrm{G}))$ of the algebra of convolution operators
$\lambda(f):\xi\in L_{2}(\mathrm{G})\mapsto f\ast\xi$ for
$f\in\mathcal{C}_{c}(\mathrm{G})$. When $\mathrm{G}$ is unimodular, its
$L_{p}$-theory is quite elementary: $\mathcal{L}\mathrm{G}$ carries a natural
semifinite trace $\tau$ given by
$\tau(\lambda(f)^{*}\lambda(f))=\int|f(g)|^{2}dg$ for every $f\in
L_{2}(\mathrm{G})$ with $\lambda(f)\in\mathcal{L}\mathrm{G}$ and
$L_{p}(\mathcal{L}\mathrm{G})$ is then defined as the completion of
$\\{x\in\mathcal{L}\mathrm{G}:\|x\|_{p}<\infty\\}$ for the norm
$\|x\|_{p}=\tau(|x|^{p})^{1/p}$. It turns out that
$L_{p}(\mathcal{L}\mathrm{G})$ contains
$\\{\lambda(f):f\in\mathcal{C}_{c}(\mathrm{G})\ast\mathcal{C}_{c}(\mathrm{G})\\}$
as a dense subspace. A bounded measurable $m\colon\mathrm{G}\to\mathbf{C}$
defines a Fourier $L_{p}$-multiplier if $\lambda(f)\mapsto\lambda(mf)$ extends
to a bounded map $T_{m}$ on $L_{p}(\mathcal{L}\mathrm{G})$. These definitions
are more involved for nonunimodular groups and will be recalled in Section 2.3
below.

When $p=1,\infty$, a bounded measurable function
$m\colon\mathrm{G}\to\mathbf{C}$ defines a completely bounded Fourier
$L_{p}$-multiplier if and only if the Schur multiplier associated to the
symbol $(g,h)\mapsto m(gh^{-1})$ —called the Herz-Schur multiplier with symbol
$m$ and denoted $S_{m}$— is completely $S_{p}$-bounded, with same norms [3].
For amenable groups, the same holds for $1<p<\infty$ [7, 32], and it is an
intriguing open problem whether this holds beyond amenable groups. We shall
use that this always holds locally. This phenomenon was discovered recently
[33, Theorem 1.4] when $p$ is an even integer and $\mathrm{G}$ unimodular, and
the following generalizes this to the general case, see [8] for other local
results of similar nature. In what follows, the Fourier support of an element
$x\in L_{p}(\mathcal{L}\mathrm{G})$ will refer to the smallest closed subset
$\Lambda$ such that $T_{m}(x)=x$ for every Fourier $L_{p}$-multiplier with
symbol $m$ which is equal to $1$ on $\Lambda$. When $\mathrm{G}$ is unimodular
and $x=\lambda(f)$ for
$f\in\mathcal{C}_{c}(\mathrm{G})\ast\mathcal{C}_{c}(\mathrm{G})$, it is easy
to see that this coincides with the support of the function $f$.

######  Theorem 2.1.

Let $\mathrm{G}$ be a locally compact group and consider a bounded measurable
function $m\colon\mathrm{G}\to\mathbf{C}$. Then, the following are equivalent
for $p\in(1,\infty)$ and $g_{0}\in\mathrm{G}\hskip-1.0pt:$

  1. (1)

There is a neighbourhood $U$ of $g_{0}$ such that the restriction $T_{m,U}$ of
$T_{m}$ to the space of elements of $L_{p}(\mathcal{L}\mathrm{G})$ Fourier
supported by $U$ is completely bounded.

  2. (2)

There exists a function $\varphi\colon\mathrm{G}\to\mathbf{C}$ which equals
$1$ on a neighbourhood of $g_{0}$ such that $\varphi m$ defines a completely
bounded Fourier multiplier on $L_{p}(\mathcal{L}\mathrm{G})$.

  3. (3)

There are open sets $V,W\subset\mathrm{G}$ with $g_{0}\in VW^{-1}$ such that
the function $(g,h)\in V\times W\mapsto m(gh^{-1})$ defines a completely
bounded Schur multiplier on the Schatten class $S_{p}(L_{2}(V),L_{2}(W))$.

When these conditions hold, we say that $m$ defines locally at $g_{0}$ a
completely bounded Fourier $L_{p}$-multiplier. The proof is given in Section
2.2. We can record the following consequence, which is immediate by looking at
condition (3).

######  Corollary 2.2.

Let $\mathrm{G}$ be a connected Lie group and denote by
$\widetilde{\mathrm{G}}$ its universal cover. Let
$\tilde{g}_{0}\in\widetilde{\mathrm{G}}$ be any lift of $g_{0}\in\mathrm{G}$.
Then $m\colon\mathrm{G}\to\mathbf{C}$ defines locally at $g_{0}$ a completely
bounded Fourier $L_{p}$-multiplier over $\mathcal{L}\mathrm{G}$ if and only
its lift $\widetilde{m}$ defines locally at $\tilde{g}_{0}$ a completely
bounded Fourier $L_{p}$-multiplier as well.

###  2.1. Idempotent multipliers

Now we are ready to prove Theorem B and also Corollaries B1 and B2 from the
Introduction. In fact, we shall prove a slightly expanded version of Theorem B
which includes non simply connected groups and Corollary B1 at once. The
groups $\mathrm{G}_{1},\mathrm{G}_{2},\mathrm{G}_{3}$ in the statement below
refer to the real line $\mathbf{R}$, $\operatorname{Aff}_{+}(\mathbf{R})$ and
$\widetilde{\mathrm{PSL}}_{2}(\mathbf{R})$ as in the Introduction.

######  Theorem 2.3.

Let $p\in(1,\infty)\setminus\\{2\\}$. Let $\mathrm{G}$ be a connected Lie
group, $\Omega\subset G$ a $\mathcal{C}^{1}$-domain and
$g_{0}\in\partial\Omega$ a point in the boundary of $\Omega$. Consider the
following conditions:

  1. (1)

$\chi_{\Omega}$ defines locally at $g_{0}$ a completely bounded Fourier
$L_{p}$-multiplier.

  2. (2)

There is a smooth action $\mathrm{G}\to\mathrm{Diff}(\mathbf{R})$ by
diffeomorphisms on the real line, such that $\Omega$ coincides on a
neighbourhood of $g_{0}$ with $\\{g\in\mathrm{G}\mid g\cdot 0>g_{0}\cdot
0\\}$.

  3. (3)

There is $j\in\\{1,2,3\\}$ and a smooth surjective homomorphism
$f\colon\mathrm{G}\to\mathrm{G}_{j}$ such that the domain $\Omega$ coincides
on a neighbourhood of $g_{0}$ with $g_{0}f^{-1}(\Omega_{j})$.

