# Nonlocal measurement of quasiparticle charge and energy relaxation in
proximitized semiconductor nanowires using quantum dots

Guanzhong Wang  Tom Dvir<EMAIL_ADDRESS>Nick van Loo  Grzegorz P. Mazur
QuTech and Kavli Institute of NanoScience, Delft University of Technology,
2600 GA Delft, The Netherlands  Sasa Gazibegovic  Ghada Badawy  Erik P. A. M.
Bakkers  Department of Applied Physics, Eindhoven University of Technology,
5600 MB Eindhoven, The Netherlands  Leo P. Kouwenhoven  Gijs de Lange
<EMAIL_ADDRESS>Microsoft Quantum Lab Delft, 2600 GA Delft, The
Netherlands

###### Abstract

The lowest-energy excitations of superconductors do not carry an electric
charge, as their wave function is equally electron-like and hole-like. This
fundamental property is not easy to study in electrical measurements that rely
on the charge to generate an observable signal. The ability of a quantum dot
to act as a charge filter enables us to solve this problem and measure the
quasiparticle charge in superconducting-semiconducting hybrid nanowire
heterostructures. We report measurements on a three-terminal circuit, in which
an injection lead excites a non-equilibrium quasiparticle distribution in the
hybrid system, and the electron or hole component of the resulting
quasiparticles is detected using a quantum dot as a tunable charge and energy
filter. The results verify the chargeless nature of the quasiparticles at the
gap edge and reveal the complete relaxation of injected charge and energy in a
proximitized nanowire, resolving open questions in previous three-terminal
experiments.

## Introduction

The elementary excitations in superconductors (SC) are Bogoliubov
quasiparticles (QPs), i.e., the superposition of an electron excitation with
amplitude $u(E)$ and a hole excitation with amplitude $v(E)$, where $E$ is the
energy of the excitation. The electron and hole components are energy-
dependent and are given by
$|u(E)|^{2}=\frac{1}{2}\left(1+\sqrt{E^{2}-\Delta_{\mathrm{SC}}^{2}}/E\right)$
and
$|v(E)|^{2}=\frac{1}{2}\left(1-\sqrt{E^{2}-\Delta_{\mathrm{SC}}^{2}}/E\right)$,
where $\Delta_{\mathrm{SC}}$ is the superconducting energy gap, and the QP
energy obeys $E>\Delta_{\mathrm{SC}}$. The charge of the excitation, given by
$q(E)=e\left(|v(E)|^{2}-|u(E)|^{2}\right)$, varies between $-e$ (electron-
like) when the excitation energy is far above the Fermi energy and $+e$ (hole-
like) when it is far below. In the vicinity of the gap,
$E\approx\Delta_{\mathrm{SC}}$, the QPs consist of nearly equal superpositions
of electron and hole parts as $|u(E)|\approx|v(E)|\approx 1/\sqrt{2}$.
Therefore, their charge approaches zero [1].

In thermal equilibrium and when $k_{\mathrm{B}}T\ll\Delta_{\mathrm{SC}}$,
where $k_{\mathrm{B}}$ is the Boltzmann constant and $T$ the electron
temperature, the presence of the energy gap ensures that almost no QP
excitations exist in the system. However, external perturbations such as
injection of charge can drive the system out of equilibrium, making its
distribution function, $f$, deviate from the Fermi-Dirac distribution,
$f_{\text{FD}}$. This departure can be decomposed into several non-equilibrium
modes [2]. The two most discussed modes in a superconductor are energy and
charge non-equilibrium. In transport experiments, injecting electrons above
the SC gap brings extra charge as well as energy into the system, exciting
both modes. Although the distribution is restored to $f_{\text{FD}}$ far away
from the perturbation, each mode relaxes to the equilibrium over different
length and time scales. This process is studied thoroughly for metallic SCs,
both theoretically and experimentally [3, 4, 5, 6, 7, 8, 9].

The discussion so far considered intrinsic superconductors where the energy
gap and the electron-hole correlations are generated by an internal pairing
mechanism, such as electron-phonon coupling. Similar effects are also found in
SC-proximitized semiconducting (SM) systems, where electron-hole correlations
are induced by the proximity effect instead. Here, similar to the formation of
Andreev bound states (ABSs) in a confined system, Andreev reflection (AR) on
the SM-SC interface pairs states above and below the Fermi surface in the SM
to create a superconducting-like band structure with an induced gap
$\Delta_{\mathrm{ind}}$. Its relative size
$\Delta_{\mathrm{ind}}/\Delta_{\mathrm{SC}}$, being between 0 and 1, is
directly related to the coupling strength between the SC and SM [10]. These
proximitized states are also superpositions of electrons and holes with
energy-dependent amplitudes $u(E)$ and $v(E)$, respectively. Similar to
intrinsic SC, at $E\approx\Delta_{\text{ind}}$, the electron and hole
components are nearly equal, driving the charge of the lowest-energy
proximitized states to zero [11, 12, 13]. This effect was measured both for
discrete ABSs [14] and proximitized semiconducting subbands [15].

