# GReS: Graphical Cross-domain Recommendation for Supply Chain Platform

Zhiwen Jing<EMAIL_ADDRESS>0000-0003-0000-8937 College of
Information and ComputerTaiyuan University of TechnologyTaiyuanChina ,
Ziliang Zhao<EMAIL_ADDRESS>Gaoling School of Artificial
IntelligenceRenmin University of ChinaBeijingChina ,  Yang Feng
<EMAIL_ADDRESS>MeituanBeijingChina ,  Xaochen Ma
<EMAIL_ADDRESS>MeituanBeijingChina ,  Nan Wu<EMAIL_ADDRESS>MeituanBeijingChina ,  Shengqiao Kang<EMAIL_ADDRESS>MeituanBeijingChina ,  Cheng Yang<EMAIL_ADDRESS>0000-0001-7507-6171 School of Computer and Information EngineeringZhejiang
Gongshang UniversityHangzhouChina ,  Yujia Zhang
<EMAIL_ADDRESS>College of Information and ComputerTaiyuan
University of TechnologyTaiyuanChina and  Hao Guo<EMAIL_ADDRESS>College
of Information and ComputerTaiyuan University of TechnologyTaiyuanChina

(2022)

###### Abstract.

Supply Chain Platforms (SCPs) provide downstream industries with raw
materials. Compared with traditional e-commerce platforms, data in SCPs is
more sparse due to limited user interests. To tackle the data sparsity
problem, one can apply Cross-Domain Recommendation (CDR) to improve the
recommendation performance of the target domain with the source domain
information. However, applying CDR to SCPs directly ignores hierarchical
structures of commodities in SCPs, which reduce recommendation performance. In
this paper, we take the catering platform as an example and propose GReS, a
graphical CDR model. The model first constructs a tree-shaped graph to
represent the hierarchy of different nodes of dishes and ingredients, and then
applies our proposed Tree2vec method combining GCN and BERT models to embed
the graph for recommendations. Experimental results show that GReS
significantly outperforms state-of-the-art methods in CDR for SCPs.111Zhiwen
Jing and Ziliang Zhao make equal contribution to this research..

Cross-Domain Recommendation, Supply Chain Platform

††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
31th ACM International Conference on Information and Knowledge Management;
October 17–21, 2022; Atlanta, GA, USA††booktitle: Proceedings of the 31th ACM
International Conference on Information and Knowledge Management (CIKM ’22),
October 17–21, 2022, Atlanta, GA, USA††price: 15.00††doi:
10.1145/3511808.3557607††isbn: 978-1-4503-9236-5/22/10††ccs: Information
systems Recommender systems

##  1\. Introduction

Figure 1. CDR for Supply Chain Platform of Catering

The development of Supply-chain Platforms (SCPs) exacerbates the information
overload of users, which emphasizes the importance of recommender systems for
SCPs. However, due to users’ concentrated interests and behaviors,
recommendations in SCPs always go through the more severe data sparsity
problem (Shapira, 2015) compared with traditional e-commerce. As one solution
to tackle the data sparsity problem, Cross-Domain Recommendation (CDR)
(Berkovsky et al., 2007) has been widely studied. The core idea of CDR is
leveraging a rich domain to improve the recommendation performance of a sparse
domain (Zhu et al., 2021). Based on this, many effective methods have been
proposed, including content-based (Winoto and Tang, 2008; Tan et al., 2014;
Fernández-Tobías and Cantador, 2014), embedding-based (Xue et al., 2017; He et
al., 2017; Liu et al., 2020b; Liu et al., 2020a), and rating pattern-based
approaches (Zhang et al., 2016; Farseev et al., 2017; Zhu et al., 2020b). Due
to the high user overlap between SCPs and downstream industries, CDR is
suitable for improving the recommendation performance of SCPs.

In this paper, we study the CDR for catering supply chain platforms. The
methods and conclusions can be directly extended to other SCPs because their
similar features. As shown in Figure 1, the CDR for catering usually contains
a catering SCP (domain _A_) and a traditional catering trading platform
(domain _B_). The common users first buy ingredients in domain _A_ , make
these ingredients into dishes, and then sell these dishes in domain _B_. In
other words, the common users are consumers in domain _A_ and suppliers in
domain _B_. Therefore, it is reasonable to improve the recommendation
performance in domain _A_ through the data in domain _B_.

However, applying CDR to SCPs is challenging. First, there is an information
gap between domain _A_ and _B_. Taking “Beef Burger” as an example, the word
segmentation and semantic understanding of “Beef Burger” in domain _B_ can
only yield two original components, “Beef” and “Burger”, but this is
incomplete and inaccurate. In fact, the beef burger of domain _B_ consist of
olive oil, onion, beef, etc. in domain _A_ , which is not well-considered.
Second, there is a hierarchical relationship between dishes in domain _B_ and
ingredients in domain _A_. In traditional CDR (Xin et al., 2015; Hu et al.,
2019; Man et al., 2017; Zhu et al., 2020a), the relationships among products
in different domains are equal. For example, if common users have watched many
sci-fi movies in the movie domain, the model would recommend sci-fi novels to
these users in the novel domain. However, in the CDR for SCPs, “Pickled
Cucumber” and several other ingredients in domain _A_ constitute the “Beef
Burger” in domain _B_ together, so the relationship between “Pickled Cucumber”
and “Beef Burger” can be described hierarchically. The hierarchical
relationship between dishes in domain _B_ and ingredients in domain _A_ is
difficult to be represented properly.

To solve the challenges above, we propose GReS, a Graphical CDR model for
SCPs. Specifically, to generate better embeddings for users and items in
domain _A_ , we construct a Heterogeneous Graph (HG) with ratings and contents
(Zhu et al., 2020a). We then apply Node2vec (Grover and Leskovec, 2016) to
embed the HG for downstream recommendations. As for domain _B_ , we build
several Tree-shaped Meta Structures (TMSs) of dishes, items (ingredients in
domain _A_), and item categories to describe the hierarchical relation among
nodes. These TMSs constitute a Tree-shaped Graph (TG). We then map the TG to
embedding by a Tree2vec method that combines GCN model (Kipf and Welling,
2016) and BERT model (Devlin et al., 2018). The GCN model focuses on
extracting relational features in the tree-shaped graph, and the BERT model is
suitable for extracting semantic features of the graph. We also use element-
wise attention (Zhu et al., 2020a) to effectively combine embeddings of users
and items from two domains. Since there is no publicly available dataset in
CDR for SCPs, we build a large dataset sampled from a commercial online
system. The experimental results and ablation studies on the dataset show that
GReS significantly outperforms the state-of-art baselines in various
evaluation metrics. The effectiveness of GReS makes it suitable to be applied
in CDR for SCPs in the future.

Overall, the contributions of this paper include:

  * •

To the best of our knowledge, GReS is the first CDR model designed to
alleviate the data sparsity problem in SCPs.

  * •

To model the hierarchy between the commodity of downstream industry and supply
chain, we design the TMS and propose a Tree2vec method to map the TMS to
embeddings.

  * •

The experimental results on a commercial dataset demonstrate the efficacy of
GReS over existing baselines.

##  2\. GReS Model

As illustrated in Figure 2, GReS consists of five layers in total. First, in
the graph structure layer, we use users and items to build two graphs, where
domain _A_ is defined as a Heterogeneous Graph (HG), and domain _B_ is defined
as a Tree-shaped Graph (TG). We then embed the two graphs using Node2vec
(Grover and Leskovec, 2016) for domain _A_ and our proposed Tree2vec for
domain _B_ in the graph embedding layer. Next, in the feature combination
layer, we use the element-wise attention mechanism to concatenate embeddings
from both domains. Finally, we apply an MLP to model the non-linear
relationship between users and items and generate final user-item interaction
predictions in the neural network layer and prediction layer.

Figure 2. The overall architecture of GReS model

###  2.1. Graph Structure Layer

As shown in Figure 2, the HG integrates information obtained from domain _A_
(SCPs). Based on existing work (Zhu et al., 2020a), we further introduce the
item category node which is the category summary of items. For example, the
item “Australian Wagyu M5 600g” belongs to the category “Beef”. Introducing
this node also adds three new edge types (user-category, category-category,
and category-item) into existing types (user-user, item-item, and user-item).
To construct the HG of domain _A_ , we apply the Doc2vec (Le and Mikolov,
2014) model to map the documents of user profiles, category details, and item
details into vectors of users, categories, and items respectively. We then use
the similarity between nodes to determine whether an edge among users,
categories, and items should be generated. Taking users as an example
(categories and items are built in a similar way), first, we calculate the
Euclidean distance among users: $sim(i,j)=|Eu(u_{i},u_{j})|$, where $u_{i}$
and $u_{j}$ are two different common users. Then we perform Min-Max
normalization on $sim(i,j)$ to get $sim_{M}(i,j)$. After that, the edge weight
between $u_{i}$ and $u_{j}$ can be calculated by:

(1) $w(i,j)=\begin{cases}0,&sim_{M}(i,j)\leq\alpha\\\
sim_{M}(i,j),&sim_{M}(i,j)>\alpha\\\ \end{cases}$

where $w(i,j)$ is the edge weight between $u_{i}$ and $u_{j}$, and $\alpha$ is
a hyper-parameter which controls the sampling rate of the edge. For user-item,
user-category, and category-item edges, we use the normalized order number of
the items, categories, and items under each category as their corresponding
edge weights.

