# Toward an Automated HPC Pipeline for Processing Large Scale Electron
Microscopy Data

Rafael Vescovi1*, Hanyu Li2*, Jeffery Kinnison3, Murat Keçeli4,
Misha Salim1, Narayanan Kasthuri2, Thomas D. Uram1 and Nicola Ferrier5
1Leadership Computing Facility, Argonne National Laboratory, Lemont, IL, USA
<EMAIL_ADDRESS>2Department of Neurobiology, University of
Chicago, Chicago, IL, USA
<EMAIL_ADDRESS>3Dept. of Computer Science and
Engineering, University of Notre Dame, Notre Dame, IN, USA
<EMAIL_ADDRESS>4Computational Science Division, Argonne National Laboratory,
Lemont, IL, USA
<EMAIL_ADDRESS>5Mathematics and Computer Science Division, Argonne National
Laboratory, Lemont, IL, USA
<EMAIL_ADDRESS>
###### Abstract

We present a fully modular and scalable software pipeline for processing
electron microscope (EM) images of brain slices into 3D visualization of
individual neurons and demonstrate an end-to-end segmentation of a large EM
volume using a supercomputer. Our pipeline scales multiple packages used by
the EM community with minimal changes to the original source codes. We tested
each step of the pipeline individually, on a workstation, a cluster, and a
supercomputer. Furthermore, we can compose workflows from these operations
using a Balsam database that can be triggered during the data acquisition or
with the use of different front ends and control the granularity of the
pipeline execution. We describe the implementation of our pipeline and
modifications required to integrate and scale up existing codes. The modular
nature of our environment enables diverse research groups to contribute to the
pipeline without disrupting the workflow, i.e. new individual codes can be
easily integrated for each step on the pipeline.

###### Index Terms:

Connectome, Workflow, Automation, Supercomputing

111These authors contributed equally to this work.

##  1 Introduction

Microscopy images are a significant source of insight and raw information for
neuroscience. Modern techniques in electron microscopy (EM) allow scientists
the ability to image at such high resolution that every single synaptic
connection can be distinguished [1, 2, 3]. Furthermore, acquisition automation
has enabled us to acquire large volumes of microscopy data spanning several
resolutions with minimal human involvement in the acquisition.

The popularization of these automated imaging systems has made acquiring large
amounts of data the standard operation for many laboratories [4] [5], and
although most of them are able to physically handle the amount of data, there
is an increasing need for streamlining the pipeline. This necessity arises
because of the growing acquisition speed of microscopes, leading to an
exponential growth in data throughput[6]. While different techniques are
emerging to solve each step of the upstream process, they still have their own
independent development communities[7, 8, 9] and there are very few
laboratories with the capability of carrying out the entire process by
themselves. Individually these algorithms contribute to the study of
neuroscience image data; it is, however, non-trivial to chain these modules
together and deploy them in one coherent environment for end-to-end
connectomics projects.

The continuous use of electron microscopes can produce single datasets that
reach multiple petabytes of data, which cannot be processed on local
workstations or small clusters and therefore require High Performance
Computing (HPC) facilities. We propose to deploy and chain these different
processing libraries into a single microscope to HPC workflow and provide a
way for the user to interact with the data and its processing in real time.
Our package is implemented in Python and is called HAPPYNeurons (HPC Automated
Pipeline for Processing Yotta Neurons) [10].

An alternative solution is the use of cloud-based services (e.g. AWS, Google
Cloud etc.), but the data size and computation time make the cost infeasible
for most laboratories. On the other hand, HPC facilities at national labs have
vast storage and computing capabilities that have not been tapped in the field
of connectomics due to accessibility or differences from established
technology stacks. Our pipeline aims to address this limitation by
incorporating state-of-the-art open source connectomics tools into an HPC
environment and building an end-to-end pipeline for EM segmentation, while
increasing accessibility to users and labs.

On a technical level, we propose to use MPI [11] as a parallelization layer
for each step in order to keep the internal mechanics of the original software
mostly intact, while achieving compatibility with most HPC infrastructure.
This allows for rapid deployment of new tools on the pipeline. It is not the
goal of our work to improve the sample preparation or acquisition, but instead
to enable the user to take advantage of large scale computing facilities in
order to process the data. This processing is currently done in several parts
that will be described in greater detail in the next chapter. Our
contributions to the field of neuroscience and HPC are: 1) deployment on HPC
of EM tools necessary to go from raw images to final reconstruction; 2)
wrapping the tools in an operation database that can be used to create custom
pipelines, 3) deployment of an computational environment that permits the user
to interact with, annotate, and visualize the data without the need to
transfer outside of the HPC facility.

##  2 Related work

The field of connectomics has been blooming with different communities trying
to understand the underlying connectivity map of neural tissue. In the
particular case of electron microscopy there have been parallel contributions
into sample preparation, data acquisition and the diverse steps on the complex
data processing involved. The sheer complexity of the problem means every step
of it can still be improved and ongoing efforts on different aspects of the
problem can be seen from various labs and groups in the community [12, 13,
14].

For alignment, computer vision methods [7, 8, 9] are still heavily used to
assemble the raw data into 3D volumes, but are slowly being surpassed by
machine learning-driven methods [15]. Segmentation has long been the rate-
limiting step and still requires a lot of human annotation. Originally it
could take weeks to trace a single neuron through a stack of images. With the
aid of deep learning algorithms, segmentation can be vastly accelerated.
Efforts on neural networks such as U-Net [16] and Flood-Filling Network (FFN)
[17] have proven successful for the task of automatic segmentation of neurons.

Human-intensive data annotation is crucial to establish datasets for machine
learning approaches. The web-based package webKnossos [18] enables
laboratories to deploy an intuitive interface to annotate datasets, without
requiring annotators to transfer up to petascale datasets between their
institution and the hosting site. We leverage webKnossos in our work to make
the increasingly large connectomics datasets available for annotation by
distant annotators.

Upon completion of reconstruction, datasets are meshed for visualization and
made available via Neuroglancer[19], a program developed by Google that
visualizes flat, black-and-white electron images, related labels, and
reconstructions as a colourful 3D forest of neurons.

##  3 Computational Pipeline

Electron microscope image processing follows a certain number of pre-defined
steps from raw data to a final scientific result. Given the data throughput of
modern microscopes, executing all those steps within the same facility as the
microscope is a challenge. Figure 1 describes the connection between the
electron microscope lab at Argonne National Laboratory and the Argonne
Leadership Computing Facility (ALCF), showing the services involved in our
pipeline environment. After the microscope finishes acquiring an image (or set
of images), it triggers an action that is stored on an external database
server. This database of actions then controls the transfer and processing of
data through the storage and computing resources. On the front end side the
user can either manipulate the data or the actions to make the pipeline unique
to each sample.

Figure 1: Model of the flow of data between the electron microscope facility
(top side) and the HPC facility (lower side). The microscope acquisition
populates the action database which controls the storage and computing
resources. The user can visualize and manipulate the data and actions through
web-based front ends.

For simplicity we will describe the basic steps involved in processing
electron microscope images individually. Even though these steps are described
in a sequential fashion, in practice the processing is conducted through an
iterative process involving automated computational steps and human-intensive
guidance.

###  3.1 Processing methods for EM data

In order to make the the interaction with the data as easy as possible for the
user, we encoded basic operations that can eventually be described as
sequential pipelines. Since human validation is necessary at multiple steps,
the user can choose where and when to interact with the pipeline. These
operations are described below:

Montage is the process of positioning and merging overlapping image tiles into
a single larger image. Given sufficient metadata about the arrangement of the
tiles, this step can be executed largely without user intervention. We
implemented a headless macro for TrakEM2 [8] and developed a Python wrapper
for MPI parallelization of this procedure. The mutual independence of image
sections makes it possible to trigger the montage operation during
acquisition, once the full set of images for a tile have been acquired, and
process it on the fly.

Alignment is the process of ensuring that neighboring images in the stack are
aligned according to their contained features, and is a crucial step for
serial electron microscopy. In our pipeline, we use AlignTK[9] to perform
elastic alignment on the montaged image stack. We implemented wrappers for
AlignTK’s core functionality to better adapt to parallel deployment on HPC,
along with a set of utility tools for image preprocessing, including contrast
normalization, scaling, and artifact thresholding. Figure 2 shows examples of
montage and alignment images.

(a) Montage
Example

(b) Alignment example

Figure 2: Example of the montage process (left) and alignment process (right).

Segmentation is the process to assign unique IDs to individual neurite objects
in a 3D volume and is one of the key challenges in connectomics analysis. The
current state of the art, FFN, has achieved great success in accuracy and
scale [17, 20, 14] over the last few years. Although it was originally
designed for distributed computing platforms and despite preliminary efforts
on distributed training[21], it has not previously been deployed on HPC
infrastructure for large scale segmentation. In this pipeline, we made
modifications to the open source release of Google’s FFN. First, we added MPI-
based parallelization for execution at large-scale HPC facilities. Second, we
added support for reading precomputed volume[22] data as input, in addition to
HDF5 which reduces repetitive data usage and seamlessly integrates with the
visualization engine Neuroglancer. Third, we implemented a reconciliation step
that merges overlapping subvolume inference results into a final segmentation
in precomputed format.

