# Litmus tests of the flat ${\Lambda\text{CDM}}$ model and model-independent
measurement of $H_{0}r_{\text{d}}$ with LSST and DESI

Benjamin L’Huillier*  | Ayan Mitra*  | Arman Shafieloo*  | Ryan E Keeley*  | and Hanwool Koo*  |

###### Abstract

In this analysis we apply a model-independent framework to test the flat
$\Lambda$CDM cosmology using simulated SNIa data from the upcoming Large
Synoptic Survey Telescope (LSST) and combined with simulated Dark Energy
Spectroscopic Instrument (DESI) five-years Baryon Acoustic Oscillations (BAO)
data. We adopt an iterative smoothing technique to reconstruct the expansion
history from SNIa data, which, when combined with BAO measurements,
facilitates a comprehensive test of the Universe’s curvature and the nature of
dark energy. The analysis is conducted under three different mock true
cosmologies: a flat $\Lambda$CDM universe, a universe with a notable curvature
($\Omega_{k,0}=0.1$), and one with dynamically evolving dark energy. Each
cosmology demonstrates different kinds and varying degrees of deviation from
the standard model predictions. We forecast that our reconstruction technique
can constrain cosmological parameters, such as the curvature ($\Omega_{k,0}$)
and $H_{0}r_{\text{d}}$, with a precision of approximately 0.52% for
$H_{0}r_{\text{d}}$ and between 0.032 to 0.037 for $\Omega_{k,0}$, competitive
with current cosmic microwave background constraints, without assuming any
form of dark energy.

##  1 Introduction

The concordance cosmological ${\Lambda\text{CDM}}$ model relies on a few
assumptions. Among them, homogeneity and isotropy lead to the Friedman-
Robertson-Lemaître-Walker (FLRW) metric. The discovery of cosmic acceleration
led to the establishment of the concordance ${\Lambda\text{CDM}}$ model. In
this model, the energy budget of the Universe is dominated by a cosmological
constant $\Lambda$, and matter by cold dark matter (CDM). The most stringent
constraints on the model have been obtained by cosmic microwave background
(CMB) measurements from the Planck satellite [1], baryon acoustic oscillations
(BAO) [2, 3, 4, 5, 6] and Type Ia Supernovae (SNIa) [7, 8, 9, 10, 11]. CMB
constrains the curvature parameter to be
$\Omega_{k}=-0.044^{+0.018}_{-0.015}$. Most of these constraints however are
direct fits to the data assuming a particular model, typically the
${\Lambda\text{CDM}}$ model or an extension. While this leverages the power of
Bayesian statistics to provide tight constraints on the model parameters, it
is important to also test the validity of the underlying hypotheses.

As opposed to model-fitting, model-independent approaches provide a healthy
check on the validity of a given model. Due to their extra flexibility
compared to model-fitting approaches, they may also reveal the presence of
unexpected features in the data and may be used to look for unaccounted for
systematics [12, 13, 14].

The rise of tensions, in particular the Hubble tension, and the unknown nature
of the main components of the model are strong hints for cosmologist to search
for evidence for beyond-${\Lambda\text{CDM}}$ physics [15, 16, 17, 18, 19, 20,
21, 22, 23].

Recent data from the Dark Energy Spectroscopic Instrument (DESI) showed
evidence for a phantom dark energy or a dynamical dark energy [5]. This was
further confirmed by studying physics-driven modifications of
${\Lambda\text{CDM}}$ [24] as well as data-driven reconstructions of the
expansion history [25].

The curvature of the Universe is another fundamental question. The Planck 2018
data are consistent with a flat universe but do not exclude a small negative
curvature [1], see also [26]. DESI DR1 finds a preference for a small positive
curvature, which disappears when combined with Planck.

In this work, we used a model-independent approach, namely, the iterative
smoothing method, to reconstruct the expansion history from SNIa and combine
it with distance measurements from BAO to perform a litmus test of the pillars
of the ${\Lambda\text{CDM}}$ model, the FLRW metric, the flatness of the
Universe, and the cosmological constant, as well to constrain
$H_{0}r_{\text{d}}/c$.

##  2 Data

In this analysis, we used simulated $3$-year LSST111www.lsst.org SNIa data and
the BAO forecast for DESI from [27] to constrain cosmology.

###  2.1 Fiducial Models

To show the potential of our model-independent approach, we simulate data for
three different cosmologies:

  * •

a flat-${\Lambda\text{CDM}}$ universe, with $\Omega_{\text{m},0}=0.315$ and a
constant equation of state $w=-1$.

  * •

a curved ${\Lambda\text{CDM}}$ universe, with
($\Omega_{\text{m},0},\Omega_{k,0})=(0.315,0.1)$. While such a large curvature
is consistent with Pantheon+ or DESI DR1 data, it is excluded by Planck 2018
data. However, it is nonetheless interesting to see the effect of curvature
and to apply our curvature test to a curved universe.

  * •

a flat universe with dynamical dark energy. We chose the phenomenologically
emergent dark energy (PEDE) model [28], which has no extra degree of freedom
relative to ${\Lambda\text{CDM}}$. The dark energy density evolves as

$\displaystyle f_{\text{PEDE}}(z)$
$\displaystyle=1-\tanh\left(\log_{10}(1+z)\right)$ (2.1a) and the equation of
state $\displaystyle w_{\text{PEDE}}(z)$ $\displaystyle=\frac{1}{3\ln
10}\left(1+\tanh\left[\log_{10}(1+z)\right]\right)-1$ (2.1b)

We note that the data and covariance matrix were initially generated for the
flat ${\Lambda\text{CDM}}$ case, and we reused the same noise and covariance
matrix for the other two cosmologies. While for a proper forecast, one should
apply the pipeline from [29] using the other two fiducial cosmologies, we
assume here that the resulting noise and covariance should be affected
minimally by the change of fiducial cosmology. On the other hand, the mock BAO
data are all generated with Gaussian white noise.

Table 1 summarizes our three fiducial cosmologies.

Fiducial Cosmology | $\Omega_{\text{m},0}$ | $h$ | $\Omega_{k,0}$ | $w$
---|---|---|---|---
flat-${\Lambda\text{CDM}}$ | 0.315 | 0.685 | 0 | -1
$\Omega_{k,0}=0.1$ | 0.315 | 0.685 | 0.1 | -1
PEDE | 0.315 | 0.685 | 0 | Eq. (2.1b)
Table 1: Fiducial cosmologies used in this work

###  2.2 LSST SNIa Simulations

####  2.2.1 Vera C. Rubin Observatory (The LSST)

The forthcoming LSST survey, scheduled to commence in 2025, represents the
most comprehensive optical survey planned to date, marking a significant
milestone for its generation. This survey will run for a decade and will
extensively explore the southern hemisphere’s sky through a large, ground-
based, wide-field observatory utilising six optical passband filters. This
combination is designed to achieve an unparalleled balance of depth, coverage
area, and observational frequency. Over the course of the survey, the LSST is
expected to catalogue millions of supernovae [30]. Operating from the Rubin
Observatory, the Simonyi Survey Telescope features an $8.4$ m mirror (with an
effective aperture of $6.7$ m) and is equipped with a state-of-the-art
$3200$-megapixel camera, providing a $9.6$ deg2 field of view. Approximately
$90\%$ of the telescope’s time will be dedicated to a deep-wide-fast survey
mode, systematically covering an area of $18,000$ square degrees roughly $800$
times across all bands over a decade, resulting in a co-added map with a depth
of $r-27.5$ (called as the Wide Fast Deep (WFD) observing strategy in the
LSST). This extensive data collection is projected to amass around $32$
trillion observations of $20$ billion galaxies and an equivalent number of
stars, primarily supporting the survey’s main scientific objectives [30]. The
remaining $10\%$ of the time will be reserved for specific initiatives, such
as the Very Deep and Very Fast time-domain surveys 222Deep Drilling Fields
(DDF), which are still in the planning stages.

####  2.2.2 Mock Data Generation

This study employs simulated SNIa data to examine dark energy results from the
forthcoming LSST observations, as detailed in [29]. The supernova data are
generated through the LSST Dark Energy Science Collaboration (DESC) Time
Domain (TD) pipeline and the SNANA software [31]. The procedure encompasses
four primary stages, depicted in Figure 4 of [29].

The simulations utilise the flat ${\Lambda\text{CDM}}$ model from § 2.1. In
particular, the Hubble constant, $H_{0}$, is set at $70.0~{}\text{km
s}^{-1}\text{Mpc}^{-1}$, aligned with the SALT2 model training. The SALT2
light curve model is employed to derive observer frame magnitudes. The
simulation noise is calculated using the following equation:

$\sigma_{\text{SIM}}^{2}=\left[F^{2}+(A\cdot
b)^{2}+(F\cdot\sigma_{\text{ZPT}})^{2}+\sigma_{0}\cdot 10^{0.4\cdot
ZPT_{\text{pe}}}+\sigma_{\text{host}}^{2}\right]S^{2}_{\text{SNR}}.$ (2.2)

where $F$ represents the simulated flux in photoelectrons, $A$ denotes the
noise equivalent area, given by
$A=\left[2\pi\int\text{PSF}^{2}(r,\theta)r\mathop{}\\!\mathrm{d}r\right]^{-1}$,
with PSF standing for the Point Spread Function, and $b$ represents the
background per unit area (inclusive of sky, CCD readouts, and dark current).
The scale factor $S_{\text{SNR}}$ is empirically determined based on the
signal-to-noise ratio. The terms represented by $\sigma$ denote uncertainties
related to zero point, flux calibration, and the underlying host galaxy,
empirically fitted to match simulated uncertainties with those derived from
the survey designs.

The LSST TD pipeline involves SN brightness standardisation via a Light Curve
(LC) fit stage, simulations for bias correction, and a BEAMS with Bias
Corrections (BBC) stage for Hubble diagram production. This pipeline yields a
redshift-binned Hubble diagram and its associated covariance matrix, which are
the data products employed for cosmological fitting.

The SNIa dataset utilised comprises both spectroscopically ($z_{\rm spec}$)
and photometrically ($z_{\rm phot}$) identified SNIa candidates. Based on
PLAsTiCC [32], the “$z_{\rm spec}$” sample includes two sets of events with
spectroscopic redshifts featuring an accuracy of $\sigma_{z}\sim 10^{-5}$. The
first subset consists of spectroscopically confirmed events with accurate
redshift predictions by the 4MOST spectrograph [33], under construction by the
European Southern Observatory and expected to be operational in 2024, situated
at a latitude similar to that of the Rubin Observatory in Chile. The second
subset includes photometrically identified events with accurate host galaxy
redshifts determined by 4MOST, with this subset being approximately $60\%$
larger than the first. For the photometric sample, host galaxy photometric
redshifts were employed as priors (adapted from [34]). The photometric
redshift and rms uncertainty derive from [35]. The entire simulation was rerun
based on the PLAsTiCC DDF data333The LSST utilises different observing
strategies, namely the deep field and the wide field, termed the DDF (Deep
Drilling Field) and the WFD (Wide Fast Deep), respectively.. Additional low
redshift spectroscopic data was sourced from the DC2 analysis, simulated with
WFD cadence. [29] restricted simulations to SNIa, omitting contamination (e.g.
core collapse, peculiar SNe, etc). The covariance matrix corresponds to the
combined statistical and systematic errors, where the latter encompasses all
individual systematics, detailed in Table 3 of [29]. The Hubble diagram is
segmented into $14$ redshift bins, containing a total of $5809$ SNIa
candidates, both spectroscopic and photometric.