Nonlocal transport is a typical experimental tool to study non-equilibrium
modes and their relaxation. Such a setup utilizes two tunnel junctions: an
injector and a detector. The injector junction injects particles into the
system under study, exciting one or more non-equilibrium modes. The detector
junction is usually unbiased and measures the response of the system at some
distance from the injection point. Electron transport is well-suited to
measure the charge non-equilibrium mode [3, 16] and can be adapted to measure
spin imbalance [17, 18]. However, the energy mode is harder to measure this
way, since electron and hole currents flowing into the unbiased probe cancel
each other out by virtue of the charge neutrality of this mode [19]. An energy
non-equilibrium mode will, however, generate a measurable charge current if
the transmission probability of the tunnel barrier, $\mathcal{T}$, is energy-
dependent and filters out only one type of carriers [20, 21]. A semiconducting
quantum dot (QD), with its single-electron orbitals having sharply-peaked
transmission amplitudes only for a particular charge, energy and spin, is such
a transmission filter. Previous works have made use of the energy filtering
effects of a QD to probe the non-equilibrium distribution of quantum Hall edge
states [22]. As demonstrated later in this text, similarly applying the charge
filtering capabilities of QDs opens up new avenues to the study of non-
equilibrium in hybrid SM-SC systems.

Nonlocal conductance (NLC) can also serve to measure the sign of the charge of
ABSs [23, 14] and other effects such as crossed Andreev reflection [24, 25,
26, 27, 23, 28]. Recently it was further suggested as a powerful tool to
measure the induced superconducting gap in semiconducting nanowires [29]. NLC
was used to differentiate between bulk induced gap closing from the presence
of local resonances in tunnel barriers [30, 31]. All reported measurements in
such geometries share common characteristic features—e.g., predominantly anti-
symmetric NLC whose global sign is heavily influenced by the tunnel barrier on
the current-receiving side. We show that these effects can arise from the
charge and energy dependence of $\mathcal{T}$, which is ubiquitous in gate-
defined tunnel barriers.

In this work, we study NLC in a hybrid SC heterostructure. Gate-defined QDs
separating the ohmic leads from the hybrid segment are used as charge and
energy filters. We detect independently the electron and hole components of
the QP wavefunction, observing charge neutrality of the excitations at the
superconducting gap edge. Applying a magnetic field that closes the induced
gap shows that the charge-to-energy conversion is independent of the presence
of an induced spectral gap and only requires Andreev reflection at the SM-SC
interface. Finally, using QDs to inject and detect QPs, we demonstrate
complete relaxation of the detected QPs into the lowest excited states.

## Results

### Qualitative description of the experiment

Fig. 1: a–c. Illustrations of non-equilibrium modes in proximitized
superconductors. a: Thermal equilibrium with no QP present. b: Injection of
electrons excites both charge and energy non-equilibrium modes. c: Pure energy
non-equilibrium mode after the charge mode has relaxed. The illustrations in
this paper are not to scale. d, e. A scheme of the nonlocal experiment: a
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$-long, grounded superconductor with a
gap $\Delta_{\text{SC}}$ is attached to the middle section of the nanowire,
inducing a proximity gap $\Delta_{\text{ind}}$. On the left side, a tunnel
junction under finite bias injects current. On the right side, a quantum dot
filters either electrons (d) or holes (e) depending on its occupation. Arrows
indicate directions of electron hopping. f. Device schematic and measurement
setup. Sketches in the InSb wire segments indicate the potential landscape
created by the finger gates. The voltages applied on each group of finger
gates are schematically represented by the height of the voltage sources in
the circuit diagram. Unlabeled voltages are kept constant within each data
set. g. False-colored scanning electron microscopy (SEM) image of a
lithographically identical device to Devices A and B presented in this work.
Bottom gate electrodes are colored in red and separated from the InSb nanowire
by a thin layer of atomic-layer-deposited (ALD) Al2O3 dielectric of around
$20\text{\,}\mathrm{n}\mathrm{m}$ (invisible in this image). The middle
section of the nanowire is covered by a thin, superconducting layer of Al film
colored in cyan. Two Au ohmic leads are colored in yellow and form the left
and right contacts. Scale bar is $500\text{\,}\mathrm{n}\mathrm{m}$.

Figure 1(a) sketches a hybrid SC heterostructure attached to a normal lead (N)
through a tunnel junction. The proximity effect from the SC opens an induced
gap $\Delta_{\mathrm{ind}}$. The system is in equilibrium, and if
$\Delta_{\mathrm{ind}}\gg k_{\mathrm{B}}T$, almost no QPs are excited in the
system. When we apply a bias $V_{\textrm{L}}$ such that
$-eV_{\textrm{L}}>\Delta_{\textrm{ind}}$ [Fig. 1(b)], electrons are injected
into the proximitized semiconductor. However, the native excitations are not
electrons, but Bogoliubov QPs with a charge generally smaller than the
elementary charge. Thus, pure electron-like excitations in the injecting lead
are eventually converted in the SM-SC hybrid to Bogoliubov quasiparticles,
i.e., excitations consisting of a superposition of electron-like and hole-like
particles with amplitudes $u(E)$ and $v(E)$, as discussed above. During this
process, the excess charge is drained to the ground through AR at the
interface with Al [Fig. 1(c)]. Until the eventual recombination of all QPs
back into the Cooper pair condensate, they carry a finite energy excitation
[9, 6].

To measure the generated non-equilibrium distribution, we attach another lead
on the right, separated by a semiconducting junction. A conventional tunnel
junction supports bi-directional currents, where electrons tunnel to
unoccupied states above $E_{\text{F}}$ and holes tunnel to occupied states
below it. When both charge carriers are present, such as in a SC, the current
is proportional to the charge of the QP, $q$ [32], which is expected to vanish
at the gap edge. A QD embedded in a junction can be tuned to only allow
current flow in a single direction. When the charging energy of the QD is far
greater than other energy scales in the circuit ($\Delta_{\mathrm{ind}}$ and
bias voltage) and it is tuned to be near a charge degeneracy between having
$N$ and $N-1$ electrons, it can be considered a single isolated Fermionic
level. If this level is at $\Delta_{\mathrm{ind}}>0$ and thus its ground state
is empty [Fig. 1(d)], it allows only electrons to flow from the SM-SC system
to the N lead, with a rate proportional to the electron component $|u|^{2}$ of
the proximitized states. Similarly, a level at $-\Delta_{\mathrm{ind}}$, being
occupied in its ground state, only allows holes to tunnel to N with a rate
proportional to $|v|^{2}$ [Fig. 1(e)]. Thus, a QD is a charge-selective probe
that couples to either the electron or hole component of the QP wavefunction
depending on its occupation.

