# On the Impact of Device-Level Techniques on Energy-Efficiency of Neural
Network Accelerators

Seyed Morteza Nabavinejad  School of Computer Science
Institute for Research in Fundamental Sciences (IPM)
Tehran, Iran
<EMAIL_ADDRESS>Behzad Salami  Barcelona Supercomputing Center (BSC)
Barcelona, Spain
<EMAIL_ADDRESS>
###### Abstract

Energy-efficiency is a key concern for neural network applications. To
alleviate this issue, hardware acceleration using FPGAs or GPUs can provide
better energy-efficiency than general-purpose processors. However, further
improvement of the energy-efficiency of such accelerators will be extremely
beneficial specially to deploy neural network in power-constrained edge
computing environments. In this paper, we _experimentally_ explore the
potential of device-level energy-efficiency techniques (e.g., supply voltage
underscaling, frequency scaling, and data quantization) for representative
off-the-shelf FPGAs compared to GPUs. Frequency scaling in both platforms can
improve the power and energy consumption but with performance overhead, e.g.,
in GPUs it improves the power consumption and $GOPs/J$ by up to 34% and 28%,
respectively. However, leveraging reduced-precision instructions improves
power (up to 13%), energy (up to 20%), and performance (up to 7%)
simultaneously, with negligible reduction in accuracy of neural network
accuracy.

###### Index Terms:

hardware accelerators, neural networks, DVFS, reduced-precision instructions,
power-efficiency

##  I Introduction

Deep Neural Networks (DNNs), especially Convolutional Neural Networks (CNNs),
are a very popular subcategory of machine learning with significant success
rates in classification problems. They are essential for cutting-edge real-
world applications, such as image and video classification, speech
recognition, and robotics. DNNs learn a model from a dataset during the
training phase and in the classification phase, make predictions about a new
dataset that has not been seen previously. The computation and the memory
requirements of these models are increasing depending on the size of the data
as well as the number of layers and the filters. The massive amount of data
movement and computational power required for such models limit their
scalability for enterprise applications specially on power-constrained
scenarios like drones and mobile devices.

Hardware acceleration using Graphics Processing Units (GPUs) [1], Field
Programmable Gate Arrays (FPGAs)) [2], or Application-Specific Integrated
Circuits (ASICs) [3] is a promising solution to overcome such energy-
efficiency drawbacks of DNNs. Among them, FPGAs are efficient data parallel
compute engines with increasing use for data centers [4]. They attract great
attention with their higher power efficiency than GPUs and better flexibility
compared to ASICs. Additionally, recent advances in High-Level Synthesis (HLS)
tools make it easier to map applications on FPGAs [5]. On the other hand, GPUs
with internal Single Instruction Multiple Thread (SIMT) structure are
convenient to accelerate these DNN models. Many of the emerging deep learning
frameworks, such as Caffe [6], Torch [7], and TensorFlow [8] are specifically
optimized to run on GPUs with CUDA programming interface. However, the high
power consumption of GPUs raises cooling concerns in data centers. Moreover,
ASIC-based accelerators are more power- and energy-efficient than FPGAs and
GPUs [9]. However, further improving the energy-efficiency of FPGA- and GPU-
based CNN accelerators can enable their deployment in power-constrained
environments, as they provide better flexibility than ASICs. Toward this goal,
this study aims to comparatively explore the potential of such off-the-shelf
hardware platforms to take advantage of energy-efficiency techniques such as
supply voltage underscaling, frequency underscaling, and reduced-precision
instructions.

We study the following techniques: (i) Undervolting, i.e., the supply voltage
level underscaling below the nominal level, reduces the total power
consumption of the underlying hardware. Thus, it directly leads to the
improved energy-efficiency. (ii) Frequency underscaling is used to prevent the
undervolting-related errors that may occur due to the increased circuit delay
when operating at reduced voltage levels. Although, frequency underscaling
(simultaneously with supply voltage reduction) may lead to reduced performance
but energy-efficiency improves while it prevents the neural network accuracy
loss. (iii) As a common software-level technique, quantization can decrease
the precision level of a numerical value used as a weight. This technique aims
to reduce the size of CNNs without a significant loss of classification
accuracy [10].

Our experimental study is based on a representative Xilinx ZCU102 Ultrascale+
(FPGA) and Nvidia Tesla P40 (GPU). We also perform our study on two state-of-
the-art image classification models in the classification phase, i.e.,
ResNet50 and Inception. On this setup, we evaluate the three above-mentioned
energy-efficiency techniques and in detail discuss the efficiency of such
techniques. For the GPU accelerator, we deployed the networks on the GPU and
applied different techniques to study their effect. For the FPGA-based
accelerator, we adopted the experimental results presented in [11]. The
experimental results indicate the surprising finding that unlike FPGA-based
accelerator, in GPU-based accelerator the highest power-efficiency and energy-
efficiency cannot be simply achieved by highest and lowest frequency, but with
careful exploration of all the frequency level options. The right combination
of frequency level and precision of the models can lead to up to 35% power-
efficiency and 28% energy-efficiency improvement. The temperature of the GPU
accelerator can also be reduced by around 20% using the lowest frequency
level, to meet certain limitations in temperature-critical situations in
embedded systems.

To our knowledge, this is the first paper to simultaneously evaluate several
energy-efficiency hardware- and software-level techniques on FPGA- and GPU-
based neural networks. We make the following major contributions in this
paper:

  * •

We explore the effect of quantization and employment of reduced-precision
instructions on the performance of DNNs as a function of computational
complexity of the DNNs. We observe that more complex DNNs benefit from the
reduced-precision instructions more than less complex ones in terms of
runtime, power, and energy consumption.

  * •

Conducting extensive set of experiments, we study the impact of DVFS on
runtime, power, energy, and temperature of the networks. We show that the
difference between runtime under lowest and highest GPU frequency is
significant. So when the DVFS is reduced, it is possible to employ reduced-
precision instructions to improve the runtime and mitigate the negative impact
of low frequency.

  * •

We evaluate the potential of energy-efficiency techniques on FPGA and GPU
architectures while running a neural network applications.

  * •

Comparing FPGA and GPU regarding frequency scaling, we find that in FPGA the
best results for various metrics can be obtained at highest or lowest level of
frequency. However, in GPU, more careful exploration of available frequency
levels is needed since the best results for some metrics is achievable at
frequency levels other than highest and lowest one.

The rest of the paper is organized as follows: In Section II, we present and
discuss the methodology and experimental results for our GPU-based CNN
accelerator. In Section III, we compare and discuss the energy-efficiency
results between FPGA- and GPU-based accelerators. In Section IV, we review the
related works and conclude the paper in Section V.

##  II GPU-based DNN

###  II-A Background

Various Machine Learning (ML) frameworks are presented such as Caffe [6],
Torch [7], and TensorFlow [8]. TensorFlow employs dataflow graphs to represent
computation pattern of ML approaches including DNNs. It maps the dataflow
graph nodes on available computing cores of a single machine or a cluster of
machines. Hence, it can support large-scale and heterogeneous clusters or
hardware with hundreds of computing cores such as GPUs. It can also work with
other hardware platforms including multi-core CPUs and ASICs. While TensorFlow
focuses on deep neural networks, it can support a variety of other ML
applications as well [8].

TensorRT platform is developed and extended on top of CUDA to optimize the
inference phase DNNs and consists of an inference optimizer and a runtime
library. A trained DNN in the form of a frozen graph with its parameters is
the input of TensorRT. TensorRT generates an optimized inference engine based
on the frozen graph of DNN. TensorRT leverages a set of methods and techniques
such as graph optimization and layer fusion for optimizing the inference
engine of DNN. It can also generate reduced-precision versions of the
inference engine that are able to employ the reduced-precision instructions of
GPU families that provide architectural support for such instructions. Deep
learning inference applications such as image classification and video
streaming can leverage various reduced-precision versions of DNNs such as
FP32, FP16, and INT8 that are offered by TensorRT to improve the inference
latency, power, and energy, provided that the GPU supports those precision
[12].

###  II-B Methodology

The steps that we follow to conduct the experiments are presented in Fig. 1.
We use the frozen graph model of DNNs that is generated for inference. The
frozen graph is generated from the pre-trained model that is trained with
ImageNet dataset [13]. To have FP32 and INT8 precision models of the network
111In our experiments, we also considered 16-bit Floating-Point (FP16)
precision. However, we saw that there is almost no difference between FP32 and
FP16 regarding inference time. The reason is that our GPU does not support
half precision arithmetic in its architecture, we first generate their
counterpart frozen graph with TensorRT and then deploy them on the GPU. After
that, the frozen graph starts execution on GPU to classify the images and tag
them with the predefined labels. The predefined set of labels includes 1000
label classes from ImageNet dataset. During the execution, the desirable
metrics such as power consumption are monitored and collected using the GPU
Monitoring Interface (NVIDIA System Management Interface- nvidia-smi [14]).
Upon completion of inference, the labels generated for each image are compared
to original labels provided by dataset to calculate the accuracy of the
results. The performance reported for the networks throughout the paper is
based on the inference of all the images. The GPU Dynamic Voltage and
Frequency Scaling (DVFS) controller is also used to control the frequency of
GPU.

####  II-B1 Quantization Technique

In our work, we use TensorRT to quantize the weights from FP32 to INT8.
TensorRT is developed to optimize the inference phase of deep learning
applications. TensorRT optimizes neural network models that are trained in
major frameworks such as TensorFlow by calibrating the weights for lower
precision with negligible accuracy loss. Symmetric linear quantization is used
to scale FP32 to INT8. A saturation threshold for scale factor is considered
and any value above (below) that threshold is mapped to +127 (-127) (max range
of INT8), and the rest of values are mapped to a value between -127 and +127.
To find the best threshold, FP32 inference is run on a calibration dataset.
After that, various quantized distributions with different saturation
thresholds are generated. The threshold that minimizes the information lost is
selected using Kullback-Leibler divergence. Next, FP32 weights are quantized
to INT8 using the selected threshold. Finally, calibration table and INT8
frozen graph are generated for inference. For the Inception network, the
accuracy for FP32 model is 88.49% and for INT8 model is 88.24%. For the
ResNet50 network, the accuracy for FP32 and INT8 models is 87.99% and 87.4%,
respectively. These results are for the GPU implementation of the networks.

Figure 1: Flow of experimental results. First, TensorRT is employed to
generate the FP32/INT8 version of the DNNs. Then, they are deployed on GPU for
execution. During the execution, the various metrics such as power consumption
are monitored using nvidia-smi tool. We use the same tool to apply DVFS on GPU
and conduct the related experiments. Finally the labels generated for images
by network are compared against the original ones provided by the dataset to
evaluate the accuracy of results.