It is noteworthy that [29] introduce a dataset with characteristics akin to
those projected in the LSST science roadmap for 1 and 10 years of SNIa
cosmology analysis, as described by [36]. However, there are significant
differences. While the analysis by [36] relies solely on SNIa with
spectroscopically confirmed host redshifts, [29] broadens the scope by
including a comprehensive end-to-end analysis that incorporates both
spectroscopic and photometric redshifts of host galaxies. In the forthcoming
LSST era, it is anticipated that the volume of photometrically observed Type
Ia supernovae will vastly outnumber the spectroscopic candidates. The
methodology presented in [29] therefore enables the use of these photometric
candidates in the cosmology analysis and hence in construction of a denser
Hubble diagram with higher redshift reaches. As a result, this presents a
critical opportunity to leverage photometric supernovae in cosmological
analyses, and potentially amplifying the dark energy constraints
significantly.

###  2.3 Mock BAO data from DESI 5-year

The Dark Energy Spectroscopic Instrument (DESI) is a stage-IV experiment
aiming to constrain dark energy by mapping the Universe in three dimensions
[6].

BAO are another probe of the expansion history from large-scale structures.
Strictly speaking, the transverse and radial modes of the BAO constrain

$\displaystyle\frac{d_{\text{A}}(z)}{r_{\text{d}}}$
$\displaystyle=\frac{c}{H_{0}r_{\text{d}}}\frac{\mathcal{D}(z)}{1+z},\text{and}$
(2.3a) $\displaystyle\frac{c}{H(z)r_{\text{d}}}$
$\displaystyle=\frac{c}{H_{0}r_{\text{d}}}\frac{1}{h(z)},$ (2.3b) where
$\displaystyle\frac{H_{0}r_{\text{d}}}{c}$
$\displaystyle=\frac{1}{\sqrt{3}}\int_{0}^{\tfrac{1}{1+z_{\mathrm{d}}}}\frac{\mathrm{d}a}{a^{2}h(a)\sqrt{1+\frac{3\omega_{b}}{4\omega_{\gamma}}a}}$
(2.3c)

and $r_{\text{d}}$ is the sound horizon at the drag epoch $z_{\mathrm{d}}$.

We generated DESI 5-year mock BAO, assuming a Gaussian white noise where the
errors are taken from [27]. While for real data, one must take care of the
correlation between $d_{\text{A}}$ and $c/Hr_{\text{d}}$, in this paper we
assume that there is no correlation. We also assume
$H_{0}=$70\text{\,}\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}\,\mathrm{M}\mathrm{p}\mathrm{c}^{-1}$$,
and $r_{\text{d}}=$147.10\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$, leading to
a fiducial value
$H_{0}r_{\text{d}}^{\mathrm{fid}}=$10\,297\text{\,}\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}\,\mathrm{M}\mathrm{p}\mathrm{c}^{-1}$=0.03435\,c$.
While such cosmology is not currently favored by CMB experiments, the exact
choice of cosmological parameters is not important for the purpose of this
study since we are focusing on model independent reconstructions and so our
results do not depend on different fiducual models.

##  3 Method

In an FLRW universe, when radiation is negligible, the expansion history reads

$\displaystyle h^{2}(z)$
$\displaystyle=\frac{H^{2}(z)}{H_{0}^{2}}=\Omega_{\text{m},0}(1+z)^{3}+\Omega_{k,0}(1+z)^{2}+(1-\Omega_{\text{m},0}-\Omega_{k,0})f_{\text{DE}}(z),$
(3.1a) where $\displaystyle f_{\text{DE}}(z)$
$\displaystyle=\exp\left(3\int_{0}^{z}\frac{1+w(x)}{1+x}\mathrm{d}x\right)$
(3.1b)

describe the time evolution of the dark energy density, $\Omega_{\text{m},0}$
and $\Omega_{k,0}$ are the matter and curvature density parameters today
respectively, and $w=p/\rho$ is the equation of state of dark energy. For the
cosmological constant $\Lambda$, $w=-1$ and $f_{\text{DE}}(z)\equiv 1$.

The dimensionless comoving distance is defined by

$\displaystyle\mathcal{D}(z)$
$\displaystyle=\frac{1}{\sqrt{-\Omega_{k,0}}}\sin\left[\sqrt{-\Omega_{k,0}}\int_{0}^{z}\frac{\mathrm{d}x}{h(x)}\right]$
(3.2)

for all signs of $\Omega_{k,0}$. In particular, for $\Omega_{k,0}=0$, eq.
(3.2) simplifies to

$\displaystyle\mathcal{D}(z)$
$\displaystyle=\int_{0}^{z}\frac{\mathrm{d}x}{h(x)}.$ (3.3)

The angular diameter and luminosity distances are related to the comoving
distance by

$\displaystyle(1+z)d_{\text{A}}(z)$
$\displaystyle=\frac{d_{\text{L}}(z)}{1+z}=d_{\text{C}}(z)=\frac{c}{H_{0}}\mathcal{D}(z).$
(3.4)

The distance modulus from supernovae is

$\displaystyle\mu(z)$
$\displaystyle=5\log_{10}\frac{d_{\text{L}}}{$1\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$}+25.$
(3.5)

Several litmus tests have been put forward to test various aspects of the
${\Lambda\text{CDM}}$ model. In particular, in an FLRW universe, we have [37,
38, 39]

$\displaystyle\mathcal{O}_{k}(z)$
$\displaystyle=\frac{\Theta^{2}(z)-1}{\mathcal{D}^{2}(z)}=\Omega_{k,0},$
(3.6a) where $\displaystyle\Theta$ $\displaystyle=h\mathcal{D}^{\prime}.$
(3.6b)

In a flat-FLRW universe, $\Theta(z)\equiv 1$.

The $O_{m}$ diagnostic, introduced by [40], is defined as

$\displaystyle O_{m}(z)$
$\displaystyle=\frac{h^{2}(z)-1}{(1+z)^{3}-1}=\Omega_{\text{m},0}$ (3.7)

in a flat-${\Lambda\text{CDM}}$ universe. These two diagnostics are litmus
tests respectively of the FLRW metric (and its curvature) and of the
flat-${\Lambda\text{CDM}}$ model. Departure from $O_{m}=$constant, or from
$\mathcal{O}_{k}=$ constant are signs of departure from
flat-${\Lambda\text{CDM}}$ and FLRW respectively.

Another widely used quantity is the deceleration parameter

$\displaystyle q$
$\displaystyle=-\frac{1}{H}\frac{\ddot{a}}{a}=-(1+z)\frac{h^{\prime}}{h}-1.$
(3.8)

$q>0$ indicate a decelerating Universe, while $q<0$ for an accelerating
universe.

###  3.1 Curvature test

The transverse and radial BAO modes respectively give us
${d_{\text{A}}}/{r_{\text{d}}}=d_{\text{C}}/((1+z)r_{\text{d}})$ and
$c/{(Hr_{\text{d}})}$, while the supernovae provide $\mathcal{D}$ and
$\mathcal{D}^{\prime}$, which are independent of both $H_{0}$ and
$r_{\text{d}}$. We can therefore combine these two measurements to estimate
$H_{0}r_{\text{d}}$ in two independent ways:

$\displaystyle\frac{H_{0}r_{\text{d}}}{c}$
$\displaystyle=\mathcal{D}\frac{r_{\text{d}}}{d_{\text{C}}}\mbox{, and}$
(3.9a) $\displaystyle\frac{H_{0}r_{\text{d}}}{c}$
$\displaystyle=\frac{1}{h}\frac{Hr_{\text{d}}}{c}=\mathcal{D}^{\prime}\frac{Hr_{\text{d}}}{c},$
(3.9b)

where the last equality holds only for a flat-FLRW universe. Therefore, we can
combine the BAO observables with $\mathcal{D}$ and $\mathcal{D}^{\prime}$
measured by SNIa to constrain $H_{0}r_{\text{d}}$. One way to do this is to
reconstruct a smooth comoving distance $\mathcal{D}(z)$, its derivative
$\mathcal{D}^{\prime}(z)$, and expansion history $h(z)$ from SNIa at any
arbitrary redshift $z$, and evaluate those curves at the BAO redshifts
$z_{\text{BAO}}$. To do the reconstructions, we apply the iterative smoothing
method to simulated LSST SNIa as described in Section 3.2. We then simply
divide the central value of the BAO measurements $d_{\text{C}}/r_{\rm d}$ and
$d_{\text{H}}/r_{\rm d}$ by the value of the reconstructed
$\mathcal{D}(z_{\text{BAO}})$, $\mathcal{D}^{\prime}(z_{\text{BAO}})$, and
$h(z_{\rm BAO)}$, and then propagate the BAO error. The results of this
calculation are seen in Fig. 4.

We can then calculate

$\displaystyle\Theta$
$\displaystyle=h\mathcal{D}^{\prime}=\frac{Hr_{\text{d}}}{H_{0}r_{\text{d}}}\mathcal{D}^{\prime}=\frac{Hr_{\text{d}}}{c}\frac{d_{\text{C}}}{r_{\text{d}}}\frac{\mathcal{D}^{\prime}}{\mathcal{D}}$
(3.10)

and $\mathcal{O}_{k}$ only with the BAO and the smooth reconstructions from
the SNIa data. Similarly to $H_{0}r_{\rm d}$, we evaluate $\Theta(z)$ by
multiplying the central values of the BAO data ($\frac{Hr_{\text{d}}}{c}$ and
$\frac{d_{\text{C}}}{r_{\text{d}}}$) with the smooth functions reconstructed
from the SNIa ($\frac{\mathcal{D}^{\prime}(z_{\rm BAO})}{\mathcal{D}(z_{\rm
BAO})}$), evaluated at the BAO redshifts.

Figure 1:  Results of the SNIa smoothing colour-coded by
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\Lambda\text{CDM}}$ for the
flat-${\Lambda\text{CDM}}$ fiducial cosmology. The top-left panel shows the
smooth residuals. The middle-left panel shows
$\mathcal{D}/\mathcal{D}_{\text{fid}}$, the bottom-left panel shows
$h_{\text{fid}}/h$, the middle-right panel shows $O_{m}$, and the bottom-right
panel $q$. The middle- and bottom-left panels also show the DESI-like BAO mock
data normalized by the fiducial cosmology.  Figure 2: Same as Fig. 1, fiducial
model: ${\Lambda\text{CDM}}$ $(\Omega_{k,0}=0.1)$ Note that in the bottom-left
panel, the BAO data are $h_{\text{fid}}/h$, but the smooth SNIa
reconstructions show $\mathcal{D}^{\prime}/\mathcal{D}^{\prime}_{\text{fid}}$.
Figure 3: Same as Fig. 1, fiducial model: PEDE

###  3.2 Iterative Smoothing

We used the iterative smoothing method to reconstruct the distance-redshift
relation from SNIa [41, 42, 43]. The strength of this method is that we make
no assumption regarding the evolution of dark energy.