  4. (4)

$\partial\Omega=g_{0}\exp(\mathfrak{h})$ locally near $g_{0}$ for some
codimension $1$ Lie subalgebra $\mathfrak{h}\subset\mathfrak{g}$.

Then $(1)\Leftrightarrow(4)\Leftarrow(2)\Leftrightarrow(3)$. If $\mathrm{G}$
is simply connected, then we also have $(4)\Rightarrow(2)$.

###### Proof.

The main difficulty is to prove the equivalence $\eqref{item:local-Hilbert-
tranform}\Leftrightarrow\eqref{item:Lie-subalg}$, which we leave to the end of
the proof. The implication
$\eqref{item:local_subgroup}\Rightarrow\eqref{item:Lie-subalg}$ is clear, with
$\mathrm{H}=\exp(\mathfrak{h})$ the stabilizer of $0$. Under the assumption
that $\mathrm{G}$ is simply connected, the converse $\eqref{item:Lie-
subalg}\Rightarrow\eqref{item:local_subgroup}$ holds by a Theorem of Mostow
[31], which implies that $\mathrm{H}$ is a closed subgroup. Therefore,
$\mathrm{G}/\mathrm{H}$ is a $1$-dimensional manifold that is simply
connected, so is diffeomorphic to $\mathbf{R}$. The implication
$\eqref{item:three-HTs}\Rightarrow\eqref{item:local_subgroup}$ is also clear
because $\mathrm{G}_{j}$ is given as a group of diffeomorphisms of
$\mathbf{R}$ with $\Omega_{j}=\\{g\in\mathrm{G}_{j}\mid g\cdot 0>0\\}$. The
converse $\eqref{item:local_subgroup}\Rightarrow\eqref{item:three-HTs}$
follows from Lie’s classification of (local) actions by diffeomorphism on the
real line [28], see also [45] for modern presentations and [16] for the global
aspect. More precisely, the fact that $g_{0}$ belongs to the boundary of
$\Omega$ implies that $0$ is not fixed by $\mathrm{G}$ —here we use that
$\mathrm{G}$ is connected— and the $\mathrm{G}$-orbit of $0$ is an open
interval, so by identifying it with $\mathbf{R}$ we can assume that the
$\mathrm{G}$-action is transitive. In that case, the image of $\mathrm{G}$ in
$\mathrm{Diff}(\mathbf{R})$ is one of the three groups in condition (3) or
Corollary B1, see [16, Section 4.1] for the details.

Next, let us focus on the equivalence $\eqref{item:local-Hilbert-
tranform}\Leftrightarrow\eqref{item:Lie-subalg}$ for general Lie groups. If we
translate $\Omega$ by $g_{0}^{-1}$, we may assume that $g_{0}=e$ and the
tangent space of $\mathrm{G}$ at $g_{0}$ identifies with its Lie algebra
$\mathfrak{g}$. Also, the tangent space of $\partial\Omega$ at $g_{0}$
identifies with a codimension $1$ subspace $\mathfrak{h}$ of $\mathfrak{g}$.
Define the $\mathcal{C}^{1}$-manifold

$\widetilde{\Omega}=\Big{\\{}(g,h)\in G\times G\mid gh\in\Omega\Big{\\}}.$

Its sections $\widetilde{\Omega}_{g}$ and $\widetilde{\Omega}^{h}$ are left
and right translates of the $\mathcal{C}^{1}$-domain $\Omega$

(2.1)
$\widetilde{\Omega}_{g}:=\big{\\{}h:(g,h)\in\widetilde{\Omega}\big{\\}}=g^{-1}\Omega\quad\mbox{and}\quad\widetilde{\Omega}^{h}:=\big{\\{}g:(g,h)\in\widetilde{\Omega}\big{\\}}=\Omega
h^{-1}.$

In particular, $\widetilde{\Omega}$ is transverse at every point of its
boundary. By Lemma 1.1 and Theorem 2.1 we know that (1) is equivalent to the
existence of a neighbourhood of the identity $U\subset\mathrm{G}$ such that
$\chi_{\widetilde{\Omega}}$ defines a Schur multiplier on $S_{p}(L_{2}(U))$.
By Theorem A this is equivalent to the existence of a neighbourhood of the
identity $V\subset\mathrm{G}$ such that both conditions below hold:

(2.2) $\displaystyle
T_{h}\partial\widetilde{\Omega}_{g_{1}}=T_{h}\partial\widetilde{\Omega}_{g_{2}}$
$\displaystyle\textrm{ for every }g_{1},g_{2},h\in V\textrm{ such that
}g_{1}h,g_{2}h\in\partial\Omega.$ (2.3) $\displaystyle
T_{g}\partial\widetilde{\Omega}^{h_{1}}=T_{g}\partial\widetilde{\Omega}^{h_{2}}$
$\displaystyle\textrm{ for every }g,h_{1},h_{2}\in V\textrm{ such that
}gh_{1},gh_{2}\in\partial\Omega.$

By the above idenfications (2.1), if we denote by
$L_{x},R_{x}\colon\mathrm{G}\to\mathrm{G}$ the left and right multiplication
by $x$, these conditions are equivalent to the existence of a neighbourhood of
the identity $W$ such that

(2.4) $\displaystyle
d_{x_{1}}L_{x_{2}x_{1}^{-1}}(T_{x_{1}}\partial\Omega)=T_{x_{2}}\partial\Omega$
$\displaystyle\textrm{ for every }x_{1},x_{2}\in\partial\Omega\cap W.$ (2.5)
$\displaystyle
d_{x_{1}}R_{x_{1}^{-1}x_{2}}(T_{x_{1}}\partial\Omega)=T_{x_{2}}\partial\Omega$
$\displaystyle\textrm{ for every }x_{1},x_{2}\in\partial\Omega\cap W.$

Indeed, taking $x_{j}=g_{j}h$ for $j=1,2$ we have

$T_{h}(g_{j}^{-1}\partial\Omega)=d_{x_{j}}L_{g_{j}^{-1}}(T_{x_{j}}\partial\Omega).$

Composing by $(d_{x_{2}}L_{g_{2}^{-1}})^{-1}=d_{h}L_{g_{2}}$, and using
$(d_{h}L_{g_{2}})\circ(d_{x_{1}}L_{g_{1}^{-1}})=d_{x_{1}}L_{x_{2}x_{1}^{-1}}$
by the chain rule, we see that (2.2) is equivalent to (2.4). The equivalence
for right multiplication maps is entirely similar. Next, recalling that
$T_{e}\partial\Omega=\mathfrak{h}$ the above conditions can be written in the
equivalent forms

(2.6) $\displaystyle d_{e}L_{x}(\mathfrak{h})=T_{x}\partial\Omega$
$\displaystyle\textrm{ for every }x\in\partial\Omega\cap W.$ (2.7)
$\displaystyle d_{e}R_{x}(\mathfrak{h})=T_{x}\partial\Omega$
$\displaystyle\textrm{ for every }x\in\partial\Omega\cap W.$