###  II-C Evaluation Platform

We run our experiments on a dual-socket Xeon server. It has two E5-2680 v4
Xeon chips where each of the chips has 28 cores running at 2.4 GHz. The server
has 128 GB of DDR4 memory. Ubuntu 16.04 with kernel 4.4 is installed on the
server with the python 2.7, CUDA 10.0, TensorFlow 1.15, and TensorRT 4.0. The
server is equipped with a PCI Express Gen3 Nvidia Tesla P40 GPU Accelerator.
The Tesla P40 leverages Nvidia Pascal architecture and has 3840 CUDA cores.
The total memory capacity of GPU is 24 GB GDDR5 memory and its maximum power
limit is 250W. The GPU accelerator supports both FP32 and INT8 instructions,
optimized for deep learning inference [15]. The GPU has 79 DVFS levels,
starting from 544 MHz to 1531 MHz. Levels are very close to each other and
moving from one level to another has negligible effect on GPU performance and
power consumption. So, we chose 10 levels that have more distance from each
other, and consequently, their effect is more significant: 544 MHz, 632 MHz,
734 MHz, 835 MHz, 949 MHz, 1063 MHz, 1189 MHz, 1303 MHz, 1430 MHz, and 1531
MHz. GPU vendor only allows controlling the frequency, but not the voltage. It
adjusts the voltage according to selected frequency via the GPU driver,
automatically.

###  II-D Experimental Results

In this section, we present the experimental results to study the impact of
DVFS on performance, accuracy, and power consumption of networks under various
precision. For each network, we consider FP32 and INT8 precision, apply
various GPU DVFS levels, and measure the performance, accuracy, and power
consumption. As mentioned in previous section, we choose 10 DVFS levels of
GPU. The performance results are depicted in Fig. 2. The performance of
networks for processing 50000 images is measured under different DVFS settings
for each precision, and normalized to the lowest performance of that precision
(performance of the lowest DVFS level, i.e., 544 Mhz in that specific
precision). The black dots indicate the performance for each DVFS level and
the horizontal lines of each box show the 25th, 50th, and 75th percentile of
dots. For both networks, we see notable performance fluctuation as we change
the frequency of GPU. However, the performance variation is more significant
in FP32 precision than INT8. We see that INT8-precision can improve the
performance that is affected by DVFS. In both networks, the INT8-precision can
mitigate the impact of DVFS remarkably.

Moreover, we see that the performance of ResNet50 is more sensitive to
frequency scaling than Inception in both precision. We can conclude that the
effect of DVFS on performance of DNNs has a direct relationship with their
architecture. Since DVFS mainly affects the computation time rather than data
transfer time, larger and more complex networks are more affected by frequency
scaling of GPU. The computation time relative to the overall execution time is
higher for larger networks in comparison to small ones, and hence, they are
more prone to performance degradation due to DVFS. We conclude that employing
reduced-precision instructions can help networks to resist against frequency
scaling and reduce its negative impact. It is very useful for systems that
have specific Service Level Agreements (SLA) and should reach a certain level
of Quality of Service (QoS) in the form of a predefined tail latency or other
similar metrics.

Obviously, the power consumption of GPU is also affected by DVFS. The results
for power consumption are depicted in Fig. 3. As expected, the power
consumption decreases with the frequency underscaling. We also observe that
the INT8-precision instructions lead to lower power consumption than FP32.
Hence, we can conclude that combining low frequency and reduced precision can
yield more power reduction than traditional frequency scaling approach.
Finally, Fig. 4 shows the effect of frequency scaling, hence the power
consumption, on GPU temperature for both precision. We see that INT8 precision
leads to decreased GPU temperature. The temperature has almost the same
pattern with power and reduced-precision can help to further decrease it, in
addition to frequency underscaling.

Similar to results presented for FPGA in Table 2 of [11], here we report Giga
Operations per second ($GOPs$), power-efficiency ($GOPs/W$), and energy-
efficiency ($GOPs/J$) for both Inception and ResNet50 in Table I and II,
respectively. All results are normalized to the highest frequency (1531 Mhz).
While the best value of $GOPs$ is obtained with the highest frequency setting,
the best value of power consumption is achieved with the lowest frequency
setting. However, for $GOPs/W$ and $GOPs/J$, a frequency setting different
than the highest and lowest ones has better results based on the network and
the precision.

Figure 2: Execution time variation of networks under different frequencies for both precision. Figure 3: Power consumption of GPU for processing the images in each DNN when applying different frequency. TABLE I: Evaluation of voltage frequency scaling of GPU for Inception for both precision. Best results for each metric is marked in blue. | GOPs | Power (W) | GOPs / W | GOPs / J
---|---|---|---|---
| Frequency
---
(Mhz)
FP32 | INT8 | FP32 | INT8 | FP32 | INT8 | FP32 | INT8
1531 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
1430 | 0.992 | 0.997 | 0.941 | 0.937 | 1.054 | 1.065 | 1.045 | 1.062
1303 | 0.99 | 0.996 | 0.857 | 0.878 | 1.154 | 1.135 | 1.143 | 1.13
1189 | 0.984 | 1 | 0.826 | 0.838 | 1.191 | 1.193 | 1.172 | 1.193
1063 | 0.977 | 0.988 | 0.772 | 0.794 | 1.265 | 1.244 | 1.236 | 1.23
949 | 0.972 | 0.986 | 0.762 | 0.778 | 1.277 | 1.267 | 1.241 | 1.249
835 | 0.96 | 0.979 | 0.742 | 0.769 | 1.294 | 1.272 | 1.243 | 1.245
734 | 0.942 | 0.97 | 0.734 | 0.757 | 1.284 | 1.282 | 1.209 | 1.244
632 | 0.93 | 0.96 | 0.728 | 0.752 | 1.277 | 1.277 | 1.188 | 1.225
544 | 0.914 | 0.954 | 0.722 | 0.748 | 1.266 | 1.276 | 1.158 | 1.217
TABLE II: Evaluation of voltage frequency scaling of GPU for ResNet50 for both precision. Best results for each metric is marked in blue. | GOPs | Power (W) | GOPs / W | GOPs / J
---|---|---|---|---
| Frequency
---
(Mhz)
FP32 | INT8 | FP32 | INT8 | FP32 | INT8 | FP32 | INT8
1531 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
1430 | 0.99 | 0.997 | 0.915 | 0.921 | 1.081 | 1.082 | 1.07 | 1.078
1303 | 0.981 | 0.99 | 0.796 | 0.852 | 1.232 | 1.161 | 1.209 | 1.149
1189 | 0.978 | 0.985 | 0.769 | 0.817 | 1.272 | 1.205 | 1.245 | 1.188
1063 | 0.964 | 0.979 | 0.734 | 0.778 | 1.314 | 1.259 | 1.266 | 1.233
949 | 0.946 | 0.97 | 0.697 | 0.756 | 1.358 | 1.284 | 1.285 | 1.246
835 | 0.926 | 0.952 | 0.694 | 0.742 | 1.333 | 1.284 | 1.234 | 1.223
734 | 0.903 | 0.944 | 0.68 | 0.727 | 1.328 | 1.297 | 1.199 | 1.224
632 | 0.875 | 0.922 | 0.665 | 0.73 | 1.318 | 1.263 | 1.154 | 1.165
544 | 0.851 | 0.911 | 0.664 | 0.727 | 1.279 | 1.253 | 1.088 | 1.141
Figure 4: Trend of GPU temperature under various frequencies.

##  III FPGA- vs. GPU-based CNN Accelerators

In this section, we comparatively evaluate the impact of the energy-efficiency
techniques for FPGA- and GPU-based CNN workloads, as described in section II.
Note that for this comparison, we use the results presented in [11] for FPGA-
based DNNs, as we find it a fair comparison case. This is because, [11] is
based on modern representative FPGA platforms of Xilinx ZCU102, uses the
similar DNN models with this paper, and evaluates the energy-efficiency
techniques, which are considered in this paper as well.

###  III-A Performance Analysis Through Voltage and Frequency Underscaling

Based on the results presented in Table 2 of [11], Table I, and Table II, we
compare the impact of voltage and frequency scaling on the performance (i.e.,
$GOPs$) of FPGA and GPU running CNN workloads. More specifically, we observe
that:

  * •

in the FPGA case, the highest frequency is nearly 50% more than the lowest
frequency. However, in the GPU case, the highest frequency is almost three
times of lowest one. The reason is that in the FPGA implementation of this
technique, the minimum supply voltage that we could practically underscale is
limited to $V_{crash}$, which FPGA stops operating below it.

  * •

for the FPGA case, the $GOPs$ varies by 30% from highest to lowest frequency.
But for the GPU case, the $GOPs$ variation is less significant. The highest
variation happens for ResNet50 in FP32 precision, where the $GOPs$ gap is
around 15%, and the lowest gap is 5% for Inception in INT8 precision. For both
networks, we see that $GOPs$ variation in INT8 is less than FP32, which means
that we can expect less performance variation in the presence of reduced-
precision instructions.

###  III-B Power Consumption Analysis Through Voltage and Frequency
Underscaling

For power consumption, again we see a wider gap in the FPGA (i.e., almost 46%)
than the GPU case (i.e., 34% highest in ResNet50). Relatively narrower gap in
INT8 than FP32 can be seen in GPU results as well. The power gap difference
observed between FPGA and GPU can justify the difference between $GOPs$ gap
discussed earlier. The wider gap in power consumption can translate to wider
computing power gap, which leads to wider $GOPs$ gap. Since the power gap
between the lowest and the highest frequency is wider in FPGA than GPU, we see
a wider gap between $GOPs$ of the lowest and the highest frequency in FPGA as
well.

###  III-C Power- and Energy-efficiency Analysis Through Voltage and Frequency
Underscaling

The first paragraph of this section would talk about the figure and the second
would explain it.

Fig. 5 shows the different behavior of GPU and FPGA regarding $GOPs/W$ and
$GOPs/J$. In this figure, we have only depicted the INT8 results for GPU.
Since the frequency levels in FPGA (7) is less than GPU (10), only seven
points for FPGA are depicted. Finally, since the value of frequency levels in
FPGA and GPU are different (for GPU from 544 MHz to 1531 MHz, and for FPGA
from 200 MHz to 333 MHz), we have used numbers from 1 to 10 to show frequency
levels, instead of the exact values.

While the best results for $GOPs$ and power is obtained by the same level of
frequency in both FPGA and GPU (the highest frequency for $GOPs$ and the
lowest one for power), for $GOPs/W$ and $GOPs/J$ the pattern is different. In
the FPGA case, the best result for $GOPs/W$ is obtained by the lowest
frequency and for $GOPs/J$ by the highest one. These results indicate that
FPGA-based NN designs with frequency scaling has the optimal energy operation
point at lowest frequency while the optimal energy-delay operation point is at
the highest frequency. In contrast with FPGA, for GPU the frequency that
yields the best performance-power and performance-energy ($GOPs/W$, $GOPs/J$)
trade-off is between the lowest and highest frequency. We see in Table I and
Table II that the best frequency for $GOPs/W$ and $GOPs/J$ depends on both
network and precision. These results indicate the non-linear behavior of GPU
regarding frequency scaling. Simply selecting the highest or lowest frequency
cannot always guarantee the best results.