Starting with some initial guess $\hat{\mu}_{0}$, we iteratively calculate the
reconstructed $\hat{\mu}_{n}+1$ at iteration $n+1$ with the inverse covariance
matrix of the SNIa data $\bm{C}_{\text{SN}}^{-1}$:

$\displaystyle\hat{\mu}_{n+1}(z)$
$\displaystyle=\hat{\mu}_{n}(z)+\frac{\bm{\delta\mu}_{n}^{\intercal}\cdot\bm{C}_{\text{SN}}^{-1}\cdot\bm{W}(z)}{\mathbbm{1}^{\intercal}\cdot\bm{C}_{\text{SN}}^{-1}\cdot\bm{W}(z)},$
(3.11a) where $\displaystyle\bm{W}_{i}(z)$
$\displaystyle={\exp{\left(-\frac{\ln^{2}\left(\frac{1+z}{1+z_{i}}\right)}{2\Delta^{2}}\right)}}$
(3.11b) is a vector of weights, $\displaystyle\mathbbm{1}$
$\displaystyle=(1,\dots,1)^{\intercal}$ (3.11c) is a column vector,
$\displaystyle\bm{\delta\mu}_{n}|_{i}$
$\displaystyle=\mu_{i}-\hat{\mu}_{n}(z_{i}).$ (3.11d) The smoothing width is
set to $\Delta=0.3$ following previous analyses in [41, 42, 39, 44, 43, 45,
46, 47].

We define the $\chi^{2}$ value of the reconstruction $\hat{y}_{n}(t)$ as

$\chi_{n}^{2}=\bm{\delta\mu}_{n}^{\intercal}\cdot\bm{C}_{\text{SN}}^{-1}\cdot\bm{\delta\mu}_{n}.$
(3.11e)

The results converge towards the solution preferred by the data, independently
from the choice of the initial condition [41, 42, 39, 43]. Therefore, starting
from different initial conditions, the results of the iterative smoothing
method will approach this solution by different paths. We collect all
reconstructions with a better $\chi^{2}$ than the best-fit
${\Lambda\text{CDM}}$, which then make up a non-exhaustive set of solutions.

We also obtained the smoothed first and second derivatives of $\hat{\mu}$ as
described in Appendix A. The initial guess for the derivative of the distance
modulus is

$\displaystyle\frac{\mathrm{d}{}}{\mathrm{d}{\ln(1+z)}}\hat{\mu}(z)$
$\displaystyle=\frac{5}{\ln
10}\left[1+\frac{1+z}{h\mathcal{D}}\sqrt{1+\Omega_{k,0}\mathcal{D}^{2}}\right],$
(3.12)
$\displaystyle\frac{\mathrm{d}{{}^{2}}}{\mathrm{d}{\ln(1+z)^{2}}}\hat{\mu}(z)$
$\displaystyle=\frac{5}{\ln
10}(1+z)\left[\frac{\mathcal{D}^{\prime}}{\mathcal{D}}\left(1-(1+z)\frac{\mathcal{D}^{\prime}}{\mathcal{D}}\right)+(1+z)\frac{\mathcal{D}^{\prime\prime}}{\mathcal{D}}\right],$
(3.13)

where ′ denotes $\mathrm{d}/\mathrm{d}z$. The formalism of the smooth
derivative is described in the appendix.

From $\hat{\mu}$, $\hat{\mu}^{\prime}$, and $\hat{\mu}^{\prime\prime}$, we can
reconstruct

$\displaystyle\mathcal{D}(z)$
$\displaystyle=\frac{H_{0}}{c}\frac{10^{\tfrac{\mu}{5}-5}}{1+z}$ (3.14)
$\displaystyle\mathcal{D}^{\prime}(z)$ $\displaystyle=\left[\frac{\ln
10}{5}\frac{\mathrm{d}\mu}{\mathrm{d}\ln(1+z)}-1\right]\frac{\mathcal{D}(z)}{1+z}\mbox{,
and}$ (3.15) $\displaystyle\mathcal{D}^{\prime\prime}(z)$
$\displaystyle=-\frac{\mathcal{D}^{\prime}}{1+z}+\left[\frac{\ln
10}{5}\frac{\mathrm{d}{{}^{2}\mu}}{\mathrm{d}{(\ln(1+z))^{2}}}\mathcal{D}\right]+\left[\frac{\ln
10}{5}\frac{\mathrm{d}{\mu}}{\mathrm{d}{\ln(1+z)}}-1\right]\frac{\mathcal{D}^{\prime}}{1+z}$
(3.16)

In practice, $\hat{\mu}$ is reconstructed to an additive constant, and one has
to normalize $\mathcal{D}$, $\mathcal{D}^{\prime}$, and
$\mathcal{D}^{\prime\prime}$ so that $\mathcal{D}^{\prime}(z=0)=1$.

From these relations, we can obtain $h$, $h^{\prime}$, $\mathcal{D}^{\prime}$
via

$\displaystyle\mathcal{D}^{\prime}$
$\displaystyle=\frac{\mathcal{D}}{1+z}\left(\frac{\ln
10}{5}\frac{\mathrm{d}{\mu}}{\mathrm{d}{\ln(1+z)}}-1\right)$ (3.17)
$\displaystyle h$
$\displaystyle=\frac{\sqrt{1+\Omega_{k,0}\mathcal{D}^{2}}}{\mathcal{D}^{\prime}}$
(3.18) and $\displaystyle h^{\prime}$
$\displaystyle=\frac{h\mathcal{D}}{\mathcal{D}^{\prime}}\left[\frac{\Omega_{k,0}}{h^{2}}-\frac{\mathcal{D}^{\prime\prime}}{\mathcal{D}}\right].$
(3.19)

It is worth noting that, while we can obtain $\mathcal{D}^{\prime}$ and
$\mathcal{D}^{\prime\prime}$ from the smoothing of the supernovae directly, to
obtain $h$ and $h^{\prime}$, one needs to know $\Omega_{k,0}$. Therefore,
wrongly assuming flatness will imply an error when inferring $h(z)$,
$h^{\prime}(z)$, and $q(z)$. This is just a restatement of the
$\mathcal{O}_{k}$ test.

Figure 4:  The bottom rows show our estimations of $H_{0}r_{\text{d}}/c$ for
our two methods at the BAO data points for the three fiducial cosmologies. The
central line is the collection of central values of $H_{0}r_{\text{d}}$ for
each reconstruction, and the coloured band is the one-sigma error from the BAO
around the central value. The top rows show likelihood for estimate (3.9b) at
fixed iteration, and the third row for estimate (3.9a). The colour bars of the
second and middle rows are the same as one of the corresponding smooth figures
(1 to 2).  Figure 5: Curvature parameter: $\Theta$. The bottom row shows
$\Theta=h\mathcal{D}^{\prime}$ for our three fiducial cosmologies. Like in
Fig. 4, the central value is obtained by combining the BAO with each iteration
of the smooth SN reconstructions, and the error-band from the $1\sigma$ errors
from the BAO. The top row shows the likelihood of $\Theta$ for each smooth
reconstruction. We note that in the middle panel of the top row, there is no
true value for $\Theta$ since $\Theta(z)=\sqrt{1+\Omega_{k}\mathcal{D}^{2}}$
varies with redshift. Figure 6: Curvature test: the $\mathcal{O}_{k}$
diagnostic. Same legend as Figs. 5. The black dotted lines in the bottom row
show the fiducial values of $\Omega_{k,0}$ in all three panels,
$\Omega_{k,0}=0$ for the left and right, and $\Omega_{k,0}=0.1$ for the
center. It is worth noting that in the $\Omega_{k,0}=0.1$ case, the
$\mathcal{O}_{k}$ diagnostic recovers the non-zero fiducial value of
$\Omega_{k,0}$.

##  4 Results

###  4.1 Supernova Smoothing

Fig. 1 shows the results of the smoothing for the flat-${\Lambda\text{CDM}}$
model. Each line is an iteration of the smoothing color-coded by its
$\Delta\chi^{2}$ with respect to the best-fit ${\Lambda\text{CDM}}$ model. The
top-left panel shows the reconstructed distance moduli with the mock SNIa
data. The middle- and bottom-left panels show the comoving distance
$\mathcal{D}(z)/\mathcal{D}_{\text{fid}}(z)$ and the expansion history
${}_{\text{fid}}(z)/hh(z)$ normalized to the fiducial values. The middle- and
bottom-right panels show $O_{m}$ and the deceleration parameter $q$. We are
able to reconstruct the fiducial quantities.

Figs. 2 and 3 and show the same results for the $\Omega_{k,0}=0.1$ and PEDE
fiducial models. In the PEDE case, we are able to reconstruct the $O_{m}$ and
$q$ parameters. However, it is interesting to look closely at the
$\Omega_{k,0}=0.1$ case. Since the assumption of flatness that goes into eq.
(3.3) is not true in this case, calculating $\mathcal{D}^{\prime}$ does not
provide $h$ (see eq. (3.18). This is turn will bias our inference of $O_{m}$
and $q$ as well as the curvature test. At high-enough redshifts $z\geq 1$, the
deviation between $\mathcal{D}^{\prime}$ and $1/h$ becomes important. This
deviation between the $h_{\rm fid}/h$ data from the BAO and the
$\mathcal{D}^{\prime}/\mathcal{D}$ reconstructions from the SNIa are the
litmus test for flatness, or signal for curvature that we are looking for.
Interestingly, the reconstructions are still consistent with the fiducial $q$.

###  4.2 $H_{0}r_{\text{d}}$

The bottom row of Fig. 4 shows our two estimations of $H_{0}r_{\text{d}}/c$
for the each model from eqs (3.9a) and (3.9b). At each BAO data point, we plot
the central value and the $\pm 1\sigma$ band for each reconstruction with
$\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\Lambda\text{CDM}}<2.3$. It is worth
stressing here that at every BAO point, we have $N$ estimations of
$H_{0}r_{\text{d}}/c$, corresponding to $N$ reconstructions of the distance-
redshift relations from the SNIa (combined with the mock BAO data point). Each
of these estimates also has an associated error bar, shown as an error band on
the figure. At each redshift point, the reconstructed $H_{0}r_{\text{d}}/c$
are re-sorted by increasing central value for aesthetic reasons. The fiducial
value $H_{0}r_{\text{d}}^{\text{fid}}/c=0.03435$ is shown as a dashed line.
For the three fiducial cosmologies, the estimations from eq. (3.9a) are
consistent with $H_{0}r_{\text{d}}^{\text{fid}}$.

The top and middle rows show the likelihood of $\Theta$ and $\mathcal{O}_{k}$
for a given iteration for eq. (3.9b) and (3.9a). This is equivalent to
calculating the weighted average over the different redshift bins weighted by
$1/\sigma_{i}^{2}$. The weighted average of a quantity
$\bm{A}=(A(z_{1}),\dots,A(z_{n}))$ with covariance matrix $C_{A}$ over the
different redshift bins and its variance are given by:

$\displaystyle\langle A\rangle$
$\displaystyle=\frac{\mathbbm{1}^{\top}\bm{C_{A}}^{-1}\bm{A}}{\mathbbm{1}^{\top}\bm{C_{A}}^{-1}\mathbbm{1}},$
(4.1a) $\displaystyle\sigma^{2}_{\langle A\rangle}$
$\displaystyle=\left(\mathbbm{1}^{\top}\bm{C_{A}}^{-1}\mathbbm{1}\right)^{-1}.$
(4.1b)

In the case of $A=H_{0}r_{\text{d}}$, the correlation between two different
redshift bins is negligible, and the covariance matrix in eq. (4.1) is
diagonal. In the case of eq. 3.9a (middle row), the three fiducial cosmologies
yield similar reconstructions, regardless of the curvature and DE. It is worth
paying attention to the $\Omega_{k,0}=0.1$ case (middle columns). While
estimator (3.9a) (third row) is totally consistent with the fiducial value,
eq. (3.9b) is centred around 0.035, and most of the reconstructions are over
$2\sigma$ away from the fiducial value, showing an inconsistency between the
two estimators. This comes from the fact that we do not calculate $1/h$ but
$\mathcal{D}^{\prime}$, which do not coincide in a non-flat FLRW universe, as
seen above. Indeed, we can see in the first row that the values of
$\mathcal{D}^{\prime}Hr_{\text{d}}/c$ diverge from the fiducial values at high
redshift, reflecting equation (3.18). The minimal and maximal values of the
precision of the weighted average are reported in table 2, with typically 0.5%
precision.