If we remember that $\operatorname{Ad}_{x}=d_{e}(R_{x^{-1}}L_{x})$, we obtain
that this system is equivalent to
$d_{e}L_{x}(\mathfrak{h})=T_{x}\partial\Omega$ and
$\operatorname{Ad}_{x}\mathfrak{h}=\mathfrak{h}$ for every
$x\in\partial\Omega\cap W$. Therefore, we have proved that (1) at $g_{0}=e$ is
equivalent to the existence of a neighbourhood of the identity $W$ such that
$T_{x}\partial\Omega=d_{e}L_{x}(\mathfrak{h})$ and
$\operatorname{Ad}_{x}\mathfrak{h}=\mathfrak{h}$ for every
$x\in\partial\Omega\cap W$. These conditions clearly hold if $\mathfrak{h}$ is
a Lie algebra and $\partial\Omega$ locally coincides with the exponential of a
neighbourhood of $0$ in $\mathfrak{h}$. Conversely, assume
$T_{x}\partial\Omega=d_{e}L_{x}(\mathfrak{h})$ and
$\operatorname{Ad}_{x}(\mathfrak{h})=\mathfrak{h}$ for every
$x\in\partial\Omega\cap W$. Making $x$ go to the identity element $e$ in the
second condition, we deduce that
$\operatorname{ad}_{\mathrm{X}}(\mathfrak{h})\subset\mathfrak{h}$ for every
$\mathrm{X}\in\mathfrak{h}$. That is, $\mathfrak{h}$ is a Lie subalgebra. By
the local uniqueness of a manifold in $\mathrm{G}$ containing $e$ and whose
tangent space at $x$ is $d_{e}L_{x}(\mathfrak{h})$ (Frobenius’ theorem), we
deduce that $\partial\Omega$ is locally the exponential of a neighbourhood of
$0$ in $\mathfrak{h}$. This completes the proof. ∎

######  Remark 2.4.

_By a partition of the unity argument, the following global form of Theorem
2.3 holds: if $p\in(1,\infty)\setminus\\{2\\}$, $\mathrm{G}$ is a connected
Lie group and $\Omega\subset\mathrm{G}$ a relatively compact
$\mathcal{C}^{1}$-domain, then $\chi_{\Omega}$ defines a Fourier
cb-$L_{p}$-multiplier if and only if the condition (4) holds for every point
$g_{0}$ in the boundary of $\Omega$._

###### Proof of Corollary  _B2_.

Assertion i) follows since the quotient of a nilpotent Lie algebra remains
nilpotent, so the nonnilpotent examples in Theorem 2.3 (3) cannot happen when
$\mathrm{G}$ is nilpotent. Assertions ii) and iii) follow immediately from
Lie’s classification [28]: up to isomorphism, there is a unique pair
$(\mathfrak{h},\mathfrak{g})$ where $\mathfrak{g}$ is a simple Lie algebra and
$\mathfrak{h}$ is a codimension $1$ subalgebra. It is given by
$\mathfrak{g}=\mathfrak{sl}_{2}$ and $\mathfrak{h}$ the subalgebra of upper-
triangular matrices. This completes the proof. ∎

###  2.2. Local Fourier-Schur transference

The rest of this paper will be devoted to justify Theorem 2.1. A form of the
following lemma was proved in [33, Lemma 1.3] for $p$ an even integer and
$\mathrm{G}$ unimodular, which was enough for the applications there. Here we
need a form valid for every $p$ and every locally compact group.

######  Lemma 2.5.

Let $V,W\subset\mathrm{G}$ be open sets and $U\subset VW^{-1}$ be a relatively
compact open subset. There are maps $J_{p}\colon
L_{p}(\mathcal{L}\mathrm{G})\to S_{p}(L_{2}(V),L_{2}(W))$ for $1\leq
p\leq\infty$ intertwining Fourier and Herz-Schur multipliers and such that

$C^{-1}\|x\|_{p}\leq\|J_{p}(x)\|_{p}\leq C\|x\|_{p}$

for some constant $C$, every $n\geq 1$ and $x\in M_{n}\otimes
L_{p}(\mathcal{L}\mathrm{G})$ Fourier supported in $U$.

This is the key point in the proof of Theorem 2.1. For the proof, we will need
the following form of the Tauberian theorem, due to Eymard [14]. Let
$A(\mathrm{G})$ be the Fourier algebra of $\mathrm{G}$. That is

$A(\mathrm{G})=\Big{\\{}g\mapsto\int\phi(gh)\psi(h)\,dh:\phi,\psi\in
L_{2}(\mathrm{G})\Big{\\}}.$

######  Lemma 2.6.

Let $f\in A(\mathrm{G})$ be a function which does not vanish on some compact
set $\mathrm{Q}\subset\mathrm{G}$. Then, there exists another function
$\rho\in A(\mathrm{G})$ such that $\rho f=1$ over $\mathrm{Q}$.

###### Proof.

The set $A_{\mid_{\mathrm{Q}}}(\mathrm{G})$ of restrictions to $\mathrm{Q}$ of
elements of $A(\mathrm{G})$ is a commutative Banach algebra —as a quotient of
the Banach algebra $A(\mathrm{G})$— that is unital because $\mathrm{Q}$ is
compact. Moreover, by [14, Théorème 3.34] its spectrum coincides with
$\mathrm{Q}$ through the evaluation maps $f\mapsto f(q)$. So by Gelfand’s
theorem, the condition that $f$ does not vanish on $\mathrm{Q}$ exactly means
that $f$’s image in $A_{\mid_{\mathrm{Q}}}(\mathrm{G})$ is invertible. ∎

For the reader’s convenience, we first prove Lemma 2.5 and Theorem 2.1 for
unimodular groups, and explain in the next paragraph how to modify the
definition of Fourier multiplier and the argument for nonunimodular groups.