### Methods

The sample was fabricated using the same methods described in Ref [33]. A 3D
illustration and a scanning electron microscope image of the device geometry
are shown in Figure 1(f,g). Ti+Pd local bottom gate electrodes were evaporated
on Si substrates followed by HSQ shadow wall structures and then atomic-layer
deposited Al2O3 dielectric. InSb nanowires were grown by MOVPE [34]. The
nanowires were then transferred using an optical manipulator to the substrate
described above. Atomic H cleaning removed the oxide on InSb, and following
in-situ transfer in the same e-beam evaporator,
$14\text{\,}\mathrm{n}\mathrm{m}$ of Al thin film was deposited at liquid N2
temperature. The film covers the nanowire middle segment, proximitizing InSb
and forming our galvanically connected middle lead (S). Finally, the N leads
were fabricated by another e-beam lithography step. After patterning, Ar
milling removes the newly formed surface oxide again, and
$140\text{\,}\mathrm{n}\mathrm{m}$ of Cr+Au contact was evaporated to form
ohmic contacts to InSb. The hybrid SC-SM segment in the devices reported here
is $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ long, and the typical
distance between the centers of the finger gates forming the QD is
$110\text{\,}\mathrm{n}\mathrm{m}$. Overall, 5 devices showing the same
qualitative behavior were measured. Here we report on detailed scans of two
such devices.

Fig. 2:  a. Circuit diagram for local conductance $G_{\mathrm{RR}}$
measurement of Device A. b. $G_{\mathrm{RR}}$ through the right N-QD-S
junction. Charge occupations numbers are labeled within corresponding Coulomb
diamonds, $N$ being odd. The white box indicates the zoomed-in area in e. c.
Circuit diagram for nonlocal conductance $G_{\mathrm{RL}}$ measurements. d.
Energy alignment between the QD and leads for the four injected vs detected
charge type possibilities when measuring $G_{\mathrm{RL}}$. Directions of
electron tunneling are indicated with arrows. e. Higher-resolution scan of the
local conductance of the right side $G_{\mathrm{RR}}$ near the QD charge
degeneracy point marked in panel b. Insets: sketches of the energy alignment
between the S, right N, and the QD level at the Coulomb-diamond tips (circled
in white). f. The NLC $G_{\mathrm{RL}}$ measured on the right side as a
function of the bias on the left side $V_{\mathrm{L}}$ and the right dot gate
$V_{\text{dot}}$, with $V_{\text{R}}=0$. The correspondence between the four
resonant features and the four situations depicted in panel d are circled and
labeled with roman numerals.

### Detection of quasiparticle charge

First, we characterize the QD defined using the three finger gates in the
right junction. Figure 2(a) illustrates the circuit used for measurement of
the local conductance
$G_{\mathrm{RR}}\equiv\mathrm{d}I_{\mathrm{R}}/\mathrm{d}V_{\mathrm{R}}$
through the QD when applying $V_{\mathrm{R}}$ and varying $V_{\mathrm{dot}}$
while keeping S and left N grounded. We observe Coulomb diamonds of varying
sizes, typical of few-electron QDs [Fig. 2(b)]. By comparing the resonance
lines with the constant interaction model of a QD [35], we extract the
capacitance parameters: gate and bias lever arms and the charging energy, thus
mapping the applied gate voltage to the QD’s chemical potential
$\mu_{\text{dot}}(N)=-e\alpha\left(V_{\text{dot}}-V_{\text{res,N}}\right)$
where $\alpha$ is the gate lever arm and $V_{\text{res,N}}$ is the gate
voltage when the $N$ and $N-1$ charge states are degenerate 111See
supplementary Figure S1 for details.

The circuit used to measure the nonlocal conductance
$G_{\mathrm{RL}}\equiv\mathrm{d}I_{\mathrm{R}}/\mathrm{d}V_{\mathrm{L}}$ is
sketched in Figure 2(c). The sign of the injecting bias determines the type of
charge carriers injected into the hybrid S, electrons if $V_{\mathrm{L}}<0$
and holes vice versa. The detected charge, as explained above, is determined
by the receiving QD’s chemical potential $\mu_{\mathrm{dot}}(N+1)$ being above
or below 0. Figure 2(d) illustrates the four possible injection versus
detection charge combinations when both the left N and QD are on resonance
with the SC gap edge.

Zooming in to the $N\rightarrow N+1$ charge transition [Fig. 2(b)], we measure
the detailed local and nonlocal conductance structures [Fig. 2(e), (f),
respectively]. The local conductance [Fig. 2(e)] shows QD diamond lines with
the exception that transport is blocked at energies smaller than
$\Delta_{\mathrm{ind}}\approx$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$
[37, 23, 13, 38, 39, 40, 41, 42, 43, 44, 45]. We note that due to the strong
SM-SC coupling $\Delta_{\mathrm{ind}}\approx\Delta_{\mathrm{SC}}$ [10]. QDs
coupled to SCs can form ABSs[13, 46]. Such formation requires two-electron
tunneling processes to take place between the QD and the proximitized segment.
By raising the tunnel barriers, we significantly suppress such two-electron
processes and inhibit the formation of ABSs in the QD. The lack of sub-gap
features confirms that the QD is not hybridized by the SC and therefore
maintains its pure electron or hole character.