Figure 5: Effects of frequency underscaling in power- and energy-efficiency of
FPGA- vs. GPU-based CNN accelerator

###  III-D Temperature Effects for FPGA- vs GPU-based CNN Accelerator

In Fig. 10 of [11], we see that the temperature has a direct impact on the
accuracy of FPGA results in critical region. As the temperature increases, the
accuracy increases too. However, for GPU, we do not observe any relation
between temperature and accuracy. While the temperature increases as the
frequency increases, the accuracy of both networks remains constant. The
reason is that in the critical region of the FPGA, the voltage is so low that
the effect of temperature on latency, and consequently the error rate, of
circuit is significant. However for GPU, the voltage is still in guardband
region even for very low frequencies, and hence, the temperature does not
affect the accuracy of the networks. As mentioned in section II-C, there is no
interface for controlling the voltage of GPU. Therefore, we cannot set the
voltage of GPU in critical region to further study the effect of temperature
on accuracy.

##  IV Related Work

Energy-efficiency is an important concern specially for power-constrained
modern applications like DNNs. Energy-efficiency, specially in the hardware-
level, is studied either using simulators or on the real underlying hardware.
For instance, about undervolting as one of the energy-efficiency techniques
evaluated in this paper, majority of the simulation-based studies target CPUs
[16, 17, 18], specifically designed for caches [19, 20, 21] or branch
predictors [22]. There are also studies for ASIC CNN accelerators [23, 24].
Additionally, there are FPGA-based designs either fully in simulation [25] or
emulation of FPGA netlists on simulation frameworks [26, 27]. Conducting
similar analysis on the real hardware requires a moderate engineering effort
regarding physical limitations, but provides relatively more accurate results.
For instance, there are several undervolting approaches targeting real
hardware system components, such as CPUs [28, 29], GPUs [30, 31], ASICs [32,
33], DRAMs [34, 35], and FPGAs [36, 11]. This paper complements the second
approach on the real hardware by a more comprehensive experimental study of
several state-of-the-art energy-efficiency techniques on GPUS and also,
compare it with FPGAs, while running DNN applications.

DVFS is a common method to manage the power consumption of a processor via
tuning the frequency of clock, and consequently, supply voltage. Controlling
the power consumption and performance of GPUs using DVFS has been studied in a
large body of research [37, 38, 39]. Jiao et al. [37] leveraged the capability
of modern GPUs for hosting concurrent kernels, along with GPU core and memory
frequency scaling to improve the performance per watt. Guerreiro et al. [39]
proposed classification models for impact of DVFS on performance and power of
GPU applications. Based on the models, they aimed to predict the impact of
different DVFS levels on the applications and set the frequency of GPU
accordingly. To reduce the energy consumption of matrix multiplication in
GPUs, GreenMM [38] proposed GPU undervolting while keeping the frequency
constant. Undervolting beyond minimum operational voltage leads to increase in
number of faults. To address this challenge, GreenMM proposes an algorithm
called Algorithm Based Fault Tolerance (ABFT). Tang et al. [40] studied the
impact of GPU DVFS on performance and energy consumption of several DNNs and
convolutional algorithms. They employed three GPUs with different
architectures and four DNNs to conduct their experiments. Based on the
results, the impact of high frequency on improving the performance and energy
inefficiency of default frequency settings are explained. None of the
aforementioned works studies the impact of reduced-precision instructions on
performance of GPUs.

##  V Conclusion

In this paper, we evaluated the energy-efficacy of CNN workloads running on
off-the-shelf FPGA and GPU accelerators, using supply voltage underscaling
techniques. We also evaluated the effect of frequency scaling, as well as
architectural quantization techniques on the performance, power, and energy of
such accelerators. We found that combining frequency scaling and reduced-
precision instructions can significantly improve the energy-efficiency of GPU,
while suffering from negligible accuracy loss. Moreover, comparing the GPU and
FPGA, we observed that unlike FPGA that achieves best results for various
metrics at highest or lowest frequency level, GPU achieves the best results
for some metrics such as GOPs/J at a frequency level between lowest and
highest. Therefore, it is necessary to explore the entire frequency state
space of different GPUs to find the best results for a specific metric.

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# Inequality on the optimal constant of Young’s convolution inequality for
locally compact groups and their closed subgroups

Takashi Satomi

###### Abstract

We define the optimal constant $Y(p_{1},p_{2};G)$ of Young’s convolution
inequality as

$\displaystyle
Y(p_{1},p_{2};G):=\sup\\{\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{p}\mid\phi_{1},\phi_{2}\colon
G\to\mathbb{C},\;\|\phi_{1}\|_{p_{1}}=\|\phi_{2}\|_{p_{2}}=1\\}$ (0.1)

for a locally compact group $G$ and $1\leq p_{1},p_{2},p\leq\infty$ with
$1/p_{1}+1/p_{2}=1+1/p$. Here $p^{\prime}$ is the Hölder conjugate of $p$,
$\|\cdot\|_{p}$ is the $L^{p}$-norm on a left Haar measure, and $\Delta\colon
G\to\mathbb{R}_{>0}$ is the modular function. The main result of this paper is
that $Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};H)$ for any closed subgroup $H\subset
G$. It follows from this inequality that $Y(p_{1},p_{2};G)\leq
Y(p_{1},p_{2};\mathbb{R})^{\dim G-r(G)}$ for any connected Lie group $G$ such
that the center of the semisimple part is a finite group such as connected
linear Lie groups and connected solvable Lie groups, where $r(G)$ is the
dimension of the maximal compact subgroups of $G$.

Keywords: locally compact group, Lie group, closed subgroup, $L^{p}$-space,
Young’s convolution inequality, maximal compact subgroup.

MSC2020: Primary 46E30; Secondary 22D15, 28C10, 39B62, 43A15.

##  1 Introduction

We write $Y(p_{1},p_{2};G)$ for the optimal constant (the optimal ratio of
both sides) of Young’s convolution inequality on a locally compact group $G$.
The main result of this paper is that $Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};H)$
holds for any closed subgroup $H\subset G$ (Theorem 1.2). As a corollary, we
generalize the estimate of $Y(p_{1},p_{2};G)$ from above by Beckner [Bec75],
Fournier [Fou77], Klein–Russo [KR78], and Nielsen [Nie94] to any connected Lie
group $G$ such that the center of the semisimple part is a finite group such
as connected linear Lie groups and connected solvable Lie groups. That is, we
have $Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};\mathbb{R})^{\dim G-r(G)}$, where
$r(G)$ is the dimension of the maximal compact subgroups of $G$ (Corollary
1.3).

We set the Hölder conjugate $1\leq p^{\prime}\leq\infty$ of $1\leq
p\leq\infty$ as

$\displaystyle\frac{1}{p}+\frac{1}{p^{\prime}}=1.$ (1.1)

The $L^{p}$-norm with respect to the left Haar measure $dg$ on the locally
compact group $G$ is written as $\|\cdot\|_{p}$. We define the convolution
$\phi_{1}*\phi_{2}$ of measurable functions $\phi_{1},\phi_{2}\colon
G\to\mathbb{C}$ as

$\displaystyle\phi_{1}*\phi_{2}(g^{\prime}):=\int_{G}\phi_{1}(g)\phi_{2}(g^{-1}g^{\prime})dg.$
(1.2)

In addition, there exists a unique continuous homomorphism $\Delta\colon
G\to\mathbb{R}_{>0}$ such that

$\displaystyle\int_{G}\phi(g^{-1})dg=\int_{G}\frac{\phi(g)}{\Delta(g)}dg.$
(1.3)

This $\Delta$ is called the modular function of $G$. Then the notion of the
optimal constant $Y(p_{1},p_{2};G)$ is defined as follows.

######  Definition 1.1 ([KR78, Section 2] [Nie94]).

For any locally compact group $G$ and any $1\leq p_{1},p_{2}\leq\infty$ with

$\displaystyle\frac{1}{p_{1}}+\frac{1}{p_{2}}\geq 1,$ (1.4)

the optimal constant $Y(p_{1},p_{2};G)$ is defined as

$\displaystyle
Y(p_{1},p_{2};G):=\sup\\{\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{p}\mid\phi_{1},\phi_{2}\colon
G\to\mathbb{C},\;\|\phi_{1}\|_{p_{1}}=\|\phi_{2}\|_{p_{2}}=1\\}.$ (1.5)

Here $\Delta$ is the modular function of $G$, and $p$ is given by

$\displaystyle\frac{1}{p_{1}}+\frac{1}{p_{2}}=1+\frac{1}{p}.$ (1.6)

The main result of this paper is the following theorem.

######  Theorem 1.2.

Suppose that $1\leq p_{1},p_{2}\leq\infty$ satisfy (1.4). Then
$Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};H)$ holds for any closed subgroup
$H\subset G$ of any locally compact group $G$.

When $H$ is a normal subgroup of $G$, Theorem 1.2 is essentially known. That
is, Cowling–Martini–Müller–Parcet proved

$\displaystyle Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};H)Y(p_{1},p_{2};G/H)$ (1.7)

[Cow+19, Proposition 2.2]. Since $Y(p_{1},p_{2};G/H)\leq 1$ by the classical
Young’s inequality (Example 2.1 (1)), Theorem 1.2 follows from (1.7). The
inequality (1.7) was essentially proved by Beckner when $G$ is abelian [Bec75,
Section IV.5] (see also Fact 2.3), and by Klein–Russo when $G$ is a semidirect
product of $H$ and $G/H$ [KR78, Lemma 2.4].

Theorem 1.2 has some interesting examples. For instance, we have the classical
Young’s inequality by applying Theorem 1.2 to the case where $H$ is trivial
(Example 2.1 (1)). In addition, if the identity component $G_{0}\subset G$ is
open (e.g. $G$ is a Lie group), then we have
$Y(p_{1},p_{2};G)=Y(p_{1},p_{2};G_{0})$ (Example 2.1 (2)) by Theorem 1.2.
Thus, it suffices to consider the identity component to determine the value of
$Y(p_{1},p_{2};G)$ for any Lie group $G$. In addition, we have
$Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};\mathbb{R})$ for any locally compact group
$G$ which has no open compact subgroup by Theorem 1.2. This claim is a
generalization of the previous results of Fournier, Nielsen, and the author
(Corollary 2.2).

As a corollary of Theorem 1.2, we bound $Y(p_{1},p_{2};G)$ for any connected
Lie group $G$ such that the center of the semisimple part is a finite group
such as connected linear Lie groups and connected solvable Lie groups by using
the dimension $r(G)$ of the maximal compact subgroups. That is, the following
corollary holds, where $\\#G$ is the cardinality of a group $G$, and $Z(G)$ is
the center of $G$.

######  Corollary 1.3.

Let $R\lhd G$ be the radical (the largest connected solvable closed normal
subgroup) of a connected Lie group $G$, and $r(G)$ be the dimension of the
maximal compact subgroups of $G$. If $\\#Z(G/R)<\infty$, then

$\displaystyle Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};\mathbb{R})^{\dim G-r(G)}$
(1.8)

holds for any $1\leq p_{1},p_{2}\leq\infty$ with (1.4).