Quantity | flat-${\Lambda\text{CDM}}$ | $\Omega_{k,0}=0.1$ | PEDE
---|---|---|---
$\langle\mathcal{D}r_{\text{d}}/d_{\text{C}}\rangle$ | 0.52% | 0.52% | 0.52%
$\langle\mathcal{D}^{\prime}Hr_{\text{d}}/c\rangle$ | 0.51% | 0.51% | 0.51%
$\mathcal{O}_{k}$ | 0.032–0.034 | 0.034–0.037 | 0.030–0.033
$\Theta$ | 0.0074–0.0075 | 0.0075–0.0077 | 0.0074–0.0076
Table 2: Precision for our three fiducial cosmologies. The precision for our
two estimates of $H_{0}r_{\text{d}}$ is relative to the central value, while
the precisions in $\mathcal{O}_{k}$ and $\Theta$ are absolute. For
$\mathcal{O}_{k}$ and $\Theta$, we quote the minimum and maximum values. For
$H_{0}r_{\text{d}}$, the difference between the minimum and maximum values of
the precision is negligible.

###  4.3 Curvature Test

The Bottom panel of Fig. 5 shows our estimate of
$\Theta(z)=h\mathcal{D}^{\prime}=\sqrt{1+\Omega_{k,0}^{2}\mathcal{D}}$ for the
three cosmologies. The top panel shows the weighted average of $\Theta(z)$
over $z$ for a given iteration of the smoothing, colour-coded by
$\Delta\chi^{2}$ of the SNIa reconstruction. For a flat FLRW universe, we have
$\Theta(z)\equiv 1$. Indeed, for the Flat-$\Lambda$ (left) and PEDE (right)
cases, our estimates are consistent with $\Theta=1$. The interesting case is
the $\Omega_{k,0}=0.1$ case, where $\Theta$ is not constant, and we can see a
trend of $\Theta(z)$ deviating from 1 as $z$ increases. Similarly, the
weighted average (peak of the likelihood) is larger than in the other two
cases, and only marginally consistent with $\Theta=1$.

Fig. 6 shows similar plots for the $\mathcal{O}_{k}$ parameter. Similarly to
$\Theta$, the Flat-$\Lambda$ and PEDE case are perfeclty consistent with
flatness ($\Omega_{k,0}=0$), while the $\Omega_{k,0}=0.1$ case is only
marginally consistent with $\Omega_{k,0}=0$ and perfectly consistent with
$\Omega_{k,0}=0.1$, showing the constraining power of this test.

In the bottom panels of both figures, at $z\gtrsim 0.5$, the error bars are
very small and the reconstructions are perfectly consistent with the fiducial
cosmology in all three cases. At low redshift, the large error bars do not
constrain $\Theta$ or $\Omega_{k,0}$.

##  5 Summary and Conclusion

We combined forecast BAO data for a DESI 5-year-like survey and a realistic
forecast for LSST SNIa to assess the ability of the joint surveys to provide
model-independent litmus tests of the flat ${\Lambda\text{CDM}}$ model. For
this purpose, we generated mock data for three cosmologies: flat
${\Lambda\text{CDM}}$ a non-flat ${\Lambda\text{CDM}}$ universe with
$\Omega_{k,0}=0.1$, and an example of dynamical dark energy, the PEDE model.

We applied the iterative smoothing method to reconstruct the expansion history
in a model-independent manner, only assuming flatness.

The expected quality of LSST SNIa will allow to reconstruct very accurately
certain diagnostics such as $Om$ and $q$. In addition, we quantify the ability
of the joint surveys to constrain $H_{0}r_{d}/c$. We then proceed to testing
the FLRW metric, measuring $\Theta$ and $O_{k}$. The combination of LSST and
DESI 5-years will allow to constrain $O_{k}$ to within $\sim 0.035$, and
$\Theta$ to within 0.0075. It is worth noting that the credible interval
reported by DESI DR1 is $(-0.078,0.068)$ for the flat ${\Lambda\text{CDM}}$
model without any external data. We show that with the expected five-year data
accuracy, a model-independent precision on the curvature parameter will be
competitive.

In the case of the $\Omega_{k,0}=0.1$ universe, the flatness assumption of eq.
(3.3) is now incorrect, which leads to a biased expansion history and
inconsistent estimations of $H_{0}r_{\text{d}}/c$ from our two methods.
Therefore, $H_{0}r_{\text{d}}$ itself can be seen as a curvature test. An
inconsistency between the two estimates of $H_{0}r_{\text{d}}$, or with
$\mathcal{O}_{k}(z)=1$ or $\Theta(z)=1$ can just be interpreted either as
evidence of departure from flatness, or inconsistency between the two data
sets.

##  Appendix A Smooth Derivatives

An interesting property of smoothing is that the derivative of the smooth
function can be readily obtained from the derivative of the smoothing kernel.

Suppose we observe a function $y_{i}=f(t)+\bm{\epsilon_{i}}$, where
$\bm{\epsilon}\sim\mathcal{N}(0,\bm{C})$. We want to reconstruct the smooth
function $\hat{f}(t)$ and its successive derivative $\hat{f}^{(i)}(t)$.

$\displaystyle\hat{y}_{n+1}(t)$ $\displaystyle=\hat{y}_{n}(t)+\frac{\bm{\delta
y}_{n}^{\intercal}\cdot\bm{C}^{-1}\cdot\bm{W}(t)}{\mathbbm{1}^{\intercal}\cdot\bm{C}^{-1}\cdot\bm{W}(t)},$
(A.1) where $\mathbbm{1}^{\intercal}=(1,\cdots,1)$, the weight $\bm{W}$ and
residual $\bm{\delta y}_{n}$ denote $\displaystyle W_{i}(t)$
$\displaystyle=\exp\left[-\frac{(t-t_{i})^{2}}{2\Delta^{2}}\right]$ (A.2)
$\displaystyle\bm{\delta y}_{n}|_{i}$ $\displaystyle=y_{i}-\hat{y}_{n}(t_{i})$
(A.3)

and $\bm{C}^{-1}$ indicates the inverse of covariance matrix of the data. The
smoothing width is set to $\Delta=0.3$ as mentioned in eq. (3.11).

Since smoothing is essentially a convolution by a smoothing kernel, one can
straightforwardly define the smooth reconstruction of the successive
derivatives as

$\displaystyle\hat{y}_{n+1}(t)$
$\displaystyle=\hat{y}_{n}(t)+\frac{B(t)}{A(t)},$ (A.4) where $\displaystyle
B$ $\displaystyle=\bm{\delta y}_{n}\cdot\bm{C}^{-1}\cdot\bm{W(t)},\text{and}$
(A.5) $\displaystyle A$
$\displaystyle=\mathbbm{1}^{\intercal}\cdot\bm{C}^{-1}\cdot\bm{W(t)}.$ (A.6)
The first and second smooth derivatives is obtained by deriving the previous
equation with respect to $t$: $\displaystyle\widehat{y}^{\prime}_{n+1}(t)$
$\displaystyle=\widehat{y^{\prime}}_{n}(t)+\frac{B^{\prime}}{A}-\frac{BA^{\prime}}{A^{2}},$
(A.7) $\displaystyle\widehat{y}^{\prime\prime}_{n+1}(t)$
$\displaystyle=\widehat{y^{\prime\prime}_{n}}(t)+\frac{B^{\prime\prime}}{A}-\frac{2A^{\prime}B^{\prime}+A^{\prime\prime}B}{A^{2}}+\frac{2A^{\prime
2}B}{A^{3}},$ (A.8) where $\displaystyle B^{\prime}$ $\displaystyle=\bm{\delta
y}_{n}^{\intercal}\cdot\bm{C}^{-1}\cdot\bm{W}^{\prime}(t),$ (A.9)
$\displaystyle A^{\prime}$
$\displaystyle=\mathbbm{1}^{\intercal}\cdot\bm{C}^{-1}\cdot\bm{W}^{\prime}(t),$
(A.10) $\displaystyle B^{\prime\prime}$ $\displaystyle=\bm{\delta
y}_{n}^{\intercal}\cdot\bm{C}^{-1}\cdot\bm{W}^{\prime\prime}(t),\text{and}$
(A.11) $\displaystyle A^{\prime\prime}$
$\displaystyle=\mathbbm{1}^{\intercal}\cdot\bm{C}^{-1}\cdot\bm{W}^{\prime\prime}(t).$
(A.12)

In this work, we apply the algorithm to the LSST simulated SNIa with
$t=\ln(1+z)$.

## Acknowledgments

B. L. acknowledges the support of the National Research Foundation of Korea
(NRF-2022R1F1A1076338 and RS-2023-00259422). AM thanks Richard Kessler and the
LSST DESC TD team for the production of the LSST dataset.