###### Proof of Lemma  _2.5_ for $\mathrm{G}$ unimodular.

Let $\phi\in\mathcal{C}_{c}(V)_{+}$ and $\psi\in\mathcal{C}_{c}(W)_{+}$ such
that $\int\phi(gh)\psi(h)dh\geq 1$ for every $g\in U$. They exist by the
assumption that the open set $U$ is relatively compact in $VW^{-1}$. Consider
the map

$J_{p}\colon L_{p}(\mathcal{L}\mathrm{G})\ni
x\mapsto\phi^{\frac{1}{p}}x\psi^{\frac{1}{p}}\in S_{p}(L_{2}(V),L_{2}(W)),$

where we identify $\phi,\psi$ with the operators of multiplication by
$\phi,\psi$. The convention is that $0^{\frac{1}{\infty}}=0$. We claim that
the maps $J_{p}$ are completely bounded with cb-norm

$\|J_{p}\|_{\mathrm{cb}(L_{p},S_{p})}\leq\|\phi\|_{L_{2}(\mathrm{G})}^{\frac{1}{p}}\|\psi\|_{L_{2}(\mathrm{G})}^{\frac{1}{p}}$

whenever $1\leq p\leq\infty$. By interpolation, it is enough to justify the
extreme cases $p=1$ and $p=\infty$. The case $p=\infty$ is clear. For the case
$p=1$, we factorize $x=x_{1}x_{2}$ so that $J_{1}(x)=\phi x_{1}\cdot
x_{2}\psi$. Take them so that $\|x\|_{1}=\|x_{1}\|_{2}\|x_{2}\|_{2}$, and it
suffices to show that both factors are bounded in
$S_{2}(L_{2}(\mathrm{G}))\simeq L_{2}(\mathrm{G}\times\mathrm{G})$ by
$\|\phi\|_{L_{2}(\mathrm{G})}\|x_{1}\|_{L_{2}(\mathcal{L}\mathrm{G})}$ and
$\|\psi\|_{L_{2}(\mathrm{G})}\|x_{2}\|_{L_{2}(\mathcal{L}\mathrm{G})}$
respectively. Using that

$J_{\infty}:x\mapsto\Big{(}\widehat{x}(gh^{-1})\Big{)},$

the expected bounds follow from Plancherel theorem
$L_{2}(\mathcal{L}\mathrm{G})\simeq L_{2}(\mathrm{G})$. Moreover when
$x=\lambda(f)$, the operator $J_{p}(x)$ has kernel
$(\phi(g)^{1/p}f(gh^{-1})\psi(h)^{1/p})$. Thus it is clear that the map
$J_{p}$ intertwines the Fourier multiplier with symbol $g\mapsto m(g)$ and the
Schur multiplier with symbol $(g,h)\mapsto m(gh^{-1})$.

The inequality $C^{-1}\|x\|_{p}\leq\|J_{p}(x)\|_{p}$ is a bit more involved.
Let $f\in M_{n}\otimes\mathcal{C}_{c}(U)$ with $\lambda(f)\in
L_{p}(M_{n}\otimes\mathcal{L}\mathrm{G})$ and assume that $x=\lambda(f)$ by
density. Let $q$ be the conjugate exponent of $p$ and $\gamma\in
M_{n}\otimes\mathcal{C}_{c}(\mathrm{G})$ with $\lambda(\gamma)\in
L_{q}(M_{n}\otimes\mathcal{L}\mathrm{G})$. By Lemma 2.6, there is $\rho\in
A(\mathrm{G})$ such that

$\rho(g)\int_{\mathrm{G}}\phi(gh)\psi(h)\,dh=1\quad\mbox{for all}\quad g\in
U.$

The reason why we require that $\rho$ belongs to $A(\mathrm{G})$ is because we
want to ensure that $\rho$ defines a completely bounded Fourier
$L_{p}$-multiplier. Then we have

$\displaystyle\operatorname{Tr}\otimes\operatorname{Tr}_{n}\big{(}J_{p}(\lambda(f))J_{q}(\lambda(\rho\gamma))^{*}\big{)}$
$\displaystyle=\operatorname{Tr}_{n}\int_{\mathrm{G}\times\mathrm{G}}\phi(g)f\gamma^{*}(gh^{-1})\rho(gh^{-1})\psi(h)dsdt$
$\displaystyle=\int_{\mathrm{G}}\operatorname{Tr}_{n}\big{(}f(g)\gamma(g)^{*}\big{)}\Big{[}\rho(g)\int_{\mathrm{G}}\phi(gh)\psi(h)\,dh\Big{]}\,dg$
$\displaystyle=\int_{\mathrm{G}}\operatorname{Tr}_{n}\big{(}f(g)\gamma(g)^{*}\big{)}\,dg=\tau\otimes\operatorname{Tr}_{n}\big{(}\lambda(f)\lambda(\gamma)^{*}\big{)}.$

In the last line, we used Lemma 2.6 and $\mathrm{supp}f\subset U$. By Hölder’s
inequality, we get

$\displaystyle\big{|}\tau\otimes\operatorname{Tr}_{n}(\lambda(f)\lambda(\gamma)^{*})\big{|}$
$\displaystyle\leq\|J_{p}(\lambda(f))\|_{p}\|J_{q}(\lambda(\rho\gamma)\|_{q}$
$\displaystyle\leq\|J_{q}\|_{\mathrm{cb}}\|J_{p}(\lambda(f))\|_{p}\|\lambda(\rho\gamma)\|_{q}$
$\displaystyle\leq\|J_{q}\|_{\mathrm{cb}}\|T_{\rho}\|_{\mathrm{cb}}\|\lambda(\gamma)\|_{q}\|J_{p}(\lambda(f))\|_{p}.$

Taking the sup over $\gamma$, we get $C^{-1}\|x\|_{p}\leq\|J_{p}(x)\|_{p}$ for
$C=\|J_{q}\|_{\mathrm{cb}}\|T_{\rho}\|_{\mathrm{cb}}<\infty$. ∎

With the same argument as in [33], we deduce:

###### Proof of Theorem  _2.1_.

The implication (1)$\Rightarrow$(2) is easy. Indeed, if (1) holds and
$\varphi\in A(\mathrm{G})$ is supported in $U$ and equal to $1$ on a
neighbourhood of $g_{0}$ (its existence is guaranteed by Lemma 2.6), then
$T_{\varphi}$ is completely bounded on $L_{p}(\mathcal{L}\mathrm{G})$ for
every $1\leq p\leq\infty$, and takes values in the space of elements Fourier
supported in $U$. In particular, $T_{m\varphi}=T_{m,U}\circ T_{\varphi}$ is
also completely bounded. The implication (2)$\Rightarrow$(3) follows from [7,
Theorem 4.2] which implies that $(g,h)\in\mathrm{G}\times\mathrm{G}\mapsto
m(gh^{-1})\varphi(gh^{-1})$ defines a completely bounded Fourier
$L_{p}$-multiplier. Thus, we get (3) if $V,W$ are chosen so that $\varphi=1$
on $VW^{-1}$. Finally, (3)$\Rightarrow$(1) follows from Lemma 2.5.∎

###  2.3. Nonunimodular groups

Let $\mathrm{G}$ be an arbitrary locally compact group with modular function
$\Delta\colon\mathrm{G}\to\mathbf{R}_{+}$. Our choice of $\Delta$ is
characterized by the following identity for all
$f\in\mathcal{C}_{c}(\mathrm{G})$

$\int_{\mathrm{G}}f(hg)\,dh=\Delta(g)^{-1}\int_{\mathrm{G}}f(h)\,dh.$

When $\mathrm{G}$ is not unimodular —that is, $\Delta$ is not the constant $1$
function— the natural weight $\lambda(f)^{*}\lambda(f)\mapsto\int|f|^{2}$ on
$\mathcal{L}\mathrm{G}$ is not tracial. Even when $\mathcal{L}\mathrm{G}$ is
semifinite, it is better to work with the general definition of $L_{p}$ spaces
associated to a von Neumann algebra. Several concrete descriptions are
possible: Haagerup’s original one [20], Kosaki’s complex interpolation [24],
Connes-Hilsum’s [12, 21]… see [41]. Here we will use the Connes-Hilsum spatial
description because we want to rely on some results from [7, 44], to which we
refer for precise definitions. In that case, $L_{p}(\mathcal{L}\mathrm{G})$ is
realized as a space of unbounded operators on $L_{2}(\mathrm{G})$.

