11institutetext:  Yizhun Medical AI Co., Ltd
11email: {anlan.sun<EMAIL_ADDRESS>22institutetext: Center for
Data Science, Peking University
22email: {zhangzh<EMAIL_ADDRESS>33institutetext: National Key
Laboratory of General Artificial Intelligence, School of Intelligence Science
and Technology, Peking University
33email: {wangdongcis<EMAIL_ADDRESS>
# Boosting Breast Ultrasound Video Classification by the Guidance of Keyframe
Feature Centers

Anlan Sun Equal contribution.11 Zhao Zhang⋆ 22 Meng Lei 22 Yuting Dai 11 Dong
Wang 33 Liwei Wang 22 3 3

###### Abstract

Breast ultrasound videos contain richer information than ultrasound images,
therefore it is more meaningful to develop video models for this diagnosis
task. However, the collection of ultrasound video datasets is much harder. In
this paper, we explore the feasibility of enhancing the performance of
ultrasound video classification using the static image dataset. To this end,
we propose KGA-Net and coherence loss. The KGA-Net adopts both video clips and
static images to train the network. The coherence loss uses the feature
centers generated by the static images to guide the frame attention in the
video model. Our KGA-Net boosts the performance on the public BUSV dataset by
a large margin. The visualization results of frame attention prove the
explainability of our method. The codes and model weights of our method will
be made publicly available.

###### Keywords:

Breast ultrasound classification Ultrasound video Coherence loss.

##  1 Introduction

Breast cancer is a life-threatening disease that has surpassed lung cancer as
leading cancer in some countries and regions [20]. Breast ultrasound is the
primary screening method for diagnosing breast cancer, and accurately
distinguishing between malignant and benign breast lesions is crucial. This
task is also an essential component of computer-aided diagnosis. Since each
frame in an ultrasound video can only capture a specific view of a lesion, it
is essential to aggregate information from the entire video to perform
accurate automatic lesion diagnosis. Therefore, in this study, we focus on the
classification of breast ultrasound videos for detecting malignant and benign
breast lesions.

Despite the fact that ultrasound videos contain more information than static
images, most previous studies have focused on static image classification [11,
2, 27]. One major difficulty in using ultrasound videos for diagnosis lies in
the collection of video data with pathology gold standard results. Firstly,
during general ultrasound examinations, sonographers usually only record
keyframe images and not entire videos. Secondly, for prospectively collected
videos, additional effort must be made to track the corresponding pathological
results. As a result, while there are many breast ultrasound image datasets
[1, 28], video datasets are scarce. Currently, there is only one breast video
dataset [15] available, which is relatively small, containing only 188 videos.

Given the difficulties in collecting ultrasound video data, we investigate the
feasibility of enhancing the performance of ultrasound video classification
using a static image dataset. To achieve this, we first analyze the
relationship between ultrasound videos and images. The images in the
ultrasound dataset are keyframes of a lesion that exhibit the clearest
appearance and most typical symptoms, making them more discriminative for
diagnosis. Although ultrasound videos provide more information, the abundance
of frames may introduce redundancy or vagueness that could disrupt
classification. From the aspect of feature distribution, as shown in Fig. 1,
the feature points of static images are more concentrated, while the feature
of video frames sometimes are away from the class centers. Frames far from the
centers are harder to classify. Therefore, it is a promising approach to guide
the video model to pay more attention to important frames close to the class
center with the assistance of static keyframe images. Meanwhile, our approach
aligns with the diagnosis of ultrasound physicians, automatically evaluates
the importance of frames, and diagnoses based on the information of key
frames. Additionally, our method provides interpretability through key frames.

Figure 1: Feature distribution of video frames from BUSV [15] and static
images from BUSI [1]. We use a 2D ResNet trained on ultrasound images to get
the features.

In this paper, we propose a novel Keyframe Guided Attention Network (KGA-Net)
to boost ultrasound video classification. Our approach leverages both image
(keyframes) and video datasets to train the network. To classify videos, we
use frame attention to predict feature weights for all frames and aggregate
them to make the final classification. The feature weights determine the
contribution of each frame for the final diagnosis. During training, we
construct category feature centers for malignant and benign examples
respectively using center loss [26] on static image inputs and use the centers
to guide the training of video frame attention. Specifically, we propose
coherence loss, which promotes the frames close to the centers to have high
attention weights and decreases the weights for frames far from the centers.
Due to the feature centers being generated by the larger scale image dataset,
it provides more accurate and discriminative feature centers which can guide
the video frame attention to focus on important frames, and finally leads to
better video classification.

Our experimental results on the public BUSV dataset [15] show that our KGA-Net
significantly outperforms other video classification models by using an
external ultrasound image dataset. Additionally, we visualized attention
values guided by the coherence loss. The frames with clear diagnostic
characteristics are given higher attention values. This phenomenon makes our
method more explainable and provides a new perspective for selecting keyframes
from video.

In conclusion, our contributions are as follows:

  1. 1.

We analyze the relationship between ultrasound video data and image data, and
propose the coherence loss to use image feature centers to guide the training
of frame attention.

  2. 2.

We propose KGA-Net, which adopts a static image dataset to boost the
performance of ultrasound video classification. KGA-Net significantly
outperforms other video baselines on the BUSV dataset.

  3. 3.

The qualitative analysis of the frame attention verifies the explainability of
our method and provides a new perspective for selecting keyframes.

##  2 Related Works

Breast Ultrasound Classification. Breast ultrasound (BUS) plays an important
supporting role in the diagnosis of breast-related diseases. Recent research
demonstrated the potential of deep learning for breast lesion classification
tasks [18, 6, 23, 27, 19]. [18, 6] design ensemble methods to integrate the
features of multiple models to obtain higher accuracy. [23, 27, 19] utilize
multi-task learning to improve the model performance. However, all of them are
based on image datasets, such as BUSI [1], while few works focus on the video
modality. [14] designed a pre-training model based on contrastive learning for
ultrasound video classification. [25, 13] develop a keyframe extraction model
for ultrasound videos and utilized the extracted keyframes to perform various
classification tasks. However, these methods rely on keyframe supervision,
which limits their applicability. Fortunately, the recent publicly available
dataset BUSV [15] has made the research on the task of BUS video-based
classification possible. In this paper, we build our model based on this
dataset.

Video recognition based on neural networks. Traditional methods are based on
Two-stream networks [10, 24, 9]. Since I3D [3] was proposed, 3D CNNs have
dominated video understanding for a long time. [22, 21] decompose 3D
convolution in different ways to reduce computation complexity without losing
performance. [8] designed two branches to focus on temporal information and
spatial features, respectively. However, 3D CNNs have a limited receptive
field, and thus struggle to capture long-range dependency. Vision Transformers
[5, 16] have become popular due to their excellent capability of aggregating
spatial-temporal information. In order to reduce computational complexity
brought by global attention, MViT [7] used hierarchical structure by reducing
spatial resolution and Video Swin [17] introduced 3D shifted window attention.
Our proposed KGA-Net is a simple framework that aggregates multi-frame
features based on the frame attention module.

##  3 Methodology

As shown in Fig. 2, our KGA-Net takes the video inputs and static image inputs
simultaneously to train the network. The coherence loss is proposed to guide
the frame attention by using the feature centers generated by the images. We
will then elaborate on each component in the following sections.

Figure 2: Overview of our proposed keyframe-guided attention network.

###  3.1 Video and Image Classification Network

The video classification network is illustrated in Fig. 2 (a). The model is
composed of a 2D CNN backbone, a frame attention module, and a classification
head. For an input video clip $V$ composed of $N$ frames, it is first
processed by the backbone network and the feature vectors of the frames
$\\{F_{i}\\}_{i=1}^{N}$ are obtained. Then, the frame attention module
predicts the attention weight for each frame using a FC and sigmoid layer, and
then the features are aggregated by the weights to form an integrated feature
vector. Formally,

$w_{i}=\text{Sigmoid}(\text{FC}(F_{i}))$ (1)

where $w_{i}$ denotes the weight for the $i_{\text{th}}$ frame and FC is the
fully-connected layer. Then, the features are aggregated by
$F_{V}=\sum_{i=1}^{N}{w_{i}\cdot F_{i}}$. Finally, the classification head is
applied to the final result of lesion classification. To train the model, the
cross-entropy loss (CE Loss) is applied to the classification prediction of
the video.

The image classification network is used to assist in training the video
model. We use the same 2D CNN as the backbone network in the video
classification network. The model weights are shared for the two backbones for
better generalization. To promote the formation of feature centers, we apply
the center loss [26] to the image model besides the cross-entropy loss. In
addition, the frame-level cross-entropy loss is also applied to the video
frames to facilitate training.

###  3.2 Training with Coherence Loss

In this section, we introduce the coherence loss to guide the frame attention
with the assistance of the category feature centers. We use the same method as
center loss [26] to obtain the feature centers for the malignancy and benign
lesions, which are denoted as $\mathcal{C}^{\text{mal}}$ and
$\mathcal{C}^{\text{benign}}$, respectively.

The distances of frame features and the feature centers can measure the
quality of the frames. The frame features close to the centers are more
discriminative for the classification task. Therefore, we use these distances
to guide the generation of frame attention. Specifically, we push the frames
close to the centers to have higher attention weights and decrease the weights
far from the centers. To do this, for each video frame with feature $F_{i}$,
we first calculate the feature distance from its corresponding class center.
Formally,

$d_{i}=\|F_{i}-\mathcal{C}^{Y}\|_{2},$ (2)

where $Y\in\\{\text{mal},\text{benign}\\}$ is the label of the video $V$ and
$d_{i}$ is the computed distance of frame $i$.

Afterward, we apply coherence loss to the attention weights
$\textbf{w}=[w_{1},w_{2},...,w_{N}]^{\intercal}$ to make them have a similar
distribution with the feature distances
$\textbf{d}=[d_{1},d_{2},...,d_{N}]^{\intercal}$. To supervise the
distribution, the coherence loss is defined as the L2 loss of the gram matrix
of these two vectors

$\text{L}_{\text{Coh}}=\|\text{Gram}_{\textbf{w}}-\text{Gram}_{\textbf{d}}\|_{2},$
(3)

where
$\text{Gram}_{\textbf{w}}=\frac{(\mathbf{1}-\textbf{w})\cdot(\mathbf{1}-\textbf{w})^{\intercal}}{\|\mathbf{1}-\textbf{w}\|_{2}^{2}}$
is the gram matrix of normalized attention weights, and
$\text{Gram}_{\textbf{d}}=\frac{\textbf{d}\cdot\textbf{d}^{\intercal}}{\|\textbf{d}\|_{2}^{2}}$
is the gram matrix of normalized feature distances. Note that lower distances
correspond to stronger attention, hence we use the opposite of w to get
$\text{Gram}_{\textbf{w}}$.

###  3.3 Total Training Loss

To summarize, the total training loss of our KGA-Net

$L_{\text{total}}=L_{\text{CE}}^{V}+L_{\text{CE}}^{I}+L_{\text{Center}}+\lambda\cdot
L_{\text{Coh}}.$ (4)

$L_{\text{CE}}^{V}$ and $L_{\text{CE}}^{I}$ denote the cross-entropy for video
classification and image and frame classification. $L_{\text{Center}}$ means
the center loss. $\lambda$ is the weight for coherence loss. Empirically, we
set $\lambda=1$ in our experiments.

During inference, to perform classification on video data, the video
classification network can be utilized individually for prediction.