Corollary 1.3 bounds $Y(p_{1},p_{2};G)$ explicitly from above because
$Y(p_{1},p_{2};\mathbb{R})$ is determined explicitly by Beckner (Fact 2.3).
Although Corollary 1.3 is known for some $G$ (Table 2.2), to the best of our
knowledge, it contains some new examples such as

$\displaystyle Y(p_{1},p_{2};SL_{2}(\mathbb{R}))\leq
Y(p_{1},p_{2};\mathbb{R})^{2}.$ (1.9)

It is known that the equality of (1.8) holds for some connected Lie groups
such as connected compact Lie groups (Corollary 2.2) and connected nilpotent
Lie groups (Fact 2.5). We prove Corollary 1.3 in Section 5 by using an
argument similar to the generalization of the Brunn–Minkowski inequality to
any Lie group by Jing–Tran–Zhang [JTZ23, Theorem 1.1].

Here is the organization of this paper. In Section 2, we compare some known
results with Theorem 1.2 and Corollary 1.3. In Section 3, we show Theorem 1.2
when $(p_{1},p_{2})$ is on the boundary of the range that satisfies the
assumption. In Section 4, we show Theorem 1.2 when $(p_{1},p_{2})$ is not on
the boundary. In Section 5, we show Corollary 1.3 by using Theorem 1.2 and the
argument of Jing–Tran–Zhang.

##  2 Comparison of some known results on the optimal constant

In this section, we compare some known results on the optimal constant
$Y(p_{1},p_{2};G)$ with Theorem 1.2 (Subsection 2.1) and Corollary 1.3
(Subsection 2.2).

###  2.1 Comparison with Theorem 1.2

In this subsection, we see some relations between known results and Theorem
1.2. Here are some examples of Theorem 1.2.

######  Example 2.1.

  1. (1)

Since the trivial group $\\{e\\}\subset G$ containing only the identity
element $e\in G$ is a closed subgroup, we have $Y(p_{1},p_{2};G)\leq
Y(p_{1},p_{2};\\{e\\})$ by Theorem 1.2. The equality
$Y(p_{1},p_{2};\\{e\\})=1$ holds by definition and hence we have the classical
Young’s inequality

$\displaystyle Y(p_{1},p_{2};G)\leq 1.$ (2.1)

There are at least two proofs of (2.1).

    1. (i)

The method to deduce the case of $p_{1}=1,p_{2}^{\prime}$ by using the
Riesz–Thorin theorem.

    2. (ii)

The direct method to use Hölder’s inequality repeatedly.

These proofs can be found in some literature listed in Table 2.1. Terp
indicated that one can prove (2.1) by the method (1)(ii) even when $G$ is not
unimodular, but there is no explicit proof in this paper. By using (2.1), Terp
generalized the Hausdorff–Young inequality to any locally compact group
[Ter17, Theorem 5.2]. The proof of Theorem 1.2 in Section 4 can be regarded as
a generalization of the method (1)(ii).

Table 2.1: Some literature in which the proof of (2.1) is mentioned |  $G$: unimodular |  $G$: general
---|---|---
(1)(i) | Weil [Wei40] | Klein–Russo [KR78, Lemma 2.1]
(1)(ii) | Hewitt–Ross [HR79, Theorem 20.18] | (Terp [Ter17, Lemma 1.1])
  2. (2)

We have $Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};G_{0})$ by Theorem 1.2, where
$G_{0}\subset G$ is the identity component. If $G_{0}$ is open (e.g. $G$ is a
Lie group), then a Haar measure on $G_{0}$ corresponds to that on $G$ and
hence $Y(p_{1},p_{2};G)\geq Y(p_{1},p_{2};G_{0})$. Thus, we obtain
$Y(p_{1},p_{2};G)=Y(p_{1},p_{2};G_{0})$ when $G_{0}$ is open.

By using Theorem 1.2, we obtain necessary and sufficient conditions for
satisfying $Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};\mathbb{R})$ as follows.

######  Corollary 2.2.

The following conditions (1)-(8) on the locally compact group $G$ are
equivalent for any $1<p_{1},p_{2}<\infty$ with

$\displaystyle\frac{1}{p_{1}}+\frac{1}{p_{2}}>1.$ (2.2)

  1. (1)

The locally compact group $G$ has no open compact subgroup.

  2. (2)

The identity component $G_{0}\subset G$ is not compact.

  3. (3)

The locally compact group $G$ has a closed subgroup which is isomorphic to
$\mathbb{R}$ as a topological group.

  4. (4)

The identity component $G_{0}$ has a closed subgroup which is isomorphic to
$\mathbb{R}$ as a topological group.

  5. (5)

One has $Y(p_{1},p_{2};G)\leq Y(p_{1},p_{2};\mathbb{R})$.

  6. (6)

One has $Y(p_{1},p_{2};G_{0})\leq Y(p_{1},p_{2};\mathbb{R})$.

  7. (7)

One has $Y(p_{1},p_{2};G)\neq 1$.

  8. (8)

One has $Y(p_{1},p_{2};G_{0})\neq 1$.

When $G$ is unimodular, $\ref{item:Young-R-compare-open-
compact}\Longleftrightarrow\ref{item:Young-R-compare-not-1-G}$ was proved by
Fournier [Fou77, Theorems 1 and 3], and $\ref{item:Young-R-compare-open-
compact}\Longleftrightarrow\ref{item:Young-R-compare-G0-non-
compact}\Longleftrightarrow\ref{item:Young-R-compare-
less-G}\Longleftrightarrow\ref{item:Young-R-compare-
less-G0}\Longleftrightarrow\ref{item:Young-R-compare-
not-1-G}\Longleftrightarrow\ref{item:Young-R-compare-not-1-G0}$ was
essentially proved in the previous paper of the author [Sat23, Corollary 1.3
and Remark 2.2].

When $G$ is not necessarily unimodular, $\ref{item:Young-R-compare-
subgroup-G}\Longleftrightarrow\ref{item:Young-R-compare-subgroup-G0}$ follows
from the connectedness of $\mathbb{R}$. The results of Iwasawa [Iwa49, Theorem
13] and the Gleason–Yamabe theorem [Gle51] [Yam53, Theorem 5’] imply
$\ref{item:Young-R-compare-G0-non-compact}\Longrightarrow\ref{item:Young-R-
compare-subgroup-G0}$. Nielsen proved $\ref{item:Young-R-compare-open-
compact}\Longleftrightarrow\ref{item:Young-R-compare-not-1-G}$ [Nie94, Theorem
1]. The author showed $\ref{item:Young-R-compare-open-
compact}\Longleftrightarrow\ref{item:Young-R-compare-G0-non-compact}$ in the
previous paper [Sat22, Remark 2.4 (3)] by using the result of Hewitt–Ross
[HR79]. Theorem 1.2 implies $\ref{item:Young-R-compare-
subgroup-G0}\Longrightarrow\ref{item:Young-R-compare-
less-G0}\Longrightarrow\ref{item:Young-R-compare-less-G}$. Theorem 1.2 and
Example 2.1 (1) imply $\ref{item:Young-R-compare-
not-1-G0}\Longrightarrow\ref{item:Young-R-compare-not-1-G}$. The implications
$\ref{item:Young-R-compare-less-G}\Longrightarrow\ref{item:Young-R-compare-
not-1-G}$ and $\ref{item:Young-R-compare-
less-G0}\Longrightarrow\ref{item:Young-R-compare-not-1-G0}$ are deduced from
$Y(p_{1},p_{2};\mathbb{R})<1$. Beckner determined the value of
$Y(p_{1},p_{2};\mathbb{R}^{n})$ explicitly as the following fact and hence
$Y(p_{1},p_{2};\mathbb{R})<1$ is essentially obtained.

######  Fact 2.3 (Beckner [Bec75, Theorem 3]).

Let $1\leq p\leq\infty$ be as in (1.6) for $1\leq p_{1},p_{2}\leq\infty$ with
(1.4). Then

$\displaystyle
Y(p_{1},p_{2};\mathbb{R}^{n})=\left(\frac{B(p_{1})B(p_{2})}{B(p)}\right)^{n/2},$
$\displaystyle B(p):=\left\\{\begin{aligned} &\frac{p^{1/p}}{p^{\prime
1/p^{\prime}}}&&\text{if $1<p<\infty$}\\\ &1&&\text{if
$p=1,\infty$}\end{aligned}\right.$ (2.3)

holds for any $n\in\mathbb{Z}_{\geq 1}$.

######  Remark 2.4.

There are some proofs of Fact 2.3 (and its generalization named the
Brascamp–Lieb inequality [BL76, Theorem 1]).

  1. (1)

Beckner reduced Fact 2.3 to the problem of finding the integrable solutions of

$\displaystyle\psi_{1}(x_{1})\psi_{2}(x_{2})=\nu_{1}(x_{1}+x_{2})\nu_{2}(x_{1}-x_{2})$
(2.4)

by using the Minkowski integral inequality. Beckner proved the existence of
rotation invariant extremal functions by using the Riesz–Sobolev rearrangement
inequality, and proved Fact 2.3 by showing that the rotation invariant
solutions of (2.4) are only Gaussian functions. Lieb proved the Brascamp–Lieb
inequality by showing implicitly that the solutions of (2.4) are only Gaussian
functions (the Darmois–Skitovich theorem [Dar53] [Ski53]) even when the
function is not necessarily rotation invariant [Lie90, Theorem 6.2].

  2. (2)

Brascamp–Lieb proved Fact 2.3 by showing
$Y(p_{1},p_{2};\mathbb{R}^{n})=Y(p_{1},p_{2};\mathbb{R})^{n}$ and bounding the
behavior of $Y(p_{1},p_{2};\mathbb{R}^{n})$ as $n\to\infty$ from above. In
addition, Brascamp–Lieb generalized Fact 2.3 to the Brascamp–Lieb inequality
by a similar argument.

  3. (3)

Barthe gave a direct proof [Bar98a, Theorem 1] utilizing the change of
variable by Henstock–Macbeath [HM53, Section 5] and the weighted AM-GM
inequality. In addition, Barthe proved that a similar argument is valid for
the Brascamp–Lieb inequality [Bar98, Theorem 1].

  4. (4)

Carlen–Lieb–Loss proved the rank-one Brascamp–Lieb inequality by showing the
property that the integral of the product of the exponentiations of the
solutions of the heat equations is increasing with time [CLL04, Theorem 3.1].
Cordero-Erausquin–Ledoux proved this property by using the estimate of
Shannon’s differential entropy [CEL10, Theorem 6].

In addition, there are many works [Bal89] [Ben+08] [Val08] [BB09] [Ben+10]
[Val10] [Bar+11] [Led14] [Leh14] [IV15] [Ben+17] [Bra17] [Gar+18] [Ben+20] and
surveys [Gar02] [Car07] [Ben14] about Fact 2.3 and the Brascamp–Lieb
inequality.