To construct the TG of domain _B_ , it should consider common users who sell
dishes in domain _B_ made by ingredients in domain _A_ (SCPs), including
dishes, item categories, and items. Therefore, we design the TMS which is the
component of TG to describe the hierarchical relationship of dishes-
categories-items. Taking the dish“Beef Burger” as an example, it is the parent
of categories “Beef”, “Tomato”, and “Pickled Cucumber”, and the category
“Beef” is the parent of item “Australian Wagyu M5 600g”. Therefore, we
logically divide dishes, categories, and items of TG into three levels to form
TMSs as shown in Figure 1. The weights in the TG will be determined in a
similar way to the HG.

###  2.2. Graph Embedding Layer

Figure 3. Tree2vec model

Based on the HG of domain _A_ , we apply Node2vec (Grover and Leskovec, 2016)
to generate an embedding for a user $u_{i}$ and an item $i_{h}$ as
$\hat{v}_{u_{i}}$ and $\hat{v}_{i_{h}}$ respectively shown in Figure 2. As for
domain _B_ , it is crucial to leverage useful features in the TG for
recommendation. In this paper, we mainly consider two types of features:
spatial feature and semantic feature. The former concentrates on extracting
spatial information to better encode the tree structure of the data. The
latter focus on the interaction of semantics in the data. Given a common user
$u_{i}$ and its TG $T_{u_{i}}$, the embedding of the TG is defined as:
$t_{i}=\text{GCN}(T_{u_{i}})\oplus\text{BERT}(T_{u_{i}})$, where “$\oplus$” is
the concatenation operator. To implement the above two features, we propose a
Tree2vec method containing a GCN model (Kipf and Welling, 2016) for extracting
spatial information and a BERT model (Devlin et al., 2018) for extracting
semantic information in the TMSs. The Tree2vec model then concatenates these
two feature vectors as the embedding of the TG, and applies the embedding for
recommendation. We ensure that a common user’s corresponding TG has at least
one TMS. That is, each merchant sells at least one dish.

####  2.2.1. Tree2vec-GCN

In Section 2.1, we have aggregated all the TMSs of each common user and
constructed an TG to represent the hierarchy of TMSs in domain _B_. To extract
spatial information, we apply GCN to encode the graph into a feature vector.
Specifically, we first represent the TG as an adjacency matrix $M_{T}^{i}$.
Each value in the matrix denotes the weight between different nodes. Since TG
is a directed graph, the $M_{T}^{i}$ is a lower triangular matrix. After that,
we implement a two-layers GCN model to encode $M_{T}^{i}$ as follows:

(2) $\displaystyle H^{(l+1)}$
$\displaystyle=\sigma\left(\tilde{D}^{-\frac{1}{2}}\tilde{M_{T}^{i}}\tilde{D}^{-\frac{1}{2}}H^{(l)}W^{(l)}\right),\tilde{M_{T}^{i}}=M_{T}^{i}+I$

where $H^{(l)}$ is the matrix of node features in the $l$th layer of the GCN,
$W$ is a trainable weight matrix, $\tilde{D}^{-\frac{1}{2}}$ is the degree
matrix, and $\tilde{M_{T}^{i}}$ is the sum of $M_{T}^{i}$ and an identity
matrix. We finally get the item embedding of the $h$th items $i_{h}$ as
$G_{i_{h}}^{I}$, and add all dish embeddings of user $u_{i}$ to obtain the
embedding of the $u_{i}$ as $G_{u_{i}}^{U}$.

####  2.2.2. Tree2vec-BERT

The GCN model can only capture spatial structural information from the TG,
while semantic information is not considered. Therefore, we further apply BERT
language model (Devlin et al., 2018) to learn semantic information of TMSs for
each common user. To form a TMS of $u_{i}$ into a sequence, we apply the DFS
strategy to traverse nodes in the entire TMS and form a node sequence
$s_{u_{i}}^{t}$. To represent the hierarchical relationship of TMSs, during
the traversal process, we add the following special tokens: (1) A “-” token
denoting the edge from dish to category. (2) A “- -” token denoting the edge
from category to item.

To distinguish different type of nodes, we further modify the position
embeddings in original BERT model to denote three different levels, and use
the pre-trained model BERT for representation learning of $s_{u_{i}}^{t}$ to
get all dish embeddings and item embeddings. We then add all dish embeddings
together to get the embedding of the common user $u_{i}$ as $B_{u_{i}}^{U}$.
We directly use the item representation obtained from BERT as the item
embedding of the $u_{i}$, denoted as $B_{i_{h}}^{I}$. Finally, we concatenate
$G_{u_{i}}^{U}$ with $B_{u_{i}}^{U}$, and $G_{i_{h}}^{I}$ with $B_{i_{h}}^{I}$
respectively to obtain $u_{i}$’s embedding $\tilde{v}_{u_{i}}$ and $i_{h}$’
embedding $\tilde{v}_{i_{h}}$.

###  2.3. Feature Combination Layer

In the feature combination layer, we use the element-wise attention mechanism
to combine the user and item embeddings generated for two domains in the graph
embedding layer. Taking the user as an example, for users with more sparse
behaviour in domain _A_ , we combine the features with a larger proportion of
the information in domain _B_. User feature combination can be expressed as
$v_{u_{i}}=W_{i}\cdot\hat{v}_{u_{i}}+(1-W_{i})\cdot\tilde{v}_{u_{i}}$, where
$v_{u_{i}}$ is the combined feature vector of the user $u_{i}$, $W_{i}$ is a
weight matrix in the attention network, and “$\cdot$” indicates element-wise
product. The calculation process for the embedding of items is the same with
that for users.

##  3\. Experiments and Analysis

###  3.1. Experiment

We conduct experiments to answer the following key questions: RQ1: Does the
TMS improve the recommendation performance? RQ2: How do GCN and BERT in
Tree2vec contribute to improving recommendation performance? RQ3: What is the
performance of GReS under different proportions of unique users?

###  3.2. Experimental Settings

Table 1. Performance comparison in HR@$K$, NDCG@$K$ and MRR@$K$ where $K=5,10,20,50$. The best scores and the second best scores are bold and underlined respectively. B-best and G-best are the best scores of baselines and variants of GReS. Model |  HR@$K$ |  NDCG@ $K$ |  MRR@$K$
---|---|---|---
$K=5$ | $K=10$ | $K=20$ | $K=50$ | $K=5$ | $K=10$ | $K=20$ | $K=50$ | $K=5$ | $K=10$ | $K=20$ | $K=50$
CTR-RBF | .0213 | .0334 | .0603 | .0811 | .0166 | .0234 | .0376 | .0462 | .0151 | .0211 | .0301 | .0350
TMH | .0224 | .0398 | .0687 | .0852 | .0179 | .0278 | .0417 | .0468 | .0169 | .0251 | .0339 | .0361
EMCDR | .0311 | .0667 | .0923 | .1026 | .0246 | .0464 | .0545 | .0572 | .0226 | .0399 | .0447 | .0452
GA-DTCDR | .0316 | .0659 | .1056 | .1173 | .0250 | .0468 | .0597 | .0666 | .0224 | .0394 | .0478 | .0508
DCDIR | .0438 | .0953 | .1320 | .1422 | .0334 | .0632 | .0794 | .0829 | .0328 | .0533 | .0631 | .0633
GReS | .0467 | .1023 | .1477 | .1577 | .0360 | .0692 | .0863 | .0889 | .0344 | .0593 | .0698 | .0707
GReS w/o tree | .0353 | .0828 | .1303 | .1388 | .0276 | .0578 | .0747 | .0811 | .0253 | .0482 | .0616 | .0623
GReS w/o GCN | .0414 | .0878 | .1403 | .1487 | .0325 | .0601 | .0808 | .0831 | .0313 | .0512 | .0631 | .0641
GReS w/o unidirectional | .0442 | .0994 | .1423 | .1533 | .0351 | .0659 | .0855 | .0872 | .0332 | .0553 | .0683 | .0688
GReS w/o BERT | .0404 | .0861 | .1421 | .1455 | .0337 | .0613 | .0821 | .0838 | .0321 | .0507 | .0627 | .0630
GReS w/o fix-position | .0454 | .1031 | .1433 | .1551 | .0352 | .0690 | .0853 | .0879 | .0331 | .0579 | .0680 | .0686
GReS vs. B-best | 6.62% | 7.35% | 11.89% | 10.90% | 7.78% | 9.49% | 8.69% | 7.24% | 4.88% | 11.26% | 10.62% | 11.69%
GReS vs. G-best | 2.86% | -0.78% | 3.07% | 1.68% | 2.27% | 0.29% | 0.94% | 1.14% | 3.61% | 2.42% | 2.20% | 2.76%

####  3.2.1. Datasets

There is no publicly available dataset in CDR for SCPs. To verify the
effectiveness of GReS, we build a dataset based on a commercial daily log for
experiments. The dataset consists of 3,000 common users, 600 unique users only
belonging to the SCP, and their purchase and sales data in domain _A_ and
domain _B_.