##  4 Experiments

###  4.1 Implementation Details

Datasets. We use the public BUSV dataset [15] for video classification and the
BUSI dataset [1] as the image dataset. BUSV consists of 113 malignant videos
and 75 benign videos. BUSI contains 445 images of benign lesions and 210
images of malignant lesions. For the BUSV dataset, we use the official data
split in [15]. All images of the BUSI dataset are adopted to train our KGA-
Net.

Model Details. ResNet-50 [12] pretrained on ImageNet [4] is used as backbone.
We use SGD optimizer with an initial learning rate of 0.005, which is reduced
by 10$\times$ at the 4,000th and 6,000th iteration. The total learning
iteration number is 8,000. The learning rate warmup is used in the first 1,000
iterations. For each batch, the video clips and static images are both sampled
and sent to the network. We use a total batchsize of 16 and the sample
probability of video clips and images is 1:1. We implement the model based on
Pytorch and train it with NVIDIA Titan RTX GPU cards.

During inference, we use the video classification network individually. In
order to satisfy the fixed video length requirement of MViT [7], we sample up
to 128 frames of each video to form a video clip and predict its
classification result using all the models in experiments.

###  4.2 Comparison with Video Models

In this section, we compare our KGA-Net with other competitive video
classification models. Comparing with ultrasound-video-based work presents
difficulty. [14, 13] is not accompanied by open-source code and relies on
private datasets, making comparisons exceedingly challenging. [25] relies on a
private dataset with keyframe annotations for supervised training. The
released code does not include keyframe detection, which makes direct
comparison impossible. Since the research on ultrasound video classification
is uncomparable, we compare our method with other strong video baselines on
natural images. The CNN-based models including I3D [3], SlowFast [8], R(2+1)D
[22] and CSN [21] are involved. Meanwhile, the recently popular transformer-
based model (MViT [7]) is also adopted. For a fair comparison, we use both the
video and image data to train these models. The images are regarded as static
videos to train the networks. During evaluation, we report the metrics on the
test set of BUSV.

As shown in Table. 1, by leveraging the guidance of the image dataset, our
KGA-Net significantly surpasses all other models on all of the metrics. The
video classification model of our KGA-Net is composed of a standard 2D
ResNet-50 and a light feature attention module, while the baseline models are
with net structures carefully designed for video analysis. Therefore, the
success of our KGA-Net lies in the correct usage of the image guidance. The
feature centers formed by the image dataset with larger data size and clear
appearance effectively improve the accuracy of frame attention hence boosting
the video classification performance.

Table 1: Comparison with other video models. Classification thresholds are determined by Youden index. Model | AUC(%) | ACC(%) | Sensitivity(%) | Specificity(%)
---|---|---|---|---
I3D [3] | 88.31 | 81.58 | 84.00 | 76.92
SlowFast [8] | 82.54 | 79.49 | 76.92 | 84.62
R(2+1)D [22] | 86.46 | 81.58 | 84.00 | 76.92
CSN [21] | 83.38 | 81.58 | 84.00 | 76.92
MViT [7] | 90.53 | 82.05 | 80.77 | 84.62
KGA-Net (Our) | 94.67 | 89.74 | 88.46 | 92.31

###  4.3 Ablation Study

In this section, we ablate the contribution of each key design in our KGA-Net.
We observe their importance by removing these key components from the whole
network. The results are shown in Table 2. The results of KGA-Net are shown in
the last row in Table 2, while the components are ablated in the first three
rows. We use the same training schedule for all of the experiments.

Image guidance is the main purpose of our method. To portray the effect of
using the image dataset, we train the KGA-Net using BUSV dataset alone in the
first row of Table 2. Without the image dataset, we generate the feature
centers from the video frames. As a result, the performance significantly
drops due to the decrease in dataset scale. It also shows that the feature
centers generated by the image dataset are more discriminative than that of
the video dataset. It is not only because the lesion number of BUSI is larger
than BUSV, but also because the images in BUSI are all the keyframes that
contain typical characteristics of lesions.

Frame attention and coherence loss are two essential modules of our KGA-Net.
We train a KGA-Net without the coherence loss in the third row of Table 2. In
the second row, we further replace the feature attention module with feature
averaging of video frames. It can be seen that both of these two modules
contribute to the overall performance according to AUC and ACC. It is worth
noting that these two models without coherence loss obtain very low
sensitivity and high specificity, which means the model predictions are
imbalanced and intend to make benign predictions. It is because that clear
malignant appearances usually only exist in limited frames in a malignant
video. Without our coherence loss or frame attention, it is difficult for the
model to focus on typical frames that possess malignant features. This
phenomenon certifies the effectiveness of our KGA-Net to prevent false
negatives in diagnosis.

Table 2: Ablation studies. Model components are removed in the first three lines to analyze their contributions in KGA-Net. Classification thresholds are determined by Youden index. Model | AUC(%) | ACC(%) | Sensitivity(%) | Specificity(%)
---|---|---|---|---
w/o image guidance | 85.21 | 76.92 | 73.08 | 84.62
w/o coherence loss & attention | 88.17 | 74.36 | 61.54 | 100.0
w/o coherence loss | 92.90 | 87.18 | 80.77 | 100.0
KGA-Net | 94.67 | 89.74 | 88.46 | 92.31

###  4.4 Visual Analysis

Figure 3: Visual Analysis. (a-d) Visualization of video frames and
corresponding frame attention weights. (e) Relationship between attention
weight and feature distance.

In Fig. 3, we illustrate video frames with their corresponding frame attention
weights predicted by KGA-Net. Overall speaking, the frames with high attention
weights do have clear image appearances for diagnosis. For example, the first
three frames in Fig. 3(b) clearly demonstrate the edge micro-lobulation and
irregular shapes, which lead to malignant judgment. Furthermore, we plot the
relationships between the predicted attention values and the feature distances
to the centers. As shown in Fig. 3(e), these two variables are linearly
related, which indicates that KGA-Net the attention weights are effectively
guided by the feature distances.

The qualitative analysis proves the interpretability of our method, which will
benefit clinical usage. Moreover, the attention weights reveal the importance
of each frame for lesion diagnosis. Therefore, it can provide a new
perspective for the keyframe extraction task of ultrasound videos.

##  5 Conclusion

We propose KGA-Net, a novel video classification model for breast ultrasound
diagnosis. Our KGA-Net takes as input both the video data and image data to
train the network. We propose the coherence loss to guide the training of the
video model by the guidance of feature centers of the images. Our method
significantly exceeds the performance of other competitive video baselines.
The visualization of the attention weights validates the effectiveness and
interpretability of our KGA-Net.

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Trigonometric chaos and $\mathrm{X}_{p}$ inequalities II

—$\mathrm{X}_{p}$ inequalities in group von Neumann algebras—

Antonio Ismael Cano-Mármol,
José M. Conde-Alonso and Javier Parcet

# Trigonometric chaos and $\mathrm{X}_{p}$ inequalities II:
$\mathrm{X}_{p}$ inequalities with sharp scaling parameter

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

In the line of previous work by Naor, we establish new forms of metric
$\mathrm{X}_{p}$ inequalities in group algebras under very general
assumptions. Our results’ applicability goes beyond the previously known
setting in two directions. In first place, we find continuous forms of the
$\mathrm{X}_{p}$ inequality in the $n$-dimensional torus. Second, we consider
transferred forms of the sharp scalar valued metric $\mathrm{X}_{p}$
inequality in the von Neumann algebra $\mathcal{L}(\mathrm{G})$ of a discrete
group $\mathrm{G}$. As a byproduct of our results, some metric consequences
and their relation with bi-Lipschitz nonembeddability of Banach spaces are
explored in the context of noncommutative $L_{p}$ spaces.

Abstract

## Introduction

In functional analysis, it is a fundamental question to know when a given
space is isomorphic to a linear subspace of another. The corresponding area,
Banach space embedding theory, dates back to the first half of the twentieth
century. For the important example of $L_{p}$ and $L_{q}$ the landscape is
well known. $L_{2}$ is isomorphic to a subspace of $L_{p}$ for all $p$ in the
Banach range, but there is no linear embedding from $L_{q}$ to $L_{p}$ if
either $q<\min\\{2,p\\}$ or $q>\max\\{2,p\\}$. Banach conjectured a positive
answer for $\min\\{2,p\\}<q<\max\\{2,p\\}$. Kadec proved it for $p<q<2$ in
[6], while Paley disproved it for $2<q<p$ in [10].

Sometimes, properties of Banach spaces that are local (that is, ones that can
be formulated in terms of vectors in finite dimensional subspaces) do not rely
on the whole strength of the linear structure. Instead, they may only need the
fact that the norm defines a metric. Determining for which properties is this
the case is known as the Ribe program, initiated after [11] and explicitly
formulated in [1]. Replacing the linear maps that yield inclusions between
Banach spaces by bi-Lipschitz maps becomes therefore natural. Regarding
$L_{p}$ spaces, the qualitative results mentioned above do not change, and
notably one gets again that $L_{q}$ cannot be embedded into $L_{p}$ in the
range $2<q<p$. This happens in spite of the fact that the classical argument
that shows that there is no bi-Lipschitz embedding from $L_{q}$ into $L_{p}$
uses explicitly the linear structure of the spaces. This is so because it goes
through differentiability —of Lipschitz maps— to reduce the metric statement
to a linear one. However, the result can be proved by other means and indeed a
different quantitative proof can be found in the work of Naor and Schechtman
[9]. One can consult the Introduction in that work for a more detailed context
and references regarding the history of the problem and its connections with
other areas.

A key innovation in [9] is the introduction of a family of inequalities,
termed $\mathrm{X}_{p}$ inequalities. Let us briefly recall the results in
[9], since they are going to be the starting point of our investigation in the
present manuscript. Denote by $[n]$ the initial segment
$[n]:=\\{1,2,\ldots,n\\}$. We view the set $\mathbb{Z}_{8m}^{n}$ as a
probability space equipped with the normalized counting measure. In the same
vein we identify $\mathbb{Z}_{2}^{n}$ with $\\{-1,1\\}^{n}$ using
multiplicative notation. A metric space $(\mathbb{X},d)$ is said to be a
metric $\mathrm{X}_{p}$ space if for each $n\in\mathbb{N}$ and $k\in[n]$,
there exists $m\in\mathbb{N}$ such that every mapping
$f:\mathbb{Z}_{8m}^{n}\to\mathbb{X}$ satisfies the following estimate for
$p\geq 2$

$\displaystyle\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\mathsf{E}$
$\displaystyle\big{[}d\big{(}\mathrm{M}_{4m\varepsilon_{\mathsf{S}}}f(x),f(x)\big{)}^{p}\big{]}$
($\mathrm{MX}_{p}$)
$\displaystyle\lesssim_{p}m^{p}\left(\frac{k}{n}\sum_{j=1}^{n}\mathsf{E}\big{[}d\big{(}\mathrm{M}_{e_{j}}f(x),f(x)\big{)}^{p}\big{]}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\mathsf{E}\big{[}d\big{(}\mathrm{M}_{\varepsilon}f(x),f(x)\big{)}^{p}\big{]}\right).$

Above, the dummy variable $(\varepsilon,x)$ belongs to
$\mathbb{Z}_{2}^{n}\times\mathbb{Z}_{8m}^{n}$, $\varepsilon_{\mathsf{S}}$
denotes the $\mathsf{S}$-truncated vector
$\sum_{j\in\mathsf{S}}\varepsilon_{j}e_{j}$, the expectation $\mathsf{E}$ is
taken in the product probability space, and the operators $\mathrm{M}_{v}$ are
just translations given by

$\mathrm{M}_{v}f(x)=f(x+v).$

They are Fourier multipliers, which will become relevant later. The main
result in [9] then asserts that $L_{p}$ is an $\mathrm{X}_{p}$ space, while
$L_{q}$ is not if $2<q\neq p$. This and the fact that being an
$\mathrm{X}_{p}$ metric space is invariant under the action of bi-Lipschitz
maps provides the quantitative nonembeddability argument alluded above. Note
that the result is not true independently of the value of $m$. It is necessary
that it scales at least so that $m\gtrsim\sqrt{n/k}$, even though the bound
was not found yet to be sharp in [9]. The above discussion justifies why
$\mathrm{X}_{p}$ inequalities are a tool to study Lipschitz nonembeddability
results between Banach spaces.