###  2.2 Comparison with Corollary 1.3

In this section, we see some relations between known results and Corollary
1.3. Table 2.2 lists the authors who proved Corollary 1.3 for some $G$. If $G$
is compact, then $G$ is unimodular and Corollary 1.3 corresponds to (2.1).
Thus, Corollary 1.3 was essentially proved by Weil (Example 2.1 (1)). The
equivalent conditions (1)-(8) in Corollary 2.2 is also equivalent to
$r(G)<\dim G$ for any connected Lie group $G$. Thus, Corollary 1.3 gives a
stronger bound than Corollary 2.2 of $Y(p_{1},p_{2};G)$ from above for any
connected Lie group $G$ with $\\#Z(G/R)<\infty$.

When $G$ is either a simply connected solvable Lie group or a connected
nilpotent Lie group, Nielsen determined the value of $Y(p_{1},p_{2};G)$ as
follows.

######  Fact 2.5 (Nielsen [Nie94, Corollaries (a) and (b)]).

Suppose that $1<p_{1},p_{2}<\infty$ satisfy (2.2). If $G$ is either a simply
connected solvable Lie group or a connected nilpotent Lie group, then

$\displaystyle Y(p_{1},p_{2};G)=Y(p_{1},p_{2};\mathbb{R})^{\dim
G-\mathrm{rank}(\ker(\tilde{G}\to G))}$ (2.5)

holds, where $\tilde{G}$ is the universal cover of $G$.

We have $r(G)=\mathrm{rank}(\ker(\tilde{G}\to G))$ for any connected solvable
Lie group $G$ (Example 5.2 (2)). Thus, if $G$ is either a simply connected
solvable Lie group or a connected nilpotent Lie group, then Corollary 1.3
follows from Fact 2.5. In addition, Bennett–Bez–Buschenhenke–Cowling–Flock
gave a stronger bound than Corollary 1.3 in the case where each support of the
functions is sufficiently small as follows.

Table 2.2: Authors who proved Corollary 1.3 for some $G$ Connected Lie group $G$ | Author
---|---
Compact group | Weil (Example 2.1 (1))
$\mathbb{R}^{n}$ | Beckner (Fact 2.3)
Simply connected nilpotent Lie group | Klein–Russo [KR78, Corollary 2.5’]
Simply connected solvable Lie group, | Nielsen (Fact 2.5)
Nilpotent Lie group

######  Fact 2.6 (Bennett–Bez–Buschenhenke–Cowling–Flock [Ben+20, Corollary
2.4]).

Let $1\leq p\leq\infty$ be as in (1.6) for $1\leq p_{1},p_{2}\leq\infty$ with
(1.4). Then for any Lie group $G$ and any
$Y_{0}>Y(p_{1},p_{2};\mathbb{R})^{\dim G}$, there exists a non-empty open set
$V\subset G$ such that

$\displaystyle\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{p}\leq
Y_{0}\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}$ (2.6)

for any measurable functions $\phi_{1},\phi_{2}\colon G\to\mathbb{C}$ whose
supports are contained in $V$.

The connectedness is assumed in the original paper of
Bennett–Bez–Buschenhenke–Cowling–Flock. Nevertheless, Fact 2.6 holds without
connectedness because $V$ can be rearranged to satisfy $V\subset G_{0}$. The
value $Y(p_{1},p_{2};\mathbb{R})^{\dim G}$ in Fact 2.6 is the best possible by
the result of Cowling–Martini–Müller–Parcet [Cow+19, Proposition 2.4 (i)].

Fact 2.6 gives a stronger bound than Theorem 1.2 with the assumption that the
supports are sufficiently small. For example, when $G=SL_{2}(\mathbb{R})$, we
have (1.9) by $\dim G=3$ and $r(G)=\dim SO(2)=1$. On the other hand, for any
$Y_{0}>Y(p_{1},p_{2};\mathbb{R})^{3}$, there exists a non-empty open set
$V\subset SL_{2}(\mathbb{R})$ such that (2.6) holds for any
$\phi_{1},\phi_{2}\colon SL_{2}(\mathbb{R})\to\mathbb{C}$ whose supports are
contained in $V$.

##  3 Optimal constant on the boundary

In this section, we show Theorem 1.2 when $(p_{1},p_{2})$ is on the boundary
of the range that satisfies the assumption. In this case, the equality of the
classical Young’s inequality (2.1) holds for any locally compact group $G$ as
follows.

######  Lemma 3.1.

Suppose that $1\leq p_{1},p_{2}\leq\infty$ with (1.4) satisfy at least one of
the following cases (1), (2), and (3).

  1. (1)

One has $p_{1}=1$.

  2. (2)

One has $p_{2}=1$.

  3. (3)

One has $1/p_{1}+1/p_{2}=1$.

Then $Y(p_{1},p_{2};G)=1$ holds for any locally compact group $G$.

Theorem 1.2 follows from Lemma 3.1 in the cases (1), (2), and (3). We show the
following lemma of the symmetry of the optimal constant to prove Lemma 3.1.

######  Lemma 3.2.

Let $G$, $p_{1}$, $p_{2}$, $p$, $\Delta$, and $Y(p_{1},p_{2};G)$ be as in
Definition 1.1.

  1. (1)

One has

$\displaystyle\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})(g^{\prime})=\left(\frac{\phi_{2}(\cdot^{-1})}{\Delta^{1/p_{2}}}\right)*\left(\frac{\phi_{1}(\cdot^{-1})}{\Delta^{1/p}}\right)(g^{\prime-1})\Delta(g^{\prime})^{-1/p}$
(3.1)

for any $g^{\prime}\in G$ and any measurable functions
$\phi_{1},\phi_{2}\colon G\to\mathbb{C}$.

  2. (2)

One has $Y(p_{1},p_{2};G)=Y(p_{2},p_{1};G)$.

###### Proof.

  1. (1)

We have

$\displaystyle\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})(g^{\prime})=\int_{G}\phi_{1}(g^{\prime}g)\phi_{2}(g^{-1})\Delta(g^{-1})^{1/p_{1}^{\prime}}dg$
(3.2)

by the left invariance of $dg$. Since

$\displaystyle\frac{1}{p}+\frac{1}{p_{1}^{\prime}}=\frac{1}{p_{1}}+\frac{1}{p_{2}}-1+\frac{1}{p_{1}^{\prime}}=\frac{1}{p_{2}}$
(3.3)

holds by (1.6), we obtain (3.1) by (3.2).

  2. (2)

It suffices to show

$\displaystyle\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{p}\leq
Y(p_{2},p_{1};G)\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}$ (3.4)

for any measurable functions $\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq
0}$. We have

$\displaystyle\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{p}^{p}$
$\displaystyle=\int_{G}\left(\frac{\phi_{2}(\cdot^{-1})}{\Delta^{1/p_{2}}}\right)*\left(\frac{\phi_{1}(\cdot^{-1})}{\Delta^{1/p}}\right)(g^{\prime-1})^{p}\Delta(g^{\prime})^{-1}dg^{\prime}$
(3.5)
$\displaystyle=\left\|\left(\frac{\phi_{2}(\cdot^{-1})}{\Delta^{1/p_{2}}}\right)*\left(\frac{\phi_{1}(\cdot^{-1})}{\Delta^{1/p}}\right)\right\|_{p}^{p}$
(3.6)

by (1.3) and the assertion (1). Since

$\displaystyle\frac{1}{p}+\frac{1}{p_{2}^{\prime}}=\frac{1}{p_{1}}$ (3.7)

holds by an argument similar to (3.3), we obtain (3.4) by (1.3) and (3.6). ∎

###### Proof of Lemma 3.1.

It suffices to show

$\displaystyle Y(p_{1},p_{2};G)\geq 1$ (3.8)

by Example 2.1 (1).

First, we show (3.8) in the case (3). We assume that an integrable function
$\phi\colon G\to\mathbb{R}_{\geq 0}$ satisfies $\|\phi\|_{1}=1$ and

$\displaystyle\phi_{1}(g)$ $\displaystyle:=\left\\{\begin{aligned}
&\phi(g)^{1/p_{1}}&&\text{if $p_{1}<\infty$}\\\ &1&&\text{if
$p_{1}=\infty$}\end{aligned}\right.,$ $\displaystyle\phi_{2}(g)$
$\displaystyle:=\left\\{\begin{aligned}
&\left(\frac{\phi(g^{-1})}{\Delta(g)}\right)^{1/p_{2}}&&\text{if
$p_{2}<\infty$}\\\ &1&&\text{if $p_{2}=\infty$}\end{aligned}\right..$ (3.9)

Since

$\displaystyle\|\phi_{1}\|_{p_{1}}$ $\displaystyle=\|\phi\|_{1}=1,$
$\displaystyle\|\phi_{2}\|_{p_{2}}$
$\displaystyle=\int_{G}\frac{\phi(g^{-1})}{\Delta(g)}dg=\|\phi\|_{1}=1$ (3.10)

hold by $1/p_{1}+1/p_{2}=1$ and (1.3), the convolution
$\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})$ is continuous. Thus, one has
$\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{\infty}=1$ by

$\displaystyle\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})(e)=\|\phi\|_{1}=1$
(3.11)

and hence we obtain (3.8).

Second, we show (3.8) in the case (1). One may assume $p_{2}<\infty$ by the
case (3). When $\phi_{2}\in L^{p_{2}}(G)$ is fixed, there exists a sequence of
integrable functions $\phi_{1,m}\colon G\to\mathbb{R}_{\geq 0}$ with
$\|\phi_{1,m}\|_{1}=1$ such that $\phi_{1,m}*\phi_{2}$ converges to $\phi_{2}$
[HR79, Theorem 20.15]. Thus, we obtain (3.8) because
$\|\phi_{1,m}*\phi_{2}\|_{p_{2}}$ converges to
$\|\phi_{1}\|_{p_{1}}\|\phi_{2}\|_{p_{2}}=\|\phi_{2}\|_{p_{2}}$.

Finally, (3.8) in the case (2) follows from the case (1) by Lemma 5.1 (2). ∎

##  4 Proof of Theorem 1.2

In this section, we prove Theorem 1.2 in the general case. In Subsection 4.1,
we prepare a lemma (Lemma 4.1) which represents the left Haar measure on $G$
by that on a closed subgroup $H\subset G$. In Subsection 4.2, we give some
inequalities (Example 4.4) to prove Theorem 1.2 by applying Hölder’s
inequality (Fact 4.3). In Subsection 4.3, we complete the proof of Theorem 1.2
by using Subsection 4.1 and Subsection 4.2.

###  4.1 Representing the Haar measure by using a closed subgroup

In this subsection, we prove a lemma (Lemma 4.1) which represents the left
Haar measure on $G$ by that on a closed subgroup $H\subset G$, and we give
some examples (Example 4.2) which is used in the proof of Theorem 1.2. We
write $X:=H\backslash G$ and $\overline{g}:=Hg\in X$ for the right coset of
$g\in G$.