In Fig. 2(f), we vary $V_{\mathrm{dot}}$ through the same resonances while
scanning $V_{\mathrm{L}}$. At sub-gap energies
($|eV_{\mathrm{L}}|<\Delta_{\mathrm{ind}}$), $G_{\mathrm{RL}}$ is similarly 0
due to the absence of sub-gap excitations. $G_{\mathrm{RL}}$ is also weak when
$V_{\text{dot}}$ is far away from the tips of the Coulomb diamonds [Fig.
2(b)]. Finite NLC is only obtained when the left bias is aligned with the
induced SC gap edges, $eV_{\text{L}}\approx\pm\Delta_{\mathrm{ind}}$, and the
QD level is inside the induced gap:
$-\Delta_{\mathrm{ind}}\leq\mu_{\text{dot}}(N+1)\leq\Delta_{\mathrm{ind}}$.
The NLC feature around the QD crossing contains four lobes that exhibit a two-
fold anti-symmetry, changing signs under either opposite bias or dot
occupation. This four-lobed structure corresponds exactly to the four charge
combinations in Figure 2(d) and shows up in almost all charge degeneracy
points we have measured, including other dot configurations and devices 222see
Supplemental Figure S2 for other devices showing the same structure.

Consider the feature marked by “I” in Figure 2(f) and the process
schematically depicted [Fig. 2(d)]. The electrons injected by the negative
left bias into the central region create both energy and charge non-
equilibrium. The resulting QPs arriving at the right end of the hybrid S must
then reach the right N lead via a QD tuned to
$\mu_{\text{dot}}(N+1)\approx-\Delta_{\mathrm{ind}}$, which only allows holes
to tunnel out. The presence of this NLC lobe is thus a result of the hole
component of the wavefunction $v$. The inversion of charge in this
$e\rightarrow h$ process results in the observed positive NLC due to the
current-direction convention. We can also tune the dot to
$\mu_{\text{dot}}(N)\approx+\Delta_{\mathrm{ind}}$ (marked “III”). Here
electrons are still injected into the central region, but now the QD allows
only electrons to tunnel out to the right lead because its ground state is an
unoccupied fermionic level. We mark this process $e\rightarrow e$. The NLC is
thus negative, with a magnitude that relates to $u$.

The NLC is also predominantly anti-symmetric with respect to the applied
voltage bias. This can be understood by considering the current-rectifying
behavior of the QD at a fixed occupancy. When $\mu_{\text{dot}}>0$, only the
flow of electrons from the S to the N lead is allowed, regardless of the
charge of the injected particles. The current passing through the dot is thus
always positive, forcing the conductance ($\textrm{d}I/\textrm{d}V$) to flip
its sign when the bias changes polarity. Similarly, when the dot is placed at
$\mu_{\text{dot}}<0$ to allow only holes to flow, the current is always
negative, and the rest follows suit. Anti-symmetric NLC is also a prevalent
feature in conventional tunnel junction measurements without any QD [30].
There, similar to our observations, the global sign of the anti-symmetric NLC
is determined by and varies with the gate on the current-detecting junction.
We argue that this ubiquitous anti-symmetry with respect to bias voltage stems
from the unintentional charge selectivity of the semiconducting tunnel
junctions 333See Supplemental Figure S3 for data supporting this
interpretation, when both sides of the device are configured to be tunnel
junctions.

We note that the amplitudes of the $G_{\mathrm{RL}}$ peaks are higher when the
ground state contains $N+1$ electrons than when it contains $N$ electrons
[Fig. 2(f)], with $N$ being odd in this setup. This difference in
$G_{\mathrm{RL}}$ can be attributed to the spin-degenerate DOS of the dot,
which gives rise to different tunneling rates for even and odd occupation
numbers [49, 50]. The opposite trend can be observed in the $N-1\rightarrow N$
transition and the application of a small Zeeman field that lifts this
degeneracy restores the electron and hole’s amplitudes to be nearly equal
444see Supplemental Figure S4 for the effect of lifting Kramer’s degeneracy on
the NLC amplitude. Spin degeneracy influences the QD transmission rates of
electrons and holes in a manner unrelated to the relative strengths of $u$ and
$v$, thus obscuring the observation of charge neutrality.

###  .1 Many-electron dots

Fig. 3:  a The local conductance $G_{\mathrm{RR}}$ of Device B as a function
of $V_{\mathrm{dot}}$, measured with fixed
$V_{\mathrm{R}}=\pm$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$ . b.
The nonlocal conductance $G_{\mathrm{RL}}$ of Device B as a function of
$V_{\mathrm{dot}}$, measured with fixed
$V_{\mathrm{L}}=\pm$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$,
respectively. The four peaks in one of the periodic structures are labeled by
their injected and detected charge states being electrons ($e$) or holes
($h$). c. The relative magnitude of $h\rightarrow h$, $h\rightarrow e$,
$e\rightarrow h$, and $e\rightarrow e$ processes to the magnitude of the local
conductance through the QD in the same configuration. See details in text.