Little after it, Naor found in [7] a different proof of the metric
$\mathrm{X}_{p}$ inequality which was quantitatively optimal. Namely, it turns
out that the necessary bound $m\gtrsim\sqrt{n/k}$ is, indeed, sufficient. This
entails a number of significant consequences in metric geometry, including
distortion estimates of discrete $\ell_{q}$-grids into $L_{p}$ and sharp
snowflake $L_{p}$-estimates. A key step towards the main result in [7] is the
inequality below for balanced averages of Fourier truncations of the Fourier-
Walsh expansions. Those are expansions of the form

$f=\sum_{\mathsf{A}\subset[n]}\widehat{f}(\mathsf{A})W_{\mathsf{A}}.$

Given $2\leq p<\infty$, $k\in[n]$, and $f\in L_{p}(\mathbb{Z}_{2}^{n})$ mean
zero, it holds

($\mathrm{N}_{p}$) $\frac{1}{{n\choose
k}}\sum_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\Big{\|}\sum_{\mathsf{A}\subset\mathsf{S}}\widehat{f}(\mathsf{A})W_{\mathsf{A}}\Big{\|}_{p}^{p}\lesssim_{p}\frac{k}{n}\sum_{j=1}^{n}\left\|\partial_{j}f\right\|_{p}^{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\|f\|_{p}^{p}.$

Above, the symbol $\partial_{j}$ denotes the $j$-th directional discrete
derivative

$\partial_{j}f(\varepsilon):=f(\varepsilon)-f(\varepsilon_{1},\ldots,\varepsilon_{j-1},-\varepsilon_{j},\varepsilon_{j+1},\ldots,\varepsilon_{n}).$

Once ($\mathrm{N}_{p}$) has been established, the metric $\mathrm{X}_{p}$
inequality for all classical $L_{p}$ spaces follows from it plus the natural
injection of the multiplicative group $\mathbb{Z}_{2}^{n}$ into
$\mathbb{Z}_{2m}^{n}$ needed to define the multipliers
$\mathrm{M}_{\varepsilon}(f)(x)=f(x+\varepsilon)$ for
$f:\mathbb{Z}_{2m}^{n}\to\mathbb{C}$. This is the starting point of our work.
We seek new metric $\mathrm{X}_{p}$ inequalities that can be proven using
appropriate versions of ($\mathrm{N}_{p}$). In [2], a wide range of
inequalities of this kind is established for the (noncommutative) $L_{p}$
spaces associated with the group von Neumann algebra $\mathcal{L}(\mathrm{G})$
of a discrete group $\mathrm{G}$. Elements in $L_{p}(\mathcal{L}(\mathrm{G}))$
are not functions for nonabelian $\mathrm{G}$ and making sense of pointwise
evaluation is not immediate, though necessary. Seeking the greatest possible
generality, we shall consider below different choices of abelian groups
$\mathrm{H}$ that will play the role of $\mathbb{Z}_{2}^{n}$ in [7].
Therefore, in the evaluation $\mathrm{M}_{\varepsilon}f(x):=f(x+\varepsilon)$,
the variable $\varepsilon\in\mathrm{H}$ but that may not be the case for $x$,
which imposes a compatibility condition between the two variables. This is a
crucial difficulty and the main contribution in this paper is to clarify when
a chaos-type inequality like ($\mathrm{N}_{p}$) —as those obtained in [2]— can
be used to deduce a new metric $\mathrm{X}_{p}$-type inequality.

Let us now explain our abstract result, we refer to [2] for notation regarding
analysis on group von Neumann algebras. Let $\mathrm{H}$ be a discrete abelian
group which can be written as a direct product:

$\mathrm{H}=\mathrm{H}_{1}\times\mathrm{H}_{2}\times\cdots\times\mathrm{H}_{n},$

equipped with its counting measure. Given $\mathsf{S}\subset[n]$, we define
the truncation of $\mathrm{H}$ on $\mathsf{S}$ as the subgroup
$\mathrm{H}_{\mathsf{S}}=\times_{j\in\mathsf{S}}\mathrm{H}_{j}\hookrightarrow\mathrm{H}$.
Truncations are transferred to the dual group $\widehat{\mathrm{H}}$, which is
a probability space with Haar measure $dx$. Assume also that $\mathrm{G}$ is
an arbitrary discrete group, and let $p\geq 2$. We say that
$(\mathrm{H},\mathrm{G})$ is an $\mathrm{X}_{p}$_-representable pair_ if the
following conditions hold:

  * (RP-1)

Balanced truncations. We have

$\hskip 20.0pt\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\big{\|}(\mathsf{E}_{[n]\setminus\mathsf{S}}\otimes\mathrm{id})h\big{\|}^{p}_{p}\lesssim_{p}\frac{k}{n}\sum\limits_{j=1}^{n}\big{\|}(\partial_{j}\otimes\mathrm{id})h\big{\|}^{p}_{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\|h\|^{p}_{p},$

for $k\in[n]$, any mean zero $h\in
L_{p}(\widehat{\mathrm{H}};L_{p}(\mathcal{L}(\mathrm{G})))$, and where

$\hskip
20.0pt\mathsf{E}_{[n]\setminus\mathsf{S}}f(x)=\int_{\widehat{\mathrm{H}}}f(x_{\mathsf{S}}+z_{[n]\setminus\mathsf{S}})\
dz$

is the conditional expectation onto functions which only depend on variables
in $\widehat{\mathrm{H}}_{\mathsf{S}}$. The $\partial_{j}$ are linear maps
which play the role of directional derivatives.

  * (RP-2)

Compatibility conditions. There exists an abelian group

$\hskip 20.0pt\Gamma=\Gamma_{1}\times\Gamma_{2}\times\cdots\times\Gamma_{n}$

and a map $\eta:\widehat{\mathrm{H}}\rightarrow\widehat{\Gamma}$ such that the
following compatibility conditions hold:

    * i)

Symmetric inclusion. Given $\mathsf{S}\subset[n]$, the variables
$\eta_{\mathsf{S}}(y):=(\eta(y))_{\mathsf{S}}$ and $-\eta_{\mathsf{S}}(y)$
(using here additive notation) have the same distribution.

    * ii)

Uniformly bounded translations. There exists a group of Fourier multipliers
$\\{\mathrm{M}_{\gamma}\\}_{\gamma\in\widehat{\Gamma}}$, completely and
uniformly bounded on $L_{p}(\mathcal{L}(\mathrm{G}))$.

######  Theorem A.

Let $p\geq 2$ and $(\mathrm{H},\mathrm{G})$ be an
$\mathrm{X}_{p}$-representable pair. Let $k\in[n]$ and suppose that
$m\in\mathbb{N}$ satisfies $m\geq\sqrt{n/k}$. Then, for every $f\in
L_{p}(\mathcal{L}(\mathrm{G}))$ there holds

$\displaystyle\frac{1}{{n\choose k}}$
$\displaystyle\sum_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}f-f\big{\|}^{p}_{p}\
dy$
$\displaystyle\lesssim_{p}m^{p}\Bigg{(}\frac{k}{n}\sum\limits_{j=1}^{n}\int_{\widehat{\mathrm{H}}}\big{\|}\partial_{j}\mathrm{M}_{2\eta(y)}f\big{\|}_{p}^{p}\
dy+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{\eta(y)}f-f\big{\|}_{p}^{p}\
dy\Bigg{)}.$

Metric $\mathrm{X}_{p}$ inequalities from [7, 9] apply for functions
$f:\mathbb{Z}_{8m}^{n}\to\mathbb{X}$. Theorem A is an inequality for $f\in
L_{p}(\mathcal{L}(\mathrm{G}))$, so that it should be regarded as a
generalization where we replace $\mathbb{Z}_{8m}^{n}$ by the dual group of
$\mathrm{G}$. Note that its dual is a quantum group rather than a classical
group when $\mathrm{G}$ is nonabelian. Additionally, the role of $\mathrm{H}$
is merely to provide an index set over which translations are performed, which
generalizes the former role of $\mathbb{Z}_{2}^{n}$. In particular, Theorem A
generalizes ($\mathrm{MX}_{p}$) with
$(\mathrm{H},\mathrm{G})=(\mathbb{Z}_{2}^{n},\mathbb{Z}_{8m}^{n})$, auxiliary
group $\Gamma=\mathbb{Z}_{8m}^{n}$ and $\eta$ being the natural inclusion
defined coordinatewise by $1\mapsto 1$ and $(-1)\mapsto 8m-1$. Note that

($\partial_{j}$)
$\int_{\mathbb{Z}_{2}^{n}}\big{\|}\partial_{j}\mathrm{M}_{2\eta(y)}f\big{\|}_{p}^{p}\
dy\lesssim_{p}\|\mathrm{M}_{e_{j}}f-f\|_{p}^{p}$

holds when $\partial_{j}$ is the aforementioned discrete derivative in the
$j$-th direction. The role of $\Gamma$ was hidden in [7, 9]. This is natural
since it is possible to formulate Theorem A without appealing to the existence
of $\eta$ and $\Gamma$. Indeed, one can find axioms on a family of multipliers
$\\{\mathrm{M}_{y}\\}_{y\in\mathrm{H}}$ and their powers —so that they act
like translations— which are formally weaker than (RP-1) and (RP-2) and
suffice to get the conclusion of Theorem A. We have chosen to formulate our
result in this manner because in all of our examples we may identify the
auxiliary objects $\eta$ and $\Gamma$, which is an efficient way of checking
that we can apply Theorem A. However the existence of weaker conditions can be
useful for future constructions and we will detail them in Section 1. In the
general case, the improved statement that one gets when an inequality like
($\partial_{j}$) holds is a true metric inequality, while the conclusion of
Theorem A is not in general. In our other examples, we shall choose
derivatives whose relation with the multipliers $\mathrm{M}_{v}$ yields the
stronger, metric statement.

The proof of Theorem A follows the strategy in [7]. Our main contribution is
the identification of the right conditions under which said strategy can be
extended to numerous contexts, including noncommutative ones. We illustrate
Theorem A with several concrete scenarios, some of which we describe next. In
each case, (RP-1) is checked using the results in [2] and we focus in getting
fully metric statements.

  * i)

Continuous metric $\mathrm{X}_{p}$ inequalities. Taking
$\mathrm{H}=\mathrm{G}=\mathbb{Z}^{n}$, we obtain two continuous statements in
the $n$-dimensional torus using two different choices for the inclusion
$\eta$. One of them can be obtained from [7] by an approximation procedure,
while the other seems to require Theorem A.

  * ii)

Cyclic extensions of metric $\mathrm{X}_{p}$ inequalities. Let $\eta_{\ell}$
be the inclusion that maps $\mathbb{Z}_{2\ell}^{n}$ into
$\\{k:\;k\in[\ell]\mbox{ or
}8m\ell-k\in[\ell]\\}\subset\mathbb{Z}_{8m\ell}^{n}$. We prove a metric
$\mathrm{X}_{p}$ inequality which reduces to ($\mathrm{MX}_{p}$) for $\ell=1$.
It is sharp in terms of the scaling parameter $m$ and entails the same metric
consequences as [7].