Mask Prediction: In practice, a prerequisite for FFN is identifying tissue
masks that would disrupt segmentation. These masks are used to omit imaging
artifacts or large objects that can make the final segmentation less accurate,
like cell-bodies and blood vessels. For cell-bodies and blood vessels, we
implemented and ran a classic 2D U-Net[23]. We created manual annotations on
every 100 images at 4x resolution and used these to train a U-net model, which
was used for patch-wise inference over the full volume. Manual seeds were
placed at the center of each cell-body, and a 3D watershed algorithm was run
to provide initial segmentation of cell bodies and blood vessels. Figure 3
shows the process of creating the cell mask from the U-Net probabilities (on
the left) to the actual cell body masks (on the right).

(a) U-Net cell body probabilities

(b) U-Net Watershed Overlay

Figure 3: Result of the U-Net segmentation of large body sizes. These can be
subsequently used to mask-out already known objects in the final segmentation.

Mesh Generation produces a mesh-based representation to support 3D
visualization of the segmented objects. Currently this step is achieved using
the Python library Igneous[24].

Skeletonization creates a point graph for every object and can also be
processed by using the TEASAR[25] implementation inside Igneous.

Manual Annotation: We used WebKnossos[18] to provide manual volumetric
annotations as a training/validation set for FFN. This step is human-
intensive, and is typically approached iteratively, with the biologist
annotating an initial sample partially, rerunning training and inference with
FFN, and making further corrections. To accommodate data format requirements
of different packages and to streamline this process, we implemented utilities
for easy transformation of data formats between WebKnossos cube, which is used
by WebKnossos, stacks of tiff images, and HDF5[26], which are traditional data
formats, and precomputed[22], which is used by CloudVolume and Neuroglancer.

User inspection of intermediate results between pipeline stages is currently
essential. To this end, we have developed a number of facilities to enable
interaction with the datasets that reside on ALCF systems. In the case of
connectomics datasets, the intermediate results can be very large, typically
involving many images in the range of hundreds of megapixels each; the ability
to view these results quickly, in-place on ALCF systems, is critical. For each
intermediate dataset, we have developed code to produce downsampled versions
of select output data, and a Jupyter notebook template which can be copied
into the target run directory to view the downsampled data. The JupyterHub
deployment at ALCF has direct access to the Theta and Cooley filesystems,
creating a highly usable environment for viewing, which can be customized with
additional Python-driven analyses.

While segmentation results can be viewed using Jupyter notebooks as above,
these results are more typically visualized in 3D using the Neuroglancer
application. We have deployed Neuroglancer at ALCF to support viewing
segmentation results, which we demonstrate later in the text. Whereas Jupyter
notebooks access Theta-resident data directly, Neuroglancer retrieves data
using web protocols; to support this, we transfer segmentation results to the
3PB Petrel community storage system[27] at ALCF, and expose the data to
Neuroglancer using standard web protocols.

###  3.2 Workflow Management

We have designed a pipeline that encompasses the individual software packages
described above, allowing them to be executed independently or assembled into
more automated workflows. These software packages are leveraged in an
iterative fashion, varying parameters to achieve desired accuracy on each
particular dataset, and software development is ongoing. To account for this
scenario, our pipeline is modular, supports multiple interfaces, and aims to
enable the entire data life cycle from raw images to final results. Figure 4
shows the electron microscope operations described above with the I/O present
on our package.

Figure 4: Schematic of the pipeline. The green box represents the data
acquisition. Orange boxes and arrows represent human interactions in the
pipeline. The white boxes represent HPC submitted jobs. Blue boxes represent
the data storage and visualization server. Green arrows represent I/O from the
computing resources.

Given the complex nature of the acquired data, we acknowledge that human
intervention is required during the process; our goal is to facilitate this
while minimizing the iteration time and making interaction with the data
easier. This is important given that any imperfection or artifact in the
sample and data acquisition can cause the steps of the pipeline to fail and
require human intervention.

The pipeline is implemented as a Python library, with an API that exposes the
individual applications to be run, together with specifications of input data
and configuration, producing a collection of jobs that will be offloaded to
our HPC facilities. This functionality is generalized such that choice of
execution machine can be made at the time of execution rather than being
tightly integrated into the job definitions, which permit users to encapsulate
the jobs on their own schedulers depending on the computational
infrastructure. On our HPC resources, we achieve further flexibility by
relying on the Balsam workflow toolkit [28]. Using Balsam’s Python programming
interface, we populate a database with the desired pipeline actions and, using
the specifics of the target computing resource, define a collection of jobs to
be run. Balsam manages the execution of these jobs on the target computing
resource, optimizing for concurrency and throughput, handling errors, and
providing monitoring and reporting details as the pipeline jobs are executed.
This level of control frees the user from laborious management of the compute
jobs and enables Balsam to systematically manage execution and data management
ensuring that the compute resources are used efficiently.

Pipeline users interact with Balsam via two interfaces: a Python API to define
steps in the workflow and a command line interface for allocating resources
and launching applications at the appropriate scale. The mapping of parallel
tasks to MPI ranks varies across applications: rank/section for montage with
TraekEM2, rank/section-pair for alignment with AlignTK, and rank/subvolume for
segmentation using FFN. This interface is mirrored in the Python/Balsam
interface. Input data and configuration details are provided via Python calls,
and passed to Balsam, which handles defining jobs in the underlying job
database.

Once the job database has been populated , one can use the Balsam command-line
interface to submit jobs to the Theta queues for execution and to monitor the
jobs as they run.

##  4 Results

In our experiments we used two HPC resources: Theta, an 11.69 PFLOPs
supercomputer and Cooley, a GPU cluster with Nvidia K80s, both at the ALCF.
Theta is composed of 4392 compute nodes, each with a 64-core, 1.3-GHz Intel
Xeon Phi 7230 processor, 192GB DDR4 RAM and 16GB high-bandwidth MCDRAM. When
needed, we also used a workstation with dual Xeon E5 2630v4, 256GB memory and
two Titan X Pascal GPUs. Our usage of FFN relied on TensorFlow 1.14.

The microscope used on our experiments was the Zeiss SEM Gemini 300 [29],
which can provide images at up to 6nm of spatial resolution. The acquisition
automation is done by the Atlas software from Zeiss.

The sample tissue was dissected from a 14 day old, post-natal mouse brain in
primary visual cortex layer 4, and was prepared according to the protocol
described by Hua et al. [30]. The tissue was then cut into 40 nm slices on
ATUM[31] and scanned with the microscope at 6 nm resolution and 3.5 µs pixel
dwell time. For each of the 1312 slices, two 10833 x 14000 pixel tiles were
scanned in sequence with 5% overlap.

###  4.1 Workflow Validation

As an initial validation step, two stages of the pipeline–montage and
alignment–were run on a subset of the data, using the Balsam execution
backend. For montage, 128 sections of data were selected, with a corresponding
Balsam job describing the input and configuration for TrakEM2. These jobs ran
on 32 Theta nodes, with Balsam managing the distribution of work to compute
nodes as they became available. We demonstrate this approach in the current
case because this flexible approach to computing will become essential in the
context of larger image stacks in the future. A similar approach was taken to
run image alignment on this 128-image stack. For alignment, the database was
populated with jobs to run on 16 nodes of the Cooley visualization cluster,
with jobs distributed to compute nodes as they became available.

To highlight user interactivity with the pipeline, we provide a Jupyter
notebook example where, given a raw dataset, the pipeline stages described
above are submitted to a Balsam database with standard (or user-provided)
configurations for execution on Theta. This approach permits the user to run
collections of jobs multiple times, such as to perform parameter sweeps. We
created Balsam jobs for the TrakEM2 montage step with multiple configurations,
varying the maximum and minimum octave used in the search for the correct
overlap between two images, which also affects the run time, and calculated
the error rate of those configurations on a given dataset (as shown on Table
I). An initial analysis of the resulting images was conducted to identify
montage failures, using the image size as a proxy. The accumulated error was
calculated by the number of images that were corrected by changing the
parameters. The final accumulated error shows the fraction of images that
could not achieve a correct montage with any of the tested parameter sets;
these, therefore, must be corrected through direct user intervention. We
continue to develop metrics for identifying montage errors, in the interest of
further automating this process.

TrakEM2 - Min | TrakEM2 - Max | RunTime | Error Rate | Accumulated Error
---|---|---|---|---
400 | 2000 | 100min | 35% | 35%
400 | 3000 | 260min | 15% | 10%
400 | 3500 | 450min | 9% | 6%
1000 | 3500 | 520min | 6% | 1%

TABLE I: Execution times of TrakEM2 headless montage macro on a test dataset
(6x2 tiles of 15000x15000 pixels and 1128 slices divided into 8 different
folders that represent the acquisition sessions). Each line shows the results
of 8 Balsam jobs with 32 nodes, 4 ranks per node and the values for the
TrakEM2 parameters for the minimum and maximum octaves used by the montage
macro script.