The few-electron QD discussed above proves to be an effective charge filter,
able to detect separately the electron and hole components of the QP
wavefunction. However, the presence of spin degeneracy modifies the tunneling
rates of electrons and holes and complicates direct comparison between $u$ and
$v$. To overcome this, we turn our attention to Device B, which has a larger
QD whose orbital level spacing is too small to be observed. Here, since
multiple orbital levels contribute to tunneling across, rendering dot and the
spin degeneracy negligible, the tunneling rates for electrons and holes are
nearly equal. Figure 3(a) shows the local conductance $G_{\mathrm{RR}}$
through the QD as a function of the gate voltage, when applying a bias of
$V_{\textrm{R}}=\pm\Delta_{\mathrm{ind}}/e=\pm$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$
between the N and S leads on either side of the QD. We observe equidistant
Coulomb oscillations typical of many-electron QDs. The magnitude of the
oscillations varies from peak to peak and between positive and negative
applied $V_{\textrm{R}}$, indicating the mesoscopic details of transport are
different under device voltage changes. We expect such differences to modulate
the NLC as well.

The NLC oscillates as a function of $V_{\textrm{dot}}$ when applying
$V_{\textrm{L}}=\pm\Delta_{\mathrm{ind}}/e=\pm$250\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$
[Fig. 3(b)]. Every period of the oscillation has an internal structure where a
positive peak follows a negative peak (green curve) or the opposite (purple
curve). Each peak in the NLC trace represents a different process in which
either electrons ($e$) or holes ($h$) are injected and either the electron
($e$) or hole ($h$) component of the wavefunction is detected. With a negative
applied bias, electrons are injected into the system. The negative conductance
peak appears first, resulting from the QD detecting the electron component of
the wavefunction ($e\rightarrow e$). After $\mu_{\textrm{dot}}$ crossed zero,
the QD detects the hole component ($e\rightarrow h$), giving rise to positive
conductance. The choice of injected charge doesn’t affect the charge
selectivity of our probe. Thus, injecting holes instead of the electrons
(positive $V_{\text{L}}$) leads to an inversion of the sign of
$G_{\text{RL}}$.

In Figure 3(c) we show the amplitude of $G_{\mathrm{RL}}$ relative to the
amplitude of $G_{\mathrm{RR}}$ through the QD, as a function of the four
processes, averaged among the multiple resonances shown in Figure 3(a,b). The
average relative amplitudes of all the processes are around 0.1. The
differences between amplitudes of the four processes are much smaller than
variations within each process. This is consistent with the BCS picture
discussed above, in which the excited states in a superconductor at the gap
edge are chargeless Bogoliubov quasiparticles, i.e., $|u|=|v|$.

Fig. 4:  a.-d. Conductance matrix $G$ as a function of in-plane magnetic field
($B_{\mathrm{X}}$) and bias. $G_{\mathrm{LL}}$ (a.) and $G_{\mathrm{RL}}$ (c.)
are measured as a function of $V_{\mathrm{L}}$. $G_{\mathrm{RR}}$ (b.) and
$G_{\mathrm{LR}}$ (d.) are measured as a function of $V_{\mathrm{R}}$. e.
$G_{\mathrm{RL}}$ as a function of $\mu_{\mathrm{dot}}(N)$ at different values
of $B_{X}$ measured in Device A. The line traces are normalized to the maximum
value of each:
$G_{RL}(B_{X})/\text{max}\left(|G_{\mathrm{RL}}(B_{X})|\right)$. f. Amplitude
of the NLC for the $h\rightarrow e$ (positive NLC, blue) and $h\rightarrow h$
(negative NLC, orange) as a function $B_{X}$. The colored markers show the
values of $B_{X}$ where the line scans in panel a are taken. The $B$ field at
which the spectral gap in InSb closes, $B_{\mathrm{C,InSb}}$ and at which the
superconductivity in Al vanishes, $B_{\mathrm{C,Al}}$, are indicated by
corresponding vertical lines.

### Presence of particle-hole correlation after the closing of the induced gap

The three-terminal setup we employ allows us to measure the induced gap
independently from possible localized ABSs in the vicinity of the junction.
The topological phase transition predicted to take place in this system would
manifest as the closing and re-opening of the induced gap in response to the
application of a magnetic field or a change in the chemical potential at a
finite magnetic field [29]. In Fig 4a-d, we show the full conductance matrix
$\boldsymbol{G}=\begin{pmatrix}G_{\text{LL}}&G_{\text{LR}}\\\
G_{\text{RL}}&G_{\text{RR}}\end{pmatrix}$ as a function of a magnetic Zeeman
field $B_{X}$ applied along the nanowire direction. While the local
conductance on the left side, $G_{\text{LL}}$, shows a local sub-gap state
crossing zero bias at a finite field, $G_{\text{RR}}$ lacks such features. The
bulk induced gap, seen in the nonlocal conductance, close at
$B=$0.8\text{\,}\mathrm{T}$$ without reopening. The lack of gap reopening is
consistent with past measurements of the nonlocal conductance in three-
terminal geometry [30] and most likely results from the presence of disorder
as the dominating energy scale in these nanowires [52].

In a conventional superconductor, the presence of chargeless excitations and
the existence of an energy gap are correlated (except for a small region in
the phase space characterized by gapless superconductivity [1]). In a
proximitized system, both effects arise separately from Andreev reflection at
the SM-SC interface. To see this, we measure the nonlocal conductance using a
detector QD at different values of $B_{X}$ [Fig. 4(e)]. Since the size of the
induced gap decreases upon increasing magnetic field, we apply a constant DC
bias of $200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$ and AC excitation
of $180\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$ RMS to ensure the
induced gap edge always lies within the measured bias window. All of the
nonlocal scans taken with $B_{X}<$2.5\text{\,}\mathrm{T}$$ show a positive and
a negative peak arising from the $h\rightarrow e$ and $h\rightarrow h$
process, respectively, as discussed above. Taking the maximal positive value
of $G_{\mathrm{RL}}(B_{X})$ as the amplitude of the $h\rightarrow e$ process,
and the maximal negative value as the amplitude of the $h\rightarrow h$
process, we track the evolution of both as a function of $B_{X}$ [Fig. 4(f)].
Both processes survive well above the gap closing field of
$0.8\text{\,}\mathrm{T}$. Only above $2\text{\,}\mathrm{T}$, the critical
field of the Al film, we observe a decay in the NLC amplitude. At higher
fields the $h\rightarrow e$ process that must involve superconductivity is
absent, and the remainder of the $h\rightarrow h$ process may be attributed to
voltage-divider effects [53]. The observation of positive nonlocal conductance
up to $2\text{\,}\mathrm{T}$ shows that electron-hole correlations persist as
long as Andreev reflection between the wire and superconducting film is
possible. This effect is independent of the presence of an induced gap in the
DOS of the proximitized system.