  * ii)

Transferred $\mathrm{X}_{p}$ inequalities. When $\mathrm{G}$ is nonabelian, we
first observe that our results in [2] hold with values on noncommutative
$L_{p}$ spaces over QWEP algebras, so that (RP-1) is satisfied. Next, using an
appropriate corepresentation for $(\mathrm{H},\mathrm{G})$ we obtain
$\mathrm{X}_{p}$ inequalities on the free group $\mathbb{F}_{n}$.

The above scenarios show the applicability of our main result. In metric terms
we observe that noncommutative $L_{p}$ spaces are metric $\mathrm{X}_{p}$
spaces. This implies that the distortion estimates for $\ell_{q}$-grids and
$L_{p}$-snowflakes in [7] still hold when considering embeddings into
noncommutative $L_{p}$. Beyond this, we have not found significantly new
applications of our main results in the context of metric geometry so far. It
would be very interesting to generalize Theorem A so as to include
$\mathrm{X}_{p}$-representable pairs $(\mathrm{H},\mathrm{G})$ admitting
_nonabelian translations_ in the index set $\mathrm{H}$.

##  1\. Metric $\mathrm{X}_{p}$ inequalities

We work in the language of noncommutative $L_{p}$ spaces over group von
Neumann algebras, as explained in [2]. Given a discrete group $\mathrm{G}$,
its von Neumann algebra is denoted by $\mathcal{L}(\mathrm{G})$ —which
coincides with $L_{\infty}(\widehat{\mathrm{G}})$ when $\mathrm{G}$ is
abelian—, while $L_{p}(\mathcal{L}(\mathrm{G}))$ is the associated
noncommutative $L_{p}$ space, $2\leq p<\infty$.
$L_{p}^{\circ}(\mathcal{L}(\mathrm{G}))$ will be the family of zero-trace
elements in $L_{p}(\mathcal{L}(\mathrm{G}))$. Throughout this section assume
that $n$ is fixed. We emphasize again that we follow the strategy of proof in
[7] for an $\mathrm{X}_{p}$-representable pair $(\mathrm{H},\mathrm{G})$, as
defined in the Introduction. Given $\mathsf{S}\subset[n]$ consider the
auxiliary operator $T_{\mathsf{S}}$ on $\mathcal{L}(\mathrm{G})$ given by

$\displaystyle
T_{\mathsf{S}}f=\int_{\widehat{\mathrm{H}}}\mathrm{M}_{2\eta_{\mathsf{S}}(y)}f\
dy,\quad\mbox{for}\quad f\in L_{p}(\mathcal{L}(\mathrm{G})).$

Proof of Theorem A. For each $\mathsf{S}\subset[n]$ with $|\mathsf{S}|=k$ and
$y\in\mathrm{H}$

$\displaystyle\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}f-f\big{\|}_{p}^{p}$
$\displaystyle\lesssim_{p}$
$\displaystyle\big{\|}T_{[n]\setminus\mathsf{S}}f-f\big{\|}_{p}^{p}$
$\displaystyle+$
$\displaystyle\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-T_{[n]\setminus\mathsf{S}}f\big{\|}_{p}^{p}$
$\displaystyle+$
$\displaystyle\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}f-\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f\big{\|}_{p}^{p}\
=:\ \mathrm{A}+\mathrm{B}+\mathrm{C}.$

First, we claim that

(1.1)
$\big{\|}T_{\mathsf{S}}f-f\big{\|}_{p}^{p}\lesssim_{p}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{\eta(y)}f-f\big{\|}_{p}^{p}\,dy.$

Indeed, by convexity of the $L_{p}$ norm

$\displaystyle\big{\|}T_{\mathsf{S}}f-f\big{\|}_{p}^{p}$ $\displaystyle=$
$\displaystyle\Big{\|}\int_{\widehat{\mathrm{H}}}\mathrm{M}_{2\eta_{\mathsf{S}}(y)}f\,dy-f\Big{\|}^{p}_{p}\
\lesssim_{p}\
\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{2\eta_{\mathsf{S}}(y)}f-f\big{\|}_{p}^{p}\,dy$
$\displaystyle\leq$ $\displaystyle
2^{p-1}\left(\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{2\eta_{\mathsf{S}}(y)}f-\mathrm{M}_{\eta(y)}f\big{\|}_{p}^{p}\,dy+\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{\eta(y)}f-f\big{\|}_{p}^{p}\,dy\right).$

Now, $\eta(y)=\eta_{S}(y)+\eta_{[n]\setminus S}(y)$ and
$\eta_{S}(y)-\eta_{[n]\setminus S}(y)$ are identically distributed by (RP-2).
Therefore, by the properties of the family
$\\{\mathrm{M}_{\varepsilon}\\}_{\varepsilon}$ —also in (RP-2)— the first term
above satisfies

$\displaystyle\int_{\widehat{\mathrm{H}}}\|\mathrm{M}_{2\eta_{\mathsf{S}}(y)}f-\mathrm{M}_{\eta(y)}f\|_{p}^{p}\
dy$
$\displaystyle=\int_{\widehat{\mathrm{H}}}\|\mathrm{M}_{\eta(y)}\mathrm{M}_{\eta_{\mathsf{S}}(y)-\eta_{[n]\setminus\mathsf{S}}(y)}f-\mathrm{M}_{\eta(y)}f\|_{p}^{p}\
dy$
$\displaystyle\lesssim\int_{\widehat{\mathrm{H}}}\|\mathrm{M}_{\eta_{\mathsf{S}}(y)-\eta_{[n]\setminus\mathsf{S}}(y)}f-f\|_{p}^{p}\
dy$
$\displaystyle=\int_{\widehat{\mathrm{H}}}\|\mathrm{M}_{\eta(y)}f-f\|_{p}^{p}\
dy,$

which implies the claim. This proves that

$\mathrm{A}+\mathrm{C}\,\lesssim\,\mathrm{A}\,\lesssim\,\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{\eta(y)}f-f\big{\|}^{p}_{p}\,dy.$

For the term $\mathrm{B}$, our claim is the following

$\displaystyle\hskip-35.0pt\frac{m^{-p}}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-T_{[n]\setminus\mathsf{S}}f\big{\|}_{p}^{p}\,dy$
$\displaystyle\lesssim_{p}$
$\displaystyle\\!\\!\\!\\!\frac{k}{n}\sum\limits_{j=1}^{n}\int_{\widehat{\mathrm{H}}}\big{\|}\partial_{j}\mathrm{M}_{2\eta(y)}f\big{\|}^{p}_{p}\,dy+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{\eta(y)}f-f\big{\|}^{p}_{p}\,dy.$

To prove (1), we start once more with (RP-2) and the triangle inequality

$\displaystyle\Big{(}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-T_{[n]\setminus\mathsf{S}}f\big{\|}_{p}^{p}\,dy\Big{)}^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\sum\limits_{j=1}^{m}\Big{(}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{4j\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-\mathrm{M}_{4(j-1)\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f\big{\|}^{p}_{p}\,dy\Big{)}^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{m}\Big{(}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{(4j-2)\eta_{\mathsf{S}}(y)}[\mathrm{M}_{2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-\mathrm{M}_{-2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f]\big{\|}_{p}^{p}\,dy\
\Big{)}^{\frac{1}{p}}$ $\displaystyle\lesssim$ $\displaystyle m\
\Big{(}\int_{\widehat{\mathrm{H}}}\big{\|}\underbrace{\mathrm{M}_{2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-\mathrm{M}_{-2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f}_{F_{\mathsf{S}}(y)}\big{\|}_{p}^{p}\,dy\
\Big{)}^{\frac{1}{p}}.$

We claim that $F_{\mathsf{S}}(y)=\mathsf{E}_{[n]\setminus\mathsf{S}}h(y)$ with

$h:\widehat{\mathrm{H}}\rightarrow L_{p}(\mathcal{L}(\mathrm{G})),$
$h(y)=\mathrm{M}_{2\eta(y)}f-\mathrm{M}_{-2\eta(y)}f.$

Indeed, recall that

$\mathsf{E}_{[n]\setminus\mathsf{S}}g(x)=\int_{\widehat{\mathrm{H}}}g(x_{\mathsf{S}}+z_{[n]\setminus\mathsf{S}})\,dz.$

Also, denote the symbol of $\mathrm{M}_{\varepsilon}$ by
$\mathrm{m}_{\varepsilon}$, so that

$\mathrm{M}_{\varepsilon}f=\sum_{w\in\mathrm{G}}\mathrm{m}_{\varepsilon}(w)\widehat{f}(w)\lambda(w),$

where $\lambda(g)$ denotes the left regular representation of $\mathrm{G}$.
Then, given that $\mathsf{E}_{[n]\setminus\mathsf{S}}$ is linear, and using
again the equidistribution property in (RP-2), we deduce that

$\displaystyle\mathsf{E}_{[n]\setminus\mathsf{S}}h(y)$ $\displaystyle=$
$\displaystyle\sum\limits_{w\in\mathrm{G}}\widehat{f}(w)\lambda(w)\
[\mathsf{E}_{[n]\setminus\mathsf{S}}\mathrm{m}_{2\eta(y)}(w)-\mathsf{E}_{[n]\setminus\mathsf{S}}\mathrm{m}_{-2\eta(y)}(w)]$
$\displaystyle=$
$\displaystyle\sum\limits_{w\in\mathrm{G}}\widehat{f}(w)\lambda(w)\
[\mathrm{m}_{2\eta_{\mathsf{S}}(y)}(w)-\mathrm{m}_{-2\eta_{\mathsf{S}}(y)}(w)]\
\int_{\widehat{\mathrm{H}}}\mathrm{m}_{2\eta_{[n]\setminus\mathsf{S}}(z)}(w)\
dz$ $\displaystyle=$
$\displaystyle\mathrm{M}_{2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-\mathrm{M}_{-2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f\
=\ F_{\mathsf{S}}(y).$

Moreover, we have

$\displaystyle\int_{\widehat{\mathrm{H}}}h(y)\ dy$
$\displaystyle=\int_{\widehat{\mathrm{H}}}\mathsf{E}_{[n]\setminus\mathsf{S}}h(y)\
dy$
$\displaystyle=\int_{\widehat{\mathrm{H}}}\big{[}\mathrm{M}_{2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-\mathrm{M}_{-2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f\big{]}\
dy$
$\displaystyle=\int_{\widehat{\mathrm{H}}}\mathrm{M}_{2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f\
dy-\int_{\widehat{\mathrm{H}}}\mathrm{M}_{2\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f\
dy=0$

since $2\eta_{S}(y)$ and $-2\eta_{S}(y)$ are identically distributed.
Therefore, $h\in
L_{p}^{\circ}(\widehat{\mathrm{H}};L_{p}(\mathcal{L}\mathrm{G}))$, so we raise
to the power $p$ and average over $\mathsf{S}\subset[n]$ in (1), and by (RP-1)
we get