## References

  * [1] Planck collaboration, __Planck 2018 results. VI. Cosmological parameters__ , _Astron. Astrophys._ 641 (2020) A6.
  * [2] D.J. Eisenstein, I. Zehavi, D.W. Hogg, R. Scoccimarro, M.R. Blanton, R.C. Nichol et al., _Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies_ , _ApJ_ 633 (2005) 560 [astro-ph/0501171].
  * [3] S. Alam, M. Aubert, S. Avila et al., _Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory_ , _Phys. Rev. D_ 103 (2021) 083533 [2007.08991].
  * [4] DESI Collaboration, A.G. Adame et al., _DESI 2024 IV: Baryon Acoustic Oscillations from the Lyman Alpha Forest_ , _arXiv e-prints_ (2024)  [2404.03001].
  * [5] DESI Collaboration, A.G. Adame et al., _DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations_ , _arXiv e-prints_ (2024)  [2404.03002].
  * [6] DESI Collaboration, A.G. Adame et al., _DESI 2024 III: Baryon Acoustic Oscillations from Galaxies and Quasars_ , _arXiv e-prints_ (2024)  [2404.03000].
  * [7] S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent et al., _Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae_, _Astrophys. J._ 517 (1999) 565 [astro-ph/9812133].
  * [8] A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks et al., _Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant [arXiv:astro-ph/9805201]_ , _Astron. J_ 116 (1998) 1009 [astro-ph/9805201].
  * [9] M. Betoule et al., _Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples_ , _Astron. Astrophys._ 568 (2014) A22 [1401.4064].
  * [10] Pan-STARRS1 collaboration, __The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample__ , _Astrophys. J._ 859 (2018) 101.
  * [11] D. Brout et al., _The Pantheon+ Analysis: Cosmological Constraints_ , _Astrophys. J._ 938 (2022) 110 [2202.04077].
  * [12] R.E. Keeley, A. Shafieloo, G.-B. Zhao, J.A. Vazquez and H. Koo, _Reconstructing the Universe: Testing the Mutual Consistency of the Pantheon and SDSS/eBOSS BAO Data Sets with Gaussian Processes_ , _AJ_ 161 (2021) 151 [2010.03234].
  * [13] S.-g. Hwang, B. L’Huillier, R.E. Keeley, M.J. Jee and A. Shafieloo, _How to use GP: effects of the mean function and hyperparameter selection on Gaussian process regression_ , _J. Cosmology Astropart. Phys_ 2023 (2023) 014 [2206.15081].
  * [14] R.E. Keeley and A. Shafieloo, _Ruling Out New Physics at Low Redshift as a Solution to the H 0 Tension_, _Phys. Rev. Lett._ 131 (2023) 111002 [2206.08440].
  * [15] P.K. Aluri et al., _Is the observable Universe consistent with the cosmological principle?_ , _Class. Quant. Grav._ 40 (2023) 094001 [2207.05765].
  * [16] L. Perivolaropoulos and F. Skara, _Challenges for $\Lambda$CDM: An update_, _New Astron. Rev._ 95 (2022) 101659 [2105.05208].
  * [17] M. Kamionkowski and A.G. Riess, _The Hubble Tension and Early Dark Energy_ , 2211.04492.
  * [18] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri et al., _In the realm of the Hubble tension—a review of solutions_ , _Class. Quant. Grav._ 38 (2021) 153001 [2103.01183].
  * [19] C. Krishnan, R. Mohayaee, E.O. Colgáin, M.M. Sheikh-Jabbari and L. Yin, _Does Hubble tension signal a breakdown in FLRW cosmology?_ , _Class. Quant. Grav._ 38 (2021) 184001 [2105.09790].
  * [20] E. Abdalla et al., _Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies_ , _JHEAp_ 34 (2022) 49 [2203.06142].
  * [21] C. Krishnan, R. Mohayaee, E.O. Colgáin, M.M. Sheikh-Jabbari and L. Yin, _Hints of FLRW breakdown from supernovae_ , _Phys. Rev. D_ 105 (2022) 063514 [2106.02532].
  * [22] T.L. Smith, M. Lucca, V. Poulin, G.F. Abellan, L. Balkenhol, K. Benabed et al., _Hints of early dark energy in Planck, SPT, and ACT data: New physics or systematics?_ , _Phys. Rev. D_ 106 (2022) 043526 [2202.09379].
  * [23] J. Sola Peracaula, _The cosmological constant problem and running vacuum in the expanding universe_ , _Phil. Trans. Roy. Soc. Lond. A_ 380 (2022) 20210182 [2203.13757].
  * [24] K. Lodha, A. Shafieloo, R. Calderon, E. Linder, W. Sohn, J.L. Cervantes-Cota et al., _DESI 2024: Constraints on Physics-Focused Aspects of Dark Energy using DESI DR1 BAO Data_ , _arXiv e-prints_ (2024) arXiv:2405.13588 [2405.13588].
  * [25] R. Calderon, K. Lodha, A. Shafieloo, E. Linder, W. Sohn, A. de Mattia et al., _DESI 2024: Reconstructing Dark Energy using Crossing Statistics with DESI DR1 BAO data_ , _arXiv e-prints_ (2024) arXiv:2405.04216 [2405.04216].
  * [26] E. Di Valentino, A. Melchiorri and J. Silk, _Cosmological constraints in extended parameter space from the Planck 2018 Legacy release_ , _J. Cosmology Astropart. Phys_ 2020 (2020) 013 [1908.01391].
  * [27] DESI Collaboration, A. Aghamousa et al., _The DESI Experiment Part I: Science,Targeting, and Survey Design_ , _arXiv e-prints_ (2016) arXiv:1611.00036 [1611.00036].
  * [28] X. Li and A. Shafieloo, _A Simple Phenomenological Emergent Dark Energy Model can Resolve the Hubble Tension_ , _ApJ_ 883 (2019) L3 [1906.08275].
  * [29] LSST Dark Energy Science collaboration, _Using Host Galaxy Photometric Redshifts to Improve Cosmological Constraints with Type Ia Supernovae in the LSST Era_ , _Astrophys. J._ 944 (2023) 212 [2210.07560].
  * [30] LSST collaboration, _LSST: from Science Drivers to Reference Design and Anticipated Data Products_ , _Astrophys. J._ 873 (2019) 111 [0805.2366].
  * [31] R. Kessler, J.P. Bernstein, D. Cinabro, B. Dilday, J.A. Frieman, S. Jha et al., _SNANA: A Public Software Package for Supernova Analysis_ , _Publ. Astron. Soc. Pac._ 121 (2009) 1028 [0908.4280].
  * [32] The PLAsTiCC team collaboration, _The Photometric LSST Astronomical Time-series Classification Challenge (PLAsTiCC): Data set_ , _arXiv:1810.00001_ (2018) .
  * [33] R.S. de Jong, O. Agertz, A.A. Berbel, J. Aird, D.A. Alexander et al., _4MOST: Project overview and information for the First Call for Proposals_ , _The Messenger_ 175 (2019) 3 [1903.02464].
  * [34] R. Kessler, D. Cinabro, B. Bassett, B. Dilday, J.A. Frieman et al., _Photometric Estimates of Redshifts and Distance Moduli for Type Ia Supernovae_ , _Astrophys. J._ 717 (2010) 40 [1001.0738].
  * [35] M.L. Graham, A.J. Connolly, Ž. Ivezić, S.J. Schmidt, R.L. Jones, M. Jurić et al., _Photometric Redshifts with the LSST: Evaluating Survey Observing Strategies_ , _Astron. J._ 155 (2018) 1 [1706.09507].
  * [36] LSST Dark Energy Science collaboration, _The LSST Dark Energy Science Collaboration (DESC) Science Requirements Document_ , _arXiv:1809.01669_ (2018)  [1809.01669].
  * [37] C. Clarkson, B. Bassett and T.H.-C. Lu, _A General Test of the Copernican Principle_ , _Physical Review Letters_ 101 (2008) 011301 [0712.3457].
  * [38] A. Shafieloo and C. Clarkson, _Model independent tests of the standard cosmological model_ , _Phys. Rev. D_ 81 (2010) 083537 [0911.4858].
  * [39] B. L’Huillier and A. Shafieloo, _Model-independent test of the FLRW metric, the flatness of the Universe, and non-local estimation of H 0 rd_, _J. Cosmology Astropart. Phys_ 2017 (2017) 015 [1606.06832].
  * [40] V. Sahni, A. Shafieloo and A.A. Starobinsky, _Two new diagnostics of dark energy_ , _Phys. Rev. D_ 78 (2008) 103502 [0807.3548].
  * [41] A. Shafieloo, U. Alam, V. Sahni and A.A. Starobinsky, _Smoothing supernova data to reconstruct the expansion history of the Universe and its age_ , _MNRAS_ 366 (2006) 1081 [astro-ph/0505329].
  * [42] A. Shafieloo, _Model-independent reconstruction of the expansion history of the Universe and the properties of dark energy_ , _MNRAS_ 380 (2007) 1573 [astro-ph/0703034].
  * [43] A. Shafieloo, B. L’Huillier and A.A. Starobinsky, _Falsifying $\Lambda$CDM : Model-independent tests of the concordance model with eBOSS DR14Q and Pantheon_, _Phys. Rev. D_ 98 (2018) 083526 [1804.04320].
  * [44] B. L’Huillier, A. Shafieloo and H. Kim, _Model-independent cosmological constraints from growth and expansion_ , _MNRAS_ 476 (2018) 3263 [1712.04865].
  * [45] H. Koo, A. Shafieloo, R.E. Keeley and B. L’Huillier, _Model-independent Constraints on Type Ia Supernova Light-curve Hyperparameters and Reconstructions of the Expansion History of the Universe_ , _ApJ_ 899 (2020) 9 [2001.10887].
  * [46] H. Koo, A. Shafieloo, R.E. Keeley and B. L’Huillier, _Model selection and parameter estimation using the iterative smoothing method_ , _J. Cosmology Astropart. Phys_ 2021 (2021) 034 [2009.12045].
  * [47] H. Koo, R.E. Keeley, A. Shafieloo and B. L’Huillier, _Bayesian vs frequentist: comparing Bayesian model selection with a frequentist approach using the iterative smoothing method_ , _J. Cosmology Astropart. Phys_ 2022 (2022) 047 [2110.10977].


* [14] M. Aizenman, J. Lebowitz, and D. Ruelle, “Some rigorous results on the Sherrington-Kirkpatrick spin glass model,” Communications in Mathematical Physics, vol. 112, pp. 3–20, 1987.
  * [15] A. Montanari and S. Sen, “Semidefinite programs on sparse random graphs and their application to community detection,” in Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing (STOC ’16), pp. 814–827, 2016.
  * [16] A. S. Bandeira, D. Kunisky, and A. S. Wein, “Computational Hardness of Certifying Bounds on Constrained PCA Problems,” 2019, arXiv:1902.07324.
  * [17] W. K. Chen, D. Gamarnik, D. Panchenko, and M. Rahman, “Suboptimality of local algorithms for a class of max-cut problems,” Annals of Probability, vol. 47, pp. 1587–1618, may 2019, 1707.05386.
  * [18] D. Gamarnik and A. Jagannath, “The Overlap Gap Property and Approximate Message Passing Algorithms for $p$-spin models,” 2019, arXiv:1911.06943.
  * [19] A. E. Alaoui and A. Montanari, “Algorithmic Thresholds in Mean Field Spin Glasses,” 2020, arXiv:2009.11481.


# Sharp constants for a class of linear integral operators on higher-
dimensional product spaces 00footnotetext: Key words and phrases: sharp
constants, linear integral, porduct spaces.
2020 Mathematics Subject Classification: Primary 42B25; Secondary 42B20,
47H60, 47B47.

Xiang Li, Zunwei Fu111Corresponding author and Zhongci Hang222Corresponding
author

Abstract: In this paper, we will study a class of linear integral operators
with the nonnegative kernels on higher-dimensional product spaces, the norms
of the operators can be obtained by integral of the product of the kernel
function and finitely many basic functions. As application, we obtain the
sharp constant for linear Hilbert operator on Higher-dimensional product
spaces.

##  1 Introduction

How to compute the sharp constants of some inequalities is a very important
question in harmonic analysis. For a class of linear integral operators with
the nonnegative kernels, which satisfy some homogeneity and the rotational
invariance condition,Beckner [3] obtained the following conclusion which is
also denoted by Stein-Weiss Lemma.

######  Lemma 1.

Suppose that $K$ is a nonnegative kernel defined on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ continuous on any domain that excludes
the point $(0,0)$, homogeneous of degree $-n$.

$K(\delta u,\delta v)=\delta^{-n}K(u,v)$ (1)

and

$K(Ru,Rv)=K(u,v)$ (2)

for any $R\in\mathrm{S}O(n)$. Then $K$ defines an integral operator

$Tf(x)=\int_{\mathbb{R}^{n}}K(x,y)f(y)dy$

which maps $L^{p}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ $i.e.$,

$\|Tf\|_{L^{p}(\mathbb{R}^{n})}\leq C\|f\|_{L^{p}(\mathbb{R}^{n})}$ (3)

holds for $1<p<\infty$, where the optimal constant is given by

$C=\int_{\mathbb{R}^{n}}K(e_{1},y)|y|^{-\frac{n}{p^{\prime}}}dy,$ (4)

where $e_{1}=(1,0,\cdots,0)$ is a unit vector in the first coordinate
direction in $\mathbb{R}^{n}$, and $\mathrm{S}O(n)$ denotes a set of the
rotation transformation on $\mathbb{R}^{n}$.