Figure 5: Final visualization using Neuroglancer. On the left, an example of a
raw image (black and white) overlapped with the inference labels(colors). On
the right, 3D rendering of the same cells.

Lastly, we simulated online processing of images from the electron microscope
by triggering a transfer of images on a schedule that approximates typical
operation. For the dataset described in this work, each section was imaged as
two separate tiles, each 8-bit tile having image dimensions 10833x14000 and
occupying 151MB. The imaging time for each tile is on the order of 10 seconds,
so a full section is imaged every 20 seconds. In this simulation, we
transferred a full section from the microscope-connected machine to Theta
every 20 seconds and added a montage job to the Balsam database, continuously,
over a period of three hours. In the current paradigm, at this rate, a wafer
of 200 sections would be imaged in about one hour, producing 30GB; this
equates to a daily rate of 720GB. It is clear from this experiment that Theta
is able to keep pace with the incoming jobs at this rate (each TrakEM2 job was
run on a single Theta node, using 64 cores per node, and 2 threads per core,
as we determined this to be the optimal configuration, with runtime averaging
440 seconds). To achieve this, we began with an initial allocation of Theta
nodes, with the Balsam executor configured to grow and shrink the pool of
nodes as needed, corresponding with the flow and ebb of incoming jobs. This
demonstration is, in itself, not a compelling demonstration of the full extent
of the current capability; it does, however, show that we have the technology
in place to trigger image transfer and job injection when a section has been
imaged, which will become a necessity in the future, where we anticipate
transfer of images from multiple microscopes simultaneously to process on ALCF
supercomputers. We are currently undertaking a scaling study to examine bounds
on throughput in this scenario, which will be the focus of a future
publication.

###  4.2 Complete Pipeline

We demonstrate the pipeline being executed from raw tiles to the final
reconstruction on a 90 x 125 x 52 $\mu m$ volume of neural tissue. Each part
of the pipeline was executed as a standalone call from a bash script using our
Python wrappers or, where appropriate, by calling applications directly.

After montage and alignment, the data size was 15000x20800x1312 voxels at
6x6x40 $nm^{3}$ resolution, in 8 bit grayscale, with a total size of 324 GB.
Segmentation was carried out at 2x lower resolution to reduce merge errors and
to increase speed. To perform training, we acquired an FFN model trained on
the Kasthuri11 [3] dataset from the authors of [17] as an initial checkpoint.
Using manual annotations of a 256x256x128 voxel volume from our own dataset,
we incrementally trained the base model until accuracy saturated at 0.91; this
training was run on a separate workstation with dual Titan X GPUs for 12
hours, as a transfer learning job that didn’t require Theta-scale computing.
Before proceeding to inference, we first performed cell-body and vessel
masking with U-Net/watershed on the workstation at 4x downsampled resolution,
and used that as an initial segmentation. We then split a total volume of
6700x9900x1312 voxels (at 12x12x40 $nm^{3}$ resolution) into 3618 512x512x128
cubes with 32x32x16 overlap in each dimension. Inference jobs with the trained
model were run on 32 nodes of the Cooley cluster each with 2 NVIDIA K80 GPUs,
with one MPI rank per GPU, for a total of 72 hours; afterwards the subvolumes
were reconciliated (recombined into a full volume) on a workstation for final
visualization and error checking. Figure 5 shows the visualization of the
reconstructed data in Neuroglancer. Our pipeline have been tested on datasets
as large as 1Tb and it can be scaled to larger volumes as long as the resource
allocation allows it. The limitations come from data sharing between different
resources and the final user. Another limitation comes from the increase in
the error rate of individual algorithms in larger volumes.

##  5 Closing Remarks

In summary, HAPPYNeurons provides a software pipeline to integrate electron
microscopes with HPC facilities. We demonstrate an end-to-end connectomics
reconstruction pipeline using HPC resources. This is achieved by wrapping
multiple libraries as a coherent set of operations and providing the ability
to chain them together using Balsam to enable more optimized scheduling on
supercomputers. Due to the modular design of the workflow, the wrapped
applications can be combined according to the needs of the current application
and dataset, and new modules can be added with ease. HPC facilities can be
used in a seamless manner, enabling the processing of large scale data without
the monetary burdens of cloud computing. This integration paves the way for
using supercomputers for connectomics reconstruction, in preparation for the
deluge of data anticipated from faster next-generation microscopes, and to
enable exascale computers to process it. This modular design also allows for
different processes of the pipeline to target different accelerators on the
HPC resources (i.e. dedicated GPU’s). We are currently studying the execution
of our pipeline at larger scale on Argonne supercomputers; these results will
appear in a future publication.

### Acknowledgements

We thank all members of the Kasthuri Laboratory at University of Chicago for
providing the insights into Electron Microscopy and biology. We thank Dr.
Shuichi Shigeno for data used in the preliminary tests. We also thank Michal
Januszewski for sharing a pre-trained FFN model and insights into FFN’s hyper-
parameters. This research is funded in part by, and used resources of, the
Argonne Leadership Computing Facility which is a DOE Office of Science User
Facility supported under Contract DE-AC02-06CH11357 and funded in part by NIH:
5R01MH110932-04.

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# Global and local maximizers for some Fourier extension estimates on the
sphere

Valentina Ciccone and Mateus Sousa  Hausdorff Center for Mathematics,
Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS>bonn.de BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14,
48009 Bilbao, Bizkaia, Spain<EMAIL_ADDRESS>
###### Abstract.

In this note we improve, for the case of low dimensions, the known range of
exponents for which constant functions are the unique maximizers for the
$L^{2}(\mathbb{S}^{d-1})$ to $L^{p}_{rad}L^{2}_{ang}(\mathbb{R}^{d})$ mixed-
norm Fourier extension estimate on the sphere. Moreover, we show that in the
same range of exponents for which constant functions are the unique maximizers
for the $L^{2}(\mathbb{S}^{d-1})$ to $L^{p}_{red}L^{2}_{ang}(\mathbb{R}^{d})$
mixed-norm Fourier extension estimates they are also local maximizers for the
$L^{p}(\mathbb{S}^{d-1})$ to $L^{p}(\mathbb{R}^{d})$ Fourier extension
estimates. As a by-product, we obtain that for the cases of dimensions $d=2,3$
constant functions are local maximizers for all $p\geq p_{\mathrm{st}}(d)$,
where $p_{\mathrm{st}}$ denotes the Stein-Tomas endpoint,
$p_{\mathrm{st}}(d):=2(d+1)/(d-1)$.

###### Key words and phrases:

Sphere, Fourier restriction, sharp inequalities, extremizers, Bessel functions

###### 2010 Mathematics Subject Classification:

42B10

##  1\. Introduction

Let $d\geq 2$ be an integer, $J_{\nu}$ denote the Bessel function of the first
kind of order $\nu$, and $k$ be a non-negative integer. It follows by the
asymptotic behaviour of Bessel functions that the weighted $L^{p}$ norms

$\Lambda_{d,p}(k):=\bigg{(}\int_{0}^{\infty}|J_{\tfrac{d}{2}-1+k}(r)r^{-\tfrac{d}{2}+1}|^{p}r^{d-1}\text{\rm
d}r\bigg{)}^{1/p}~{},$ $\Lambda_{d,\infty}(k):=\sup_{r\geq
0}|J_{\tfrac{d}{2}-1+k}(r)r^{d-1}|~{},$

are bounded whenever $\tfrac{2d}{d-1}<p\leq\infty$.

The problem of determining for which $k$ such weighted norms are maximized,
which is a problem of independent interest in the theory of special functions,
has been studied in [COeSS19] in connection with certain mixed-norm sharp
Fourier extension problems. In particular, in [COeSS19] the authors have
studied the problem of computing

$\sup_{k\geq 0}\;\Lambda_{d,p}(k).$ (P1)

The properties of Bessel functions (see e.g. [St00]) guarantee that such
supremum is a maximum. It has been shown in [COeSS19] that such maximum is
achieved at $k=0$ (and only at $k=0$) whenever $p$ is an even exponent and
that the set of exponents for which the maximum is achieved at $k=0$ is open
and it contains a neighborhood of infinity $(p_{0}(d),\infty]$, providing some
upper-bounds for $p_{0}(d)$. In particular, they obtained the following upper-
bounds in low dimensions:

$p_{0}(2)\leq 6.76,\quad p_{0}(3)\leq 5.45,\quad p_{0}(4)\leq 5.53,\quad
p_{0}(5)\leq 6.07,\quad p_{0}(6)\leq 6.82,$ $p_{0}(7)\leq 7.70,\quad
p_{0}(8)\leq 8.69,\quad p_{0}(9)\leq 9.78,\quad p_{0}(10)\leq 10.95,$

and, more in general, they showed that

$\displaystyle p_{0}(d)\leq(\tfrac{1}{2}+o(1))d\log d.$ (1)

Problem (P1) is related to several problems in sharp Fourier restriction
theory.