$\displaystyle\hskip-20.0pt\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}T_{[n]\setminus\mathsf{S}}f-T_{[n]\setminus\mathsf{S}}f\big{\|}_{p}^{p}\,dy$
$\displaystyle\lesssim_{p}$ $\displaystyle\frac{m^{p}}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\widehat{\mathrm{H}}}\big{\|}\mathsf{E}_{[n]\setminus\mathsf{S}}h(y)\big{\|}^{p}_{p}\,dy$
$\displaystyle\lesssim_{p}$ $\displaystyle
m^{p}\Bigg{(}\frac{k}{n}\sum\limits_{j=1}^{n}\int_{\widehat{\mathrm{H}}}\big{\|}(\partial_{j}\otimes\mathrm{id})h(y)\big{\|}_{p}^{p}\,dy+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\int_{\widehat{\mathrm{H}}}\|h(y)\|_{p}^{p}\,dy\Bigg{)}$
$\displaystyle\lesssim$ $\displaystyle
m^{p}\Bigg{(}\frac{k}{n}\sum\limits_{j=1}^{n}\int_{\widehat{\mathrm{H}}}\big{\|}\partial_{j}\mathrm{M}_{2\eta(y)}f\big{\|}_{p}^{p}\,dy+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\int_{\widehat{\mathrm{H}}}\|\mathrm{M}_{\eta(y)}f-f\|_{p}^{p}\,dy\Bigg{)}.$

Therefore, putting altogether we get

$\displaystyle\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{4m\eta_{\mathsf{S}}(y)}f-f\big{\|}_{p}^{p}\,dy$
$\displaystyle\lesssim$ $\displaystyle
m^{p}\Bigg{(}\frac{k}{n}\sum\limits_{j=1}^{n}\int_{\widehat{\mathrm{H}}}\big{\|}\partial_{j}\mathrm{M}_{2\eta(y)}f\big{\|}^{p}_{p}\,dy+\Big{(}\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}+\frac{1}{m^{p}}\Big{)}\int_{\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{\eta(y)}f-f\big{\|}^{p}_{p}\,dy\Bigg{)}.$

This completes the proof for any $m\geq\sqrt{n/k}$, as imposed in the
statement. $\square$

######  Remark 1.1.

_A close inspection of the proof of Theorem A shows that one can exchange
requirement (RP-2) by the existence of a family of invertible operators
$\\{\mathrm{M}_{y}\\}_{y\in\widehat{\mathrm{H}}}$ satisfying the following
properties:_

  * •

Product structure.
$\mathrm{M}_{y}=\mathrm{M}_{y_{\mathsf{S}}}\mathrm{M}_{y_{[n]\setminus\mathsf{S}}}$.

  * •

Uniform boundedness. $\displaystyle\max_{|k|\leq
4m}\sup_{y\in\widehat{\mathrm{H}}}\big{\|}\mathrm{M}_{y}^{k}\big{\|}_{p\to
p}<\infty$.

  * •

Symmetry. _$\mathrm{M}_{y}^{k}f$ and $\mathrm{M}_{y}^{-k}f$ have the same
distribution function for each $f$._

_The above conditions are formally weaker than(RP-2), and more importantly,
they clarify the fact that the auxiliary group $\Gamma$ is not necessary in
the definition of $\mathrm{X}_{p}$-representable pairs. However, in the paper
we have chosen to work with the stronger condition (RP-2) because in all of
our examples we are able to identify the associated group $\Gamma$ and this
makes checking the conditions of Theorem A simpler. _

######  Remark 1.2.

_In many of the cases of interest there holds_

(1.4)
$\int_{\widehat{\mathrm{H}}}\big{\|}\partial_{j}\mathrm{M}_{2\eta(y)}f\big{\|}_{L_{p}(\mathcal{L}(\mathrm{G}))}^{p}\,dy\lesssim_{p}\big{\|}\mathrm{M}_{e_{j}}f-f\big{\|}_{L_{p}(\mathcal{L}(\mathrm{G}))}^{p},$

_for each $j=1,\ldots,n$. This improves the conclusion of Theorem A to a truly
metric inequality, which resembles the one in [7] more closely. This
highlights the importance of the choice of the family of derivatives when
checking conditions (RP-1)-(RP-2). One case in which this holds is when
derivatives and conditional expectations are related by_

(1.5) $\partial_{j}=\mathrm{id}-\mathsf{E}_{\\{j\\}}\quad\mbox{for}\quad 1\leq
j\leq n.$

_In turn, ( 1.5) holds when
$\partial_{j}\mbox{\large$\chi$}_{w}(x)=\delta_{w_{j}\neq
0}\,\mbox{\large$\chi$}_{w}(x)$ for any character $\mbox{\large$\chi$}_{w}$ on
$\widehat{\mathrm{H}}$._

##  2\. Intrinsic $\mathrm{X}_{p}$ inequalities

In this section we explore intrinsic $\mathrm{X}_{p}$ inequalities in the
framework provided by Theorem A. The term “intrinsic” means here that there
exists a natural inclusion of the index group $\mathrm{H}$ into the
$\mathrm{X}_{p}$-group $\mathrm{G}$, so there is no need to use an auxiliary
group $\Gamma$ other than $\mathrm{G}$. In our setting, this unfortunately
forces us to work with $\mathrm{G}=\Gamma$ abelian. An extension of Theorem A
including noncommutative translations indexed by nonabelian groups
$\mathrm{H}$ would open a door to potential applications in the metric
geometry of noncommutative $L_{p}$-spaces. Intrinsic $\mathrm{X}_{p}$
inequalities will be complemented below with “transferred $\mathrm{X}_{p}$
inequalities” which refer to those which admit some $\Gamma\neq\mathrm{G}$,
including nonabelian $\mathrm{X}_{p}$-groups in the picture. We will
illustrate this scenario in the context of free groups below. In the language
of quantum group theory, intrinsic $\mathrm{H}$-translations are given by
$\mathrm{G}$-comultiplications, while transferred $\mathrm{H}$-translations
require more general $\mathrm{G}$-corepresentations in terms of $\Gamma$.

###  2.1. Continuous $\mathrm{X}_{p}$ inequalities

Let $\mathrm{H}=\mathrm{G}=\mathbb{Z}^{n}$, so that the dual groups are
$n$-dimensional tori, which we identify with $[-1/2,1/2)^{n}$ for convenience.
We also pick $\Gamma=\mathbb{Z}^{n}$, and we explore two different choices for
$\eta:\mathbb{T}^{n}\to\mathbb{T}^{n}$. First, we take the map
$\eta(y)_{j}=y_{j}/4m$, so each component of $\eta$ maps $\mathbb{T}^{n}$ to
$[-1/(8m),1/(8m))$. The following encodes what follows from Theorem A in this
case.

######  Proposition 2.1.

If $p\geq 2$, $k\in[n]$ and $m\geq\sqrt{n/k}$, every
$f:\mathbb{T}^{n}\rightarrow\mathbb{C}$ satisfies

$\displaystyle\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\big{|}f(x+y_{\mathsf{S}})-f(x)\big{|}^{p}\,dydx$
$\displaystyle\lesssim_{p}$
$\displaystyle\\!\\!\\!\\!m^{p}\hskip-3.0pt\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\hskip-3.0pt\Bigg{\\{}\frac{k}{n}\sum\limits_{j=1}^{n}\Big{|}f\big{(}x\hskip-1.0pt+\hskip-1.0pt\frac{y_{j}e_{j}}{4m}\big{)}-f(x)\Big{|}^{p}\hskip-3.0pt+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\Big{|}f\big{(}x\hskip-1.0pt+\hskip-1.0pt\frac{y}{4m}\big{)}-f(x)\Big{|}^{p}\Bigg{\\}}dydx.$

Proof. We start checking that $(\mathbb{Z}^{n},\mathbb{Z}^{n})$ is an
$\mathrm{X}_{p}$-representable pair. Taking the group $\Gamma=\mathbb{Z}^{n}$,
it is clear that (RP-2) holds with our choice of $\eta$ and multipliers given
by the usual translations $\mathrm{M}_{\gamma}f(x)=f(x+\gamma)$ which are
unitary modulations at the Fourier side. Second, if $\chi_{w}=\exp(2\pi
i\langle w,\cdot\rangle)$ is the character associated with
$w\in\mathbb{Z}^{n}$, we consider the differential operators

$\partial_{j}\chi_{w}=\delta_{w_{j}\neq 0}\,\chi_{w}\quad\mbox{for}\quad
j\in[n].$

By [2, Subsection 2.1], we have

$\frac{1}{{n\choose k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\big{\|}\mathsf{E}_{[n]\setminus\mathsf{S}}f\big{\|}^{p}_{L_{p}(\mathbb{T}^{n})}\lesssim_{p}\frac{k}{n}\sum\limits_{j=1}^{n}\|\partial_{j}f\|_{L_{p}(\mathbb{T}^{n})}^{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\|f\|^{p}_{L_{p}(\mathbb{T}^{n})}.$

In particular, by Fubini’s theorem we get (RP-1). Then, Theorem A yields

$\displaystyle\hskip-5.0pt\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\big{|}f(x+y_{\mathsf{S}})-f(x)\big{|}^{p}\,dydx$
$\displaystyle\lesssim_{p}$
$\displaystyle\\!\\!\\!\\!m^{p}\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\frac{k}{n}\sum\limits_{j=1}^{n}\Big{|}\partial_{j}^{y}f\big{(}x+\frac{y}{2m}\big{)}\Big{|}^{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\Big{|}f\big{(}x+\frac{y}{4m}\big{)}-f(x)\Big{|}^{p}dydx,$

using the superscripts in the partial derivatives to indicate the variable
over which differentiation is performed. It only remains to estimate (2.1) to
establish the result. In order to do that, observe that (1.5) holds in this
case. Applying it to $h(y)=f(x+y/2m)$ and using the properties of translations
we get

$\displaystyle\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\Big{|}\partial_{j}^{y}f\Big{(}x+\frac{y}{2m}\Big{)}\Big{|}^{p}dydx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\Big{|}f\Big{(}x+\frac{y}{2m}\Big{)}-\mathsf{E}_{\\{j\\}}^{y}f\Big{(}x+\frac{y}{2m}\Big{)}\Big{|}^{p}dydx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\Big{|}f\Big{(}x+\frac{y}{2m}\Big{)}-\int_{\mathbb{T}}f\Big{(}x+\frac{y_{[n]\setminus\\{j\\}}+te_{j}}{2m}\Big{)}dt\Big{|}^{p}dydx$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}}\Big{|}f\Big{(}x+\frac{y}{2m}\Big{)}-f\Big{(}x+\frac{y_{[n]\setminus\\{j\\}}+te_{j}}{2m}\Big{)}\Big{|}^{p}dtdydx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}}\Big{|}f\Big{(}x+\frac{y_{j}e_{j}}{2m}\Big{)}-f\Big{(}x+\frac{te_{j}}{2m}\Big{)}\Big{|}^{p}dtdydx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\Big{|}f\Big{(}x+\frac{y_{j}e_{j}}{2m}\Big{)}-f(x)\Big{|}^{p}dydx\leq
2\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\Big{|}f\Big{(}x+\frac{y_{j}e_{j}}{4m}\Big{)}-f(x)\Big{|}^{p}dydx.$

Inserting that in the outcome of Theorem A yields the assertion. $\square$

######  Remark 2.2.

_A different choice of $\eta:\mathbb{T}^{n}\to\mathbb{T}^{n}$ is given by
$\eta(y)_{j}=\mathrm{sgn}(y_{j})/8m$ which leads to an inequality closer to
($\mathrm{MX}_{p}$). Recalling that we identify $\mathbb{T}$ with
$[-1/2,1/2)$, it turns out that $4m\eta(y)\equiv-e/2$ for $e=(1,1,\ldots,1)$,
and one can prove the following statement for every
$f:\mathbb{T}^{n}\rightarrow\mathbb{C}$_