The inequality (1) was given by Stein and Weiss [1], then Beckner [2, 3, 4]
pointed out that the constant $C$ is the best constant and gave the sharp
constant for the Hardy-Littlewood-Sobolev inequality with some power weights
£¬more information readers can refer to [5, 6] . Our main result is inspired
by Lemma 1. For more information about higher-dimensional product spaces, the
reader can refer to [7, 8, 9]. Now , we give the definition of a class of
linear integral operator with nonnegative kernel on higher-dimensional product
spaces.

######  Definition 1.

Let $m\in\mathbb{N},n_{i}\in\mathbb{N},x_{i}\in\mathbb{R}^{n_{i}},1\leq i\leq
m$, and $f$ be a nonnegative measurable function on
$\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$.Then
operator $T_{m}$ with the kernel $K$ is defined by

$T_{m}f(x)=\int_{\mathbb{R}^{n_{1}}}\cdots\int_{\mathbb{R}^{n_{m}}}K(x_{1},y_{1})\cdots
K(x_{m},y_{m})f(y_{1},y_{2}\cdots y_{m})dy_{1}dy_{2}\cdots dy_{m}.$ (5)

It is not difficult to find that our work is actually to generalize the Stein-
Weiss Lemma in higher-dimensional product space. Now, we give our main result.

######  Theorem 1.

Let $1<p<\infty$,
$m\in\mathbb{N},n_{i}\in\mathbb{N},x_{i}\in\mathbb{R}^{n_{i}},1\leq i\leq m$.
If $f\in
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})$,
then we have

$\|T_{m}f\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})}\leq
C\|f\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})}.$
(6)

Moreover,

$\|T_{m}f\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})\rightarrow
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})}=C,$
(7)

where

$C=\int_{\mathbb{R}^{n_{1}}}\cdots\int_{\mathbb{R}^{n_{m}}}K(e_{1},y_{1})\cdots
K(e_{m},y_{m})|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}\cdots|y_{m}|^{-\frac{n_{m}}{p}}dy_{1}dy_{2}\cdots
dy_{m}.$ (8)

To prove above Theorem, we first provide some definitions and Lemmas which
will be used in the following.Some Lemmas are well known in many classic books
so we omit their proofs.

##  2 Auxiliary lemmas

######  Definition 2.

A topological group $\mathcal{G}$ is Hausdorff topological spaces that is also
a group with law $(x,y)\rightarrow xy$ such that the maps $(x,y)\rightarrow
xy$ and $x\rightarrow x^{-1}$ are continuous.

######  Definition 3.

A topological group $\mathcal{G}$ is locally compact, if each point has a
compact neighborhood.

######  Lemma 2.

A topological group $\mathcal{G}$ is locally compact if and only if the
identity element has a compact neighborhood.

######  Lemma 3.

The product space $X=X_{1}\times\cdots\times X_{n}$ is a locally compact
topological group if and only if each $X_{i},i=1,2,\ldots,n$ ,is a locally
compact topological group.

######  Lemma 4.

Suppose that $\mathcal{G}$ is a locally compact topological group. Then
$\mathcal{G}$ possesses a positive measure $\mu$ on the Borel sets such that
$\mu$ is nonzero on all nonempty open sets and is left invariant, which means
$\mu(sA)=\mu(A)$ for all $s\in\mathcal{G}$ and any measurable set A.
Furthermore, left Haar measure is unique up to positive multiplicative
constants. A locally compact topological group $\mathcal{G}$ with a Haar
measure $\mu$ is denoted by $(\mathcal{G},\mu)$.

The proofs of Lemmas 2-4 can be found in [10, 11] .

######  Lemma 5.

(1) The multiplicative group $\mathbb{R}^{+}=(0,\infty)$ is a locally compact
group, and $G=(0,\infty)^{n}$.
(2) For the multiplicative group $G$ with group operation

$xy=(x_{1}y_{1},\ldots,x_{n}y_{n}),$

the Haar measure $\mu$ on $G$ is $\frac{dx}{x_{1}x_{2}\cdots x_{n}}$, and
$\mu$ is a $\sigma$-finite measure.
(3) If $A^{-1}={x^{-1}:x\in A}$, then $\mu(A^{-})=\mu(A)$.

###### Proof.

The proof of the first assertion of Lemma 5 is trivial.We omit it here. Let us
turn to the proof of $(2)$ and $(3)$. For all $c=(c_{1},c_{2},\ldots,c_{n})\in
G$, set $y=(y_{1},y_{2},\ldots,y_{n})$ and $y_{i}=$
$c_{i}x_{i},i=1,2,\ldots,n$. We have

$dy=\prod_{i=1}^{n}c_{i}dx_{i}.$

For any measurable set $A$ in $G$, we have

$\mu(cA)=\int_{cA}\frac{dx}{x_{1}\cdots x_{n}}=\int_{A}\frac{dy}{y_{1}\cdots
y_{n}}=\mu(A),$

where we use the change of variables $x=cy$. Assume that
$A_{m}=\left(\frac{1}{m},m\right)^{n},m\in\mathbb{Z}^{+}$, then
$\mu(A_{m})<\infty$. Since

$G=\bigcup_{m=1}^{\infty}\left(\frac{1}{m},m\right)^{n},$

then $\mu$ is a $\sigma$-finite measure on G. This has finished the proof of
(2).

Assume that $0\leq c_{i}<d_{i}\leq\infty,i=1,2,\ldots,n,1/0=\infty$, and
$1/\infty=0$. If $A=(c_{1},d_{1})\times\cdots\times(c_{n},d_{n})$, then
$A^{-}=(d_{1}^{-},c_{1}^{-})\times\cdots\times(d_{n}^{-},c_{n}^{-})$. By using
the change of variables $x=cy$, we have

$\mu(A^{-})=\int_{A^{-}}\frac{dx}{x_{1}\cdots
x_{n}}=\int_{A}\frac{dy}{y_{1}\cdots y_{n}}=\mu(A).$

By the classical method in measure theory, one concludes $\mu(A^{-})=\mu(A)$
holds for any Borel measurable $A$ in $G$. This has finished the proof of (3).
∎

######  Definition 4.

Define

$L^{p}(\mathcal{G},\mu)=\left\\{f:\|f\|_{L^{p}(\mathcal{G},\mu)}=\left(\int_{\mathcal{G}}|f(x)|^{p}d\mu(x)\right)^{1/p}<+\infty\right\\},\quad
1\leq p<+\infty,$

when $p=\infty$, we have

$L^{\infty}(\mathcal{G},\mu)={f:\|f\|_{L^{\infty}}<+\infty}.$

It is well known that Hölder’s inequality holds in $L^{p}(\mathcal{G},\mu)$.
Moreover, Fubini’s theorem also holds, provided that $\mu$ is a
$\sigma$-finite measure on $\mathcal{G}$. In this paper, $L^{p}(G,\mu)$
denotes the function space on the locally compact group $G=(0,\infty)^{n}$
with respect to the Haar measure $\mu=\frac{dx}{x_{1}x_{2}\cdots x_{n}}$.
Others such as $L^{p}(\mathbb{R}^{+}),L^{p}(\mathbb{R}^{n})$ and $L^{p}(G)$
are ones with respect to the Lebesgue measure.

######  Lemma 6.

(see[10]) Let $(X,\mu)$ and $(Y,v)$ be two $\sigma$-finite measure spaces.Let
$T$ be a linear operator defined the set of all finitely simple functions on X
and taking values in the set of measurable functions on $Y$.Let $1\leq
p_{0},p_{1},q_{0},q_{1}\leq\infty$ and we assume that

$\displaystyle\|T(f)\|_{L^{q_{0}}}\leq M_{0}\|f\|_{L^{p_{0}}},$
$\displaystyle\|T(f)\|_{L^{q_{1}}}\leq M_{1}\|f\|_{L^{p_{1}}}$

hold for all finitely simple functions $f$ on $X$. Then for all $0<\theta<1$ ,
we have

$\|T(f)\|_{L^{q}}\leq M_{0}^{1-\theta}M_{1}^{\theta}\|f\|_{L^{p}}$

hold for all finitely simple functions $f$ on $X$, where

$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}\quad\text{ and
}\quad\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}\text{. }$ (9)

Consequently, when $p<\infty$, by density, $T$ has a unique bounded extension
from $L^{p}(X,\mu)$ to $L^{q}(Y,v)$ where $p$ and $q$ are as in Theroem 9.

######  Lemma 7.

Let $(X,\mu)$ and$(Y,v)$ be two $\sigma$-finite measure spaces. Let $T$ be a
linear operator defined on the set of all finitely simple functions on $X$ and
taking values in the set of measurable functions on $Y$. If $T$ satisfies

$\|T\|_{L^{\infty}(X,\mu)\rightarrow L^{\infty}(Y,v)}\leq C$

and

$\|T\|_{L^{1}(X,\mu)\rightarrow L^{1}(Y,v)}\leq C,$

then the operator $T$ maps $L^{p}$ to $L^{p}$, and

$\|T\|_{L^{p}\rightarrow L^{p}}\leq C,$

for $1\leq p\leq\infty$ with

$0\leq\frac{1}{p}\leq 1.$

######  Lemma 8.

Suppose that a function $K:\mathbb{R}^{n}\rightarrow[0,\infty)$, and each $f$
is a measure function on $\mathbb{R}^{n}$. Define linear integral operator as

$\widetilde{T}_{K}(f)(x)=:\int_{\mathbb{R}^{n}}K(y-x)f(y)dy,$

where $x,y\in\mathbb{R}^{n}$. If $1\leq p\leq\infty$ and

$0\leq\frac{1}{p}\leq 1,$

then we have

$\|\widetilde{T}_{K}\|_{L^{p}\rightarrow
L^{p}}=\|K\|_{L^{1}(\mathbb{R}^{n})}.$ (10)

######  Corollary 1.

Suppose that a function $H:\mathbb{G}^{n}\rightarrow[0,\infty)$, and $f$ is a
measure function on $\mathbb{G}^{n}$ with respect to Haar measure. Define
linear integral operator as

$\widetilde{T}_{H}(f)(x):=\int_{\mathbb{G}^{n}}H(yx^{-1})f(y)d\mu(y),$

where $x,y\in\mathbb{G}^{n}$. If $1\leq p\leq\infty$ and

$0\leq\frac{1}{p}\leq 1,$

then we have

$\|\widetilde{T}_{H}\|_{L^{p}\rightarrow
L^{p}}=\|H\|_{L^{1}(\mathbb{G}^{n})}.$

##  3 Proof of Theorem 1

We merely give the proof with the case $m=2$ , and the same is true for the
general case $m>2$. Firstly, we prove the sufficiency of Theorem 1. Since
$T_{2}$ is a positive operator, without loss of generality,we take $f\leq 0$.
By the polar coordinates integral formula,we have

$\displaystyle\|T_{2}(f)\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}$
(11) $\displaystyle=$
$\displaystyle\left\\{\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\left(\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(x_{1},y_{1})K(x_{2},y_{2})f(y_{1},y_{2})dy_{1}dy_{2}\right)^{p}dx_{1}dx_{2}\right\\}^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\left\\{\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}\left[\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}\left(\int_{\mathbb{S}^{n_{1}-1}}\int{\mathbb{S}^{n_{2}-1}}K(t_{1}x_{1}^{\prime},t_{1}y_{1}^{\prime})K(t_{2}x_{2}^{\prime},t_{2}y_{2}^{\prime})f(t_{1}y_{1}^{\prime},t_{2}y_{2}^{\prime})\right.\right.\right.$
$\displaystyle\left.\left.\left.\times
d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime})\right)t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right]^{p}d\sigma(x_{1}^{\prime})d\sigma(x_{2}^{\prime})t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right\\}^{\frac{1}{p}}.$