In [7], Caspers and the second-named author defined Fourier
$L_{p}$-multipliers for symbols that ensure that the Fourier multiplier is
completely bounded for every $1\leq p\leq\infty$. Here we extend the
definition, allowing to talk about Fourier multipliers for a single $p$, and
possibly only bounded. The shortest way to do so properly in this context is
by using Terp’s Hausdorff-Young inequality [44].

Informally, a typical element of $L_{p}(\mathcal{L}\mathrm{G})$ is of the form
$\lambda(f)\Delta^{\frac{1}{p}}$ —where we are identifying the function
$\Delta$ with the densely defined operator of multiplication by $\Delta$ on
$L_{2}(\mathrm{G})$— for some _suitable_ function $f$. Keeping at the informal
level, the Fourier multiplier with symbol $m\colon\mathrm{G}\to\mathbf{C}$
should be, whenever it exists, the operator acting as follows

$\lambda(f)\Delta^{\frac{1}{p}}\mapsto\lambda(mf)\Delta^{\frac{1}{p}}.$

Making this definition precise requires some lengthy and unpleasant
discussions about domains/cores of unbounded operators, but fortunately we can
rely on the results from [44], where these discussions have been performed. We
shall need to distinguish the cases $p\geq 2$ and $p\leq 2$. Let
$q=\frac{p}{p-1}$ be the conjugate exponent of $p$. When $p\geq 2$, the
Fourier transform

$\mathcal{F}_{q}\colon L_{q}(\mathrm{G})\to L_{p}(\mathcal{L}\mathrm{G})$

is an injective norm $1$ linear map with dense image, where
$\mathcal{F}_{q}(f)$ is defined as a suitable extension of
$\lambda(f)\Delta^{1/p}$, see [44, Theorem 4.5]. When $p\leq 2$, the adjoint
of $\mathcal{F}_{p}$ gives a norm $1$ injective map with dense image
$\overline{\mathcal{F}}_{p}\colon L_{p}(\mathcal{L}\mathrm{G})\to
L_{q}(\mathrm{G})$. If $I_{q}$ denotes the isometry of $L_{q}(\mathrm{G})$
defined by $I_{q}(f)(g)=f(g^{-1})\Delta(g)^{-1/q}$, we know from [44,
Proposition 1.15] that every element $x$ of $L_{p}(\mathcal{L}\mathrm{G})$ is
a suitable extension of $\lambda(f)\Delta^{1/p}$ for
$f=I_{q}\circ\overline{\mathcal{F}}_{p}(x)\in L_{q}(\mathrm{G})$. In the
particular case $p=2$, these two statements together yield Plancherel’s
formula : $\mathcal{F}_{2}$ is a unitary. If $p=1$, the image of
$\overline{\mathcal{F}}_{1}$ is the Fourier algebra $A(\mathrm{G})$, and
following standard notation we write

(2.8) $\operatorname{tr}(x)=\varphi(e)\quad\textrm{ if
}\quad\overline{\mathcal{F}}_{1}(x)=\varphi.$

######  Definition 2.7.

Let $1<p<\infty$ and $m\in L_{\infty}(\mathrm{G})$. We say that $m$ defines a
bounded Fourier $L_{p}$-multiplier when the condition below holds according to
the value of $p\hskip-1.0pt:$

  * •

Case $p\geq 2$. The map

$\mathcal{F}_{q}(f)\mapsto\mathcal{F}_{q}(mf)$

(densely defined on $\mathcal{F}_{q}(L_{q}(\mathrm{G}))$) extends to a bounded
map $T_{m}$ on $L_{p}(\mathcal{L}\mathrm{G})$.

  * •

Case $p\leq 2$. The multiplication by $m$ preserves the image of
$I_{q}\circ\overline{\mathcal{F}}_{p}$ when the map
$T_{m}:x\mapsto(I_{q}\circ\overline{\mathcal{F}}_{p})^{-1}(m(I_{q}\circ\overline{\mathcal{F}}_{p}(x)))$
is a bounded map on $L_{p}(\mathcal{L}\mathrm{G})$.

We say that $m$ defines a completely bounded Fourier $L_{p}$-multiplier when
$m$ defines a bounded Fourier $L_{p}$-multiplier and the Fourier multiplier
$T_{m}$ is completely bounded.

It follows from the above definition that $m$ defines a (completely) bounded
$L_{p}$ multiplier if and only it defines a (completely) bounded $L_{q}$
multiplier, and in that case

$\operatorname{tr}(T_{m}(x)y^{*})=\operatorname{tr}(x(T_{\overline{m}}(y))^{*})\quad\mbox{for
all}\quad x\in L_{p}(\mathcal{L}\mathrm{G}),y\in
L_{q}(\mathcal{L}\mathrm{G}).$

Once we have polished the definition of Fourier $L_{p}$-multipliers in
nonunimodular group von Neumann algebras, we can extend a Cotlar identity from
[18] to arbitrary locally compact groups.

######  Example 2.8.

Let $\mathrm{G}\to\mathrm{Homeo}_{+}(\mathbf{R})$ be a continuous action of a
connected Lie group. Then, the indicator function $m$ of
$\\{g\in\mathrm{G}\mid g\cdot 0>0\\}$ defines a completely bounded $L_{p}$
Fourier multiplier on $G$ with completely bounded norm $\leq
2\max\\{p,\frac{p}{p-1}\\}$.

###### Proof.

Let $m$ be the indicator function of $\\{g\in\mathrm{G}\mid g\cdot 0>0\\}$. It
suffices to prove the following implication for every $2\leq p<\infty$: if $m$
defines a completely bounded Fourier $L_{p}$-multiplier with norm $\leq
C_{p}$, then it defines a completely bounded Fourier $L_{2p}$-multiplier with
norm $\leq 2C_{p}$. Indeed, using that $C_{2}=\|m\|_{\infty}=1$, we deduce
$C_{2^{N}}\leq 2^{N}$ for every integer $N$, so by interpolation $C_{p}\leq
2p$ for all $p\geq 2$. By duality the conclusion also holds for $p\leq 2$.