$\displaystyle\hskip 30.0pt\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\mathbb{T}^{n}}\Big{|}f\big{(}x+\frac{e_{\mathsf{S}}}{2}\big{)}-f(x)\Big{|}^{p}dx$
$\displaystyle\lesssim_{p}$
$\displaystyle\\!\\!\\!\\!m^{p}\int_{\mathbb{T}^{n}}\frac{k}{n}\sum\limits_{j=1}^{n}\Big{|}f\big{(}x+\frac{e_{j}}{8m}\big{)}-f(x)\Big{|}^{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\frac{1}{2^{n}}\sum_{\varepsilon\in\mathbb{Z}_{2}^{n}}\Big{|}f\big{(}x+\frac{\varepsilon}{8m}\big{)}-f(x)\Big{|}^{p}dx.$

_The proof of ( 2.2) follows the same lines as that of Proposition 2.1. It is
also worth noting that inequality (2.2) can be discretized to recover
($\mathrm{MX}_{p}$) and a limiting procedure allows one to pass from
($\mathrm{MX}_{p}$) to (2.2), we omit the details. This means that this _semi-
continuous_ inequality is equivalent to Naor’s original $\mathrm{X}_{p}$
inequality._

######  Remark 2.3.

_It is also possible to consider the usual differential structure on
$\mathbb{T}^{n}$ which can be thought of as the most natural. In that case,
$L_{p}$ valued versions of estimates for balanced truncations of Fourier
series hold by Fubini’s theorem [2] and one gets the following_

$\displaystyle\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\int_{\mathbb{T}^{n}}\int_{\mathbb{T}^{n}}\big{|}f(x+y_{\mathsf{S}})-f(x)\big{|}^{p}\,dydx$
$\displaystyle\lesssim_{p}$
$\displaystyle\\!\\!\\!\\!m^{p}\int_{\mathbb{T}^{n}}\hskip-3.0pt\int_{\mathbb{T}^{n}}\frac{k}{n}\sum\limits_{j=1}^{n}\Big{|}\partial_{y_{j}}f\big{(}x+\frac{y}{2m}\big{)}\Big{|}^{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\Big{|}f\big{(}x+\frac{y}{4m}\big{)}-f(x)\Big{|}^{p}dydx,$

_which can be seen to be weaker than those from Proposition 2.1, see [2] for
details. _

###  2.2. Cyclic groups with the word length

We now focus on a discrete inequality that is a strict generalization of
($\mathrm{MX}_{p}$). Consider the inclusion
$\eta_{\ell}:\mathbb{Z}_{2\ell}^{n}\to\mathbb{Z}_{8\ell m}^{n}$ given by

(2.3)
$\displaystyle\eta_{\ell}(x)=\big{(}\beta_{\ell}(x_{1}),\ldots,\beta_{\ell}(x_{n})\big{)},$

where $\beta_{\ell}(y)=y-\ell$ when $0\leq y\leq\ell-1$ and
$\beta_{\ell}(y)=y-(\ell-1)$ when $\ell\leq y\leq 2\ell-1$.

######  Proposition 2.4.

If $p\geq 2$, $k\in[n]$ and $m\geq\sqrt{n/k}$, every $f:\mathbb{Z}_{8\ell
m}^{n}\rightarrow\mathbb{C}$ satisfies

$\displaystyle\hskip-60.0pt\frac{m^{-p}}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\frac{1}{(8\ell
m)^{n}(2\ell)^{n}}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}\big{|}f\big{(}x+4m\eta_{\ell}(y)_{\mathsf{S}}\big{)}-f(x)\big{|}^{p}$
$\displaystyle\lesssim_{p}$
$\displaystyle(4\ell)^{p-1}\frac{k}{n}\sum\limits_{j=1}^{n}\frac{1}{(8\ell
m)^{n}}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\big{|}f(x+e_{j})-f(x)\big{|}^{p}$ $\displaystyle+$
$\displaystyle\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\frac{1}{(8\ell
m)^{n}(2\ell)^{n}}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}\big{|}f(x+\eta_{\ell}(y))-f(x)\big{|}^{p}.$

Proof. Choose $\ell\geq 1$ and take
$(\mathrm{H},\mathrm{G})=(\mathbb{Z}_{2\ell}^{n},\mathbb{Z}_{8\ell m}^{n})$
with auxiliary group $\Gamma=\mathrm{G}$. We want to check that this is an
$\mathrm{X}_{p}$-representable pair. It is clear that the range of
$\beta_{\ell}$ is $-[\ell]\cup[\ell]\subset\mathbb{Z}_{8\ell m}^{n}$. When
$\ell=1$, it recovers the map that sends $0$ to $-1$ and $1$ to $1$. In
addition, a simple observation yields that
$-\beta_{\ell}(y)=\beta_{\ell}(2\ell-1-y)$ which implies condition (RP-2) i).
Next, denoting again characters by $\chi_{w}$ for each
$w\in\mathbb{Z}_{2\ell}^{n}$, we choose conditional expectations given by

$\mathsf{E}_{[n]\setminus\mathsf{S}}\chi_{w}=\delta_{w\in\mathbb{Z}_{2\ell}^{\mathsf{S}}}\chi_{w}.$

By [2, Subsection 2.2] and Fubini’s theorem, (RP-1) holds with derivatives
given again by $\partial_{j}:\chi_{w}\mapsto\delta_{w_{j}\neq 0}\chi_{w}$.
Finally, the map $\mathrm{M}_{\gamma}:\chi_{w}\mapsto\chi_{w}(\gamma)\chi_{w}$
is a completely bounded multiplier on $L_{p}(\mathbb{Z}_{8\ell m}^{n})$, so
the family $\\{\mathrm{M}_{\gamma}\\}_{\gamma\in\Gamma}$ satisfies (RP-2) ii).
The application of Theorem A yields

$\displaystyle\hskip-40.0pt\frac{m^{-p}}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\frac{1}{(8\ell
m)^{n}(2\ell)^{n}}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}\big{|}f\big{(}x+4m\eta_{\ell}(y)_{\mathsf{S}}\big{)}-f(x)\big{|}^{p}$
$\displaystyle\lesssim_{p}$
$\displaystyle\frac{k}{n}\sum\limits_{j=1}^{n}\frac{1}{(8\ell
m)^{n}(2\ell)^{n}}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{m}}\big{|}\partial_{j}^{y}f\big{(}x+2\eta_{\ell}(y)\big{)}\big{|}^{p}$
$\displaystyle+$
$\displaystyle\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\frac{1}{(8\ell
m)^{n}(2\ell)^{n}}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}\big{|}f(x+\eta_{\ell}(y))-f(x)\big{|}^{p}.$

Since (1.5) holds in this setting too, we get

$\displaystyle\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}$
$\displaystyle\big{|}\partial_{j}^{y}f(x+2\eta_{\ell}(y))\big{|}^{p}$
$\displaystyle=\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}\big{|}f(x+2\eta_{\ell}(y))-\mathsf{E}_{\\{j\\}}^{y}f(x+2\eta_{\ell}(y))\big{|}^{p}$
$\displaystyle=\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}\Big{|}\frac{1}{2\ell}\sum\limits_{t\in\mathbb{Z}_{2\ell}}f(x+2\eta_{\ell}(y))-f(x+2\eta_{\ell}(y_{[n]\setminus\\{j\\}}+te_{j}))\Big{|}^{p}$
$\displaystyle\leq\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y\in\mathbb{Z}_{2\ell}^{n}}\frac{1}{2\ell}\sum\limits_{t\in\mathbb{Z}_{2\ell}}\big{|}f(x+2\eta_{\ell}(y))-f(x+2\eta_{\ell}(y_{[n]\setminus\\{j\\}}+te_{j}))\big{|}^{p}$
$\displaystyle=(2\ell)^{n-2}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y_{j}\in\mathbb{Z}_{2\ell}}\sum\limits_{t\in\mathbb{Z}_{2\ell}}\big{|}f(x+2\beta_{\ell}(y_{j})e_{j})-f(x+2\beta_{\ell}(t)e_{j})\big{|}^{p},$

by the definition of $\mathsf{E}_{[n]\setminus\mathsf{S}}$ and convexity.
Since $|\beta_{\ell}(y_{j})-\beta_{\ell}(t)|\leq 2\ell$, the translation
invariance of the Haar measure yields

$\displaystyle\hskip-100.0pt(2\ell)^{n-2}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\sum\limits_{y_{j},t\in\mathbb{Z}_{2\ell}}\big{|}f(x+2\beta_{\ell}(y_{j})e_{j})-f(x+2\beta_{\ell}(t)e_{j})\big{|}^{p}$
$\displaystyle\leq$
$\displaystyle(2\ell)^{n}(4\ell)^{p-1}\sum\limits_{x\in\mathbb{Z}_{8\ell
m}^{n}}\big{|}f(x+e_{j})-f(x)\big{|}^{p},$

which yields the result when inserted in (2.2). This completes the proof.
$\square$

######  Remark 2.5.

_In our cyclic metric $\mathrm{X}_{p}$ inequalities, a similar computation as
the one in [9, Proposition 1.4] shows that the condition $m\geq\sqrt{n/k}$ is
necessary in Proposition 2.4._

##  3\. Transferred $\mathrm{X}_{p}$ inequalities

Now we apply Theorem A to some pairs $(\mathrm{H},\mathrm{G})$ with
$\mathrm{H}=\mathbb{Z}_{2}^{n}$ and nonabelian $\mathrm{G}$. Clearly, the
choices of the auxiliary group $\Gamma$ and $\mathrm{G}$ in each case must be
related. We give the details for one particular choice and comment on a few
more later. We choose $\Gamma=\mathbb{Z}_{8m}^{n}$ and

$\mathrm{G}=\mathbb{Z}_{8m}^{*n}:=\mathbb{Z}_{8m}*\ldots*\mathbb{Z}_{8m},$

the free product of $n$ copies of $\mathbb{Z}_{8m}$. For $j\in[n]$, denote the
generator of its $j$-th factor by $g_{j}$. Elements $w\in\mathbb{Z}_{8m}^{*n}$
can and will be described as reduced words in the form

$w=g_{k_{1}}^{\ell_{1}}g_{k_{2}}^{\ell_{2}}\ldots
g_{k_{r}}^{\ell_{r}}\quad\mbox{with}\quad k_{i}\neq
k_{i+1}\quad\mbox{and}\quad\ell_{i}\in\mathbb{Z}\setminus 8m\mathbb{Z}.$

Given $u\in\mathbb{Z}_{8m}^{n}$, we consider the character
$\mbox{\large$\chi$}_{u}:\mathbb{Z}_{8m}^{*n}\to\mathbb{C}$ determined by

$\mbox{\large$\chi$}_{u}(w)=\exp\Bigg{(}\frac{2\pi
i}{8m}\sum\limits_{j=1}^{n}u_{j}\sum\limits_{t:k_{t}=j}\ell_{t}\Bigg{)}\quad\mbox{for}\quad
w=g_{k_{1}}^{\ell_{1}}g_{k_{2}}^{\ell_{2}}\ldots g_{k_{r}}^{\ell_{r}}.$

It is clear that
$\mbox{\large$\chi$}_{u}(w)\mbox{\large$\chi$}_{v}(w)=\mbox{\large$\chi$}_{u+v}(w)$.
Next, we define the Fourier multipliers
$\mathrm{M}_{u}:\mathcal{L}(\mathbb{Z}_{8m}^{*n})\rightarrow\mathcal{L}(\mathbb{Z}_{8m}^{*n})$
determined by $\lambda(w)\mapsto\mbox{\large$\chi$}_{u}(w)\lambda(w)$ for the
left regular representation $\lambda$ on $\mathbb{Z}_{8m}^{*n}$.

######  Lemma 3.1.

$\mathrm{M}_{u}$ is a normal unital trace-preserving $*-$homomorphism on
$\mathcal{L}(\mathbb{Z}_{8m}^{*n})$.