####  3.2.2. Comparative Baselines and Evaluation Metrics

To prove the effectiveness of GReS, in our experiments, it is compared with
five baselines which can be divided into three groups. They are Single-Domain
Recommendation (SDR) (CTR-RBF (Xin et al., 2015), TMH (Hu et al., 2019)), CDR
(EMCDR (Man et al., 2017), GA-DTCDR), and CDR with knowledge graph (DCDIR (Bi
et al., 2020)). We also study some variants of GReS for Ablation Study. In
order to be consistent with the previous work (Xin et al., 2015; Hu et al.,
2019; Man et al., 2017; Zhu et al., 2020a; Bi et al., 2020), all models are
evaluated by HR@$K$, NDCG@$K$, and MRR@$K$.

####  3.2.3. Implementation Details

We randomly select 80% data for training, 10% for validation, and 10% for
testing. All parameters of the model are tuned by fitting the validation set
and determined with an early-stop strategy. In the graph structure layer, we
set the hyper-parameters of Doc2vec and Node2vec models as suggested in (Le
and Mikolov, 2014; Grover and Leskovec, 2016), and the sampling rate $\alpha$
as 0.05. For the performance of GReS under different proportions $M$ of unique
users, the parameter $M\in$ {0.010, 0.015, 0.020,0.025, 0.030, 0.035, 0.040,
0.045, 0.050}. For three evaluation metrics HR@$K$, NDCG@$K$, and MRR@$K$,
$K\in\\{5,10,20,50\\}$. The neural network is trained using Adam (Kingma and
Ba, 2014), the learning rate is set to 0.0015, and the batch size is set to
30.

###  3.3. Experimental Performance

To verify the performance of GReS on SCPs (RQ1), we select five baselines for
comparison. For Dual-Target CDR methods, we only discuss its performance on
the sparsity domain to ensure the fairness of the comparison. In DCDIR (Bi et
al., 2020), we choose dish-category-item as the meta-path. Overall, GReS
outperforms all baselines and is even 9.03% averagely on three evaluation
metrics better than the best DCDIR, indicating the effectiveness of our
proposed Tree2vec model. The results also manifest that, (1) All CDR methods
significantly outperform SDR methods, which proves that introducing relevant
external features can alleviate the data sparsity problem of SDR. (2) The
recommendation performance of DCDIR is generally better than other baselines
with the benefit of representation of domain _B_ by knowledge graph. (3) With
the increase of $K$, the speed of the improvement of all metrics slows down.

###  3.4. Ablation Study

To answer RQ2, we compare GReS with its five variants. We directly use the
embedding of the TG without TMSs and denote it as GReS w/o tree. We removed
GCN part and BERT part in Tree2vec as GReS w/o GCN and GReS w/o BERT
respectively. GReS w/o unidirectional means allowing nodes in domain _B_
passing message bidirectionally in the GCN. GReS w/o fix-position means that
the position embedding of BERT will not be modified. The results show that,
first, GReS w/o tree is inferior to GReS with an average of 17.36%, which
proves that the TMSs have a positive effect on the recommendation performance
of SCPs. Second, benefit from the extraction of structural information from
each user’s directed graph, the average performance of GReS is 9.57% higher
than GReS w/o GCN. And because the unidirectional passing of message depicts
the hierarchical relationship of different nodes to a certain extent, the
average performance of GReS is 3.25% better than GReS w/o unidirectional.
Finally, since GReS w/o BERT is difficult to capture the semantic information,
its average performance is inferior to GReS by 9.68%, which highlights the
importance of BERT. Furthermore, for GReS w/o fix-position, its average
performance drops by 1.80% compared to GReS. It can be seen that setting fixed
position embeddings for nodes at different levels is more conducive to node
representation learning.

##  4\. Conclusion

To alleviate the data sparsity problem in supply chain platforms, in this
paper, we take catering SCP as an example and propose the GReS model.
Specifically, for the target domain, we use a heterogeneous graph with three
kinds of nodes to model users’ behaviour data. For the source domain, we
represent the hierarchical relationship by a tree-shaped graph composed of
tree-shaped meta structures, and use the element-attention mechanism to
concatenate the features of the source and target domains. Furthermore, we
designed experiments to verify the performance of GReS on SCPs. On the other
hand, due to the high similarity between different SCPs, we also believe that
GReS still has an excellent performance in other SCP recommendation
environments, such as industrial accessories, medical care, and commodity
wholesale.

###### Acknowledgements.

The work is supported by National Natural Science Foundation of China
(61876124, 61873178) and Shanxi Provincial Department of Science and
Technology Basic Research Project (20210302123129).

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# Reverse Stein-Weiss, Hardy-Littlewood-Sobolev, Hardy, Sobolev and
Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups

Aidyn Kassymov  Aidyn Kassymov: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium and Al-Farabi Kazakh National
University 71 Al-Farabi avenue 050040 Almaty Kazakhstan and Institute of
Mathematics and Mathematical Modeling 125 Pushkin str. 050010 Almaty
Kazakhstan E-mail address<EMAIL_ADDRESS>and<EMAIL_ADDRESS>,
Michael Ruzhansky  Michael Ruzhansky: Department of Mathematics: Analysis,
Logic and Discrete Mathematics Ghent University, Belgium and School of
Mathematical Sciences Queen Mary University of London United Kingdom E-mail
address<EMAIL_ADDRESS>and  Durvudkhan Suragan  Durvudkhan
Suragan: Department of Mathematics, Nazarbayev University 53 Kabanbay Batyr
Ave, Nur-Sultan 010000 Kazakhstan E-mail address<EMAIL_ADDRESS>
###### Abstract.

In this note we prove the reverse Stein-Weiss inequality on general
homogeneous Lie groups. The obtained results extend previously known
inequalities. Special properties of homogeneous norms and the reverse integral
Hardy inequality play key roles in our proofs. Also, we show reverse Hardy,
Hardy-Littlewood-Sobolev, $L^{p}$-Sobolev and $L^{p}$-Caffarelli-Kohn-
Nirenberg inequalities on homogeneous groups.

###### Key words and phrases:

Riesz potential, fractional operator, reverse Hardy-Littlewood-Sobolev
inequality, reverse Stein-Weiss inequality, reverse Hardy inequality, reverse
Sobolev inequality, reverse Caffarelli-Kohn-Nirenberg inequality, homogeneous
Lie group.

###### 1991 Mathematics Subject Classification:

22E30, 43A80.

The authors were supported in parts by the FWO Odysseus Project, the
Leverhulme Grant RPG-2017-151 and by EPSRC Grant EP/R003025/1, as well as NU
CRG 091019CRP2120 and NU FDCRG 240919FD3901.

##  1\. Introduction

In one of their pioneering work [24], Hardy and Littlewood considered the one
dimensional fractional integral operator on $(0,\infty)$ given by

(1.1)
$T_{\lambda}u(x)=\int_{0}^{\infty}\frac{u(y)}{|x-y|^{\lambda}}dy,\,\,\,\,0<\lambda<1,$

and proved the following theorem:

######  Theorem 1.1.

Let $1<p<q<\infty$ and $u\in L^{p}(0,\infty)$ with
$\frac{1}{q}=\frac{1}{p}+\lambda-1$. Then

(1.2) $\|T_{\lambda}u\|_{L^{q}(0,\infty)}\leq C\|u\|_{L^{p}(0,\infty)},$

where $C$ is a positive constant independent of $u$.

The $N$-dimensional analogue of (1.1) can be written by the formula:

(1.3)
$I_{\lambda}u(x)=\int_{\mathbb{R}^{N}}\frac{u(y)}{|x-y|^{\lambda}}dy,\,\,\,\,0<\lambda<N.$

The $N$-dimensional case of Theorem 1.1 was extended by Sobolev in [44]:

######  Theorem 1.2.

Let $1<p<q<\infty$, $u\in L^{p}(\mathbb{R}^{N})$ with
$\frac{1}{q}=\frac{1}{p}+\frac{\lambda}{N}-1$. Then

(1.4) $\|I_{\lambda}u\|_{L^{q}(\mathbb{R}^{N})}\leq
C\|u\|_{L^{p}(\mathbb{R}^{N})},$

where $C$ is a positive constant independent of $u$.

Later, in [46] Stein and Weiss obtained the following two-weight extention of
the Hardy-Littlewood-Sobolev inequality, which is known as the Stein-Weiss
inequality or weighted Hardy-Littlewood-Sobolev inequality.

######  Theorem 1.3.

Let $0<\lambda<N$, $1<p<\infty$, $\alpha<\frac{N(p-1)}{p}$,
$\beta<\frac{N}{q}$, $\alpha+\beta\geq 0$ and
$\frac{1}{q}=\frac{1}{p}+\frac{\lambda+\alpha+\beta}{N}-1$. If $1<p\leq
q<\infty$, then

(1.5) $\||x|^{-\beta}I_{\lambda}u\|_{L^{q}(\mathbb{R}^{N})}\leq
C\||x|^{\alpha}u\|_{L^{p}(\mathbb{R}^{N})},$

where $C$ is a positive constant independent of $u$.

The Hardy-Littlewood-Sobolev inequality on Euclidean spaces and the regularity
of fractional integrals was studied in [10], [12], [26] and [30]. On the
Heisenberg group, Folland and Stein in [14] obtained the Hardy-Littlewood-
Sobolev inequality and in [23] the authors also proved an analogue of the
Stein-Weiss inequality. In [18] the authors studied the Stein-Weiss inequality
on the Carnot groups. On homogeneous Lie groups, the Hardy-Littlewood-Sobolev
and Stein-Weiss inequalities were obtained in [43] and [21]. In [8] the author
proved the Stein-Weiss inequality on the Euclidean half-space.