The Fourier restriction problem for the sphere asks for which pairs of
exponents $(p,q)$ the inequality

$\displaystyle\|\widehat{f\sigma}\|_{L^{p}(\mathbb{R}^{d})}\leq
C_{d,p,q}\|f\|_{L^{q}(\mathbb{S}^{d-1})}$ (2)

holds. Here $\sigma=\sigma_{d-1}$ is the surface measure on $\mathbb{S}^{d-1}$
and $\widehat{f\sigma}$ is the Fourier transform of the measure $f\sigma$,

$\widehat{f\sigma}(x)=\int_{\mathbb{S}^{d-1}}e^{-ix\cdot\xi}f(\xi)d\sigma(\xi).$

The Fourier restriction problem has been fully solved only in dimension $d=2$
and for the case $q=2$ for which a complete answer is given by the Stein-Tomas
inequality. A mixed-norm version of the problem has been studied in [Ve92,
Ve88] showing that the mixed-norm Fourier extension inequality

$||\widehat{f\sigma}||_{L^{p}_{rad}L^{2}_{ang}(\mathbb{R}^{d})}\leq
C_{d,p}||f||_{L^{2}(\mathbb{S}^{d-1})}$ (3)

holds when $\frac{2d}{d-1}<p$, where

$\displaystyle||\widehat{f\sigma}||_{L^{p}_{rad}L^{2}_{ang}(\mathbb{R}^{d})}=\left(\int_{0}^{\infty}\left(\int_{\mathbb{S}^{d-1}}|\widehat{f\sigma}(r\omega)|^{2}d\sigma(\omega)\right)^{p/2}r^{d-1}\text{\rm
d}r\right)^{1/p}.$

The problem of determining the sharpest constant for (3) has been studied in
[COeSS19]. Namely, in [COeSS19] the authors have studied the problem of
computing

$\sup_{f\in L^{2}(\mathbb{S}^{d-1}),\;f\neq
0}\Phi^{\rm{mn}}_{p,d}(f),\quad\Phi^{\rm{mn}}_{p,d}(f):=\frac{||\widehat{f\sigma}||_{L^{p}_{rad}L^{2}_{ang}(\mathbb{R}^{d})}}{||f||_{L^{2}(\mathbb{S}^{d-1})}}.$
(P2)

It was observed in [COeSS19] that the studying of such problem can be
restricted to functions $f$ which are spherical harmonics. In other words

$\sup_{f\in L^{2}(\mathbb{S}^{d-1}),\,f\neq
0}\Phi_{d,p}^{\rm{mn}}(f)=\sup_{Y_{k},\,Y_{k}\neq
0}\Phi_{p,d}^{\rm{mn}}(Y_{k}),$

where $Y_{k}$ denotes a spherical harmonics of degree $k$. Due to the identity

$\displaystyle\widehat{Y_{k}\sigma}(x)=(2\pi)^{\frac{d}{2}}i^{-k}|x|^{-\frac{d}{2}+1}J_{\frac{d}{2}-1+k}(|x|)Y_{k}\big{(}\frac{x}{|x|}\big{)}$
(4)

we have that

$\|\widehat{Y_{k}\sigma}\|_{L^{p}_{rad}L^{2}_{ang}(\mathbb{R}^{d})}=(2\pi)^{\tfrac{d}{2}}\bigg{(}\int_{0}^{\infty}|J_{\tfrac{d}{2}-1+k}(r)r^{-\tfrac{d}{2}+1}|^{p}r^{d-1}\text{\rm
d}r\bigg{)}^{1/p}\|Y_{k}\|_{L^{2}(\mathbb{S}^{d-1})}.$

Hence, the problem of establishing the sharpest constant for (3), namely (P2),
is equivalent to the problem of determining for which non-negative integer $k$
the maximum in (P1) is achieved.

Problem (P1) has been addressed in [COeSS19] by relating the integrals
$\Lambda_{d,p}(k)$’s to integration on spheres using delta-calculus. Our
approach, on the other hand, relies on some sharper estimates (with an
improved constant) between weighted norms of Bessel functions inspired by
those obtained in [CG22, Lemma 2] for the case of dimension $d=2$, see the
forthcoming inequality (6).

Our first result lowers, for the case of low dimensions, the upper bounds for
$p_{0}(d)$ established in [COeSS19], hence extending the ranges of exponents
for which the maximum in (P1) is achieved when $k=0$. We use the notation
$p_{\rm st}(d)$ to denote the Stein-Tomas endpoint exponent in dimension $d$,
$p_{\rm st}(d):=\frac{2(d+1)}{(d-1)}$.

######  Theorem 1.

It holds that

$p_{0}(2)<6,\quad p_{0}(3)<4,\quad p_{0}(4)<3.48,\quad p_{0}(5)<3.50,$
$p_{0}(6)<3.58,\quad p_{0}(7)<3.7,\quad p_{0}(8)<3.86,\quad
p_{0}(9)<4.06,\quad p_{0}(10)<4.46.$

A direct consequence of this result is that $p_{0}(d)<p_{\rm st}(d)$ when
$d=2,3$, which is of interest because constant functions are natural
candidates to be extremizers for the full range of exponents of the
Stein–Tomas Fourier extension inequality. This has been verified only when
$p\geq 4$ is an even number and $d\in\\{3,4,5,6,7\\}$ (see [COeS15, Fo15,
OeSQ21]), but it is open for all other cases. If this were true, then by
Hölder inequality, constant functions would be also maximizers for
$\Phi_{d,p}^{\rm mn}$ when $p\geq p_{\rm{st}}(d)$.

As mentioned above Problems (P1) and (P2) are equivalent. Next, we observe
that the same holds true if one considers the problem of finding extremizers
among functions of the form $aY_{k}\in L^{2}(\mathbb{S}^{d-1})$, with
$a\in\mathbb{C}$ and $Y_{k}$ a spherical harmonic of degree $k$, for
$L^{p}(\mathbb{S}^{d-1})$ to $L^{p}(\mathbb{R}^{d})$ Fourier extension
estimates. Namely, if one considers the problem of computing

$\sup_{Y_{k},\,Y_{k}\neq
0}\Phi_{p,d}(Y_{k}),\quad\Phi_{p,d}(f):=\frac{||\widehat{f\sigma}||_{L^{p}(\mathbb{R}^{d})}}{||f||_{L^{p}(\mathbb{S}^{d-1})}}~{}.$
(P3)

In fact,

$\|\widehat{Y_{k}\sigma}\|_{L^{p}(\mathbb{R}^{d})}=(2\pi)^{\tfrac{d}{2}}\bigg{(}\int_{0}^{\infty}|J_{\tfrac{d}{2}-1+k}(r)r^{-\tfrac{d}{2}+1}|^{p}r^{d-1}\text{\rm
d}r\bigg{)}^{1/p}\|Y_{k}\|_{L^{p}(\mathbb{S}^{d-1})}~{},$

and, therefore,

$\sup_{Y_{k},\,Y_{k}\neq 0}\Phi_{p,d}(Y_{k})=\sup_{f\in
L^{2}(\mathbb{S}^{d-1}),\,f\neq
0}\Phi_{d,p}^{\rm{mn}}(f)=(2\pi)^{\tfrac{d}{2}}\sup_{k\geq
0}\Lambda_{d,p}(k).$

The following corollary is an immediate consequence of the above
considerations.

######  Corollary 2.

For all $p\in(p_{0}(d),\infty]$ we have that

$\displaystyle\sup_{f\in L^{2}(\mathbb{S}^{d-1}),\,f\neq
0}\Phi_{p,d}^{\rm{mn}}$
$\displaystyle(f)\leq\Phi_{p,d}^{\rm{mn}}(\mathbf{1})~{},$
$\displaystyle\sup_{Y_{k},\,Y_{k}\neq 0}\Phi_{p,d}$
$\displaystyle(Y_{k})\leq\Phi_{p,d}(Y_{0})~{}.$

That is, for all such $p$’s, constant functions are maximizers for (P2) and
(P3).

Note that the fact that constant functions are extremizers for (P3) is a
necessary condition for this to be the case also for the more general problem
of computing

$\sup_{f\in L^{p}(\mathbb{S}^{d-1}),\,f\neq 0}\Phi_{p,d}(f).$ (P4)

Extremizers for (P4) are known only when $p$ is an even admissible exponent,
in which case it has been shown in [COeS15] that constant functions are
maximizers, and when $p=\infty$ in which case the same conclusion holds
[FS22]. Except for these cases, even the question of the existence of global
extremizers for Problem (P4) is open [FS22]. Due to symmetry, constant
functions would be natural candidate to be extremizers. Also, it was noted in
[CQ14] that constant functions are always solutions to the corresponding
Euler–Lagrange equations for any admissible pair of exponents $(p,q)$ for the
Fourier extension inequality (2), so, in particular, for any admissible pair
$(p,p)$.