Let $r$ be the dual exponent of $2p$. Let $f\in\mathcal{C}_{c}(\mathrm{G})$
and consider $X=\mathcal{F}_{r}(f)$ and $Y=\mathcal{F}_{r}(mf)$; these are
well-defined elements of $L_{2p}(\mathcal{L}G)$ by [44]. Then, we claim that
the equality below holds

(2.9) $Y^{*}Y=T_{m}(Y^{*}X)+T_{m}(Y^{*}X)^{*}.$

Indeed, this inequality is equivalent to the almost everywhere equality

$(mf)^{*}\ast(mf)=m\big{(}(mf)^{*}\ast f\big{)}+\big{(}m\big{(}(mf)^{*}\ast
f\big{)}\big{)}^{*},$

which follows for the fact that
$m(g^{-1})m(g^{-1}h)=m(h)m(g^{-1})+m(h^{-1})m(g^{-1}h)$ for almost every
$g,h\in\mathrm{G}$. If the whole group $\mathrm{G}$ fixes $0$, this is obvious
because $m$ is identically $0$. Otherwise, the stabilizer of $0$ is a closed
subgroup, so it has measure $0$ and it is enough to justify the equality for
$h\cdot 0\neq 0$. Set $(\alpha,\beta)=(g\cdot 0,h\cdot 0)$ and observe that
$m(g^{-1})m(g^{-1}h)=1$ if and only if $\alpha<\min\\{0,\beta\\}$. Similarly,
we have $m(h)m(g^{-1})=1$ iff $\alpha<0<\beta$ and $m(h^{-1})m(g^{-1}h)=1$ iff
$\alpha<\beta<0$. Therefore the expected identity reduces to the trivial one
$\chi_{\alpha<0\wedge\beta}=\chi_{\alpha<0<\beta}+\chi_{\alpha<\beta<0}$. This
justifies (2.9), both sides of which are in $L_{p}(\mathcal{L}\mathrm{G})$.
Thus, taking the norm and applying the triangle inequality, the hypothesis and
Hölder’s inequality leads to

$\|Y\|_{2p}^{2}\leq 2C_{p}\|X\|_{2p}\|Y\|_{2p}.$

We deduce $\|Y\|_{2p}\leq 2C_{p}\|X\|_{2p}$. Since
$\mathcal{C}_{c}(\mathrm{G})$ is dense in $L_{r}(\mathrm{G})$, we obtain that
$m$ defines a Fourier $L_{2p}$-multiplier with norm $\leq 2C_{p}$. A similar
argument gives the same bound for the completely bounded norm, which concludes
the proof.∎

######  Remark 2.9.

_The Cotlar-type identity from[18] is refined in some cases by (2.9)._

The following summarizes the properties that we need.

######  Lemma 2.10.

Let $1\leq p\leq\infty$ and consider functions $\phi,\psi\in
L_{2p}(\mathrm{G})$, which we identify with $($possibly unbounded$)$
multiplication operators on $L_{2}(\mathrm{G})$. Then

  * •

Given $x\in L_{2p}(\mathcal{L}\mathrm{G})$, $x\phi$ is densely defined and
closable. In fact, its closure $[x\phi]$ belongs to
$S_{2p}(L_{2}(\mathrm{G}))$ and has $S_{2p}$-norm
$\leq\|\phi\|_{L_{2p}(\mathrm{G})}\|x\|_{L_{2p}(\mathcal{L}\mathrm{G})}$.

  * •

There exists a bounded linear map $L_{p}(\mathcal{L}\mathrm{G})\to
S_{p}(L_{2}(\mathrm{G}))$†††That, with a slight abuse of notation, we denote
$z\mapsto\psi z\phi$. sending $y^{*}x$ to $[y\psi^{*}]^{*}[x\phi]$ for every
$x,y\in L_{2p}(\mathcal{L}\mathrm{G})$. It has norm
$\leq\|\phi\|_{L_{2p}(\mathrm{G})}\|\psi\|_{L_{2p}(\mathrm{G})}$.

  * •

If $q$ denotes the conjugate exponent of $p$, consider
$\phi^{\prime},\psi^{\prime}\in L_{2q}(\mathrm{G})$ and $y\in
L_{q}(\mathcal{L}\mathrm{G})$. Then, we have

(2.10) $\operatorname{Tr}\big{(}\phi
x\psi(\phi^{\prime}y\psi^{\prime})^{*}\big{)}=\operatorname{tr}(T_{m}(x)y^{*}),$

where $m\in A(\mathrm{G})$ is the function $\displaystyle
m(g)=\int_{\mathrm{G}}(\phi\overline{\phi^{\prime}})(h)(\psi\overline{\psi^{\prime}})(g^{-1}h)\,dh$.

###### Proof.

When $\phi=\psi$ are indicator functions, the first two points were proved in
[7, Proposition 3.3, Theorem 5.2]. The same argument applies in our case. Let
us justify identity (2.10). First observe that $\phi\overline{\phi^{\prime}}$
and $\psi\overline{\phi^{\prime}}$ belong to $L_{2}(\mathrm{G})$ by Hölder’s
inequality, so that $m$ indeed belongs to $A(\mathrm{G})$. In particular, $m$
defines a completely bounded $L_{1}$ and $L_{\infty}$ Fourier multiplier.
Thus, it also defines a Fourier $L_{p}$-multiplier [7, Definition-Proposition
3.5]. Therefore, by interpolation [7, Section 6] it suffices to prove (2.10)
for $p=1$ and $p=\infty$. These two cases are formally equivalent and we just
consider $p=\infty$. In that case, $y\in L_{1}(\mathcal{L}\mathrm{G})$
corresponds to an element $f\in A(\mathrm{G})$ and
$\phi^{\prime}y\psi^{\prime}$ is the trace class operator with kernel

$\Big{(}\phi^{\prime}(g)f(hg^{-1})\psi^{\prime}(h)\Big{)}_{g,h\in\mathrm{G}},$

see [7, Lemma 3.4]. By a weak-$*$ density argument, it is enough to prove
(2.10) for $x=\lambda(g_{0})$ for some $g_{0}\in\mathrm{G}$. In that case,
$T_{m}(x)=m(g_{0})\lambda(g_{0})$ and we obtain
$\operatorname{tr}(T_{m}(x)y^{*})=m(g_{0})\overline{f(g_{0}^{-1})}$. We can
compute

$\displaystyle\operatorname{Tr}\big{(}\phi
x\psi(\phi^{\prime}y\psi^{\prime})^{*}\big{)}$ $\displaystyle=$
$\displaystyle\operatorname{Tr}\big{(}\lambda(g_{0})\psi\overline{\psi^{\prime}}y^{*}\phi\overline{\phi^{\prime}}\big{)}$
$\displaystyle=$
$\displaystyle\operatorname{Tr}\Big{[}\Big{(}\psi\overline{\psi^{\prime}})(g_{0}^{-1}g)\overline{f(g_{0}^{-1}gh^{-1})}(\phi\overline{\phi^{\prime}})(h)\Big{)}_{g,h\in\mathrm{G}}\Big{]}$
$\displaystyle=$
$\displaystyle\int_{\mathrm{G}}(\psi\overline{\psi^{\prime}})(g_{0}^{-1}g)\overline{f(g_{0}^{-1})}(\phi\overline{\phi^{\prime}})(g)\,dg=m(g_{0})\overline{f(g_{0}^{-1})}.$

This justifies the identity (2.10) and completes the proof of the lemma. ∎

Lemma 2.10 allows us to adapt the proof of Lemma 2.5 from the unimodular case.