###  .2 Detecting energy relaxation using QDs

Fig. 5:  a.  A sketch of the energy relaxation experiment when the device is
operated under negative bias voltage on the left side. (The situation of a
positive bias voltage, as the case with the data shown in the following
panels, can be similarly represented using a vertically mirrored sketch.) Dots
are formed on both sides of the nanowire. The bias is fixed on the right side,
and injection energy is determined by the energy of the left dot. The nonlocal
current is collected by the right dot. The applied bias is fixed to be
$V_{\text{inj}}$ = $1\text{\,}\mathrm{m}\mathrm{V}$. b., c. Dependence of
$I_{\text{inj}}$ (b.) and $I_{\text{det}}$ (c.) vs. $\mu_{\text{inj}}$ for
fixed
$\mu_{\text{det}}\approx$-240\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}$$
(see corresponding color in panel d). d. $I_{\text{det}}$ as functions of the
chemical potentials of the QDs on the left $\mu_{\text{inj}}$ and right
$\mu_{\text{det}}$. The red vertical bar marks the gap edge and detector QD
voltage under which curves in panels b,c are taken. Horizontal bars mark the
line cuts shown in panel e. e.  Normalized line scans of $I_{\text{det}}$ vs.
$\mu_{\text{det}}$ for different injection energies (see corresponding color
in panel d). Lines are vertically offset for clarity.

A commonly observed feature in the three-terminal setup we study is the
presence of significant NLC when applying a bias greater than the gap of the
parent SC, $|eV|>\Delta_{\text{SC}}$ [30, 14]. This is unexpected since the
injected particles can directly drain to the ground via the available states
in the Al film at such energies, and thus should not be able to emerge at the
other side of a long enough nanowire. To study the behavior of the injected
QPs at high energies, we tune the finger gates in Device A to transform the
current-injecting junction from a tunnel barrier into a second QD, so that QPs
are injected only at controlled energies into the SC [Fig. 5(a)]. We fix the
bias on the injecting side to $V_{\text{inj}}\equiv
V_{\text{L}}=$1\text{\,}\mathrm{m}\mathrm{V}$$ and vary the energy of the
injected electrons by scanning the potential of the injecting dot
$\mu_{\text{inj}}$. We verify that both inelastic tunneling and elastic co-
tunneling across the injector dot are negligible in our device 555see
Supplemental Figures S5, S6 for discussions on the effects of inelastic
tunneling and elastic co-tunneling. Thus, the QDs are operated not only as
charge filters, but also as energy filters for both injected and detected
electrons.

Next, we show the simultaneously measured currents through the injector dot
($I_{\text{inj}}$) and the detector dot ($I_{\text{det}}$) as a function of
the injection filter energy $\mu_{\text{inj}}$ [Fig. 5(b,c)], when the
detector dot is fixed to the gap edge. The local current, $I_{\text{inj}}$, is
higher at $\mu_{\text{inj}}\approx\Delta_{\mathrm{ind}}$ and decays to a
plateau, a result of the high DOS at the gap edge combined with the tunneling
out of the QD being primarily elastic. $I_{\text{inj}}$ depends only on
$\mu_{\text{inj}}$, showing that coherent effects such as crossed Andreev
reflection are negligible [24, 23, 26, 25, 27, 28] 666see the dependence of
$I_{\text{inj}}$ on $\mu_{\text{det}}$ in Supplementary Figure S7. The
nonlocal current [$I_{\text{det}}$, Fig. 5(c)] is smaller than
$I_{\text{inj}}$ by two orders of magnitude. It also has a peak when current
is injected directly at the gap edge, but persists when injection energy is
higher.

The nonlocal current depends strongly on both $\mu_{\text{inj}}$ and
$\mu_{\text{det}}$ [Fig. 5(d)]. Remarkably, we see that while the injection
energy modulates the magnitude of the detector current, it does not influence
the energy range in which finite $I_{\text{det}}$ can be measured. Fig 5(e)
shows $I_{\text{det}}$ vs. $\mu_{\text{det}}$ for different values of
$\mu_{\text{inj}}$, normalized by their maximal values. They all follow the
same trend—regardless of the injected electrons’ energy, the nonlocal signal
is only collected at energies comparable to or smaller than
$|\Delta_{\mathrm{ind}}|$. The observation that electrons injected at energies
larger than $\Delta_{\mathrm{ind}}$ are detected only at the gap edge implies
inelastic relaxation plays an important role in nonlocal conductance.
Electrons injected above the gap are either drained by the grounded SC or
decay to the gap edge, after which they are free to diffuse and reach the
detector junction. This observation explains why we do not observe a finite QP
charge even though they are only neutral at the gap edge while the QDs have a
finite energy broadening — as long as the linewidth of the QD is smaller than
$\Delta_{\mathrm{ind}}$, the only QPs available for transport are those with
energy $\Delta_{\mathrm{ind}}$.