An intermediate step toward a solution of Problem (P4) is to understand the
behavior of local extremizers. Local extremizers have been studied before for
the case of the endpoint Stein-Tomas inequalities in [CFOeST15], [CS12], and
[GN22] showing, respectively, that constant functions are local maximizers for
such inequalities when $d=2$, when $d=3$, and when $2\leq d\leq 60$. The
question that we would like to address here is whether constant functions are
local maximizers also for the case of the $L^{p}(\mathbb{S}^{d-1})$ to
$L^{p}(\mathbb{R}^{d})$ Fourier extension inequalities, namely for (P4).

Our second main result is the following.

######  Theorem 3.

Let $d\geq 2$ and $p>\tfrac{2d}{d-1}$. Assume that the
$L^{p}(\mathbb{S}^{d-1})$ to $L^{p}(\mathbb{R}^{d})$ Fourier extension
inequality holds and that the maximum in (P1) is achieved at $k=0$. Then there
exists $\delta>0$ such that whenever
$\|f-\mathbf{1}\|_{L^{p}(\mathbb{S}^{d-1})}<\delta$,

$\Phi_{p,d}(f)\leq\Phi_{p,d}(\mathbf{1}).$ (5)

That is, constant functions are local maximizers for (P4).

As an immediate consequence we have that constant functions are local
maximizers for the case of the $L^{p}(\mathbb{S}^{d-1})$ to
$L^{p}(\mathbb{R}^{d})$ Fourier extension inequalities whenever $p\geq
p_{0}(d)$ where an upper bound on $p_{0}(d)$ is provided by Theorem 1 for the
cases of dimensions $2\leq d\leq 10$, and, in general, by (1) for greater
dimensions.

The proof of Theorem 3 is contained in Section 4, the proof of Theorem 1 is
the content of Section 3, while some auxiliary results about hierarchies
between weighted norms of Bessel functions are presented in Section 2.

The topic of sharp spherical restriction has received much attention over the
last years, in particular for the case of inequalities in the Stein-Tomas
range [FVV11, CS12, Fo15, COeS15, FLS16, Sh16, CFOeST15, OeSTZK19, BTZK20,
OeSQ21, CG22, Be23]. We refer to the survey [NOeST23] for an up-to-date
description of the state of the art.

##  2\. Hierarchies between weighted norms of Bessel functions

It is known that when $p\in 2\mathbb{N}$, $p>\frac{2d}{d-1}$, or when
$p=\infty$ then

$\frac{\Lambda_{d,p}(k)}{\Lambda_{d,p}(0)}<1$

for all positive integers $k$, see [COeSS19]. In this section, we are
interested in obtaining sharper estimates for such ratio, at least for certain
values of the exponent $p$.

In this direction, for the case of dimension $d=2$ and exponent $p=6$ it has
been shown in [CG22] that

$\displaystyle\Lambda_{2,6}^{6}(k)<\frac{1}{3}\Lambda_{2,6}^{6}(0)$ (6)

for all $k\geq 1$.

Moreover, in [COeSS19] combining the identity

$\Lambda_{d,\infty}(0)=\frac{1}{2^{\frac{d}{2}-1}\Gamma(\frac{d}{2})},$ (7)

with a decreasing upper-bound (with respect to the order $k$) for
$\Lambda_{d,p}(k)$, it has been shown that

$\frac{\Lambda_{d,\infty}(k)}{\Lambda_{d,\infty}(0)}\leq\bigg{(}{\rm{L}}^{6}\frac{2^{3d-6}\Gamma(\frac{d}{2})}{d^{3d-4}}\bigg{)}^{\frac{1}{3d+2}}$

for all $k\geq 1$, where the constant ${\rm{L}}$ is defined as

${\rm{L}}:=\sup_{\nu>0,\,r>0}|r^{1/3}J_{\nu}(r)|=0.785746...$ (8)

and it has been found by Landau [La00].

Our first result of this section establishes a hierarchy between the
$\Lambda_{d,\infty}(k)$’s, hence determining the sharpest upper-bound on the
ratio $\frac{\Lambda_{d,\infty}(k)}{\Lambda_{d,\infty}(0)}$.

######  Proposition 4.

For all positive integers $k$ it holds that

$\Lambda_{d,\infty}(k-1)>\Lambda_{d,\infty}(k).$

In particular,

$\Lambda_{d,\infty}(k)\leq{\rm C}_{\infty}(d)\Lambda_{d,\infty}(0)$

for all positive integers $k$, where ${\rm
C}_{\infty}(d):=\frac{\Lambda_{d,\infty}(1)}{\Lambda_{d,\infty}(0)}$, and
equality is attained if and only if $k=1$.

###### Proof.

We begin with the case $d=2$. In such case $\Lambda_{d,\infty}(k)=\sup_{r\geq
0}|J_{k}(r)|$. It has been shown in [La00] that $\sup_{r>0}|J_{k}(r)|$ is a
strictly decreasing function of $k$. In particular, if we denote by
$j^{\prime}_{k,1}$ the first positive zero of $J_{k}^{\prime}$ with $k$ a
positive real number, then

$\displaystyle\sup_{r>0}|J_{k}(r)|=J_{k}(j_{k,1}^{\prime}),$

and therefore

$\displaystyle\sup_{r>0}|J_{k}(r)|=J_{k}(j_{k,1}^{\prime})>\sup_{r>0}|J_{k+1}(r)|=J_{k}(j_{k+1,1}^{\prime}).$

As $\sup_{r\geq 0}|J_{0}(k)|=J_{0}(0)=1>J_{1}(j_{1,1}^{\prime})$ the claim in
the statement is verified for the case $d=2$.

We turn to the case of $d\geq 3$. In these cases
$\Lambda_{d,\infty}(k)=\sup_{r\geq
0}|r^{-\tfrac{d}{2}+1}J_{\tfrac{d}{2}-1+k}|$. We start by observing that as
$r^{-\tfrac{d}{2}+1}$ is a strictly decreasing function of $r$ and
$\sup_{r>0}|J_{\nu}(r)|=J_{\nu}(j_{\nu,1}^{\prime})$ it holds that

$\displaystyle\sup_{r>0}|r^{-\tfrac{d}{2}+1}J_{\tfrac{d}{2}-1+k}(r)|=\sup_{0<r<j_{d/2-1+k,1}^{\prime}}|r^{-\tfrac{d}{2}+1}J_{\tfrac{d}{2}-1+k}(r)|.$

Hence to conclude it would be enough to show that

$\displaystyle J_{\nu}(r)>J_{\nu+1}(r)\qquad\text{for all
}r\in(0,j_{\nu,1}^{\prime}).$ (9)

We recall that $j_{\nu,1}^{\prime}<j_{\nu+1,1}^{\prime}$, see e.g. [Wa66]. In
particular, $J_{\nu},\;J_{\nu+1},\,J_{\nu}^{\prime},\,J_{\nu+1}^{\prime}$ are
strictly positive in $(0,j^{\prime}_{\nu,1})$, and
$J_{\nu}(j_{\nu,1}^{\prime})>J_{\nu+1}(j_{\nu,1}^{\prime})$. Hence to prove
(9) it suffices to show that there exist no
$\overline{r}\in(0,j_{\nu,1}^{\prime})$ such that
$J_{\nu}(\overline{r})=J_{\nu+1}(\overline{r})$. We argue by contradiction.
Consider the recursive relations for Bessel functions

$\displaystyle\tfrac{2\nu}{r}J_{\nu}(r)=J_{\nu-1}(r)+J_{\nu+1}(r),$ (10)
$\displaystyle 2J^{\prime}_{\nu}(r)=J_{\nu-1}(r)-J_{\nu+1}(r).$ (11)

By taking the sum of (10) and (11) we obtain the identity

$J_{\nu}^{\prime}(r)=J_{\nu-1}(r)-\tfrac{(\nu)}{r}J_{\nu}(r),$

and shifting $\nu\mapsto\nu+1$ we obtain

$J_{\nu+1}^{\prime}(r)=J_{\nu}(r)-\tfrac{(\nu+1)}{r}J_{\nu+1}(r).$

Assume there exist $\overline{r}\in(0,j_{\nu,1}^{\prime})$ such that
$J_{\nu}(\overline{r})=J_{\nu+1}(\overline{r})$. Evaluating the last display
at $\overline{r}$ we get

$J_{\nu+1}^{\prime}(\overline{r})=(1-\tfrac{(\nu+1)}{\overline{r}})J_{\nu+1}(\overline{r})$

and, as $J_{\nu+1}^{\prime}$ and $J_{\nu+1}$ are strictly positive on
$(0,j_{\nu,1}^{\prime})$, we have that necessarily

$\displaystyle\overline{r}>\nu+1.$ (12)

Next, we take the difference between (11) and (10) obtaining the identity

$J_{\nu}(r)^{\prime}=\tfrac{\nu}{r}J_{\nu}(r)-J_{\nu+1}(r).$

Evaluating it at $\overline{r}$ we get that

$J_{\nu}^{\prime}(\overline{r})=(\tfrac{\nu}{\overline{r}}-1)J_{\nu}(\overline{r})$

and, as both $J_{\nu}^{\prime}$ and $J_{\nu}$ are strictly positive on
$(0,j_{\nu,1}^{\prime})$, we have that necessarily

$\nu>\overline{r}.$

Comparing this with (12) yields the contradiction. ∎

The values of $\Lambda_{d,\infty}(1)$ can be computed by using Mathematica.
For the case of $2\leq d\leq 10$ one obtains, with 6 significant figures
(s.f.),

$\displaystyle\begin{split}\Lambda_{2,\infty}(1)=0.581865,\quad\Lambda_{3,\infty}(1)&=0.348023,\quad\Lambda_{4,\infty}(1)=0.179963,\\\
\Lambda_{5,\infty}(1)=0.0830013,\quad\Lambda_{6,\infty}(1)&=0.0348492,\quad\Lambda_{7,\infty}(1)=0.0135129,\\\
\Lambda_{8,\infty}(1)=0.00489072,\quad\Lambda_{9,\infty}(1)&=0.00166575,\quad\Lambda_{10,\infty}(1)=0.000537364.\end{split}$
(13)

By combining them with (7) one can obtain a numerical evaluation for ${\rm
C}_{\infty}(d)$.