###### Proof of Lemma  _2.5_ , general case.

We take $\phi,\psi\in\mathcal{C}_{c}(\mathrm{G})$ as in the proof in the
unimodular case, and set $m(g)=\int\phi(h)\psi(g^{-1}h)dh$. Let $\rho\in
A(\mathrm{G})$ such that $\rho m=1$ on $U$, as provided by the Tauberian
theorem in Lemma 2.6. By Lemma 2.10, we can define completely bounded maps

$J_{p}\colon L_{p}(\mathcal{L}\mathrm{G})\ni
x\mapsto\phi^{\frac{1}{p}}x\psi^{\frac{1}{p}}\in S_{p}(L_{2}(V),L_{2}(W)),$

which intertwine Fourier and Schur multipliers. Now, if $x\in
L_{p}(M_{n}\otimes\mathcal{L}\mathrm{G})$ is Fourier supported in $U$ and
$y\in L_{q}(M_{n}\otimes\mathcal{L}\mathrm{G})$ —for $q$ being the dual
exponent of $p$— we get

$\displaystyle\operatorname{tr}(xy^{*})$
$\displaystyle=\operatorname{tr}\big{(}T_{m\rho}(x)y^{*}\big{)}=\operatorname{tr}\big{(}T_{m}(x)(T_{\rho}(y))^{*}\big{)}=\operatorname{Tr}\big{(}J_{p}(x)J_{q}(T_{\rho}(y))^{*}\big{)}$
$\displaystyle\leq\|J_{p}(x)\|_{S_{p}}\|J_{q}(T_{\rho}(y))\|_{S_{q}}\leq\|J_{q}\|_{\mathrm{cb}}\|\rho\|_{A(\mathrm{G})}\|y\|_{L_{q}(\mathcal{L}\mathrm{G})}\|J_{p}(x)\|_{S_{p}}.$

The first line is by definition of $x$ being Fourier supported in $U$, the
duality for Fourier multipliers, and (2.10). The last line is Hölder’s
inequality. Taking suprema over $y$ in the unit ball of
$L_{q}(\mathcal{L}\mathrm{G})$ gives
$\|x\|_{L_{p}(\mathcal{L}\mathrm{G})}\leq\|J_{q}\|_{\mathrm{cb}}\|\rho\|_{A(\mathrm{G})}\|J_{p}(x)\|_{S_{p}}$.
∎

###  2.4. The group $\mathrm{SL}_{2}(\mathbf{R})$

Consider the symbol

$m_{0}\Big{[}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Big{]}=\frac{1}{2}\Big{(}1+\mathrm{sgn}(ac+bd)\Big{)}.$

This was identified in [18] as the canonical Hilbert transform in
$\mathrm{SL}_{2}(\mathbf{Z})$. Its complete $L_{p}$-boundedness follows for
$1<p<\infty$ from a Cotlar-type identity. The same problem in
$\mathrm{SL}_{2}(\mathbf{R})$ was left open in [18, Problem A]. Now this is
solved by condition (4) in Theorem 2.3, which disproves cb-$L_{p}$-boundedness
for any $p\neq 2$. On the other hand, according to Corollary B2, the map

$m(g)=m\Big{[}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\Big{]}:=\frac{1}{2}\big{(}1+\mathrm{sgn}(c)\big{)}=m_{0}(gg^{\mathrm{t}})$

does define, locally at every point of its boundary, a completely bounded
Fourier $L_{p}$-multiplier for every $1<p<\infty$. But, is it globally
$L_{p}$-bounded? Is it completely $L_{p}$-bounded as well? We leave these
problems open for future attempts.

###  2.5. Stratified Lie groups

A Lie algebra $\mathfrak{g}$ is called graded when there exists a finite
family of subspaces $\mathrm{W}_{1},\mathrm{W}_{2},\ldots,\mathrm{W}_{N}$ of
the Lie algebra satisfying conditions below

$\mathfrak{g}=\bigoplus_{j=1}^{N}\mathrm{W}_{j}\qquad\mbox{and}\qquad[\mathrm{W}_{j},\mathrm{W}_{k}]\subset\mathrm{W}_{j+k}.$

A simply connected Lie group $\mathrm{G}$ is called stratified when its Lie
algebra $\mathfrak{g}$ is graded and the first stratum $\mathrm{W}_{1}$
generates $\mathfrak{g}$ as an algebra. Stratified Lie groups are nilpotent
and include, among many other examples, Heisenberg groups. According to
Corollary B2 Hilbert transforms are of the form $H\circ\varphi$, for the
classical Hilbert transform $H$ and some continuous homomorphism
$\varphi:\mathrm{G}\to\mathbf{R}$. A quick look at Theorem 2.3 shows that
$\varphi$ corresponds on the Lie algebra with the projection onto any
1-dimensional subspace of the first stratum, since codimension 1 Lie
subalgebras are exactly those codimension 1 subspaces leaving out a vector in
the first stratum.

Acknowledgement. The research of JP and ET was partially supported by the
Spanish Grant PID2022-141354NB-I00 “Fronteras del Análisis Armónico” (MCIN) as
well as Severo Ochoa Grant CEX2019-000904-S (ICMAT), funded by MCIN/AEI
10.13039/501100011033. The research of MdlS was partially supported by the
Charles Simonyi Endowment at the Institute for Advanced Study, and the ANR
project ANCG Project-ANR-19-CE40-0002. ET was supported as well by Spanish
Ministry of Universities with a FPU Grant with reference FPU19/00837.