## Discussion

Summing up our observations of charge and energy relaxation above, the
emerging microscopic picture of nonlocal charge transport in three-terminal
nanowire devices thus consists of four (possibly simultaneous) processes.
First, a charge is injected at some given energy into the nanowire. Second,
some of the injected electrons/holes are drained to the ground via the
superconducting lead and the remaining relax to the lowest available state at
the induced gap edge. Third, through Andreev reflection, the charged electrons
are converted to chargeless Bogoliubov QPs. Finally, the QPs diffuse toward
the other exit lead, where they are projected onto a charge polarity
determined by the receiving junction.

In contrast to the QP continuum investigated here, previous works have
examined NLC produced by transport through discrete sub-gap Andreev bound
states in similar N-S-N hybrid devices [14, 32]. Distinctly from chargeless
Bogoliubov QPs at the gap edge, the BCS charge of these sub-gap states is in
general nonzero and varies drastically with gates. These works concluded that
the NLC produced by such states is proportional to their BCS charge, implying
$G_{\mathrm{LR}}=G_{\mathrm{RL}}=0$ when $|u|=|v|$. However, as observed here
and in accordance with other works detecting NLC of the QP continuum [30],
anti-symmetric NLC is still consistently detected even at the gap edge where
BCS charge of the QPs is expected to vanish. We argue that finite NLC of
chargeless QPs stems from non-ideal tunnel barriers that transmit electrons
and holes with non-equal, energy-dependent probabilities, a generic property
of semiconducting junctions. As the N-S tunnel junctions also exhibit a
preference for a certain charge (see Supplemental Figure S3), even when
$|u|=|v|$, the tunnel barrier transmits more electrons than holes. Thus, the
resulting NLC, being proportional to the difference between these two
transmission amplitudes [32], becomes finite and antisymmetric. To put the
above into the scattering formalism, the commonly used framework to describe
electron transport in this system, the NLC is given by:

$G_{ij}(E)=\frac{e^{2}}{h}\left(T^{eh}_{ij}-T^{ee}_{ij}\right)$ (1)

where $T^{ee}_{ij}$ and $T^{eh}_{ij}$ are the energy-dependent transmission
amplitudes from an electron in lead $i$ to an electron or a hole in lead
$j\neq i$ [29]. The presence of charge and energy filters, such as the QDs
employed in this work, significantly modifies these transmission amplitudes.
When the QD’s chemical potential is tuned above the Fermi level, $T^{eh}_{ij}$
is suppressed, whereas when the QD’s chemical potential is tuned below the
Fermi level, $T^{ee}_{ij}$ is suppressed. We further note that although one is
tempted to associate the NLC observed in our system directly with the relevant
transmission amplitudes expressed here, the scattering formalism itself is
insufficient to model some important aspects of the actual transport. The
scattering formalism assumes that the motion of the QPs in the system
conserves energy [56], but this assumption does not hold in our system since
we observe that energy relaxation plays an important role in NLC. We thus
conclude that while the present or absence of NLC can serve as a useful tool
in the determination of the induced gap, a quantitative analysis of NLC should
go beyond the scattering formalism.

## Conclusions

In conclusion, by measuring the nonlocal conductance in a three-terminal
device with well controlled QDs at the ends, we can detect the electron and
hole components of non-equilibrium quasiparticle wavefunctions. Our results
reveal a population of fully charge-relaxed neutral BCS excitations at the gap
edge in a proximitized nanowire under nonlocal charge injection. We further
show that the conversion of injected charge to correlated electron-hole
excitations does not require an induced gap. By injecting particles at
specific energies, we observe the inelastic decay of injected charges to the
lowest excited states, the gap edge. We show that the combined effect of
charge neutral excitation and a tunnel barrier with energy selectivity leads
to a current-rectifying effect. These results allow us to understand the
ubiquitous anti-symmetry of the nonlocal conductance observed in previous
reports [30] and suggest that the correct framework to discuss such
experiments is in terms of non-equilibrium superconductivity. Crucially, we
show that inelastic decay of injected quasiparticles dominate nonlocal
transport and therefore should be taken into account when attempting to model
the system.

The results observed here are in very good agreement with the results obtained
by Denisov et al. [15], measuring the nonlocal response of Al-covered InAs
nanowires. There, the problem of detecting the charge-neutral mode was
resolved by measuring the nonlocal shot noise response, showing the charge
neutrality of the excitations within the hybrid nanowire. The alternative
approach presented here, utilizing QDs as as energy- and charge-selective
injectors and detectors supplements the shot noise measurements by uncovering
the relaxation processes taking place in the transport process. It can further
allow the study of non-equilibrium distribution functions in proximitized
semiconducting systems with spectral resolution. In the presence of discrete
ABSs or QPs occupying a wide energy range, the energy resolution of a QD
allows one to excite and probe different energies as desired. We further
propose that in the presence of magnetic fields, QDs can also serve as
efficient bipolar spin filters [57], allowing us to directly measure the spin-
polarized density of states of proximitized SC, triplet SC correlations, and
extract the relevant relaxation rates and mechanisms.

###### Acknowledgements.

We wish to acknowledge useful discussions with Marco Aprili, Torsten Karzig,
Jelena Klinovaja, Daniel Loss, Filip Malinowski, Dimitry Pikulin, Charis Quay,
Hadar Steinberg and Bernard van Heck. This work has been supported by the
Dutch Organization for Scientific Research (NWO), the Foundation for
Fundamental Research on Matter (FOM) and Microsoft Corporation Station Q. G.
W. and T. D. contributed equally to this work. All raw data in the publication
and the analysis code used to generate figures are available at
https://zenodo.org/record/5534254.