Proof. $\mathrm{M}_{u}$ is clearly unital and

$\mathrm{M}_{u}(\lambda(w)^{*})=\mathrm{M}_{u}(\lambda(w^{-1}))=\overline{\mbox{\large$\chi$}_{u}(w)}\
\lambda(w)^{*}=\mathrm{M}_{u}(\lambda(w))^{*}.$

Next we claim that
$\mathrm{M}_{u}(\lambda(ww^{\prime}))=\mathrm{M}_{u}(\lambda(w))\mathrm{M}_{u}(\lambda(w^{\prime}))$.
It is clear that this identity holds when both $w$ and $w^{\prime}$ are powers
of a fixed generator $g_{j}$. More precisely, we have

$\mathrm{M}_{u}(\lambda(g_{j}^{\ell_{1}+\ell_{2}}))=\mathrm{M}_{u}(\lambda(g_{j}^{\ell_{1}}))\
\mathrm{M}_{u}(\lambda(g_{j}^{\ell_{2}})).$

Now, let us consider two reduced words given by

$w=g_{k_{1}}^{\ell_{1}}g_{k_{2}}^{\ell_{2}}\ldots
g_{k_{r}}^{\ell_{r}}\quad\mbox{and}\quad
w^{\prime}=g_{k_{1}^{\prime}}^{\ell_{1}^{\prime}}g_{k_{2}^{\prime}}^{\ell_{2}^{\prime}}\ldots
g_{k_{s}^{\prime}}^{\ell_{s}^{\prime}}.$

If $k_{r}\neq k_{1}^{\prime}$ the claim trivially holds. If
$k_{r}=k_{1}^{\prime}$ and $\ell_{r}+\ell_{1}^{\prime}\not\equiv 0\
(\mbox{mod}\ 8m)$, then

$\mathrm{M}_{u}(\lambda(ww^{\prime}))=\mathrm{M}_{u}(\lambda(g_{k_{1}}^{\ell_{1}}\ldots
g_{k_{r-1}}^{\ell_{r-1}}))\
\mathrm{M}_{u}(\lambda(g_{k_{r}}^{\ell_{r}+\ell_{1}^{\prime}}))\
\mathrm{M}_{u}(\lambda(g_{k_{2}^{\prime}}^{\ell_{2}^{\prime}}\ldots
g_{k_{s}^{\prime}}^{\ell_{s}^{\prime}}))$

which yields the same conclusion. For the remaining case, we may write
$ww^{\prime}$ as

$ww^{\prime}=\underbrace{g_{k_{1}}^{\ell_{1}}\ldots
g_{k_{r-1}}^{\ell_{r-1}}}_{\rho}\underbrace{g_{k_{2}^{\prime}}^{\ell_{2}^{\prime}}\ldots
g_{k_{s}^{\prime}}^{\ell_{s}^{\prime}}}_{\rho^{\prime}}.$

Arguing as above, if $k_{r-1}\neq k^{\prime}_{2}$ or
$\ell_{r-1}+\ell_{2}^{\prime}\not\equiv 0\ (\mbox{mod}\ 8m)$ we get

$\displaystyle\mathrm{M}_{u}(\lambda(ww^{\prime}))$ $\displaystyle=$
$\displaystyle\mathrm{M}_{u}(\lambda(\rho))\mathrm{M}_{u}(\lambda(\rho^{\prime}))$
$\displaystyle=$
$\displaystyle\mathrm{M}_{u}(\lambda(\rho))\mathrm{M}_{u}(\lambda(g_{k_{r}}^{\ell_{r}+\ell_{1}^{\prime}}))\mathrm{M}_{u}(\lambda(\rho^{\prime}))\
=\ \mathrm{M}_{u}(\lambda(w))\mathrm{M}_{u}(\lambda(w^{\prime})).$

One can iterate this process to deduce the claim. Thus, $\mathrm{M}_{u}$ is a
$*-$homomorphism on $\mathrm{span}\\{\lambda(w):w\in\mathbb{Z}_{8m}^{*n}\\}$.
In particular, it is a completely positive unital map. Moreover, it is trace-
preserving since
$\tau(\mathrm{M}_{u}(\lambda(w)))=\mbox{\large$\chi$}_{u}(w)\tau(\lambda(w))$
and $\mbox{\large$\chi$}_{u}(e)=1$. Finally, note that for any
$f,g\in\mathcal{L}(\mathbb{Z}_{8m}^{*n})$ there holds

$\displaystyle\tau(\mathrm{M}_{u}(f)g^{*})\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\tau\Big{[}\Big{(}\sum\limits_{w\in\mathbb{Z}_{8m}^{*n}}\widehat{f}(w\
\mbox{\large$\chi$}_{u}(w\
\lambda(w)\Big{)}\Big{(}\sum\limits_{\eta\in\mathbb{Z}_{8m}^{*n}}\widehat{g}(\eta)\
\lambda(\eta)\Big{)}^{*}\Big{]}$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\sum\limits_{w\in\mathbb{Z}_{8m}^{*n}}\widehat{f}(w)\mbox{\large$\chi$}_{u}(w)\overline{\widehat{g}(w)}\,=\sum\limits_{w\in\mathbb{Z}_{8m}^{*n}}\widehat{f}(w)\overline{\mbox{\large$\chi$}_{-u}(w)}\overline{\widehat{g}(w)}\,=\,\tau(f(\mathrm{M}_{-u}g)^{*}).$

Since $\mathrm{M}_{-u}$ extends to a bounded map on
$L_{1}(\mathcal{L}(\mathbb{Z}_{8m}^{*n}))$, $\mathrm{M}_{u}$ is
$w^{*}-$continuous. $\square$

Before exploring $\mathrm{X}_{p}$ inequalities for free groups, we shall need
to use Theorem B in [2] with values in a noncommutative $L_{p}$-space over a
QWEP von Neumann algebra $\mathcal{M}$. More precisely, given a mean-zero
$f:\mathbb{Z}_{2}^{n}\to\mathcal{M}$ we claim that

$\displaystyle\hskip-30.0pt\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\big{\|}(\mathsf{E}_{[n]\setminus\mathsf{S}}\otimes\mathrm{id})f\big{\|}^{p}_{L_{p}(\mathbb{Z}_{2}^{n};L_{p}(\mathcal{M}))}$
$\displaystyle\lesssim_{p}$
$\displaystyle\frac{k}{n}\sum\limits_{j=1}^{n}\big{\|}(\partial_{j}\otimes\mathrm{id})f\big{\|}^{p}_{L_{p}(\mathbb{Z}_{2}^{n};L_{p}(\mathcal{M}))}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\|f\|^{p}_{L_{p}(\mathbb{Z}_{2}^{n};L_{p}(\mathcal{M}))},$

where $\mathsf{E}_{[n]\setminus\mathsf{S}}$ and $\partial_{j}$ stand for the
original conditional expectation and directional derivative used by Naor in
[7]. To check that (3) holds, one can see that the proof in [2] goes through
in the operator valued setting as long as dimension-free bounds hold for
operator valued Riesz transforms [4], which just requires to use the same
argument together with Fubini’s theorem from [3].

######  Proposition 3.2.

If $p\geq 2$, $k\in[n]$ and $m\geq\sqrt{n/k}$, every $f\in
L_{p}(\mathcal{L}(\mathbb{Z}_{8m}^{*n}))$ satisfies

$\displaystyle\hskip-40.0pt\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\frac{1}{2^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}\mathrm{M}_{4m\varepsilon_{\mathsf{S}}}f-f\big{\|}^{p}_{L_{p}(\mathcal{L}(\mathbb{Z}_{8m}^{*n}))}$
$\displaystyle\lesssim_{p}$ $\displaystyle
m^{p}\Bigg{(}\frac{k}{n}\sum\limits_{j=1}^{n}\big{\|}\mathrm{M}_{e_{j}}f-f\big{\|}_{p}^{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\frac{1}{2^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}\mathrm{M}_{\varepsilon}f-f\big{\|}_{p}^{p}\Bigg{)}.$

Proof. We apply Theorem A to
$(\mathrm{H},\mathrm{G})=(\mathbb{Z}_{2}^{n},\mathbb{Z}_{8m}^{*n})$, so we
need to check that it is an $\mathrm{X}_{p}$-representable pair. Condition
(RP-1) follows from (3) since $\mathcal{L}(\mathbb{Z}_{8m}^{*n})$ is a QWEP
von Neumann algebra. Consider the auxiliary group $\Gamma=\mathbb{Z}_{8m}^{n}$
with the inclusion $\eta$ defined as in (2.3) for $\ell=1$. Condition (RP-2)
i) is then automatically satisfied. Finally, according to Lemma 3.1 we know
that $\mathrm{M}_{\gamma}$ extends to a completely contractive map on
$L_{p}(\mathcal{L}(\mathbb{Z}_{8m}^{*n}))$ for any $1\leq p\leq\infty$.
Therefore, the family
$\\{\mathrm{M}_{\gamma}\\}_{\gamma\in\mathbb{Z}_{8m}^{n}}$ can be used to
check (RP-2) ii) and Theorem A leads to

$\displaystyle\frac{1}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}$
$\displaystyle\frac{1}{2^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}\mathrm{M}_{4m\varepsilon_{\mathsf{S}}}f-f\big{\|}^{p}_{L_{p}(\mathcal{L}(\mathbb{Z}_{8m}^{*n}))}$
$\displaystyle\lesssim_{p}m^{p}\Bigg{(}\frac{k}{n}\sum\limits_{j=1}^{n}\frac{1}{2^{n}}\sum_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}(\partial_{j}\otimes\mathrm{id})\mathrm{M}_{2\varepsilon}f\big{\|}^{p}_{p}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\frac{1}{2^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}\mathrm{M}_{\varepsilon}f-f\big{\|}_{p}^{p}\Bigg{)},$

for every $f\in L_{p}(\mathbb{Z}_{8m}^{n};L_{p}(\mathbb{Z}_{8m}^{*n}))$.
Finally, the computations for discrete derivatives from Subsection 2.2 imply
that condition (1.4) holds, yielding the result. $\square$

The example above is clearly not the only possible choice that we can make for
the groups $\Gamma$ and $\mathrm{G}$ in the pair. In general, given abelian
groups $\Gamma_{1},\ldots,\Gamma_{n}$ and
$\Gamma=\Gamma_{1}\times\Gamma\times\cdots\times\Gamma_{n}$ so that there is a
nice map

$\eta:\mathbb{Z}_{2}^{n}\to\widehat{\Gamma},$

one can take $\mathrm{G}=\Gamma_{1}*\ldots*\Gamma_{n}$ and construct a family
of multipliers $\\{\mathrm{M}_{\gamma}\\}_{\gamma\in\widehat{\Gamma}}$ as
above. Once the properties of the family are checked, the above scheme yields
a metric $\mathrm{X}_{p}$ inequality for the pair
$(\mathbb{Z}_{2}^{n},\Gamma,\mathrm{G})$. For example, the interested reader
can check that a representation in the spirit of Lemma 3.1 for
$\Gamma=\mathbb{Z}^{n}$ and $\mathrm{G}=\mathbb{F}_{n}$ (the free group of $n$
generators) holds. This yields the following free metric $\mathrm{X}_{p}$
inequality in the free group algebra

$\displaystyle\frac{1}{{n\choose k}}$
$\displaystyle\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\frac{1}{2^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}\mathrm{M}_{4m\varepsilon_{\mathsf{S}}}f-f\big{\|}^{p}_{L_{p}(\mathcal{L}(\mathbb{F}_{n}))}$
$\displaystyle\lesssim_{p}m^{p}\Bigg{(}\frac{k}{n}\sum\limits_{j=1}^{n}\big{\|}\mathrm{M}_{e_{j}}f-f\big{\|}^{p}_{L_{p}(\mathcal{L}(\mathbb{F}_{n}))}+\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\frac{1}{2^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}\mathrm{M}_{\varepsilon}f-f\big{\|}_{L_{p}(\mathcal{L}(\mathbb{F}_{n}))}^{p}\Bigg{)}.$