The reverse Hardy-Littlewood-Sobolev inequality in the Euclidean space was
obtained in the works [9], [11] and [27]. In [5] and [4], the authors obtained
the reverse Stein-Weiss inequality on the Euclidean space and half-space,
respectively. In this paper, we first show the reverse Stein-Weiss inequality
on the homogeneous groups. In the proof we use special properties of
homogeneous norms of the homogeneous Lie groups and reverse integral Hardy
inequality, which are playing key roles in our calculations. Thus, in Theorem
2.5 we establish the reverse Stein-Weiss inequality on general homogeneous
groups based on the reverse integral Hardy inequalities with one negative
exponent. In particular, the obtained result recovers the previously known
results of Abelian (Euclidean), Heisenberg, Carnot groups since the class of
the homogeneous Lie groups contains those and since we can work with an
arbitrary homogeneous quasi-norm. Note that in this direction systematic
studies of different functional inequalities on general homogeneous (Lie)
groups were initiated by the paper [37]. We refer to this and other papers by
the authors (e.g. [39]) for further discussions.

We also note that the best constant in the Hardy-Littlewood-Sobolev inequality
on the Heisenberg group is now known, see Frank and Lieb [16] (in the
Euclidean case this was done earlier by Lieb in [25]). The expression for the
best constant depends on the particular quasi-norm used and may change for a
different choice of a quasi-norm.

A multidimensional version of one of the well-known inequalities of G.H. Hardy
is

(1.6)
$\left\|\frac{f}{|x|}\right\|_{L^{p}(\mathbb{R}^{N})}\leq\frac{p}{N-p}\|\nabla
f\|_{L^{p}(\mathbb{R}^{N})},\,\,\,\,1<p<N,$

where $f\in C^{\infty}_{0}(\mathbb{R}^{N})$, $\nabla$ is the Euclidean
gradient and constant $\frac{p}{N-p}$ is sharp. The Hardy inequality was
intensively studied. For example, the Hardy inequalities were considered on
the Euclidean space in [19], [20], on the Heisenberg group in [6], [7], on
stratified groups in [3], [31, 32, 33], [36], [42], for the vector fields in
[34] and [38], on homogeneous groups in [37], [39]. In [37] (see e.g., [35]),
authors showed the Hardy inequality with radial derivative on homogeneous
groups in the following form:

######  Theorem 1.4.

Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q$. Let
$|\cdot|$ be a homogeneous quasi-norm on $\mathbb{G}$. Let $1<p<Q$. Let $f\in
C^{\infty}_{0}(\mathbb{G}\setminus\\{0\\})$ be a complex-valued function. Then

(1.7)
$\left\|\frac{f}{|x|}\right\|_{L^{p}(\mathbb{G})}\leq\frac{p}{Q-p}\|\mathcal{R}f\|_{L^{p}(\mathbb{G})},\,\,\,\,\,1<p<Q,$

where $\mathcal{R}=\frac{d}{d|x|}$ is the radial derivative. The constant
$\frac{p}{Q-p}$ is sharp.

In Abelian case $(\mathbb{R}^{N},+)$ with $Q=N$ and $|\cdot|=|\cdot|_{E}$
where $|\cdot|_{E}$ is the standard Euclidean distance, from (1.7) we have

(1.8)
$\left\|\frac{f}{|x|_{E}}\right\|_{L^{p}(\mathbb{R}^{N})}\leq\frac{p}{N-p}\left\|\frac{x}{|x|_{E}}\cdot\nabla
f\right\|_{L^{p}(\mathbb{R}^{N})},\,\,\,\,\,1<p<N,$

where $\nabla$ is the standard Euclidean gradient. By using the Cauchy-
Schwartz inequality from the inequality (1.8), we get (1.6). In this note, we
show the reverse Hardy inequality on homogeneous general Lie groups.

We are also interested in Sobolev and Caffarelli-Kohn-Nirenberg inequalities,
let us recall them briefly. In the classical work of Sobolev, he showed the
following inequality:

(1.9) $\|u\|_{L^{p^{*}}(\mathbb{R}^{N})}\leq C\|\nabla
u\|_{L^{p}(\mathbb{R}^{N})},\,\,\,1<p<N,$

where $p^{*}=\frac{Np}{N-p}$. The Sobolev inequalities on stratified groups
were obtained in [39, 40, 41]. In [41], on homogeneous groups, the authors
showed $L^{p}$-Sobolev inequality in the following form:

######  Theorem 1.5.

Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q$. Let
$|\cdot|$ be a homogeneous quasi-norm on $\mathbb{G}$. Let $1<p<Q$. Let $f\in
C^{\infty}_{0}(\mathbb{G}\setminus\\{0\\})$ be a complex-valued function. Then

(1.10)
$\left\|f\right\|_{L^{p}(\mathbb{G})}\leq\frac{p}{Q}\|\mathbb{E}f\|_{L^{p}(\mathbb{G})},\,\,\,\,\,1<p<\infty,$

where $\mathbb{E}=|x|\frac{d}{d|x|}$ is the Euler operator. The constant
$\frac{p}{Q}$ is sharp.

In Abelian case $(\mathbb{R}^{N},+)$ with $Q=N$ and $|\cdot|=|\cdot|_{E}$
where $|\cdot|_{E}$ is the standard Euclidean distance, from (1.7) we have

(1.11)
$\left\|f\right\|_{L^{p}(\mathbb{R}^{N})}\leq\frac{p}{N-p}\left\|x\cdot\nabla
f\right\|_{L^{p}(\mathbb{R}^{N})},\,\,\,\,\,1<p<\infty,$

where $\nabla$ is the standard Euclidean gradient. In the Euclidean case, this
$L^{p}$-Sobolev inequality has been considered in [28].

In their fundamental work [2], L. Caffarelli, R. Kohn and L. Nirenberg
established:

######  Theorem 1.6.

Let $N\geq 1$, and let $l_{1}$, $l_{2}$, $l_{3}$,
$a,\,b,\,d,\,\delta\in\mathbb{R}$ be such that $l_{1},l_{2}\geq 1$,
$l_{3}>0,\,\,0\leq\delta\leq 1,$ and

(1.12)
$\frac{1}{l_{1}}+\frac{a}{N},\,\,\,\frac{1}{l_{2}}+\frac{b}{N},\,\,\,\frac{1}{l_{3}}+\frac{\delta
d+(1-\delta)b}{N}>0.$

Then,

(1.13) $\||x|^{\delta d+(1-\delta)b}u\|_{L^{l_{3}}(\mathbb{R}^{N})}\leq
C\||x|^{a}\nabla
u\|^{\delta}_{L^{l_{1}}(\mathbb{R}^{N})}\||x|^{b}u\|^{1-\delta}_{L^{l_{2}}(\mathbb{R}^{N})},\,\,\,u\in
C^{\infty}_{c}(\mathbb{R}^{N}),$

if and only if

(1.14) $\frac{1}{l_{3}}+\frac{\delta
d+(1-\delta)b}{N}=\delta\left(\frac{1}{l_{1}}+\frac{a-1}{N}\right)+(1-\delta)\left(\frac{1}{l_{2}}+\frac{b}{N}\right),\\\
a-d\geq 0,\,\,\,\,\text{if}\,\,\,\delta>0,\\\ a-d\leq
1,\,\,\,\,\text{if}\,\,\,\delta>0\,\,\,\text{and}\,\,\,\frac{1}{l_{3}}+\frac{\delta
d+(1-\delta)b}{N}=\frac{1}{l_{1}}+\frac{a-1}{N},$

where $C$ is a positive constant independent of $u$.

On homogeneous groups, $L^{p}$-Caffarelli-Kohn-Nirenberg inequality was
obtained in [29].

######  Theorem 1.7.

Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q$ and let
$\alpha,\,\beta\in\mathbb{R}$. Then for all complex-valued functions $f\in
C^{\infty}_{0}(\mathbb{G}\backslash\\{0\\}),$ $1<p<\infty,$ and any
homogeneous quasi-norm $|\cdot|$ on $\mathbb{G}$ we have

(1.15)
$\frac{|Q-\gamma|}{p}\left\|\frac{f}{|x|^{\frac{\gamma}{p}}}\right\|^{p}_{L^{p}(\mathbb{G})}\leq\left\|\frac{1}{|x|^{\alpha}}\mathcal{R}f\right\|_{L^{p}(\mathbb{G})}\left\|\frac{f}{|x|^{\frac{\beta}{p-1}}}\right\|^{p-1}_{L^{p}(\mathbb{G})},$

where $\gamma=\alpha+\beta+1$. If $\gamma\neq Q$ then the constant
$\frac{|Q-\gamma|}{p}$ is sharp.

The Caffarelli-Kohn-Nirenberg inequalities, on stratified and homogeneous
groups were obtained in [39, 40, 41]. On homogeneous groups in [29], a general
result which was obtained in particular cases gives the Hardy inequality (1.7)
and $L^{p}$-Sobolev inequality (1.10).

In this note, we show the reverse Hardy, Hardy-Littlewood-Sobolev,
$L^{p}$-Sobolev, $L^{p}$-Caffarelli-Kohn-Nirenberg inequalities on homogeneous
Lie groups. The main idea for proving such reverse inequalities was developed
in [37]. In addition, to the best of our knowledge, such reverse inequalities
above on a homogeneous group $\mathbb{G}$ are new even in the Euclidean case.