Our second observation is for the case of exponent $p=4$ and dimensions $3\leq
d\leq 10$.

######  Proposition 5.

Let $3\leq d\leq 10$. Then

$\Lambda_{d,4}(k)\leq{\rm C}_{4}(d)\,\Lambda_{d,4}(0)$

holds for all positive integers $k$, where ${\rm
C}_{4}(d):=\tfrac{\Lambda_{d,4}(1)}{\Lambda_{d,4}(0)}.$ Equality is attained
if and only if $k=1$.

To prove Proposition 5 we need the following upper-bound; see also [GN22].

######  Lemma 6.

Let $d\geq 2$. In the range of exponents
$\tfrac{6d-2}{3d-4}<p<\tfrac{12d+4}{3d-4}$ it holds that

$\Lambda_{d,p}^{p}(k)\leq{\rm
L}^{p-2}\frac{\Gamma(\lambda)\Gamma(\tfrac{d}{2}-1+k+\frac{1-\lambda}{2})}{2^{\lambda}\Gamma(\tfrac{1+\lambda}{2})^{2}\Gamma(\tfrac{d}{2}-1+k+\tfrac{1+\lambda}{2})}$

for all positive integers $k$, where
$\lambda=p(\tfrac{d}{2}-\tfrac{2}{3})-d+\tfrac{1}{3}$.

To establish the upper-bound in the lemma we rely on the following identity
which can be found in [GR14, Equation 6.574-2]

$\displaystyle\int_{0}^{\infty}J_{\nu}^{2}(r)r^{-\lambda}\text{\rm
d}r=\frac{\Gamma(\lambda)\Gamma(\nu+\tfrac{(1-\lambda)}{2})}{2^{\lambda}\Gamma\big{(}\tfrac{1+\lambda}{2}\big{)}^{2}\Gamma\big{(}\nu+\tfrac{1+\lambda}{2}\big{)}}$
(14)

for $0<\lambda<2\nu+1$.

###### Proof.

We use (8) to obtain the upper-bound

$\Lambda_{d,p}(k)^{p}\leq{\rm
L}^{p-2}\int_{0}^{\infty}J_{\tfrac{d}{2}-1+k}^{2}(r)r^{-p\big{(}\tfrac{d}{2}-\tfrac{2}{3}\big{)}+d-\tfrac{1}{3}}\text{\rm
d}r.$

By applying identity (14) to the right hand side of the last display we
further obtain

$\Lambda_{d,p}^{p}(k)\leq{\rm
L}^{p-2}\frac{\Gamma(\lambda)\Gamma(\tfrac{d}{2}-1+k+\frac{1-\lambda}{2})}{2^{\lambda}\Gamma(\tfrac{1+\lambda}{2})^{2}\Gamma(\tfrac{d}{2}-1+k+\tfrac{1+\lambda}{2})}$

where $\lambda=p(\tfrac{d}{2}-\tfrac{2}{3})-d+\tfrac{1}{3}$. Such upper-bound
holds whenever $0<\lambda<2\big{(}\tfrac{d}{2}-1+k\big{)}+1.$ In particular,
for a fixed dimension $d\geq 2$ the upper-bound holds for all positive
integers $k$ whenever $\tfrac{6d-2}{3d-4}<p<\tfrac{12d+4}{3d-4}$. ∎

Note that both the case of $p=4$ and the case of Stein-Tomas endpoint $p_{\rm
st}(d)$ are included in the range of exponents covered by Lemma 6. Also, note
that, for a fixed exponent $p$ and a fixed dimension $d$, the above upper
bound is a decreasing function of $k$. Throughout, we use the notation
$U_{d,p}(k)$ to denote the upper bound for $\Lambda_{d,p}^{p}(k)$ in Lemma 6.

Proof of Proposition 5. We compare the upper bound $U_{d,4}(k)$ for
$\Lambda_{d,4}^{4}(k)$ established in Lemma 6 with a (lower) estimate for
$\Lambda_{d,4}^{4}(1)$. To this end, we rely on Mathematica to evaluate the
integrals

$\displaystyle\int_{0}^{40}|J_{\frac{d}{2}}(r)r^{-\tfrac{d}{2}+1}|^{4}r^{d-1}\text{\rm
d}r$ (15)

for $3\leq d\leq 10$ obtaining, respectively, the following values (with 6
s.f.)

$\displaystyle 0.144681\quad 0.0337263$ $\displaystyle\quad 0.00661348\quad
0.00107217$ $\displaystyle 0.000146318\quad 0.0000171549$ $\displaystyle\quad
1.75867\times 10^{-6}\quad 1.59953\times 10^{-7}.$

By comparison, one can see that $U_{d,4}(k)<\Lambda_{d,4}^{4}(1)$ for all
integers $k\geq 2$ when $d\in\\{5,6,7,8,9\\}$, for all integers $k\geq 3$ when
$d\in\\{4,10\\}$, and for all integers $k\geq 5$ when $d=3$. We check the
remaining cases separately. We rely on Mathematica to evaluate the integrals

$\int_{0}^{200}|J_{\tfrac{d}{2}-1+k}(r)r^{-\tfrac{d}{2}+1}|^{4}r^{d-1}\text{\rm
d}r$

for the cases of interest obtaining, for the case of $d=3$ and $k=2,3,4$, the
values (6 s.f.)

$0.0992828\quad 0.0757045\quad 0.0615859,$

respectively, and for the cases $d=4$ and $k=2$, and $d=10$ and $k=2$, the
values (6 s.f.)

$0.0172602\quad 4.00184\times 10^{-8},$

respectively. Then, we use the estimate

$\displaystyle|J_{\nu}(r)|\leq r^{-1/2}$ (16)

which holds for all $\nu\geq\tfrac{1}{2}$ and $r\geq\tfrac{3}{2}\nu$ (see
[COeSS19, Lemma 8] and [Kr14, Theorem 3]) to upper bound the tails obtaining
that

$\int_{200}^{\infty}|J_{\tfrac{d}{2}-1+k}(r)r^{-\tfrac{d}{2}+1}|^{4}r^{d-1}\text{\rm
d}r\leq\frac{200^{-d+2}}{d-2}.$

Hence, by comparison, we see that also for these cases it holds that
$\Lambda_{d,4}^{4}(k)<\Lambda_{d,4}^{4}(1)$. As it is known from [COeS15,
COeSS19] that $\Lambda_{d,4}^{4}(1)<\Lambda_{d,4}^{4}(0)$ when $d\geq 3$, the
result in the statement follows. ∎

To evaluate ${\rm C}_{4}(d)$ one can rely on the identity

$\int_{0}^{\infty}|J_{\nu}(r)|^{4}r^{-2\nu+1}\text{\rm
d}r=\frac{\Gamma(\nu)\Gamma(2\nu)}{2\pi\Gamma(\nu+\tfrac{1}{2})^{2}\Gamma(3\nu)},$

which can be found, for example, in [COeSS19, Lemma 7] (see also [GR14,
Equation 6.5793-3]) and which provides an explicit expression for
$\Lambda_{d,4}^{4}(0)$, together with a numerical estimates for
$\Lambda_{d,4}^{4}(1)$.

Our last result of this section is for the case of the Stein-Tomas endpoint,
$p_{\rm st}=p_{\rm{st}}(d)$.

######  Proposition 7.

Let $4\leq d\leq 10$. Then the following inequality holds for all positive
integers $k$

$\Lambda_{d,p_{\rm{st}}}(k)<{\rm C}_{p_{\rm
st}}(d)\;\Lambda_{d,p_{\rm{st}}}(0),$

where ${\rm C}_{p_{\rm st}}(d):=\tfrac{\Lambda_{d,p_{\rm
st}}(1)}{\Lambda_{d,p_{\rm st}}(0)}$. Equality is attained if and only if
$k=1$.