##  4\. Metric consequences

We conclude by collecting a few metric consequences of the results in the
previous sections. In [7], Naor gave two sufficient hypothesis for a Banach
space $\mathbb{X}$ to be a metric $\mathrm{X}_{p}$ space: a Banach
$\mathrm{X}_{p}$ inequality and operator valued dimension-free estimates for
Riesz transforms. In particular, any mean-zero
$f:\mathbb{Z}_{2}^{n}\to\mathbb{X}$ must satisfy

$\frac{1}{2^{n}}\sum_{\varepsilon\in\mathbb{Z}_{2}^{n}}\Big{\|}\sum\limits_{j=1}^{n}\varepsilon_{j}(\partial_{j}\otimes\mathrm{id})f\Big{\|}^{p}_{L_{p}(\mathbb{Z}_{2}^{n};\mathbb{X})}\simeq_{p}\big{\|}(\Delta^{-1/2}\otimes\mathrm{id})f\big{\|}^{p}_{L_{p}(\mathbb{Z}_{2}^{n};\mathbb{X})}.$

When $\mathbb{X}=L_{p}(\mathcal{M})$, the Banach $\mathrm{X}_{p}$ inequality
is [2, Theorem 1.2], while the Riesz transform estimates can be found in [4]
when $\mathcal{M}$ is QWEP, as explained in the previous section. This latter
fact had previously been announced for the Schatten classes $S_{p}$ in [7, 8].
The result that we get is the following:

######  Theorem 4.1 (Noncommutative $L_{p}$ spaces are metric $\mathrm{X}_{p}$
spaces).

Let $\mathcal{M}$ be a _QWEP_ von Neuman algebra. Then, if $p\geq 2$,
$k\in[n]$ and $m\geq\sqrt{n/k}$, every $f:\mathbb{Z}_{8m}^{n}\rightarrow
L_{p}(\mathcal{M})$ satisfies

$\displaystyle\hskip-60.0pt\frac{m^{-p}}{{n\choose
k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\frac{1}{(16m)^{n}}\sum\limits_{x\in\mathbb{Z}_{8m}^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}f(x+4m\varepsilon_{\mathsf{S}})-f(x)\big{\|}^{p}_{L_{p}(\mathcal{M})}$
$\displaystyle\lesssim_{p}$
$\displaystyle\frac{k}{n}\sum\limits_{j=1}^{n}\frac{1}{(8m)^{n}}\sum\limits_{x\in\mathbb{Z}_{8m}^{n}}\big{\|}f(x+e_{j})-f(x)\big{\|}^{p}_{L_{p}(\mathcal{M})}$
$\displaystyle+$
$\displaystyle\Big{(}\frac{k}{n}\Big{)}^{\frac{p}{2}}\frac{1}{(16m)^{n}}\sum\limits_{x\in\mathbb{Z}_{8m}^{n}}\sum\limits_{\varepsilon\in\mathbb{Z}_{2}^{n}}\big{\|}f(x+\varepsilon)-f(x)\big{\|}^{p}_{L_{p}(\mathcal{M})}.$

Theorem 4.1 provides a large class of examples of metric $\mathrm{X}_{p}$
spaces beyond the class of (commutative) $L_{p}$ spaces. Also, as shown in
[7], being a metric $\mathrm{X}_{p}$ space implies lower estimates on the
distorsion of bi-Lipschitz embeddings of nonlinear sets. Indeed, let
$c_{L_{p}(\mathcal{M})}(\mathbf{X})$ denote the smallest norm of a bi-
Lipschitz map $\mathbf{X}\to L_{p}(\mathcal{M})$. Then we have the following:

######  Corollary 4.2.

Let $\mathcal{M}$ be _QWEP_ and $2<q<p$. Then

  * _i)_

$\displaystyle
c_{L_{p}(\mathcal{M})}\big{(}[m]_{q}^{n}\big{)}\,\asymp_{p,q}\,\min\Big{\\{}n^{\frac{(p-q)(q-2)}{q^{2}(p-2)}},m^{1-\frac{2}{q}}\Big{\\}}$.

  * _ii)_

If $c_{L_{p}(\mathcal{M})}\big{(}(L_{q},\|x-y\|_{q}^{\theta})\big{)}<\infty$,
then necessarily $\theta\leq q/p$.

We refer to [7, 9] for precise definitions on $[m]_{q}^{n}$ and
$\theta$-snowflakes. We end with a note about the optimality of the discrete
cyclic inequalities from Section 2. Given $m,n\geq 2$, we know from [9, Lemma
3.1] that there exists
$h_{m}^{n}:\mathbb{Z}_{m}^{n}\rightarrow\\{0,\ldots,4m\\}^{2n}$ such that

$\Bigg{(}\sum\limits_{j=1}^{n}\Big{|}\exp\Big{(}\frac{2\pi
ix_{j}}{m}\Big{)}-\exp\Big{(}\frac{2\pi
iy_{j}}{m}\Big{)}\Big{|}^{q}\Bigg{)}^{\frac{1}{q}}\,\sim\,\big{\|}h_{m}^{n}(x)-h_{m}^{n}(y)\big{\|}_{q}$

holds (up to absolute constants) for any $x,y\in\mathbb{Z}_{m}^{n}$ and $q\geq
2$. Given a bi-Lipschitz map $g:[16m]^{2n}_{q}\rightarrow L_{p}$ with bi-
Lipschitz norm $D$, set $F=g\circ h_{4m,n}$. Then, arguing as in [9, Theorem
1.14] we get

$\frac{1}{{n\choose k}}\sum\limits_{\begin{subarray}{c}\mathsf{S}\subset[n]\\\
|\mathsf{S}|=k\end{subarray}}\frac{1}{(4m)^{2n}}\sum\limits_{x,y\in\mathbb{Z}_{4m}^{n}}\big{\|}F(x+y_{\mathsf{S}})-F(x)\big{\|}_{L_{p}}^{p}\,\gtrsim_{p}\,m^{p}k^{\frac{p}{q}}\big{|}e^{\frac{2\pi
i}{4m}}-1\big{|}.$

Next, we make the choices $k=\lceil n^{\frac{p(q-2)}{q(p-2)}}\rceil$ and
$m=\lceil n^{\frac{p-q}{q(p-2)}}\rceil$, which ensure that
$m\gtrsim\sqrt{n/k}$, and so we can apply Proposition 2.4 to $F$ in order to
get the following statement: if $2<q<p$ and $m,n,\ell\in\mathbb{N}$, there
holds

$c_{L_{p}}([m]_{q}^{n})\gtrsim_{p,q}\min\Big{\\{}|e^{i\pi/\ell}-1|\,n^{\frac{(p-q)(q-2)}{q^{2}(p-2)}},m^{1-\frac{2}{q}}\Big{\\}}\gtrsim_{p,q,\ell}\min\Big{\\{}n^{\frac{(p-q)(q-2)}{q^{2}(p-2)}},m^{1-\frac{2}{q}}\Big{\\}}.$

Thus, up to constants (which get worse as $\ell\to\infty$) we recover the
optimal distortions found in [7]. Also, Naor’s result for $\theta$-snowflakes
in [7] follow from Proposition 2.4.

######  Problem 4.3.

According to [8, Theorem 12]

(4.1)
$c_{S_{p}}(\mathrm{X})\leq\dim(\mathrm{X})^{\frac{q}{2}(\frac{1}{q}-\frac{1}{p})}$

for every linear subspace $\mathrm{X}\subset S_{q}^{n}$ and any
$2<q<p<\infty$. Is there an embedding of $\mathrm{X}$ into
$S_{p}(\ell_{q}^{m})$ with $m=\dim(\mathrm{X})$ and constants independent of
$m$? This is trivially true for row, column or diagonal subspaces and we know
from [5, Theorem 2] that it holds for the full space $\mathrm{X}=S_{q}^{n}$
with optimal $m$ being $\dim(\mathrm{X})=n^{2}$. If this was the case for
general $\mathrm{X}$, we would get $c_{S_{p}}(\mathrm{X})\lesssim
c_{S_{p}}(S_{p}(\ell_{q}^{m}))$ whose rightmost term is dominated by an
operator space $($matrix amplification$)$ form of $c_{S_{p}}(\ell_{q}^{m})$.
Using that $S_{p}$ is a metric $\mathrm{X}_{p}$-space and arguing as above

(4.2) $c_{S_{p}}(\ell_{q}^{m})\asymp_{p,q}m^{\frac{(p-q)(q-2)}{q^{2}(p-2)}}.$

Does $c_{S_{p}}(S_{p}(\ell_{q}^{m}))$ behave like $c_{S_{p}}(\ell_{q}^{m})$?
If so, one could guess that

(4.3)
$c_{S_{p}}(\mathrm{X})\leq_{p,q}\dim(\mathrm{X})^{\frac{(p-q)(q-2)}{q^{2}(p-2)}}\quad\mbox{for
every}\quad\mathrm{X}\subset S_{q}^{n}?$

This would already improve (4.1) and it is best possible for
$\mathrm{X}=\ell_{q}^{n}=\mathrm{diag}(S_{q}^{n})$. It is still open to find
the optimal distortion $c_{S_{p}}(\mathrm{X})$ for the full space
$\mathrm{X}=S_{q}^{n}$. In this case, we have the following bounds

(4.4) $\mathrm{A}:=n^{\frac{(p-q)(q-2)}{q^{2}(p-2)}}\leq
c_{S_{p}}(S_{q}^{n})\leq\min\Big{\\{}n^{\frac{1}{2}-\frac{1}{q}},n^{\frac{1}{q}-\frac{1}{p}}\Big{\\}}.$

Indeed, the lower bound follows from (4.2) and the inclusion
$\ell_{q}^{n}\subset S_{q}^{n}$. The upper bound follows easily using Hölder
inequality. Note that (4.3) gives $c_{S_{p}}(S_{q}^{n})\leq\mathrm{A}^{2}$ but
(4.4) implies that no bound $c_{S_{p}}(S_{q}^{n})\leq\mathrm{A}^{\beta}$ could
be optimal for any $\beta>1$.

#### Acknowledgement.

The third-named author wants to express his gratitude to Alexandros Eskenazis
and Assaf Naor for some valuable comments on a preliminary version, which led
to an improved presentation. The three authors were partly supported by the
Spanish Grant PID 2019-107914GB-I00 (MCIN / PI J. Parcet) as well as Severo
Ochoa Grant CEX2019-000904-S (ICMAT), funded by MCIN/AEI
10.13039/501100011033. José M. Conde-Alonso was also supported by the Madrid
Government Program V under PRICIT Grant SI1/PJI/2019-00514. Antonio Ismael
Cano-Mármol was supported by the Grant SEV-2015-0554-19-3 which is funded by
MCIN/AEI/10.13039/501100011033 and by ESF Investing in your future.

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Antonio Ismael Cano Mármol
Instituto de Ciencias Matemáticas
Consejo Superior de Investigaciones Científicas
<EMAIL_ADDRESS>
José M. Conde-Alonso
Universidad Autónoma de Madrid
Instituto de Ciencias Matemáticas
<EMAIL_ADDRESS>
Javier Parcet
Instituto de Ciencias Matemáticas
Consejo Superior de Investigaciones Científicas
<EMAIL_ADDRESS>