######  Lemma 4.1.

Let $H\subset G$ be a closed subgroup of a locally compact group $G$.

  1. (1)

There exists a continuous function $\delta\colon G\to\mathbb{R}_{>0}$ such
that $\delta|_{H}$ is the modular function of $H$ and

$\displaystyle\int_{H}\phi(hg)dh\delta(g)$ (4.1)

is left $H$-invariant for any measurable function $\phi\colon G\to\mathbb{C}$.

  2. (2)

We fix $\delta$ in (1). Then there exists a Borel measure $d\overline{g}$ on
$X$ such that

$\displaystyle\int_{X}\int_{H}\phi(hg)dh\delta(g)d\overline{g}=\int_{G}\phi(g)dg$
(4.2)

for any integrable function $\phi\colon G\to\mathbb{C}$.

###### Proof.

  1. (1)

There exists a continuous function $\rho\colon G\to\mathbb{R}_{>0}$ such that

$\displaystyle\rho(gh)=\frac{\Delta_{H}(h)\rho(g)}{\Delta(h)}$ (4.3)

for any $g\in G$ and $h\in H$ [Fol16, Proposition 2.56], where $\Delta_{H}$ is
the modular function of $H$. By scaling, we may assume $\rho(e)=1$. When the
continuous function $\delta\colon G\to\mathbb{R}_{>0}$ is defined as

$\displaystyle\delta(g):=\frac{1}{\Delta(g^{-1})\rho(g^{-1})},$ (4.4)

we have

$\displaystyle\delta(hg)=\frac{1}{\Delta(g^{-1}h^{-1})\rho(g^{-1}h^{-1})}=\frac{\Delta_{H}(h)}{\Delta(g^{-1})\rho(g^{-1})}=\Delta_{H}(h)\delta(g)$
(4.5)

by (4.3). In particular, we have
$\delta(h)=\Delta_{H}(h)\delta(e)=\Delta_{H}(h)$. Thus, it follows from (4.5)
that

$\displaystyle\int_{H}\phi(hh^{\prime}g)dh\delta(h^{\prime}g)=\int_{H}\phi(hh^{\prime}g)dh\delta(h^{\prime})\delta(g)=\int_{H}\phi(hg)dh\delta(g)$
(4.6)

for any $h^{\prime}\in H$ and any measurable function $\phi\colon
G\to\mathbb{C}$ and hence (4.1) is left $H$-invariant.

  2. (2)

For any $g\in G$ and $h^{\prime}\in H$, we have

$\displaystyle\delta(h^{\prime}g)\int_{H}\phi(hg)dh=\delta(g)\int_{H}\phi(hh^{\prime-1}g)dh=\delta(g)\delta(h^{\prime})\int_{H}\phi(hg)dh$
(4.7)

for any measurable function $\phi\colon G\to\mathbb{C}$ by the left invariance
of (4.1) and hence

$\displaystyle\delta(h^{\prime}g)=\delta(h^{\prime})\delta(g).$ (4.8)

Since

$\displaystyle\rho(g):=\frac{1}{\Delta(g)\delta(g^{-1})}$ (4.9)

is a rho-function by (4.8), there exists a Borel measure $dgH$ on $G/H$ such
that

$\displaystyle\int_{G/H}\int_{H}\omega(gh)dhdgH=\int_{G}\omega(g)\rho(g)dg$
(4.10)

for any integrable function $\omega\colon G\to\mathbb{C}$ [Fol16, Theorem
2.58]. By (1.3) and applying

$\displaystyle\omega(g)=\frac{\phi(g^{-1})}{\Delta(g)\rho(g)}=\phi(g^{-1})\delta(g^{-1})$
(4.11)

to (4.10), we have

$\displaystyle\int_{G}\phi(g)dg=\int_{G/H}\int_{H}\phi((gh)^{-1})\delta((gh)^{-1})dhdgH.$
(4.12)

Since $\delta|_{H}$ is the modular function of $H$, we have

$\displaystyle\int_{H}\phi((gh)^{-1})\delta((gh)^{-1})dh=\int_{H}\phi(hg^{-1})dh\delta(g^{-1})$
(4.13)

by (1.3) and (4.8). Thus, by setting the Borel measure $d\overline{g}$ on
$X:=H\backslash G$ as

$\displaystyle\int_{X}\alpha(\overline{g})d\overline{g}=\int_{G/H}\alpha(Hg^{-1})dgH,$
(4.14)

we obtain (4.2) by (4.12) and (4.13). ∎

Now we give some examples of Lemma 4.1 to prove Theorem 1.2.

######  Example 4.2.

Let $\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq 0}$ be measurable functions.
We suppose $1<p_{1},p_{2},p<\infty$ satisfy (1.6).

  1. (1)

Let $s(h,g):=\phi_{1}(hg)\delta(g)^{1/p_{1}}$ for $g\in G$ and $h\in H$. Then

$\displaystyle
S(\overline{g}):=\int_{H}s(h,g)^{p_{1}}dh=\int_{H}\phi_{1}(hg)^{p_{1}}dh\delta(g)$
(4.15)

is well-defined by Lemma 4.1 (1), and we have

$\displaystyle\int_{X}S(\overline{g})d\overline{g}=\|\phi_{1}\|_{p_{1}}^{p_{1}}$
(4.16)

by Lemma 4.1 (2).

  2. (2)

Let
$t(g,g^{\prime}):=(\phi_{2}(g^{-1}g^{\prime})^{p_{2}}\delta(g^{\prime}))^{1/p}$
for $g,g^{\prime}\in G$. Since

$\displaystyle\int_{H}t(h^{-1}h^{\prime}g,g^{\prime})^{p}dh=\int_{H}t(h^{-1}g,g^{\prime})^{p}dh$
(4.17)

for any $h^{\prime}\in H$, the function

$\displaystyle
T(\overline{g},\overline{g^{\prime}}):=\int_{H}t(h^{-1}g,g^{\prime})^{p}dh=\int_{H}\phi_{2}(g^{-1}hg^{\prime})^{p_{2}}dh\delta(g^{\prime})$
(4.18)

is well-defined by Lemma 4.1 (1). Thus, we have

$\displaystyle\int_{X}T(\overline{g},\overline{g^{\prime}})d\overline{g^{\prime}}=\int_{X}\int_{H}\phi_{2}(g^{-1}hg^{\prime})^{p_{2}}dh\delta(g^{\prime})d\overline{g^{\prime}}=\int_{G}\phi_{2}(g^{-1}g^{\prime})^{p_{2}}dg^{\prime}=\|\phi_{2}\|_{p_{2}}^{p_{2}}$
(4.19)

by lemma 4.1 (2).

  3. (3)

Let

$\displaystyle
u(g,h,g^{\prime}):=\left(\frac{\phi_{2}(g^{-1}hg^{\prime})^{p_{2}}\Delta(g^{-1}hg^{\prime})\delta(g)}{\delta(h)}\right)^{1/p_{1}^{\prime}}$
(4.20)

for $g,g^{\prime}\in G$ and $h\in H$. Since $\delta|_{H}$ is the modular
function of $H$ by Lemma 4.1 (1),

$\displaystyle\int_{H}u(g,h,h^{\prime}g^{\prime})^{p_{1}^{\prime}}dh$
$\displaystyle=\int_{H}\frac{\phi_{2}(g^{-1}hh^{\prime}g^{\prime})^{p_{2}}\Delta(g^{-1}hh^{\prime}g^{\prime})\delta(g)}{\delta(h)}dh$
(4.21)
$\displaystyle=\int_{H}\phi_{2}(g^{-1}h^{-1}h^{\prime}g^{\prime})^{p_{2}}\Delta(g^{-1}h^{-1}h^{\prime}g^{\prime})dh\delta(g)$
(4.22)
$\displaystyle=\int_{H}\phi_{2}(g^{-1}h^{-1}g^{\prime})^{p_{2}}\Delta(g^{-1}h^{-1}g^{\prime})dh\delta(g)$
(4.23)

is independent of $h^{\prime}\in H$ by (1.3). Thus, the function

$\displaystyle
U(\overline{g},\overline{g^{\prime}}):=\int_{H}u(g,h,g^{\prime})^{p_{1}^{\prime}}dh$
(4.24)

is well-defined by Lemma 4.1 (1). Now we prove

$\displaystyle\int_{X}U(\overline{g},\overline{g^{\prime}})d\overline{g}=\|\phi_{2}\|_{p_{2}}^{p_{2}}$
(4.25)

for any $g^{\prime}\in G$. We have

$\displaystyle\int_{X}U(\overline{g},\overline{g^{\prime}})d\overline{g}=\int_{X}\int_{H}\phi_{2}(g^{-1}h^{-1}g^{\prime})^{p_{2}}\Delta(g^{-1}h^{-1}g^{\prime})dh\delta(g)d\overline{g}$
(4.26)

by (4.23) and

$\displaystyle\int_{X}\int_{H}\phi_{2}(g^{-1}h^{-1}g^{\prime})^{p_{2}}\Delta(g^{-1}h^{-1}g^{\prime})dh\delta(g)d\overline{g}$
$\displaystyle=\int_{G}\phi_{2}(g^{-1}g^{\prime})^{p_{2}}\Delta(g^{-1}g^{\prime})dg$
(4.27) $\displaystyle=\int_{G}\phi_{2}(g^{-1})^{p_{2}}\Delta(g^{-1})dg$ (4.28)

by Lemma 4.1 (2). Since

$\displaystyle\int_{G}\phi_{2}(g^{-1})^{p_{2}}\Delta(g^{-1})dg=\int_{G}\phi_{2}(g)^{p_{2}}dg=\|\phi_{2}\|_{p_{2}}^{p_{2}}$
(4.29)

by (1.3), we obtain (4.25).