Let

$F(f)(t_{1},t_{2},x_{1}^{\prime},x_{2}^{\prime}):=\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K(t_{1}x_{1}^{\prime},t_{1}y_{1}^{\prime})K(t_{2}x_{2}^{\prime},t_{2}y_{2}^{\prime})f(t_{1}y_{1}^{\prime},t_{2}y_{2}^{\prime})d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime}).$
(12)

Using Minkowski’s inequality and (2),we have

$\displaystyle\|T_{2}$
$\displaystyle(f)\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}$ (13)
$\displaystyle=$
$\displaystyle\left\\{\int_{0}^{\infty}\int_{0}^{\infty}\left\\{\left[\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}\left(\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{1}}F(f)(t_{1},t_{2},x_{1}^{\prime},x_{2}^{\prime})t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right)^{p}\right.\right.\right.$
$\displaystyle\left.\left.\left.\times
d\sigma\left(x_{1}^{\prime}\right)d\sigma(x_{2}^{\prime})\right]^{\frac{1}{p}}\right\\}^{p}t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right\\}^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\left\\{\int_{0}^{\infty}\int_{0}^{\infty}\left\\{\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}\left(\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}\left|F\left(f\right)\left(t_{1},t_{2},x_{1}^{\prime},x_{2}^{\prime}\right)\right|^{p}d\sigma(x_{1}^{\prime})d\sigma(x_{2}^{\prime})\right)^{\frac{1}{p}}\right.\right.$
$\displaystyle\left.\left.\times
t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right\\}^{p}t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right\\}^{\frac{1}{p}}.$

Let $t_{1},t_{2}$ be fixed temporarily.By the definition of $F$ in (12), we
may regard $F$ as a linear operator from
$L^{p}(\mathbb{S}^{n_{1}}\times\mathbb{S}^{n_{2}})$ to
$L^{p}(\mathbb{S}^{n_{1}}\times\mathbb{S}^{n_{2}})$.Next , we consider the
boundedness of the operator $F$.By the condition of
(2),$K(t_{1}x_{1},t_{1},y_{1}),K(t_{1},x_{1},t_{2}y_{2})$ as a function of the
variables $x_{1},x_{2},y_{1},y_{2}$ is rotation-invarint.

Define

$A(t_{1},t_{2})=\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K(t_{1}e_{1},t_{1}y_{1})K(t_{2}e_{2},t_{2}y_{2})d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime}),$
(14)

where $e_{i}=(1,0),i=1,2$. By the rotation invariance of $K$, we have

$\displaystyle\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K(t_{1}e_{1},t_{1}y_{1})K(t_{2}x_{e},t_{2}y_{2})d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime})$
$\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K(t_{1}x_{1},t_{1}y_{1})K(t_{2}x_{2},t_{2}y_{2})d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime}).$

The linear operator $F$ defined in (12) satisfies two conditions (1), (2) in
Lemma 7. Thus, we have

$\displaystyle\left(\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}\left|F(f)(t_{1},t_{2},x_{1}^{\prime},x_{2}^{\prime})\right|^{p}d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime})\right)^{\frac{1}{p}}$
(15) $\displaystyle\quad\leq$ $\displaystyle
A(t_{1},t_{2})\left\\{\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}\left|f(t_{1}y_{1}^{\prime},t_{2}y_{2}^{\prime})\right|^{p}d\sigma(y_{1}^{\prime})d\sigma(y_{1}^{\prime})\right\\}^{\frac{1}{p_{i}}}.$

Set

$h(t_{1},t_{2}):=\left(\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}|f(t_{1}y_{1},t_{2}y_{2})|^{p}d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime})\right)^{\frac{1}{p}},$

for $0<t_{1}<\infty,0<t_{2}<\infty$. By a computation ,we have

$\|h\|_{L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})}=\|f\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}.$
(16)

Combining the inequalities (13),(15) with (16) yields that

$\displaystyle\|T_{2}(f)\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}$
(17) $\displaystyle\leq$
$\displaystyle\left\\{\int_{0}^{\infty}\int_{0}^{\infty}\left\\{\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}A(t_{1},t_{2})\left\\{\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}\left|f(t_{1}y_{1}^{\prime},t_{2}y_{2}^{\prime})\right|^{p}d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime})\right\\}^{\frac{1}{p}}\right.\right.$
$\displaystyle\left.\left.\times
t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right\\}^{p}\times
t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right\\}^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\left\\{\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}h(t_{1},t_{2})\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K(t_{1}x_{1}^{\prime},t_{1}y_{1}^{\prime})K(t_{2}x_{2}^{\prime},t_{2}y_{2}^{\prime})d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime})\right.\right.$
$\displaystyle\left.\left.\times
t_{1}^{\frac{n_{1}}{p^{\prime}}-1}t_{2}^{\frac{n_{2}}{p^{\prime}}-1}dt_{1}dt_{2}\right]^{p}t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right\\}^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\left\\{\int_{0}^{\infty}\int_{0}^{\infty}\left[\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}h(t_{1},t_{2})\left(\frac{t_{1}}{t}\right)^{\frac{n_{1}}{p^{\prime}}}\left(\frac{t_{2}}{t}\right)^{\frac{n_{2}}{p^{\prime}}}\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K\left(x_{1}^{\prime},\frac{t_{1}}{t}y_{1}^{\prime}\right)K\left(x_{2}^{\prime},\frac{t_{2}}{t}y_{2}^{\prime}\right)\right.\right.$
$\displaystyle\left.\left.\times
d\sigma\left(y_{1}^{\prime}\right)d\sigma\left(y_{2}^{\prime}\right)\frac{dt_{1}}{t_{1}}\frac{dt_{2}}{t_{2}}\right]^{p}\frac{dt_{1}}{t_{1}}\frac{dt_{2}}{t_{2}}\right\\}^{\frac{1}{p}},$

where we use the homogeneity of degree $-n$ of the kernel $K$.Notice that the
integral

$\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K\left(x_{1}^{\prime},\frac{t_{1}}{t}y_{1}^{\prime}\right)K\left(x_{2}^{\prime},\frac{t_{2}}{t}y_{2}^{\prime}\right)d\sigma(y_{1}^{\prime})d\sigma(y_{2}^{\prime})$

is independent of $x_{1}^{\prime},x_{2}^{\prime}$ because of the rotation of
$K$,i.e.,it only depends on $t_{1},t_{2}$. Set

$H(t_{1},t_{2}):=t_{1}^{\frac{n_{1}}{p^{\prime}}}t_{2}^{\frac{n_{2}}{p^{\prime}}}\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K(x_{1}^{\prime},t_{1}y_{1}^{\prime})K(x_{2}^{\prime},t_{2}y_{2}^{\prime})d\sigma(y_{1}^{\prime})d\sigma(y_{1}^{\prime}),$
(18)

simple computation of integral implies that

$\|H\|_{L^{1}(\mathbb{G}^{1}\times\mathbb{G}^{2})}=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(e_{1},y_{1})K(e_{2},y_{2})|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}dy_{1}dy_{2}.$

Regarding $H$ in (18) as a integral kernel,we define a new operator

$\widetilde{T}_{H}(h_{1},h_{2})(t):=\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}H\left(\frac{t_{1}}{t},\frac{t_{2}}{t}\right)h\left(t_{1},t_{2}\right)\frac{dt_{1}}{t_{1}}\frac{dt_{2}}{t_{2}},$
(19)

where $h(t_{1},t_{2})$ defined (16).We have

$\displaystyle\widetilde{T}_{H}\left(h\right)(t_{1},t_{2})$ (20)
$\displaystyle\quad=$
$\displaystyle\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}h\left(t_{1},t_{2}\right)\left(\frac{t_{1}}{t}\right)^{\frac{n_{1}}{p^{\prime}}}\left(\frac{t_{2}}{t}\right)^{\frac{n_{2}}{p^{\prime}}}\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K\left(x_{1}^{\prime},\frac{t_{1}}{t}y_{1}^{\prime}\right)K\left(x_{2}^{\prime},\frac{t_{2}}{t}y_{2}^{\prime}\right)$
$\displaystyle\times
d\sigma\left(y_{1}^{\prime}\right)d\sigma\left(y_{2}^{\prime}\right)\frac{dt_{1}}{t_{1}}\frac{dt_{2}}{t_{2}}.$

It is easy to check that operator $\widetilde{T}_{H}$ satisfies the all
conditions of Corollary 1. Thus it follows from (16) and (19) that

$\displaystyle\|\widetilde{T}_{H}(h)\|_{L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})}$
$\displaystyle\leq\|H\|_{L^{1}(\mathbb{G}^{1}\times\mathbb{G}^{2})}\|h\|_{L^{p}{(\mathbb{G}^{1}\times\mathbb{G}^{2})}}$
(21)
$\displaystyle=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(e_{1},y_{1})K(e_{2},y_{2})\left|y_{1}\right|^{-\frac{n_{1}}{p}}\left|y_{2}\right|^{-\frac{n_{2}}{p}}dy_{1}dy_{2}\left\|f\right\|_{L^{p}\left(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\right)}.$

Consequently,it implies from (17),(20) and (21) that

$\displaystyle\|{T}_{2}(f)\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}$
$\displaystyle\leq\|\widetilde{T}_{H}(h)\|_{L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})}$
(22)
$\displaystyle\leq\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(e_{1},y_{1})K(e_{2},y_{2})|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}dy_{1}dy_{2}\|f\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}.$

The inequality (22) implies that

$\|T_{2}\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})\rightarrow
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}\leq\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(e_{1},y_{1})K(e_{2},y_{2})|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}dy_{1}dy_{2}.$
(23)

Next,we will prove the reverse inequality.In fact, it suffices to prove

$\|T_{2}\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})\rightarrow
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}\geq\|\tilde{T}_{H}\|_{L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})\rightarrow
L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})},$

where $H$ is defined as(19). For any function $h$ :
$\mathbb{G}\rightarrow[0,\infty)$, we define the corresponding function
$f:\mathbb{R}^{n}\rightarrow[0,\infty)$ as

$f(y_{1},y_{2})=\omega_{n_{1}-1}^{-\frac{1}{p}}\omega_{n_{2}-1}^{-\frac{1}{p}}|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}h(|y_{1}|,|y_{2}|).$

Clearly , $f$ is a radial nonnegative function and it easily implies that

$\displaystyle\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}K(t_{1}x_{1}^{\prime},t_{1}y_{1}^{\prime})K(t_{2}x_{2}^{\prime},t_{2}y_{2}^{\prime})f(t_{1}y_{1},t_{2}y_{2})d\sigma(y_{1})d\sigma(y_{2})$
(24) $\displaystyle=$
$\displaystyle\omega_{n_{1}-1}^{-\frac{1}{p}}\omega_{n_{2}-1}^{-\frac{1}{p}}A(t_{1},t_{2})t_{1}^{-\frac{1}{p}}t_{2}^{-\frac{1}{p}}h(t_{1},t_{2}).$