###### Proof.

We compare the upper bound $U_{d,p_{\rm st}}(k)$ for $\Lambda_{d,p_{\rm
st}}^{p_{\rm st}}(k)$ established in Lemma 6 with a (lower) estimate for
$\Lambda_{d,p_{\rm st}}^{p_{\rm st}}(1)$. To this end, we rely on Mathematica
to evaluate the integrals

$\displaystyle\int_{0}^{50}|J_{\frac{d}{2}}(r)r^{-\tfrac{d}{2}+1}|^{p_{\rm
st}}r^{d-1}\text{\rm d}r$ (17)

for $4\leq d\leq 10$ obtaining, respectively, the following values (with 6
s.f.)

$0.143391\quad 0.131693\quad 0.118941\quad 0.10719\quad 0.0969753\quad
0.088279\quad 0.0807943.$

By comparison, one can see that $U_{d,p_{\rm st}}(k)<\Lambda_{d,p_{\rm
st}}^{p_{\rm st}}(1)$ for all integers $k\geq 3$ when
$d\in\\{5,6,7,8,9,10\\}$, and for all integers $k\geq 4$ when $d=4$. We check
the remaining cases separately. We use Mathematica to evaluate the integral

$\int_{0}^{200}|J_{\tfrac{d}{2}-1+k}(r)r^{-\tfrac{d}{2}+1}|^{p_{\rm
st}}r^{d-1}\text{\rm d}r$

obtaining for the cases $k=2$ and $5\leq d\leq 10$ the values (6 s.f.)

$0.0998066\quad 0.0938562\quad 0.0875322\quad 0.0814907\quad 0.075952\quad
0.0709569$

and for the cases $d=4$ and $k=2,3$ the values (6 s.f.)

$0.103492\quad 0.080522.$

We use the estimate (16) to upper bound the tail of the integrals obtaining

$\int_{200}^{\infty}|J_{\tfrac{d}{2}-1+k}(r)r^{-\tfrac{d}{2}+1}|^{p_{\rm
st}}r^{d-1}\text{\rm d}r\leq\frac{1}{200}.$

Hence, by comparison, it follows that $\Lambda_{4,p_{\rm st}}^{p_{\rm
st}}(2)<\Lambda_{4,p_{\rm st}}^{p_{\rm st}}(1)$.

We are left to show that $\Lambda_{d,p_{\rm st}}^{p_{\rm
st}}(1)<\Lambda_{d,p_{\rm st}}^{p_{\rm st}}(0)$ whenever $4\leq d\leq 10$. The
cases of $d=4,5$ have already be verified in [COeSS19]. To verify the
remaining cases $6\leq d\leq 10$ we compare the bound for $\Lambda_{d,p_{\rm
st}}^{p_{\rm st}}(1)$ obtained by combining the numerical evaluation of the
truncated integral (17) and an upper bound for the tail obtained using (16)
with a (lower) estimate for $\Lambda_{d,p_{\rm st}}^{p_{\rm st}}(0)$. To this
end we numerically evaluate the integral

$\int_{0}^{50}|J_{\frac{d}{2}+1}(r)r^{-\tfrac{d}{2}+1}|^{p_{\rm
st}}r^{d-1}\text{\rm d}r$

for $6\leq d\leq 10$ obtaining the values (6 s.f.)

$0.173201\quad 0.147926\quad 0.1286\quad 0.113331\quad 0.101086.$

By comparison, we see that $\Lambda_{d,p_{\rm st}}^{p_{\rm
st}}(1)<\Lambda_{d,p_{\rm st}}^{p_{\rm st}}(0)$ also for $6\leq d\leq 10$
hence concluding the proof. ∎

##  3\. Proof of Theorem 1

###  3.1. Case $d=2$

We combine the estimate (6) from [CG22] and the estimate in Proposition 4
(here for the case $d=2$) with the interpolation strategy utilized in
[COeSS19]. Let $p\geq 6$ and $k$ be a positive integer. It follows from Hölder
inequality that

$\Lambda_{2,p}(k)\leq\Lambda_{2,6}(k)^{6/p}\Lambda_{2,\infty}(k)^{1-6/p}.$

Using (6) and the sharp estimate from Proposition 4 we further obtain that

$\Lambda_{2,p}(k)\leq\frac{1}{3^{1/p}}\Lambda_{2,6}(0)^{6/p}\Lambda_{2,\infty}(1)^{1-6/p}.$

We need the following lower bound on $\Lambda_{d,p}(0)$ which has been
established in [COeSS19, Equation 4.8]

$\displaystyle\Lambda_{d,p}(0)>\frac{(2^{d-1}(\tfrac{d}{2})^{d/2})^{1/p}}{2^{d/2-1}\Gamma(\tfrac{d}{2})}\bigg{(}\frac{\Gamma(p+1)\Gamma(\tfrac{d}{2})}{\Gamma(p+\tfrac{d}{2}+1)}\bigg{)}^{1/p}.$
(18)

Then, we rely on standard numerical evaluation to determine for which $p\geq
6$ it holds that

$\frac{1}{3^{1/p}}\Lambda_{2,6}(0)^{6/p}\Lambda_{2,\infty}(1)^{1-6/p}\leq
2^{1/p}\frac{\Gamma(p+1)}{\Gamma(p+2)}.$

We obtain that such inequality is satisfied for all $p\geq 6$. Hence,
$p_{0}(2)<6$ as claimed.

###  3.2. Case $d\geq 3$

We proceed in two steps. First, we combine the estimates in Proposition 5 and
Proposition 4 with the interpolation strategy utilized in [COeSS19]. This will
establish the upper bound on $p_{0}(d)$ in the statement of Theorem 1 for the
cases of $d=3,9,10$. Second, we use the estimates in Proposition 5 and
Proposition 7 and interpolation to establish the upper bound on $p_{0}(d)$ for
the cases of $d=4,5,6,7,8$.

#### Step 1

Let $p\geq 4$ and $k$ be a positive integer. It follows from Hölder inequality
that

$\Lambda_{d,p}(k)\leq\Lambda_{d,4}(k)^{4/p}\Lambda_{d,\infty}(k)^{1-4/p}.$

Using the sharp estimate from Proposition 4 and Proposition 5 we further
obtain that

$\Lambda_{d,p}(k)\leq\Lambda_{d,4}(1)^{4/p}\Lambda_{d,\infty}(1)^{1-4/p}.$

Then, we compare the right-hand side of the last display with the lower bound
for $\Lambda_{d,p}(0)$ in equation (18) to determine for which $p\geq 4$ the
following inequality is satisfied

$\Lambda_{d,4}(1)^{4/p}\Lambda_{d,\infty}(1)^{1-4/p}\leq\frac{(2^{d-1}(\tfrac{d}{2})^{d/2})^{1/p}}{2^{d/2-1}\Gamma(\tfrac{d}{2})}\bigg{(}\frac{\Gamma(p+1)\Gamma(\tfrac{d}{2})}{\Gamma(p+\tfrac{d}{2}+1)}\bigg{)}^{1/p}.$

We use the numerical values for $\Lambda_{d,\infty}(1)$ in (13) and the bound
for $\Lambda_{d,4}(1)$ obtained by combining the numerical evaluation for the
truncated integral in (15) with an upper bound for the tail obtained using
(16). Via a standard numerical evaluation, we obtain that
$\Lambda_{d,p}(k)<\Lambda_{d,p}(0)$ for all $p\geq 4$ for the cases of
dimensions $d=3,4,5,6,7,8$, for all $p\geq 4.06$ for the case of $d=9$, and
for all $p\geq 4.46$ for the case of $d=10$.

#### Step 2

Let $4\leq d\leq 8$, $p_{\rm st}(d)\leq p\leq 4$ and $k$ be a positive
integer. It follows from Hölder inequality that

$\Lambda_{d,p}(k)\leq\Lambda_{d,p_{\rm
st}}(k)^{(1-\theta)}\Lambda_{d,4}(k)^{\theta},$

with $\theta:=\tfrac{4}{p}\tfrac{(p-p_{\rm st})}{(4-p_{\rm st})}$. Using the
estimates of Proposition 5 and Proposition 7 we further obtain that

$\Lambda_{d,p}(k)\leq\Lambda_{d,p_{\rm
st}}(1)^{(1-\theta)}\Lambda_{d,4}(1)^{\theta}.$

As before, we bound $\Lambda_{d,4}(1)$ by combining the numerical evaluation
for the truncated integral in (15) with an upper bound for the tail obtained
using (16) and we proceed analogously for $\Lambda_{d,p_{\rm st}}(1)$. Then,
we compare this upper bound with the lower bound for $\Lambda_{d,p}(0)$ in
equation (18) to determine, for a fixed $4\leq d\leq 8$ , for which $p_{\rm
st}(d)\leq p\leq 4$ the former is greater than the latter. We obtain that
$\Lambda_{d,p}(k)<\Lambda_{d,p}(0)$ for all $p\geq p(d)$ with

$p(4)=3.48,\quad p(5)=3.5,\quad p(6)=3.58\quad,p(7)=3.7,\quad p(8)=3.86.$

∎

##  4\. Proof of Theorem 3

Consider the deficit functional

$\displaystyle\zeta_{p}[f]=\Phi_{p,d}(\mathbf{1})^{p}\|f\|_{L^{p}(\mathbb{S}^{d-1})}^{p}-\|\widehat{f\sigma}\|_{L^{p}(\mathbb{R}^{d})}^{p}.$

Inequality (5) is equivalent to

$\zeta_{p}[f]\geq 0,$ (19)

therefore it is enough to prove that there is a $\delta>0$ such that
$\zeta_{p}[f]>0$ when $\|f-\mathbf{1}\|_{L^{p}(\mathbb{S}^{d-1})}<\delta$ and
$f$ is not constant, which we proceed to do.