Summarising our main results of the present note, we prove the following
facts:

  * •

The reverse Stein-Weiss and Hardy-Littlewood-Sobolev inequalities on
homogeneous groups;

  * •

The reverse Hardy inequality on homogeneous groups;

  * •

The reverse $L^{p}$-Sobolev inequality on homogeneous groups;

  * •

The reverse $L^{p}$-Caffarelli-Kohn-Nirenberg inequality on homogeneous
groups.

##  2\. Main results

Let us recall that a Lie group (on $\mathbb{R}^{N}$) $\mathbb{G}$ with the
dilation

$D_{\lambda}(x):=(\lambda^{\nu_{1}}x_{1},\ldots,\lambda^{\nu_{N}}x_{N}),\;\nu_{1},\ldots,\nu_{n}>0,\;D_{\lambda}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N},$

which is an automorphism of the group $\mathbb{G}$ for each $\lambda>0,$ is
called a homogeneous (Lie) group. For simplicity, throughout this paper we use
the notation $\lambda x$ for the dilation $D_{\lambda}.$ The homogeneous
dimension of the homogeneous group $\mathbb{G}$ is denoted by
$Q:=\nu_{1}+\ldots+\nu_{N}.$ Also, in this note we denote a homogeneous quasi-
norm on $\mathbb{G}$ by $|x|$, which is a continuous non-negative function

(2.1) $\mathbb{G}\ni x\mapsto|x|\in[0,\infty),$

with the properties

  * i)

$|x|=|x^{-1}|$ for all $x\in\mathbb{G}$,

  * ii)

$|\lambda x|=\lambda|x|$ for all $x\in\mathbb{G}$ and $\lambda>0$,

  * iii)

$|x|=0$ iff $x=0$.

Moreover, the following polarisation formula on homogeneous Lie groups will be
used in our proofs: there is a (unique) positive Borel measure $\sigma$ on the
unit quasi-sphere $\mathfrak{S}:=\\{x\in\mathbb{G}:\,|x|=1\\},$ so that for
every $f\in L^{1}(\mathbb{G})$ we have

(2.2)
$\int_{\mathbb{G}}f(x)dx=\int_{0}^{\infty}\int_{\mathfrak{S}}f(ry)r^{Q-1}d\sigma(y)dr.$

The quasi-ball centred at $x\in\mathbb{G}$ with radius $R>0$ can be defined by

$B(x,R):=\\{y\in\mathbb{G}:|x^{-1}y|<R\\}.$

We refer to [15] for the original appearance of such groups, and to [13, 35]
for a recent comprehensive treatment.

Let us consider the integral operator

(2.3)
$I_{\lambda,|\cdot|}u(x)=|x|^{\lambda}*u=\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy,\,\,\,\lambda>0,$

where $*$ is the convolution. Let us recall briefly the reverse Hölder’s
inequality.

######  Theorem 2.1 ([1], Theorem 2.12, p. 27)).

Let $p\in(0,1)$, so that $p^{\prime}=\frac{p}{p-1}<0$. If non-negative
functions satisfy $f\in L^{p}(\mathbb{G})$ and
$0<\int_{\mathbb{G}}g^{p^{\prime}}(x)dx<+\infty,$ we have

(2.4)
$\int_{\mathbb{G}}f(x)g(x)dx\geq\left(\int_{\mathbb{G}}f^{p}(x)dx\right)^{\frac{1}{p}}\left(\int_{\mathbb{G}}g^{p^{\prime}}(x)dx\right)^{\frac{1}{p^{\prime}}}.$

Let us also recall a well-known fact about quasi-norms.

######  Proposition 2.2 ([13], Theorem 3.1.39).

Let $\mathbb{G}$ be a homogeneous Lie group. Then there exists a homogeneous
quasi-norm on $\mathbb{G}$ which is a norm, that is, a homogeneous quasi-norm
$|\cdot|$ which satisfies the triangle inequality

(2.5) $|xy|\leq|x|+|y|,\,\,\,\forall x,y\in\mathbb{G}.$

Furthermore, all homogeneous quasi-norms on $\mathbb{G}$ are equivalent.

The next theorem is the reverse integral version of Hardy inequalities on
general homogeneous groups that will be instrumental in our proof.

######  Theorem 2.3 ([22]).

Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q$. Assume
that $p\in(0,1)$ and $q<0$. Suppose that $W,U\geq 0$ are locally integrable
functions on $\mathbb{G}$. Then the inequality

(2.6)
$\left[\int_{\mathbb{G}}\left(\int_{B(0,|x|)}f(y)dy\right)^{q}W(x)dx\right]^{\frac{1}{q}}\geq
C_{1}(p,q)\left(\int_{\mathbb{G}}f^{p}(x)U(x)dx\right)^{\frac{1}{p}}$

holds for some $C_{1}(p,q)>0$ and all non-negative measurable functions $f,$
if and only if

(2.7) $0<A_{1}:=\inf_{x\neq a}\left[\left(\int_{\mathbb{G}\setminus
B(0,|x|)}W(y)dy\right)^{\frac{1}{q}}\left(\int_{B(0,|x|)}U^{1-p^{\prime}}(y)dy\right)^{\frac{1}{p^{\prime}}}\right].$

Moreover, the biggest constant $C_{1}(p,q)$ in (2.6) has the following
relation to $A_{1}$:

(2.8) $A_{1}\geq
C_{1}(p,q)\geq\left(\frac{p^{\prime}}{p^{\prime}+q}\right)^{-\frac{1}{q}}\left(\frac{q}{p^{\prime}+q}\right)^{-\frac{1}{p^{\prime}}}A_{1}.$

Also, inequality

(2.9) $\left[\int_{\mathbb{G}}\left(\int_{\mathbb{G}\setminus
B(0,|x|)}f(y)dy\right)^{q}W(x)dx\right]^{\frac{1}{q}}\geq
C_{2}(p,q)\left(\int_{\mathbb{G}}f^{p}(x)U(x)dx\right)^{\frac{1}{p}}$

holds for some $C_{2}(p,q)>0$ and all non-negative measurable functions $f,$
if and only if

(2.10) $0<A_{2}:=\inf_{x\neq
a}\left[\left(\int_{B(0,|x|)}W(y)dy\right)^{\frac{1}{q}}\left(\int_{\mathbb{G}\setminus
B(0,|x|)}U^{1-p^{\prime}}(y)dy\right)^{\frac{1}{p^{\prime}}}\right].$

Moreover, the biggest constant $C_{2}(p,q)$ in (2.9) has the following
relation to $A_{2}$:

(2.11) $A_{2}\geq
C_{2}(p,q)\geq\left(\frac{p^{\prime}}{p^{\prime}+q}\right)^{-\frac{1}{q}}\left(\frac{q}{p^{\prime}+q}\right)^{-\frac{1}{p^{\prime}}}A_{2}.$

######  Remark 2.4.

In our sense, the negative exponent $q<0$ of $0$, we understand in the
following form:

$0^{q}=(+\infty)^{-q}=+\infty,\,\,\,\,\text{and}\,\,\,\,\,\,0^{-q}=(+\infty)^{q}=0.$

Now we formulate the reverse Stein-Weiss inequality on $\mathbb{G}$.

######  Theorem 2.5.

Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q\geq 1$ and
let $|\cdot|$ be an arbitrary homogeneous quasi-norm on $\mathbb{G}$. Let
$\lambda>0$, $p,q^{\prime}\in(0,1)$, $0\leq\alpha<-\frac{Q}{q}$,
$0\leq\beta<-\frac{Q}{p^{\prime}}$,
$\frac{1}{q^{\prime}}+\frac{1}{p}=\frac{\alpha+\beta+\lambda}{Q}+2$, where
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $\frac{1}{q}+\frac{1}{q^{\prime}}=1$.
Then for all non-negative functions $f\in L^{q^{\prime}}(\mathbb{G})$ and
$h\in L^{p}(\mathbb{G})$ we have

(2.12)
$\int_{\mathbb{G}}\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}f(x)h(y)|y|^{\beta}dxdy\geq
C\|f\|_{L^{q^{\prime}}(\mathbb{G})}\|h\|_{L^{p}(\mathbb{G})},$

where $C$ is a positive constant independent of $f$ and $h$.

######  Corollary 2.6.

By setting $\alpha=\beta=0$ we get the reverse Hardy-Littlewood-Sobolev
inequality on the homogeneous groups, in the following form:

(2.13) $\int_{\mathbb{G}}\int_{\mathbb{G}}|y^{-1}x|^{\lambda}f(x)h(y)dxdy\geq
C\|f\|_{L^{q^{\prime}}(\mathbb{G})}\|h\|_{L^{p}(\mathbb{G})},$

for all non-negative functions $f\in L^{q^{\prime}}(\mathbb{G})$ and $h\in
L^{p}(\mathbb{G})$ with $\lambda>0$, $p,q^{\prime}\in(0,1)$,
$\frac{1}{q^{\prime}}+\frac{1}{p}=\frac{\lambda}{Q}+2$, where
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $\frac{1}{q}+\frac{1}{q^{\prime}}=1$.