  4. (4)

Let $s$, $t$, and $u$ be as in (1), (2), and (3), respectively. Now we show

$\displaystyle\quad\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})(h^{\prime}g^{\prime})$
(4.30)
$\displaystyle=\int_{X}\int_{H}\frac{s(h,g)t(h^{\prime-1}hg,g^{\prime})u(g,h^{-1}h^{\prime},g^{\prime})\delta(h^{-1}h^{\prime})^{1/p_{1}^{\prime}}}{\delta(g^{\prime})^{1/p}}dhd\overline{g}$
(4.31)

for any $h^{\prime}\in H$ and $g^{\prime}\in G$. Since

$\displaystyle\quad\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})(h^{\prime}g^{\prime})$
(4.32)
$\displaystyle=\int_{G}\phi_{1}(g)\phi_{2}(g^{-1}h^{\prime}g^{\prime})\Delta(g^{-1}h^{\prime}g^{\prime})^{1/p_{1}^{\prime}}dg$
(4.33)
$\displaystyle=\int_{X}\int_{H}\phi_{1}(hg)\phi_{2}(g^{-1}h^{-1}h^{\prime}g^{\prime})\Delta(g^{-1}h^{-1}h^{\prime}g^{\prime})^{1/p_{1}^{\prime}}dh\delta(g)d\overline{g}$
(4.34)

by Lemma 4.1, we obtain (4.31) by

$\displaystyle\quad\phi_{1}(hg)\phi_{2}(g^{-1}h^{-1}h^{\prime}g^{\prime})\Delta(g^{-1}h^{-1}h^{\prime}g^{\prime})^{1/p_{1}^{\prime}}\delta(g)$
(4.35)
$\displaystyle=\frac{s(h,g)t(h^{\prime-1}hg,g^{\prime})u(g,h^{-1}h^{\prime},g^{\prime})\delta(h^{-1}h^{\prime})^{1/p_{1}^{\prime}}}{\delta(g^{\prime})^{1/p}}.$
(4.36)

  5. (5)

Since

$\displaystyle\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{p}^{p}=\int_{X}\int_{H}\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})(h^{\prime}g^{\prime})^{p}dh^{\prime}\delta(g^{\prime})d\overline{g^{\prime}}$
(4.37)

by Lemma 4.1, we have

$\displaystyle\quad\|\phi_{1}*(\phi_{2}\Delta^{1/p_{1}^{\prime}})\|_{p}^{p}$
(4.38)
$\displaystyle=\int_{X}\int_{H}\left(\int_{X}\int_{H}s(h,g)t(h^{\prime-1}hg,g^{\prime})u(g,h^{-1}h^{\prime},g^{\prime})\delta(h^{-1}h^{\prime})^{1/p_{1}^{\prime}}dhd\overline{g}\right)^{p}dh^{\prime}d\overline{g^{\prime}}$
(4.39)

by (4).

###  4.2 Hölder’s inequality

In this subsection, we obtain some inequalities (Example 4.4) by using
Hölder’s inequality (Fact 4.3) to prove Theorem 1.2.

######  Fact 4.3 (Hölder’s inequality).

Let $k,l\in\mathbb{Z}_{\geq 1}$ and $p_{i,j},c_{i}>0$ for $i=1,\cdots,k$ and
$j=1,\cdots,l$. Then

$\displaystyle\quad\left(\int_{G}\phi_{1}(g)^{p_{1}}\cdots\phi_{l}(g)^{p_{l}}dg\right)^{c}$
(4.40)
$\displaystyle\leq\left(\int_{G}\phi_{1}(g)^{p_{1,1}}\cdots\phi_{l}(g)^{p_{1,l}}dg\right)^{c_{1}}\cdots\left(\int_{G}\phi_{1}(g)^{p_{k,1}}\cdots\phi_{l}(g)^{p_{k,l}}dg\right)^{c_{k}}$
(4.41)

holds for any measurable functions $\phi_{1},\cdots,\phi_{l}\colon
G\to\mathbb{R}_{\geq 0}$ on a measure space $G$, where

$\displaystyle c:=c_{1}+\cdots+c_{k},$ $\displaystyle
p_{j}:=\frac{p_{1,j}c_{1}+\cdots+p_{k,j}c_{k}}{c}$ (4.42)

for $j=1,\cdots,l$.

######  Example 4.4.

Let $\phi_{1}$, $\phi_{2}$, $S$, $t$, $T$, $u$, and $U$ be as in Example 4.2.
Suppose that $1<p_{1},p_{2},p<\infty$ satisfy (1.6).

  1. (1)

We have

$\displaystyle\left(\int_{H}(t(h^{-1}g,g^{\prime})u(g,h,g^{\prime}))^{p_{2}}dh\right)^{1/p_{2}}\leq
T(\overline{g},\overline{g^{\prime}})^{1/p}U(\overline{g},\overline{g^{\prime}})^{1/p_{1}^{\prime}}$
(4.43)

by (3.3) and Fact 4.3.

  2. (2)

We show

$\displaystyle\quad\left(\int_{X}S(\overline{g})^{1/p_{1}}T(\overline{g},\overline{g^{\prime}})^{1/p}U(\overline{g},\overline{g^{\prime}})^{1/p_{1}^{\prime}}d\overline{g}\right)^{p}$
(4.44)
$\displaystyle\leq\left(\|\phi_{1}\|_{p_{1}}^{p_{1}/p_{2}^{\prime}}\|\phi_{2}\|_{p_{2}}^{p_{2}/p_{1}^{\prime}}\right)^{p}\int_{X}S(\overline{g})T(\overline{g},\overline{g^{\prime}})d\overline{g}.$
(4.45)

One has

$\displaystyle
p\left(\frac{1}{p_{1}^{\prime}}+\frac{1}{p_{2}^{\prime}}\right)+1=p\left(2-\frac{1}{p_{1}}-\frac{1}{p_{2}}+\frac{1}{p}\right)=p$
(4.46)

by (1.6). Thus, it follows from (3.7) and Fact 4.3 that

$\displaystyle\quad\left(\int_{X}S(\overline{g})^{1/p_{1}}T(\overline{g},\overline{g^{\prime}})^{1/p}U(\overline{g},\overline{g^{\prime}})^{1/p_{1}^{\prime}}d\overline{g}\right)^{p}$
(4.47)
$\displaystyle\leq\left(\int_{X}S(\overline{g})d\overline{g}^{1/p_{2}^{\prime}}\int_{X}U(\overline{g},\overline{g^{\prime}})d\overline{g}^{1/p_{1}^{\prime}}\right)^{p}\int_{X}S(\overline{g})T(\overline{g},\overline{g^{\prime}})d\overline{g}$
(4.48)

and hence we obtain (4.45) by Example 4.2 (1) and (3).

###  4.3 Completion of the proof

In this subsection, we complete the proof of Theorem 1.2 by using Example 4.2
and Example 4.4.

###### Proof of Theorem 1.2.

It suffices to consider the case of $1<p_{1},p_{2},p<\infty$ by Lemma 3.1. Let
$s$, $S$, $t$, $T$, $u$, and $U$ be as in Example 4.2 for
$\phi_{1},\phi_{2}\colon G\to\mathbb{R}_{\geq 0}$ with
$\|\phi_{1}\|_{p_{1}}=\|\phi_{2}\|_{p_{2}}=1$. Then it suffices to show

$\displaystyle\int_{X}\int_{H}\left(\int_{X}F(g,h^{\prime},g^{\prime})d\overline{g}\right)^{p}dh^{\prime}d\overline{g^{\prime}}\leq
Y(p_{1},p_{2};H)^{p}$ (4.49)

by Example 4.2 (5), where

$\displaystyle
F(g,h^{\prime},g^{\prime}):=\int_{H}s(h,g)t(h^{\prime-1}hg,g^{\prime})u(g,h^{-1}h^{\prime},g^{\prime})\delta(h^{-1}h^{\prime})^{1/p_{1}^{\prime}}dh.$
(4.50)

We have

$\displaystyle\int_{H}\left(\int_{X}F(g,h^{\prime},g^{\prime})d\overline{g}\right)^{p}dh^{\prime}\leq\left(\int_{X}\left(\int_{H}F(g,h^{\prime},g^{\prime})^{p}dh^{\prime}\right)^{1/p}d\overline{g}\right)^{p}$
(4.51)

by the Minkowski integral inequality. Since $\delta|_{H}$ is the modular
function of $H$ by Lemma 4.1 (1), we have

$\displaystyle\left(\int_{H}F(g,h^{\prime},g^{\prime})^{p}dh^{\prime}\right)^{1/p}$
$\displaystyle\leq
Y(p_{1},p_{2};H)S(\overline{g})^{1/p_{1}}\left(\int_{H}(t(h^{-1}g,g^{\prime})u(g,h,g^{\prime}))^{p_{2}}dh\right)^{1/p_{2}}$
(4.52) $\displaystyle\leq
Y(p_{1},p_{2};H)S(\overline{g})^{1/p_{1}}T(\overline{g},\overline{g^{\prime}})^{1/p}U(\overline{g},\overline{g^{\prime}})^{1/p_{1}^{\prime}}$
(4.53)

by Example 4.4 (1). Thus, it follows from (4.51) that

$\displaystyle\int_{H}\left(\int_{X}F(g,h^{\prime},g^{\prime})d\overline{g}\right)^{p}dh^{\prime}\leq\left(Y(p_{1},p_{2};H)\int_{X}S(\overline{g})^{1/p_{1}}T(\overline{g},\overline{g^{\prime}})^{1/p}U(\overline{g},\overline{g^{\prime}})^{1/p_{1}^{\prime}}d\overline{g}\right)^{p}.$
(4.54)

Since

$\displaystyle\left(\int_{X}S(\overline{g})^{1/p_{1}}T(\overline{g},\overline{g^{\prime}})^{1/p}U(\overline{g},\overline{g^{\prime}})^{1/p_{1}^{\prime}}d\overline{g}\right)^{p}\leq\int_{X}S(\overline{g})T(\overline{g},\overline{g^{\prime}})d\overline{g}$
(4.55)

holds by $\|\phi_{1}\|_{p_{1}}=\|\phi_{2}\|_{p_{2}}=1$ and Example 4.4 (2), we
have

$\displaystyle\int_{H}\left(\int_{X}F(g,h^{\prime},g^{\prime})d\overline{g}\right)^{p}dh^{\prime}\leq
Y(p_{1},p_{2};H)^{p}\int_{X}S(\overline{g})T(\overline{g},\overline{g^{\prime}})d\overline{g}.$
(4.56)

Thus, it follows that

$\displaystyle\int_{X}\int_{H}\left(\int_{X}F(g,h^{\prime},g^{\prime})d\overline{g}\right)^{p}dh^{\prime}d\overline{g^{\prime}}\leq
Y(p_{1},p_{2};H)^{p}\int_{X}\int_{X}S(\overline{g})T(\overline{g},\overline{g^{\prime}})d\overline{g}d\overline{g^{\prime}}.$
(4.57)

Since

$\displaystyle\int_{X}\int_{X}S(\overline{g})T(\overline{g},\overline{g^{\prime}})d\overline{g}d\overline{g^{\prime}}=\int_{X}\int_{X}T(\overline{g},\overline{g^{\prime}})d\overline{g^{\prime}}S(\overline{g})d\overline{g}=\int_{X}S(\overline{g})d\overline{g}=1$
(4.58)

by $\|\phi_{1}\|_{p_{1}}=\|\phi_{2}\|_{p_{2}}=1$ and Example 4.2 (1) (2), we
obtain (4.49). ∎

##  5 Proof of Corollary 1.3

In this section, we show Corollary 1.3 by using Theorem 1.2 and the argument
of Jing–Tran–Zhang [JTZ23]. We write $\mathcal{A}$ for the set of the
connected Lie groups $G$ satisfying the assumption in Corollary 1.3 (i.e. the
center of the semisimple part of $G$ is a finite group). We note that any
connected solvable Lie group is an element of $\mathcal{A}$. Let $r(G)$ be the
dimension of the maximal compact subgroups for a connected Lie group $G$. We
show the following lemma to prove Corollary 1.3.