## References

  * [1] G. Bennett G, Schur multipliers. Duke Math J. 44 (1977), 603-639.
  * [2] M. Bożejko and G. Fendler, Herz–Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group. Boll. Un. Mat. Ital. A 3 (1984), 297-302.
  * [3] M. Bożejko and G. Fendler, A note on certain partial sum operators. Banach Center Publications 73 (2006) 117-225.
  * [4] J. de Cannière and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107 (1985), 455-500.
  * [5] M. Caspers, A Sobolev estimate for radial $L_{p}$-multipliers on a class of semi-simple Lie groups. Trans. Amer. Math. Soc. To appear.
  * [6] M. Caspers, J. Parcet, M. Perrin and É. Ricard, Noncommutative de Leuuw theorems. Forum Math. $\Sigma$ 3 (2015), e21.
  * [7] M. Caspers and M. de la Salle, Schur and Fourier multipliers of an amenable group acting on non-commutative $L_{p}$-spaces. Trans. Amer. Math. Soc. 367 (2015), 6997-7013.
  * [8] M. Caspers, B. Janssens, A. Krishnaswamy-Usha and L Miaskiwskyi, Local and multilinear noncommutative de Leeuw theorems. Math. Ann. (2023).
  * [9] C.Y. Chuah, Z-C. Liu and T. Mei, A Marcinkiewicz multiplier theory for Schur multipliers. ArXiv: 2209.13108.
  * [10] J.M. Conde-Alonso, A.M. González-Pérez, J. Parcet and E. Tablate, Schur multipliers in Schatten-von Neumann classes. Ann. of Math. 198 (2023), 1229-1260.
  * [11] J.M. Conde-Alonso, A.M. González-Pérez, J. Parcet and E. Tablate, A Hörmander-Mikhlin theorem for higher rank simple Lie groups. ArXiv: 2201.08740.
  * [12] A. Connes, On the spatial theory of von Neumann algebras, J. Funct. Anal. 35 (1980), no. 2, 153–164.
  * [13] M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96 (1989), 507-549.
  * [14] P. Eymard, L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181–236.
  * [15] C. Fefferman, The multiplier problem for the ball. Ann. of Math. 94 (1971), 330-336.
  * [16] E. Ghys, Groups acting on the circle. Enseign. Math. (2) 47 (2001), no. 3-4, 329-407.
  * [17] A.M. González-Pérez, M. Junge and J. Parcet. Smooth Fourier multipliers in group algebras via Sobolev dimension. Ann. Sci. École Norm. Sup. 50 (2017), 879-925.
  * [18] A.M. González-Pérez, J. Parcet and R. Xia. Noncommutative Cotlar identities for groups acting on tree-like structures. ArXiv: 2209.05298.
  * [19] U. Haagerup, An example of a non nuclear $C^{*}$-algebra, which has the metric approximation property. Invent. Math. 50 (1979), 279-293.
  * [20] U. Haagerup, $L^{p}$-spaces associated with an arbitrary von Neumann algebra. Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), pp. 175–184.
  * [21] M. Hilsum, Les espaces $L^{p}$ d’une algèbre de von Neumann définies par la derivée spatiale, J. Funct. Anal. 40 (1981), no. 2, 151–169.
  * [22] M. Junge, T. Mei and J. Parcet, Smooth Fourier multipliers on group von Neumann algebras. Geom. Funct. Anal. 24 (2014), 1913-1980.
  * [23] M. Junge, T. Mei and J. Parcet, Noncommutative Riesz transforms – Dimension free bounds and Fourier multipliers. J. Eur. Math. Soc. 20 (2018), 529-595.
  * [24] H. Kosaki, Applications of the complex interpolation method to a von Neumann algebra: noncommutative $L^{p}$-spaces, J. Funct. Anal. 56 (1984), no. 1, 29–78
  * [25] T. de Laat and M. de la Salle, Approximation properties for noncommutative $L_{p}$ of high rank lattices and nonembeddability of expanders. J. Reine Angew. Math. 737 (2018), 46-69.
  * [26] V. Lafforgue and M. de la Salle, Noncommutative $L_{p}$-spaces without the completely bounded approximation property. Duke. Math. J. 160 (2011), 71-116.
  * [27] K. de Leeuw, On $L_{p}$ multipliers. Ann. of Math. 81 (1965), 364-379.
  * [28] S. Lie, Theorie der Transformationsgruppen. Dritter (und letzten) Abschnitt. Unter Mitwirkung von Fr. Engel bearbeitet., Leipzig. B. G. Teubner. XXVII + 830 S. $8^{\circ}$ (1893).
  * [29] T. Mei and É. Ricard, Free Hilbert transforms. Duke Math. J. 166 (2017), 2153-2182.
  * [30] T. Mei, É. Ricard and Q. Xu, A Hörmander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras. Adv. Math. 403 (2022), 108394.
  * [31] G. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces. Ann. of Math. (2) 52 (1950), 606–636.
  * [32] S. Neuwirth and É. Ricard, Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group. Canad. J. Math. 63 (2011), 1161-1187.
  * [33] J. Parcet, É. Ricard and M. de la Salle, Fourier multipliers in $S\hskip-1.0ptL_{n}(\mathbf{R})$. Duke Math. J. 171 (2022), 1235-1297.
  * [34] J. Parcet and K. Rogers, Twisted Hilbert transforms vs Kakeya sets of directions. J. Reine Angew. Math. 710 (2016), 137-172.
  * [35] G. Pisier, Multipliers and lacunary sets in non-amenable groups. Amer. J. Math. 117 (1995), 337-376.
  * [36] G. Pisier, Non-commutative vector valued $L_{p}$-spaces and completely $p$-summing maps. Asrérisque 247. Soc. Math. France, 1998.
  * [37] G. Pisier, Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics 1618. Springer-Verlag, 2001.
  * [38] G. Pisier, Introduction to Operator Space Theory. Cambridge University Press, 2003.
  * [39] G. Pisier, Grothendieck’s Theorem, past and present. Bull. Amer. Math. Soc. 49 (2012), 237-323.
  * [40] G. Pisier and D. Shlyakhtenko, Grothendieck’s theorem for operator spaces. Invent. Math. 150 (2002), 185-217.
  * [41] G. Pisier and Q. Xu, Non-commutative Lp-spaces. Handbook of the Geometry of Banach Spaces II (Eds. W.B. Johnson and J. Lindenstrauss) North-Holland (2003), 1459-1517.
  * [42] D. Potapov and F. Sukochev, Operator-Lipschitz functions in Schatten-von Neumann classes. Acta Math. 207 (2011), 375-389.
  * [43] E.M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Math. Ser. 43. Princeton Univ. Press., NJ, 1993.
  * [44] M. Terp, $L_{p}$ Fourier transformation on non-unimodular locally compact groups. Adv. Oper. Theory 2 (2017), no.4, 547–583.
  * [45] J. Tits, Sur une classe de groupes de Lie résolubles, Bull. Soc. Math. Belg. 11 (2) (1959), 100–115.

Javier Parcet | Mikael de la Salle | Eduardo Tablate
---|---|---
CSIC | CNRS | CSIC
Instituto de Ciencias Matemáticas | Institut Camille Jordan | Instituto de Ciencias Matemáticas
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