######  Remark 2.7.

In the Abelian (Euclidean) case ${\mathbb{G}}=(\mathbb{R}^{N},+)$, hence $Q=N$
and $|\cdot|$ can be any homogeneous quasi-norm on $\mathbb{R}^{N}$, in
particular with the usual Euclidean distance, i.e. $|\cdot|=\|\cdot\|_{E}$,
this was investigated in [4].

###### Proof of Theorem 2.5.

By using reverse Hölder’s inequality with $\frac{1}{q}+\frac{1}{q^{\prime}}=1$
(Theorem 2.1) in (2.12), we calculate,

$\begin{split}\int_{\mathbb{G}}\int_{\mathbb{G}}|x|^{\alpha}f(x)|y^{-1}x|^{\lambda}&h(y)|y|^{\beta}dydx=\int_{\mathbb{G}}\left(\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}h(y)|y|^{\beta}dy\right)f(x)dx\\\
&\stackrel{{\scriptstyle\eqref{Holin}}}{{\geq}}\left(\int_{\mathbb{G}}\left(\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}h(y)|y|^{\beta}dy\right)^{q}dx\right)^{\frac{1}{q}}\|f\|_{L^{q^{\prime}}(\mathbb{G})}.\end{split}$

So for (2.12), it is enough to show that

$\left(\int_{\mathbb{G}}\left(\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}h(y)|y|^{\beta}dy\right)^{q}dx\right)^{\frac{1}{q}}\geq
C\|h\|_{L^{p}(\mathbb{G})},$

and by changing $u(y)=h(y)|y|^{\beta}$, this is equivalent to

$\int_{\mathbb{G}}\left(\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\leq
C\||y|^{-\beta}u\|^{q}_{L^{p}(\mathbb{G})}.$

We have that

$\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}u(y)dy\geq\int_{B\left(0,\frac{|x|}{2}\right)}|x|^{\alpha}|y^{-1}x|^{\lambda}u(y)dy,$

then

$\left(\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}\stackrel{{\scriptstyle
q<0}}{{\leq}}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|x|^{\alpha}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}.$

Therefore, we obtain

(2.14) $\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}\\\
\stackrel{{\scriptstyle q<0}}{{\geq}}\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}:=I^{\frac{1}{q}}_{1}.$

Similarly with (2.14), we have

(2.15) $\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}\\\
\stackrel{{\scriptstyle q<0}}{{\geq}}\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{\mathbb{G}}\setminus
B(0,2|x|)}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}:=I^{\frac{1}{q}}_{2}.$

By summarising above facts, from (2.14)-(2.15), we have

(2.16) $\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}\geq\frac{I^{\frac{1}{q}}_{1}}{2}+\frac{I^{\frac{1}{q}}_{2}}{2}.$

From now on, in view of Proposition 2.2 we can assume that our quasi-norm is
actually a norm.

Step 1. Let us consider $I_{1}$. By using Proposition 2.2 and the properties
of the quasi-norm with $|y|\leq\frac{|x|}{2}$, we get

(2.17)
$|x|=|x^{-1}|=|x^{-1}yy^{-1}|\stackrel{{\scriptstyle\eqref{tri}}}{{\leq}}|x^{-1}y|+|y^{-1}|=|y^{-1}x|+|y|\leq|y^{-1}x|+\frac{|x|}{2}.$

Then for any $\lambda>0$, we have

$2^{-\lambda}|x|^{\lambda}\leq|y^{-1}x|^{\lambda}.$

It means,

$2^{-\lambda}\int_{B\left(0,\frac{|x|}{2}\right)}|x|^{\lambda}u(y)dy\leq\int_{B\left(0,\frac{|x|}{2}\right)}|y^{-1}x|^{\lambda}u(y)dy,$

so that

$\left(\int_{B\left(0,\frac{|x|}{2}\right)}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}\leq
2^{-\lambda
q}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|x|^{\lambda}u(y)dy\right)^{q}.$

Therefore, we get

$\begin{split}I_{1}&=\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\\\
&\leq 2^{-\lambda
q}\int_{\mathbb{G}}|x|^{(\alpha+\lambda)q}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}dx.\end{split}$

If condition (2.7) in Theorem 2.3 with $W(x)=|x|^{(\alpha+\lambda)q}$ and
$U(y)=|y|^{-\beta p}$ in (2.6) is satisfied, then we have

$I_{1}\leq 2^{-\lambda
q}\int_{\mathbb{G}}\left(\int_{B(0,\frac{|x|}{2})}u(y)dy\right)^{q}|x|^{(\alpha+\lambda)q}dx\leq
C_{1}\||y|^{-\beta}u\|^{q}_{L^{p}(\mathbb{G})}.$

Let us verify condition (2.7). By using assumption
$\beta<-\frac{Q}{p^{\prime}}$, we obtain

$\frac{1}{p}+\frac{1}{q^{\prime}}=\frac{\alpha+\beta+\lambda}{Q}+2<\frac{\alpha+\lambda}{Q}-\frac{1}{p^{\prime}}+2,$

that is, $\frac{Q+(\alpha+\lambda)q}{Qq}>0,$ then $Q+(\alpha+\lambda)q<0$ and
by using the polar decomposition (2.2):

$\begin{split}\left(\int_{\mathbb{G}\setminus
B(0,|x|)}W(y)dy\right)^{\frac{1}{q}}&=\left(\int_{\mathbb{G}\setminus
B(0,|x|)}|y|^{(\alpha+\lambda)q}dy\right)^{\frac{1}{q}}\\\
&=\left(\int_{|x|}^{\infty}\int_{\mathfrak{S}}r^{Q-1}r^{(\alpha+\lambda)q}drd\sigma(\omega)\right)^{\frac{1}{q}}\\\
&=\left(|\mathfrak{S}|\int_{|x|}^{\infty}r^{Q-1+(\alpha+\lambda)q}dr\right)^{\frac{1}{q}}\\\
&=\left(-\frac{|\mathfrak{S}|}{Q+(\alpha+\lambda)q}|x|^{Q+(\alpha+\lambda)q}\right)^{\frac{1}{q}}\\\
&=\left(\frac{|\mathfrak{S}|}{|Q+(\alpha+\lambda)q|}\right)^{\frac{1}{q}}|x|^{\frac{Q+(\alpha+\lambda)q}{q}}.\end{split}$

Since $\beta<-\frac{Q}{p^{\prime}}$, we have

$-\beta p(1-p^{\prime})+Q>-\beta p(1-p^{\prime})-\beta p^{\prime}=0.$

So, $-\beta p(1-p^{\prime})+Q>0$. Then, let us consider

(2.18)
$\begin{split}\left(\int_{B(0,|x|)}U^{1-p^{\prime}}(y)dy\right)^{\frac{1}{p^{\prime}}}&=\left(\int_{B(0,|x|)}|y|^{-\beta
p(1-p^{\prime})}dy\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(\int^{|x|}_{0}\int_{\mathfrak{S}}r^{-\beta
p(1-p^{\prime})}r^{Q-1}drd\sigma(\omega)\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(|\mathfrak{S}|\int^{|x|}_{0}r^{-\beta
p(1-p^{\prime})+Q-1}dr\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(\frac{|\mathfrak{S}|}{-\beta p(1-p^{\prime})+Q}|x|^{-\beta
p(1-p^{\prime})+Q}\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(\frac{|\mathfrak{S}|}{Q-\beta
p(1-p^{\prime})}\right)^{\frac{1}{p^{\prime}}}|x|^{\frac{-\beta
p(1-p^{\prime})+Q}{p^{\prime}}}.\end{split}$

Therefore, we have

(2.19) $\begin{split}A_{1}&=\inf_{x\neq a}\left(\int_{\mathbb{G}\setminus
B(0,|x|)}W(y)dx\right)^{\frac{1}{q}}\left(\int_{B(0,|x|)}U^{1-p^{\prime}}(y)dy\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(\frac{|\mathfrak{S}|}{|Q+(\alpha+\lambda)q|}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{Q-\beta
p(1-p^{\prime})}\right)^{\frac{1}{p^{\prime}}}\inf_{x\neq
a}|x|^{\frac{(\alpha+\lambda)q+Q}{q}+\frac{-\beta
p(1-p^{\prime})+Q}{p^{\prime}}}\\\
&=\left(\frac{|\mathfrak{S}|}{|Q+(\alpha+\lambda)q|}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{Q-\beta
p(1-p^{\prime})}\right)^{\frac{1}{p^{\prime}}}\inf_{x\neq
a}|x|^{Q\left(\frac{1}{q}+\frac{1}{p^{\prime}}+\frac{\alpha+\beta+\lambda}{Q}\right)}\\\
&=\left(\frac{|\mathfrak{S}|}{|Q+(\alpha+\lambda)q|}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{Q-\beta
p(1-p^{\prime})}\right)^{\frac{1}{p^{\prime}}}\inf_{x\neq
a}|x|^{Q\left(2-\frac{1}{q^{\prime}}-\frac{1}{p}+\frac{\alpha+\beta+\lambda}{Q}\right)}\\\
&=\left(\frac{|\mathfrak{S}|}{|Q+(\alpha+\lambda)q|}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{Q-\beta
p(1-p^{\prime})}\right)^{\frac{1}{p^{\prime}}}>0.\end{split}$