######  Lemma 5.1.

Suppose that a real number $d(G)$ is defined for each $G\in\mathcal{A}$, and

$\displaystyle d(G)\geq d(H)+d(G/H)$ (5.1)

holds for any connected closed normal subgroup $H\in\mathcal{A}$ of any
$G\in\mathcal{A}$ with $G/H\in\mathcal{A}$. For $G\in\mathcal{A}$, we denote
by $I(G)$ the inequality

$\displaystyle d(G)\geq d(\mathbb{R})\dim
G+(d(\mathbb{R}/\mathbb{Z})-d(\mathbb{R}))r(G).$ (5.2)

  1. (1)

Suppose that a connected closed normal subgroup $H\in\mathcal{A}$ of
$G\in\mathcal{A}$ satisfies $G/H\in\mathcal{A}$. If $I(H)$ and $I(G/H)$ hold,
then $I(G)$ also holds.

  2. (2)

Every non-trivial connected solvable Lie group $G$ satisfies $I(G)$.

  3. (3)

Furthermore, we assume that

$\displaystyle d(\mathbb{R}/\mathbb{Z})=0\leq d(H)\leq d(G)$ (5.3)

for any connected closed subgroup $H\in\mathcal{A}$ of any $G\in\mathcal{A}$.
Then every $G\in\mathcal{A}$ satisfies $I(G)$, that is,

$\displaystyle d(G)\geq d(\mathbb{R})(\dim G-r(G)).$ (5.4)

Jing–Tran–Zhang generalized the Brunn–Minkowski inequality to any Lie group by
essentially using Lemma 5.1 [JTZ23, Theorem 1.1]. Now we give some examples of
Lemma 5.1 (2).

######  Example 5.2.

  1. (1)

If the equality of (5.1) holds for any connected closed normal subgroup
$H\in\mathcal{A}$ of any $G\in\mathcal{A}$, then the equality of $I(G)$ also
holds for any connected solvable Lie group $G$. Actually, if $G$ is not
trivial, then this claim can be shown by replacing $d(G)$ with $-d(G)$ and
applying Lemma 5.1 (2). If $G$ is trivial, we have $d(G)=d(G)+d(G)$ because
the equality of (5.1) holds. Thus, we have $d(G)=r(G)=0$ and hence the
equality of $I(G)$ holds.

  2. (2)

Let $d(G):=\mathrm{rank}(\ker(\tilde{G}\to G))$ for connected Lie group $G$,
where $\tilde{G}$ is the universal covering of $G$. Now we show $d(G)=r(G)$
for any connected solvable Lie group $G$. The kernel $\ker(\tilde{G}\to G)$ is
isomorphic to the fundamental group $\pi_{1}(G)$ of $G$ [HN12, Theorem 9.5.4].
Since $\pi_{1}(H)\to\pi_{1}(G)$ is injective and $\pi_{1}(G)/\pi_{1}(H)$ is
isomorphic to $\pi_{1}(G/H)$ for any $G\in\mathcal{A}$ and any connected
closed normal subgroup $H\in\mathcal{A}$ of $G$ [HN12, Remark 11.1.17], we
have

$\displaystyle
d(G)=\mathrm{rank}(\pi_{1}(G))=\mathrm{rank}(\pi_{1}(H))+\mathrm{rank}(\pi_{1}(G/H))=d(H)+d(G/H).$
(5.5)

Thus, the equality of $I(G)$ holds for any connected solvable Lie group by
(1). In this case, we obtain

$\displaystyle d(G)=d(\mathbb{R})\dim
G+(d(\mathbb{R}/\mathbb{Z})-d(\mathbb{R}))r(G)=r(G)$ (5.6)

by $d(\mathbb{R})=0$ and $d(\mathbb{R}/\mathbb{Z})=1$.

Now we show the following lemma to prove Lemma 5.1.

######  Lemma 5.3.

Let $G$ be a connected Lie group.

  1. (1)

One has $r(G)=r(H)+r(G/H)$ for any closed normal subgroup $H\lhd G$.

  2. (2)

Every solvable Lie group $G$ with $\dim G\geq 2$ satisfies the following
condition (2)(i).

    1. (i)

There exists a closed normal subgroup $H\in\mathcal{A}$ of $G$ such that
$G/H\in\mathcal{A}$ and $1\leq\dim H<\dim G$.

  3. (3)

Suppose that $G\in\mathcal{A}$ and $\dim G\geq 2$. If $G$ does not satisfy the
condition (2)(i), then there exists a closed subgroup $H\in\mathcal{A}$ of $G$
with $\dim H<\dim G$ such that

$\displaystyle\dim H-r(H)=\dim G-r(G).$ (5.7)

###### Proof.

  1. (1)

Let $K\subset G$ be a maximal compact subgroup of $G$. Then $K\cap H$ and
$K/(K\cap H)$ are maximal compact subgroups of $H$ and $G/H$, respectively
[HN12, Theorem 14.3.13 (i) (a)]. Thus, we obtain

$\displaystyle r(G)=\dim K=\dim(K\cap H)+\dim(K/(K\cap H))=r(H)+r(G/H).$ (5.8)

  2. (2)

Since $G$ is a connected solvable Lie group, there exists a connected closed
solvable normal subgroup $H\lhd G$ such that $\dim(G/H)=1$. Then $1\leq\dim
H<\dim G$ holds by $\dim G\geq 2$, and we have

$\displaystyle\dim G=\dim H+\dim(G/H).$ (5.9)

Since $G/H$ is abelian by $\dim(G/H)=1$, we have $G/H\in\mathcal{A}$. Thus,
$G$ satisfies the condition (2)(i).

  3. (3)

Let $R\lhd G$ be the radical (the largest connected solvable closed normal
subgroup) of $G$. Since $\dim G\geq 2$ and $G$ does not satisfy the condition
(2)(i), $G$ is not a solvable Lie group by (2). Thus, we have $\dim R<\dim G$.
Since $G/R\in\mathcal{A}$ by $G\in\mathcal{A}$, the connected Lie group $G$ is
semisimple (i.e. $\dim R=0$) by the assumption that $G$ does not satisfy the
condition (2)(i). Let $G=KAN$ be the Iwasawa decomposition. Then the closed
subgroup $H:=AN\subset G$ is a simply connected solvable Lie group [Kna02,
Theorem 6.46]. Thus, we have $\dim H<\dim G$, $H\in\mathcal{A}$, and $r(H)=0$.
Since $K\subset G$ is a maximal compact subgroup of $G$ by $G\in\mathcal{A}$
[Kna02, Theorem 6.31 (g)], the equality

$\displaystyle\dim H-r(H)=\dim G-\dim K=\dim G-r(G)$ (5.10)

is obtained. ∎

###### Proof of Lemma 5.1.

  1. (1)

Since

$\displaystyle d(G)\geq d(\mathbb{R})(\dim
H+\dim(G/H))+(d(\mathbb{R}/\mathbb{Z})-d(\mathbb{R}))(r(H)+r(G/H))$ (5.11)

by $I(H)$, $I(G/H)$, and (5.1), we obtain $I(G)$ by (5.9) and Lemma 5.3 (1).

  2. (2)

We prove it by induction on $\dim G$. If $\dim G=1$, then either
$G=\mathbb{R}$ or $G=\mathbb{R}/\mathbb{Z}$ holds. We obtain $I(\mathbb{R})$
by $r(\mathbb{R})=0$, and $I(\mathbb{R}/\mathbb{Z})$ by
$r(\mathbb{R}/\mathbb{Z})=1$.

Now we show $I(G)$ when $\dim G\geq 2$. Then $G$ is a solvable Lie group and
hence $G$ satisfies the condition (2)(i) by Lemma 5.3 (2). Then $H$ and $G/H$
are connected solvable Lie groups, we have $I(H)$ and $I(G/H)$ by the
induction hypothesis. Thus, we also have $I(G)$ by (1).

  3. (3)

We prove it by induction on $\dim G$. If $\dim G=0$, then one has $r(G)=0$ and
hence

$\displaystyle d(G)\geq 0=d(\mathbb{R})(\dim G-r(G)).$ (5.12)

If $\dim G=1$, then $G$ is a solvable Lie group and hence $I(G)$ follows from
(2).

Now we show $I(G)$ when $\dim G\geq 2$. If $G$ satisfies the condition (2)(i),
then $I(G)$ follows from (1). Thus, it suffices to show $I(G)$ when $G$ does
not satisfy the condition (2)(i). By Lemma 5.3 (3), there exists a closed
subgroup $H\in\mathcal{A}$ of $G$ such that $\dim H<\dim G$ and (5.7). We have
$I(H)$ by the induction hypothesis and hence

$\displaystyle d(G)\geq d(H)\geq d(\mathbb{R})(\dim H-r(H))$ (5.13)

by (5.3) and (5.4). Thus, we obtain $I(G)$ by (5.7). ∎

###### Proof of Corollary 1.3.

Let $d(G):=-\ln(Y(p_{1},p_{2};G))$ for locally compact group $G$. Then

$\displaystyle d(G)$ $\displaystyle:=-\ln(Y(p_{1},p_{2};G))$ (5.14)
$\displaystyle\geq-\ln(Y(p_{1},p_{2};H)Y(p_{1},p_{2};G/H))$ (5.15)
$\displaystyle=-\ln(Y(p_{1},p_{2};H))-\ln(Y(p_{1},p_{2};G/H))$ (5.16)
$\displaystyle=d(H)+d(G/H)$ (5.17)

holds for any connected closed normal subgroup $H\lhd G$ and hence we have
(5.1). In addition,

$\displaystyle d(G):=-\ln(Y(p_{1},p_{2};G))\geq-\ln(Y(p_{1},p_{2};H))=d(H)$
(5.18)

holds for any closed subgroup $H\subset G$ by Theorem 1.2. We have

$\displaystyle d(H):=-\ln(Y(p_{1},p_{2};H))\geq 0$ (5.19)

by Example 2.1 (1), and the equality holds for $H:=\mathbb{R}/\mathbb{Z}$ by
Corollary 2.2. Since (5.3) follows from (5.18) and (5.19), we have (5.4) for
$G\in\mathcal{A}$ by Lemma 5.1 (3). Thus, the inequality

$\displaystyle Y(p_{1},p_{2};G)=e^{-d(G)}\leq e^{-d(\mathbb{R})(\dim
G-r(G))}=Y(p_{1},p_{2};\mathbb{R})^{\dim G-r(G)}$ (5.20)

is obtained. ∎

## Acknowledgement

This work was supported by JSPS KAKENHI Grant Number JP19J22628 and Leading
Graduate Course for Frontiers of Mathematical Sciences and Physics (FMSP). The
author would like to thank his advisor Toshiyuki Kobayashi for his support.
The author is also grateful to Yuichiro Tanaka, Toshihisa Kubo, and the
anonymous referees for their careful comments.

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Takashi Satomi: Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan.

E-mail<EMAIL_ADDRESS>