According to (11) and (24) , we have

$\displaystyle\|T_{2}(f)\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}$
$\displaystyle\quad=$
$\displaystyle\left(\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{S}^{n_{1}-1}}\int_{\mathbb{S}^{n_{2}-1}}\left(\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}\omega_{n_{1}-1}^{-\frac{1}{p}}\omega_{n_{2}-1}^{-\frac{1}{p}}A(t_{1},t_{2})t_{1}^{-\frac{n_{1}}{p}}t_{2}^{-\frac{n_{2}}{p}}h(t_{1},t_{2})t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right)^{p}\right.$
$\displaystyle\left.\times
d\sigma(x_{1}^{\prime})d\sigma(x_{2}^{\prime})t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right)^{\frac{1}{p}}$
$\displaystyle\quad=$
$\displaystyle\left(\int_{0}^{\infty}\int_{0}^{\infty}\left(\int_{\mathbb{G}^{1}}\int_{\mathbb{G}^{2}}A(t_{1},t_{2})h(t_{1},t_{2})t_{1}^{\frac{n_{1}}{p^{\prime}}}t_{2}^{\frac{n_{2}}{p^{\prime}}}\frac{dt_{1}}{t_{1}}\frac{dt_{1}}{t_{1}}\right)^{p}t_{1}^{n_{1}-1}t_{2}^{n_{2}-1}dt_{1}dt_{2}\right)^{\frac{1}{p}}.$

Combining (14),(18) with the homogeneity of $-n$, we obtain

$A(t_{1},t_{2})t_{1}^{\frac{n_{1}}{p}}t_{2}^{\frac{n_{2}}{p}}=H(\frac{t_{1}}{t},\frac{t_{2}}{t})t_{1}^{\frac{n_{1}}{p}}t_{2}^{\frac{n_{2}}{p}}.$

Consequently, it follows that

$\|T_{2}(f)\|_{L^{P}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}=\|\widetilde{T}_{H}(h)\|_{L^{P}(\mathbb{G}^{1}\times\mathbb{G}^{2})}.$
(25)

Since operator $\widetilde{T}_{H}$ satisfies all the conditions of Corollary
10, we have

$\|\widetilde{T}\|_{L^{p}\rightarrow
L^{p}}=\|H\|_{L^{1}(\mathbb{G}^{1}\times\mathbb{G}^{2})}=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(e_{1},y_{1})K(e_{2},y_{2})|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}dy_{1}dy_{2}.$
(26)

Since $H$ is nonnegative , we conclude from (25) that

$\displaystyle\|\widetilde{T}_{H}\|_{L^{p}\rightarrow
L^{p}}=\sup_{\begin{subarray}{c}\|h\|\end{subarray}_{L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})}\neq
0,h\geq
0}\frac{\|\widetilde{T}_{H}(h)\|_{L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})}}{\|h\|_{L^{p}(\mathbb{G}^{1}\times\mathbb{G}^{2})}}$
(27)
$\displaystyle\leq\sup_{\begin{subarray}{c}\|f\|\end{subarray}_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}\neq
0}\frac{\|T_{2}(f)\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}}{\|f\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}}$
$\displaystyle=\|T_{2}\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})\rightarrow
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}.$

It immediately follows from (26) and (27) that

$\|T_{2}\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})\rightarrow
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}\geq\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(e_{1},y_{1})K(e_{2},y_{2})|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}dy_{1}dy_{2}.$
(28)

Both the inequalities (23) and (28) , yield

$\|T_{2}\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})\rightarrow
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})}=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}K(e_{1},y_{1})K(e_{2},y_{2})|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{2}}{p}}dy_{1}dy_{2}.$

So far, we have completed the proof of the Theorem 1.

##  4 Sharp constant for linear Hilbert operator

In this section, we will consider linear Hilbert operator which can be found
in [10]. Applying Theorem 1, we can easily obtain the sharp constant on
higher-dimensional product spaces.

######  Definition 5.

Let $m\in\mathbb{N},n_{i}\in\mathbb{N},x_{i}\in\mathbb{R}^{n_{i}},1\leq i\leq
m$, and f be a nonnegative measurable function on
$\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}}$.
Then the linear Hilbert operator $T^{*}_{m}$ on higher-dimensional product
spaces is defined by

$T^{*}_{m}f(x)=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\cdots\int_{\mathbb{R}^{nm}}\frac{f(y_{1},\ldots,y_{m})}{\prod_{i=1}^{m}(|x_{i}|^{n_{i}}+|y_{i}|^{n_{i}})}dy_{1}dy_{2}\cdots
dy_{m}.$ (29)

######  Theorem 2.

Let
$1<p<\infty,m\in\mathbb{N},n_{i}\in\mathbb{N},x_{i}\in\mathbb{R}^{n_{i}},1\leq
i\leq m$. If $f\in
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})$,
then we have

$\|T^{*}_{m}f\|_{L^{p}(\mathbb{R}^{n_{1}}\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})}\leq
A.$

Morever,

$\|T^{*}_{m}f\|_{L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})\rightarrow
L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\times\cdots\times\mathbb{R}^{n_{m}})}=A,$

where

$A=\prod_{i=1}^{m}\frac{2\pi^{\frac{n_{i}}{2}}}{\Gamma(\frac{n_{i}}{2})n_{i}}m\Gamma(1-\frac{1}{p})\Gamma(\frac{1}{p}).$

###### Proof of Theorem 2.

Obviously, the operator $T^{*}_{m}$ satisfies the condition of Theorem 1, so
we have

$\displaystyle T^{*}_{m}f(x)=$
$\displaystyle\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\cdots\int_{\mathbb{R}^{n_{m}}}\frac{f(y_{1},\ldots,y_{m})}{\prod_{i=1}^{m}(|x_{i}|^{n_{i}}+|y_{i}|^{n_{i}})}dy_{1}dy_{2}\cdots
dy_{m}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\cdots\int_{\mathbb{R}^{nm}}\frac{|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{1}}{p}}\cdots|y_{m}|^{\frac{n_{m}}{p}}}{\prod_{i=1}^{m}(1+|y_{i}|^{n_{i}})}dy_{1}dy_{2}\cdots
dy_{m}.$

We merely give the proof with the case $m=2$ , and the same is true for the
general case $m>2$. By the polar coordinates integral formula,we have

$\displaystyle\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\frac{|y_{1}|^{-\frac{n_{1}}{p}}|y_{2}|^{-\frac{n_{1}}{p}}}{(1+|y_{1}|^{n_{1}})(1+|y_{2}|^{n_{2}})}dy_{1}dy_{2}$
$\displaystyle=$
$\displaystyle\omega_{n_{1}-1}\omega_{n_{2}-1}\int_{0}^{\infty}\int^{\infty}_{0}\frac{r_{1}^{\frac{n_{1}}{p}-1}r_{2}^{\frac{n_{2}}{p}-1}}{(1+r_{1}^{n_{1}})(1+r_{2}^{n_{2}})}dr_{1}dr_{2}$
$\displaystyle=$
$\displaystyle\frac{\omega_{n_{1}-1}}{n_{1}}\frac{\omega_{n_{2}-1}}{n_{2}}\int_{0}^{\infty}\int_{0}^{\infty}\frac{s_{1}^{-\frac{1}{p}}s_{2}^{-\frac{1}{p}}}{(1+s_{1})(1+s_{2})}ds_{1}ds_{2}.$

Define

$F(\beta):=\int_{0}^{\infty}\int_{0}^{\infty}\frac{s_{1}^{-\beta}}{1+s_{1}}\frac{s_{2}^{-\beta}}{1+s_{2}}ds_{1}ds_{2},$

then we have

$\displaystyle F(\beta)$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\frac{s_{1}^{-\beta}}{1+s_{1}}\frac{s_{2}^{-\beta}}{1+s_{2}}ds_{1}ds_{2}$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{(1+s_{1})s_{1}^{\beta}}\frac{1}{(1+s_{2})s_{2}^{\beta}}ds_{1}ds_{2}$
$\displaystyle=\int_{0}^{\infty}\frac{1}{(1+s_{1})s_{1}^{\beta}}ds_{1}\int_{0}^{\infty}\frac{1}{(1+s_{2})s_{2}^{\beta}}ds_{2}$
$\displaystyle=\int_{0}^{1}(1-t_{1})^{-\beta}t_{1}^{\beta-1}dt_{1}\int_{0}^{1}(1-t_{2})^{-\beta}t_{2}^{\beta-1}dt_{2}$
$\displaystyle=2B(1-\beta,\beta)$
$\displaystyle=2\Gamma\left(1-\frac{1}{p}\right)\Gamma\left(\frac{1}{p}\right).$

Thus, we can obtain

$\displaystyle T^{*}_{2}f(x)$
$\displaystyle=\frac{\omega_{n_{1}-1}}{n_{1}}\frac{\omega_{n_{2}-1}}{n_{2}}2\Gamma\left(1-\frac{1}{p}\right)\Gamma\left(\frac{1}{p}\right)$
$\displaystyle=\frac{2\pi^{\frac{n_{1}}{2}}}{\Gamma\left(\frac{n_{1}}{2}\right)n_{1}}\frac{2\pi^{\frac{n_{2}}{2}}}{\Gamma\left(\frac{n_{2}}{2}\right)n_{2}}2\Gamma\left(1-\frac{1}{p}\right)\Gamma\left(\frac{1}{p}\right).$

This completes the proof of Theorem 2. ∎

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant
No. 12071473) and Shandong Jianzhu University Foundation (Grant No.
XNBS20099).

## References

  * [1] Stein E M ,Weiss G . Fractional integral analysis(in Russian).Mat Sbornik, 1 ,1938, 46: 471-497.
  * [2] Beckner W. Geometric inequalities in Fourier analysis. In: Essays on Fourier Analysis in Honor of Elias M. Stein. Princeton Mathematical Series, vol. 42. Princeton: Princeton University Press, 1995, 36¨C68.
  * [3] Beckner W. Pitt¡¯s inequality with sharp convolution estimates. Proc Amer Math Soc, 2008, 136: 1871¨C1885.
  * [4] Beckner W. Weighted inequalities and Stein-Weiss potentials. Forum Math, 2008, 20: 587¨C606.
  * [5] Benyi A, Oh T. Best constants for certain multilinear integral operators. J Inequal Appl, 2006: 1¨C12.
  * [6] Di W U , Yan D Y , Science M C , et al. Sharp constants for a class of multilinear integral operators and some applications[J]. Science China(Mathematics), 2016.
  * [7] ShiMo, et al. ”Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces.” Science China Mathematics ,2012.
  * [8] Q.J. He, X. Li and D.Y. Yan, Sharp bounds for Hardy type operators on higher-dimensional product spaces. Front. Math. China 13(6) ,2018, 1341–1353.
  * [9] Lu S Z,Yan D Y,Zhao F Y.Sharp bounds for Hardy type operators on higher-dimensional product spaces.J Inequal Appl,2013(1):148-159.
  * [10] Grafakos L. Classical Fourier Analysis, 2nd ed. New York: Springer Science, Business Media LLC, 2008
  * [11] Lang S. Real Analysis, 2nd ed. Reading, MA: Addison-Wesley, 1983.

Xiang Li
School of Science
Shandong Jianzhu University
Jinan, 250000
P. R. China

_E-mail address_<EMAIL_ADDRESS>
Zunwei Fu
Department of Mathmatics
Linyi University
Linyi, 270005
P. R. China

_E-mail address_<EMAIL_ADDRESS>
Zhongci Hang
School of Science
Shandong Jianzhu University
Jinan, 250000
P. R. China

_E-mail address_<EMAIL_ADDRESS>