Then by using (2.6), we obtain

(2.20) $I_{1}\leq 2^{-\lambda
q}\int_{\mathbb{G}}|x|^{(\alpha+\lambda)q}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}dx\leq
2^{-\lambda q}C^{q}_{1}\||y|^{-\beta}u\|^{q}_{L^{p}(\mathbb{G})},$

so that

(2.21) $I^{\frac{1}{q}}_{1}\geq
2^{-\lambda}C_{1}\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})}=2^{-\lambda}C_{1}\|h\|_{L^{p}(\mathbb{G})}.$

Step 2. As in the previous case $I_{1}$, now we consider $I_{2}$. From
$2|x|\leq|y|$, we calculate

$|y|=|y^{-1}|=|y^{-1}xx^{-1}|\stackrel{{\scriptstyle\eqref{tri}}}{{\leq}}|y^{-1}x|+|x|\leq|y^{-1}x|+\frac{|y|}{2},$

that is,

$\frac{|y|}{2}\leq|y^{-1}x|.$

Then, if condition (2.10) with $W(x)=|x|^{\alpha q}$ and
$U(y)=|y|^{-(\beta+\lambda)p}$ is satisfied, then we have

$I_{2}=\int_{\mathbb{G}}\left(\int_{\mathbb{G}\setminus
B(0,2|x|)}|x|^{\alpha}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\\\ \leq
2^{-\lambda q}\int_{\mathbb{G}}|x|^{\alpha q}\left(\int_{\mathbb{G}\setminus
B(0,2|x|)}u(y)|y|^{\lambda}dy\right)^{q}dx\leq 2^{-\lambda
q}\||y|^{-\beta}u\|^{q}_{L^{p}(\mathbb{G})}.$

Now let us check condition (2.10). We have

$\begin{split}\left(\int_{B(0,|x|)}W(y)dy\right)^{\frac{1}{q}}&=\left(\int_{B(0,|x|)}|y|^{\alpha
q}dy\right)^{\frac{1}{q}}\\\
&=\left(\int_{0}^{|x|}\int_{\mathfrak{S}}r^{\alpha
q}r^{Q-1}drd\sigma(\omega)\right)^{\frac{1}{q}}\\\
&=\left(\frac{|\mathfrak{S}|}{Q+\alpha
q}\right)^{\frac{1}{q}}|x|^{\frac{Q+\alpha q}{q}},\end{split}$

where $Q+\alpha q>0$. From $\alpha<-\frac{Q}{q}$, we have

$\begin{split}\frac{1}{q^{\prime}}+\frac{1}{p}=\frac{\alpha+\beta+\lambda}{Q}+2<-\frac{1}{q}+\frac{\beta+\lambda}{Q}+2=\frac{\beta+\lambda}{Q}+1+\frac{1}{q^{\prime}},\end{split}$

then

(2.22) $(\beta+\lambda)p^{\prime}+Q<0.$

By using this fact, we have

$\begin{split}\left(\int_{\mathbb{G}\setminus
B(0,|x|)}U^{1-p^{\prime}}(y)dy\right)^{\frac{1}{p^{\prime}}}&=\left(\int_{\mathbb{G}\setminus
B(0,|x|)}|y|^{-(\beta+\lambda)(1-p^{\prime})p}dy\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(\int_{|x|}^{\infty}\int_{\mathfrak{S}}r^{Q-1}r^{-(\beta+\lambda)(1-p^{\prime})p}drd\sigma(\omega)\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(|\mathfrak{S}|\int_{|x|}^{\infty}r^{-(\beta+\lambda)(1-p^{\prime})p+Q-1}dr\right)^{\frac{1}{p^{\prime}}}\\\
&\stackrel{{\scriptstyle\eqref{betalambda}}}{{=}}\left(-\frac{|\mathfrak{S}|}{Q-(\beta+\lambda)(1-p^{\prime})p}|x|^{Q-(\beta+\lambda)(1-p^{\prime})p}\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(-\frac{|\mathfrak{S}|}{Q+(\beta+\lambda)p^{\prime}}|x|^{Q+(\beta+\lambda)p^{\prime}}\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(\frac{|\mathfrak{S}|}{|Q+(\beta+\lambda)p^{\prime}|}\right)^{\frac{1}{p^{\prime}}}|x|^{\frac{Q+(\beta+\lambda)p^{\prime}}{p^{\prime}}}.\end{split}$

Combining these facts we have

(2.23) $\begin{split}A_{2}&=\inf_{x\neq
a}\left(\int_{B(0,|x|)}W(y)dx\right)^{\frac{1}{q}}\left(\int_{\mathbb{G}\setminus
B(0,|x|)}U^{1-p^{\prime}}(y)dx\right)^{\frac{1}{p^{\prime}}}\\\
&=\left(\frac{|\mathfrak{S}|}{Q+\alpha
q}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{|Q+(\beta+\lambda)p^{\prime}|}\right)^{\frac{1}{p^{\prime}}}\inf_{x\neq
a}|x|^{\frac{Q+\alpha q}{q}+\frac{Q+(\beta+\lambda)p^{\prime}}{p^{\prime}}}\\\
&=\left(\frac{|\mathfrak{S}|}{Q+\alpha
q}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{|Q+(\beta+\lambda)p^{\prime}|}\right)^{\frac{1}{p^{\prime}}}\inf_{x\neq
a}|x|^{\frac{Q}{q}+\alpha+\frac{Q}{p^{\prime}}+\beta+\lambda}\\\
&=\left(\frac{|\mathfrak{S}|}{Q+\alpha
q}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{|Q+(\beta+\lambda)p^{\prime}|}\right)^{\frac{1}{p^{\prime}}}\inf_{x\neq
a}|x|^{Q\left(\frac{1}{q}+\frac{1}{p^{\prime}}+\frac{\alpha+\beta+\lambda}{Q}\right)}\\\
&=\left(\frac{|\mathfrak{S}|}{Q+\alpha
q}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{|Q+(\beta+\lambda)p^{\prime}|}\right)^{\frac{1}{p^{\prime}}}\inf_{x\neq
a}|x|^{Q\left(2-\frac{1}{q^{\prime}}-\frac{1}{p}+\frac{\alpha+\beta+\lambda}{Q}\right)}\\\
&=\left(\frac{|\mathfrak{S}|}{Q+\alpha
q}\right)^{\frac{1}{q}}\left(\frac{|\mathfrak{S}|}{|Q+(\beta+\lambda)p^{\prime}|}\right)^{\frac{1}{p^{\prime}}}>0.\end{split}$

Therefore, we have

$I_{2}=\int_{\mathbb{G}}\left(\int_{\mathbb{G}\setminus
B(0,2|x|)}|x|^{\alpha}u(y)|y^{-1}x|^{\lambda}dy\right)^{q}dx\leq 2^{-\lambda
q}C^{q}_{2}\||y|^{-\beta}u\|^{q}_{L^{p}(\mathbb{G})}.$

Then, we have

(2.24) $I^{\frac{1}{q}}_{2}\geq
2^{-\lambda}C_{2}\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})}=2^{-\lambda}C_{2}\|h\|_{L^{p}(\mathbb{G})}.$

Finally, by using (2.21) and (2.24) in (2.16), we obtain

(2.25) $\begin{split}\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}&\geq\frac{I^{\frac{1}{q}}_{1}}{2}+\frac{I^{\frac{1}{q}}_{2}}{2}\\\
&\geq\frac{2^{-\lambda}(C_{1}+C_{2})}{2}\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})}\\\
&=\frac{2^{-\lambda}(C_{1}+C_{2})}{2}\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})}\\\
&=C_{3}\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})},\end{split}$

where $C_{3}=\frac{2^{-\lambda}(C_{1}+C_{2})}{2}>0$.

Theorem 2.5 is proved. ∎

Let us give improved reverse Stein-Weiss inequality.

######  Theorem 2.8.

Let $\mathbb{G}$ be a homogeneous group of homogeneous dimension $Q\geq 1$ and
let $|\cdot|$ be an arbitrary homogeneous quasi-norm on $\mathbb{G}$. Let
$\lambda>0$, $p,q^{\prime}\in(0,1)$ and
$\frac{1}{q^{\prime}}+\frac{1}{p}=\frac{\alpha+\beta+\lambda}{Q}+2$, where
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$ , $\frac{1}{q}+\frac{1}{q^{\prime}}=1$.
Then for all non-negative functions $f\in L^{q^{\prime}}(\mathbb{G})$ and
$h\in L^{p}(\mathbb{G})$, inequality (2.12) holds, that is,

$\int_{\mathbb{G}}\int_{\mathbb{G}}|x|^{\alpha}|y^{-1}x|^{\lambda}f(x)h(y)|y|^{\beta}dxdy\geq
C\|f\|_{L^{q^{\prime}}(\mathbb{G})}\|h\|_{L^{p}(\mathbb{G})},$

if one of the following conditions is satisfied:

  * (a)

$0\leq\alpha<-\frac{Q}{q}$.

  * (b)

$0\leq\beta<-\frac{Q}{p^{\prime}}$.