We recall that here $p>2$ and we are assuming that the Fourier extension
operator is bounded from $L^{p}(\mathbb{S}^{d-1})$ to $L^{p}(\mathbb{R}^{d})$.
We compute

$\displaystyle\begin{split}\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)+\varepsilon\widehat{g\sigma}(x)|^{p}\text{\rm
d}x=&\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)|^{p}\text{\rm
d}x+p\varepsilon\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)|^{p-2}\mathfrak{R}(\widehat{\mathbf{1}\sigma}(x){\widehat{g\sigma}(x)})\text{\rm
d}x\\\
&+\frac{p(p-2)\varepsilon^{2}}{4}\mathfrak{R}\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)|^{p-4}(\widehat{\mathbf{1}\sigma}(x){\widehat{g\sigma}(x)})^{2}\text{\rm
d}x\\\
&+\frac{p\varepsilon^{2}}{4}\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)|^{p-2}|\widehat{g\sigma}(x)|^{2}\text{\rm
d}x\\\ &+o(\varepsilon^{2}).\end{split}$ (20)

and

$\displaystyle\begin{split}\int_{\mathbb{S}^{d-1}}|{\mathbf{1}}+\varepsilon{g(x)}|^{p}{\mathrm{d}}\sigma(x)=&\|{\mathbf{1}}\|_{L^{p}(\mathbb{S}^{d-1})}^{p}+p\varepsilon\int_{\mathbb{S}^{d-1}}\mathfrak{R}(g(x)){\mathrm{d}}\sigma(x)\\\
&+\frac{p(p-2)\varepsilon^{2}}{4}\mathfrak{R}\int_{\mathbb{S}^{d-1}}{g(x)}^{2}{\mathrm{d}}\sigma(x)\\\
&+\frac{p\varepsilon^{2}}{4}\int_{\mathbb{S}^{d-1}}|{g(x)}|^{2}{\mathrm{d}}\sigma(x)+o(\varepsilon^{2}).\end{split}$
(21)

We take $f$ to be of the form $f=\mathbf{1}+\varepsilon g$, with
$0<\varepsilon\leq\delta$ and $\|g\|_{L^{p}(\mathbb{R}^{d})}=1$. By applying
(20) and (21) one has

$\displaystyle\begin{split}\zeta_{p}[f]=&\zeta_{p}[\mathbf{1}+\varepsilon
g]=\,p\varepsilon\,\left(\Phi_{p,d}(\mathbf{1})^{p}\int_{\mathbb{S}^{d-1}}\mathfrak{R}(g(x)){\mathrm{d}}\sigma(x)-\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)|^{p-2}\mathfrak{R}(\widehat{\mathbf{1}\sigma}(x){\widehat{g\sigma}(x)})\text{\rm
d}x\right)\\\
&+\frac{\varepsilon^{2}}{4}p(p-2)\left(\Phi_{p,d}(\mathbf{1})^{p}\mathfrak{R}\int_{\mathbb{S}^{d-1}}{g(x)}^{2}{\mathrm{d}}\sigma(x)-\mathfrak{R}\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)|^{p-2}(\widehat{g\sigma}(x))^{2}\text{\rm
d}x\right)\\\
&+\frac{\varepsilon^{2}}{4}p\left(\Phi_{p,d}(\mathbf{1})^{p}\int_{\mathbb{S}^{d-1}}|{g(x)}|^{2}{\mathrm{d}}\sigma(x)-\int_{\mathbb{R}^{d}}|\widehat{\mathbf{1}\sigma}(x)|^{p-2}|\widehat{g\sigma}(x)|^{2}\text{\rm
d}x\right)+o(\varepsilon^{2}).\end{split}$ (22)

Furthermore, due to the aforementioned observation that $\mathbf{1}$ is a
critical point of $\Phi_{p,d}$, the first order terms in $\varepsilon$ of (22)
all vanish. To deal with the second order terms, we use the fact that
$L^{p}(\mathbb{S}^{d-1})\subset L^{2}(\mathbb{S}^{d-1})$ since $p>2$ in order
to expand $g$ in spherical harmonics. For that purpose we choose for each $k$
an orthonormal basis $\\{Y_{j,k}\\}_{j}$ of $\mathcal{H}_{k}^{d}$ where each
$Y_{j,k}$ is a real-valued spherical harmonic of degree $k$. Then

$\displaystyle\begin{split}&g=\sum_{k,j}a_{j,k}Y_{j,k}.\end{split}$

By combining identity (4) with the observation that the first order terms
vanish at (22) we can integrate in polar coordinates to obtain

$\displaystyle\begin{split}\zeta_{p}[f]=&\frac{\varepsilon^{2}}{4}p(p-2)\bigg{(}\Phi_{p,d}(\mathbf{1})^{p}\sum_{k,j}\mathfrak{R}(a_{j,k})^{2}\\\
&-(2\pi)^{pd/2}\sum_{k,j}(-1)^{k}\mathfrak{R}(a_{j,k})^{2}\int_{0}^{\infty}|J_{\frac{d}{2}-1}(r)|^{p-2}|J_{\frac{d}{2}-1+k}(r)|^{2}r^{d-1-p(1-\frac{d}{2})}{\mathrm{d}}{r}\bigg{)}\\\
&+\frac{\varepsilon^{2}}{4}p\sum_{k,j}|a_{j,k}|^{2}\bigg{(}\Phi_{p,d}(\mathbf{1})^{p}-(2\pi)^{pd/2}\int_{0}^{\infty}|J_{\frac{d}{2}-1}(r)|^{p-2}|J_{\frac{d}{2}-1+k}(r)|^{2}r^{d-1-p(1-\frac{d}{2})}{\mathrm{d}}{r}\bigg{)}\\\
&+o(\varepsilon^{2}).\end{split}$ (23)

Lastly, using Hölder inequality and the fact that by hypothesis
$\Lambda_{d,p}(k)<\Lambda_{d,p}(0)$ for all positive integers $k$ we observe
that

$\displaystyle\int_{0}^{\infty}|$ $\displaystyle
J_{\frac{d}{2}-1}(r)|^{p-2}|J_{\frac{d}{2}-1+k}(r)|^{2}r^{d-1-p(1-\frac{d}{2})}{\mathrm{d}}{r}$
$\displaystyle<\bigg{(}\int_{0}^{\infty}|J_{\tfrac{d}{2}-1}(r)|^{p}r^{d-1-p(1-\frac{d}{2})}{\mathrm{d}}{r}\bigg{)}^{(p-2)/p}\bigg{(}\int_{0}^{\infty}|J_{\tfrac{d}{2}-1+k}(r)|^{p}r^{d-1-p(1-\frac{d}{2})}{\mathrm{d}}{r}\bigg{)}^{2/p}$
$\displaystyle<\int_{0}^{\infty}|J_{\tfrac{d}{2}-1}(r)|^{p}r^{d-1-p(1-\frac{d}{2})}{\mathrm{d}}{r}=(2\pi)^{-pd/2}\Phi_{p,d}(\mathbf{1})^{p},$

hence concluding the proof of Theorem 3.

## Acknowledgements

This project was partially carried out during V. Ciccone’s research visits to
BCAM - Basque Center for Applied Mathematics and to the University of the
Basque Country UPV/EHU. Her visits have been partially supported by the
Hausdorff Center for Mathematics in Bonn through the Global Math Exchange
Program and BIGS, and by the project
PID2021-122156NB-I00/AEI/10.13039/501100011033 funded by Agencia Estatal de
Investigación and acronym “HAMIP” and grant IT1615-2 of the Basque Government.
She thanks BCAM - Basque Center for Applied Mathematics, the University of the
Basque Country UPV/EHU, and her hosts, Odysseas Bakas and Ioannis Parisssis,
for their warm hospitality.

M. Sousa is supported by the grant Juan de la Cierva Incorporación
IJC2019-039753-I, the Basque Government through the BERC 2022-2025 program, by
the Spanish State Research Agency project
PID2020-113156GB-I00/AEI/10.13039/501100011033 and through BCAM Severo Ochoa
excellence accreditation SEV-2023-2026.

The authors thank Christoph Thiele and Emanuel Carneiro for valuable comments
during the preparation of this manuscript.

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