###### Proof.

Let us prove (a). By using some notations from proof of Theorem 2.5 and
(2.16), we obtain

(2.26) $\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}\geq
I^{\frac{1}{q}}_{2},$

and by using Step 2 in the proof of Theorem 2.5 and from (2.24), we have
$I_{2}^{\frac{1}{q}}\geq C\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})},$ then we get

(2.27) $\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}\geq
I^{\frac{1}{q}}_{2}\geq C\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})}.$

Let us prove (b). By (2.16), we obtain

(2.28) $\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}\geq
I^{\frac{1}{q}}_{1},$

and by using Step 1 in the proof of Theorem 2.5 and from (2.14), we have
$I_{1}^{\frac{1}{q}}\geq C\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})},$ then we get

(2.29) $\left(\int_{\mathbb{G}}|x|^{\alpha
q}\left(\int_{\mathbb{G}}|y^{-1}x|^{\lambda}u(y)dy\right)^{q}dx\right)^{\frac{1}{q}}\geq
I^{\frac{1}{q}}_{1}\geq C\||y|^{-\beta}u\|_{L^{p}(\mathbb{G})}.$

∎

Let us give reverse Hardy, $L^{p}$-Sobolev and $L^{p}$-Caffarelli-Kohn-
Nirenberg inequalities on $\mathbb{G}$. Assume now that $f$ is a radially
decreasing function, i.e., $\mathcal{R}f:=\frac{d}{d|x|}f<0$. Let us give the
reverse Hardy inequality on homogeneous Lie groups, the reverse to Theorem
1.4.

######  Theorem 2.9 (Reverse Hardy inequality).

Let $\mathbb{G}$ be a homogeneous Lie group with homogeneous dimension $Q\geq
1$. Assume that $p\in(0,1)$. Then for any non-negative, real-valued and
radially decreasing function $f\in
C^{\infty}_{0}(\mathbb{G}\setminus\\{0\\})$, we have

(2.30)
$\left\|\frac{f}{|x|}\right\|_{L^{p}(\mathbb{G})}\geq\frac{p}{Q-p}\|\mathcal{R}f\|_{L^{p}(\mathbb{G})}.$

###### Proof.

Let us denote $\mathcal{R}_{1}=-\mathcal{R}$, so that we have
$\mathcal{R}_{1}f>0$. By using polar decomposition (2.2), integration by parts
and reverse Hölder’s inequality, we obtain

(2.31)
$\begin{split}\int_{\mathbb{G}}\frac{f^{p}(x)}{|x|^{p}}dx&=\int_{0}^{\infty}\int_{\mathfrak{S}}\frac{f^{p}(ry)}{r^{p}}r^{Q-1}drd\sigma(y)\\\
&=-\frac{p}{Q-p}\int_{\mathbb{G}}\frac{f^{p-1}(x)}{|x|^{p-1}}\mathcal{R}f(x)dx\\\
&=\frac{p}{Q-p}\int_{\mathbb{G}}\frac{f^{p-1}(x)}{|x|^{p-1}}\mathcal{R}_{1}f(x)dx\\\
&\geq\frac{p}{Q-p}\left\|\frac{f}{|x|}\right\|^{p-1}_{L^{p}(\mathbb{G})}\|\mathcal{R}_{1}f\|_{L^{p}(\mathbb{G})}.\end{split}$

This gives

(2.32)
$\left\|\frac{f}{|x|}\right\|_{L^{p}(\mathbb{G})}\geq\frac{p}{Q-p}\|\mathcal{R}_{1}f\|_{L^{p}(\mathbb{G})},$

implying (2.30). ∎

Let us define by $\mathbb{E}=|x|\mathcal{R}$ the Euler operator. Then we have
the reverse $L^{p}$-Sobolev inequality, the reverse to Theorem 1.5.

######  Theorem 2.10 (Reverse $L^{p}$-Sobolev inequality).

Let $\mathbb{G}$ be a homogeneous Lie group with homogeneous dimension $Q\geq
1$. Assume that $p\in(0,1)$. Then for any non-negative, real-valued and
radially decreasing function $f\in
C^{\infty}_{0}(\mathbb{G}\setminus\\{0\\})$, we have

(2.33)
$\left\|f\right\|_{L^{p}(\mathbb{G})}\geq\frac{p}{Q}\|\mathbb{E}f\|_{L^{p}(\mathbb{G})}.$

###### Proof.

Let us denote $\mathbb{E}_{1}=|x|\mathcal{R}_{1}$, so that
$\mathbb{E}_{1}f>0.$ By using polar decomposition (2.2), integration by parts
and reverse Hölder’s inequality, we obtain

(2.34)
$\begin{split}\int_{\mathbb{G}}f^{p}(x)dx&=\int_{0}^{\infty}\int_{\mathfrak{S}}f^{p}(ry)r^{Q-1}drd\sigma(y)\\\
&=-\frac{p}{Q}\int_{\mathbb{G}}f^{p-1}(x)|x|\mathcal{R}f(x)dx\\\
&=\frac{p}{Q}\int_{\mathbb{G}}f^{p-1}(x)|x|\mathcal{R}_{1}f(x)dx\\\
&=\frac{p}{Q}\int_{\mathbb{G}}f^{p-1}(x)\mathbb{E}_{1}f(x)dx\\\
&\geq\frac{p}{Q}\left\|f\right\|^{p-1}_{L^{p}(\mathbb{G})}\|\mathbb{E}_{1}f\|_{L^{p}(\mathbb{G})}.\end{split}$

This gives

(2.35)
$\left\|f\right\|_{L^{p}(\mathbb{G})}\geq\frac{p}{Q}\|\mathbb{E}_{1}f\|_{L^{p}(\mathbb{G})},$

implying (2.33). ∎

Let us give the reverse $L^{p}$-Caffarelli-Kohn-Nirenberg inequality on
$\mathbb{G}$, that is, the reverse to Theorem 1.7.

######  Theorem 2.11 (Reverse $L^{p}$-Caffarelli-Kohn-Nirenberg inequality).

Let $\mathbb{G}$ be a homogeneous Lie group with homogeneous dimension $Q\geq
1$. Assume that $p\in(0,1)$. Then for any nonnegative, real-valued and
radially decreasing function $f\in
C^{\infty}_{0}(\mathbb{G}\setminus\\{0\\})$, we have

(2.36)
$\left\|\frac{f}{|x|^{\frac{\gamma}{p}}}\right\|^{p}_{L^{p}(\mathbb{G})}\geq\frac{p}{Q-\gamma}\left\|\frac{\mathcal{R}f}{|x|^{\alpha}}\right\|_{L^{p}(\mathbb{G})}\left\|\frac{f}{|x|^{\frac{\beta}{p-1}}}\right\|^{p-1}_{L^{p}(\mathbb{G})},$

for all $\alpha,\beta\in\mathbb{R}$ and $\gamma=\alpha+\beta+1$, such that
$Q>\gamma$.

###### Proof.

By using polar decomposition (2.2), integration by parts and reverse Hölder’s
inequality, we obtain

(2.37)
$\begin{split}\int_{\mathbb{G}}\frac{f^{p}(x)}{|x|^{\gamma}}dx&=\int_{0}^{\infty}\int_{\mathfrak{S}}\frac{f^{p}(ry)}{r^{\gamma}}r^{Q-1}drd\sigma(y)\\\
&=-\frac{p}{Q-\gamma}\int_{\mathbb{G}}\frac{f^{p-1}(x)}{|x|^{\gamma-1}}\mathcal{R}f(x)dx\\\
&=\frac{p}{Q-\gamma}\int_{\mathbb{G}}\frac{f^{p-1}(x)}{|x|^{\alpha+\beta}}\mathcal{R}_{1}f(x)dx\\\
&=\frac{p}{Q-\gamma}\int_{\mathbb{G}}\frac{f^{p-1}(x)}{|x|^{\beta}}\frac{\mathcal{R}_{1}f(x)}{|x|^{\alpha}}dx\\\
&\geq\frac{p}{Q-\gamma}\left\|\frac{f}{|x|^{\frac{\beta
p^{\prime}}{p}}}\right\|^{p-1}_{L^{p}(\mathbb{G})}\left\|\frac{\mathcal{R}_{1}f}{|x|^{\alpha}}\right\|_{L^{p}(\mathbb{G})}\\\
&=\frac{p}{Q-\gamma}\left\|\frac{f}{|x|^{\frac{\beta\frac{p}{p-1}}{p}}}\right\|^{p-1}_{L^{p}(\mathbb{G})}\left\|\frac{\mathcal{R}_{1}f}{|x|^{\alpha}}\right\|_{L^{p}(\mathbb{G})}\\\
&=\frac{p}{Q-\gamma}\left\|\frac{f}{|x|^{\frac{\beta}{p-1}}}\right\|^{p-1}_{L^{p}(\mathbb{G})}\left\|\frac{\mathcal{R}_{1}f}{|x|^{\alpha}}\right\|_{L^{p}(\mathbb{G})}.\end{split}$

This gives

(2.38)
$\left\|\frac{f}{|x|^{\frac{\gamma}{p}}}\right\|^{p}_{L^{p}(\mathbb{G})}\geq\frac{p}{Q-\gamma}\left\|\frac{\mathcal{R}_{1}f}{|x|^{\alpha}}\right\|_{L^{p}(\mathbb{G})}\left\|\frac{f}{|x|^{\frac{\beta}{p-1}}}\right\|^{p-1}_{L^{p}(\mathbb{G})},$

which implies (2.36). ∎

######  Remark 2.12.

In (2.36), if we take $\gamma=p$ and $\alpha=0$, then we have the reverse
Hardy inequality. Also, if we take $\gamma=0$ and $\beta=0$, then we have the
reverse $L^{p}$-Sobolev inequality.

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