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Next we compute whether $x<z$ controlled on $\mathsf{1}[y<x]$, and store the
output in our target. We know that
$\mathbf{1}[x\in(y,z)]\equiv(\mathbf{1}[y<x]\cdot\mathbf{1}[x<z])$, and since
we have both the comparator values stored in memory
$\displaystyle\ket{0}_{s-1}\ket{x}_{n}\ket{y}_{n}\ket{z}_{n}\ket{\mathbf{1}[y<x]}\ket{t}_{1}\xmapsto{\mathsf{Q^{\prime}_{C\text{-}COMP}}}\ket{0}_{s-1}\ket{x}_{n}$
$\displaystyle\ket{y}_{n}\ket{z}_{n}\ket{\mathbf{1}[y<x]}_{1}\ket{t\oplus\mathbf{1}[x\in(y,z)]}_{1}~{}.$
We finally uncompute the intermediate value $\mathsf{1}[y<x]$ using
$\mathsf{Q_{COMP}}$
$\displaystyle\ket{0}_{s-1}\ket{x}_{n}\ket{y}_{n}\ket{z}_{n}\ket{\mathbf{1}[y<x]}_{1}\ket{t\oplus\mathbf{1}[x\in(y,z)]}_{1}\xmapsto{\mathsf{Q_{COMP}}}$
$\displaystyle\ket{0}_{s}\ket{x}_{n}\ket{y}_{n}\ket{z}_{n}\ket{t\oplus(\mathbf{1}[y<x]\cdot\mathbf{1}[x<z])}_{1}~{}.$

Therefore, we get a cost of
$r=2r_{\mathsf{COMP}}+r^{\prime}_{\mathsf{C\text{-}COMP}}$. This cost can be
decreased by applying the MBU lemma on the $\mathsf{Q_{COMP}}$ uncomputing
$\mathbf{1}[y<x]$, thus leading to a cost of
$r=1.5r_{\mathsf{COMP}}+r^{\prime}_{\mathsf{C\text{-}COMP}}$. ∎

### Acknowledgement

This work started when AS was working at Inveriant Pte. Ltd. AS is supported
by Innovate UK under grant _10004359_. AMM has received funding from the
European Union’s _Horizon 2020 research and innovation programme_ under the
grant agreement _N°101034324_ and has been partially supported by the _Italian
Group for Algebraic and Geometric Structures and their Application_
(GNSAGA–INdAM). This work is supported by the National Research Foundation,
Singapore, and A*STAR under its CQT Bridging Grant. We also acknowledge
funding from the Quantum Engineering Programme (QEP 2.0) under grants
_NRF2021-QEP2-02-P05_ and _NRF2021-QEP2-02-P01_. We thank Anupam Chattopadhyay
and Zeyu Fan for useful discussions. AMM and AL thank Niels Benedikter for the
hospitality at Università degli Studi di Milano.

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##  Appendix A Background on binary arithmetics

In this appendix, we briefly survey some standard results on bit string
arithmetics. In particular, we recall how bit strings can be interpreted as
unsigned or signed integers. We refer to the section 1.3 for our notation on
bit strings arithmetics.

######  Proposition A.1.

The bit string subtraction can be expressed in terms of $2$’s complement as

$\mathbf{x}-\mathbf{y}=\mathbf{x}+\overline{\overline{\mathbf{y}}}~{}.$

###### Proof.

According to definition 1.2 and 1.4, the string on the right-hand side is
defined bitwise as $r_{k}=x_{k}\oplus y_{k}\oplus 1\oplus z_{k}$, where
$z_{k}$ is defined recursively as follows

$\displaystyle z_{0}$ $\displaystyle=1$ $\displaystyle z_{i+1}$
$\displaystyle=\textrm{maj}(x_{i},y_{i}\oplus 1,z_{i})~{}.$

The thesis follows by showing inductively that $z_{k}=b_{k}\oplus 1$, where
$b_{k}$ is the borrowing defined in definition 1.5. Namely, assume
$z_{k}=b_{k}\oplus 1$, then

$\displaystyle z_{k+1}$ $\displaystyle=~{}x_{k}y_{k}\oplus x_{k}y_{k}\oplus
x_{k}z_{k}\oplus y_{k}z_{k}\oplus x_{k}\oplus z_{k}$
$\displaystyle=~{}x_{k}y_{k}\oplus x_{k}y_{k}\oplus x_{k}(b_{k}\oplus 1)\oplus
y_{k}(b_{k}\oplus 1)\oplus x_{k}\oplus b_{k}\oplus 1$
$\displaystyle=~{}x_{k}y_{k}\oplus x_{k}y_{k}\oplus x_{k}b_{k}\oplus
y_{k}b_{k}\oplus y_{k}\oplus b_{k}\oplus 1$ $\displaystyle=~{}b_{k+1}\oplus
1~{}.$

∎

######  Remark A.2 (Binary encoding of a positive integer).

Any bit string can be understood as the binary representation of a given
positive integer number via the following bijective function

$\displaystyle\\{0,1\\}^{n}$
$\displaystyle\leftrightarrow\\{0,1,\dots,2^{n}-1\\}$
$\displaystyle\mathbf{x}$ $\displaystyle\mapsto\sum_{k=0}^{n-1}x_{k}2^{k}$

If not stated differently, we shall often equate the decimal and binary
representation of a given positive integer number. Note that bit string
addition is precisely built in such a way that the string
$\mathbf{x}+\mathbf{y}$ corresponds to the positive natural number $x+y$; the
extra bit is added to take into account the overflow of the carry when
$(x+y)\geq 2^{n}$. Moreover, one can easily check that
$\mathbf{x}+\overline{\mathbf{x}}=\mathbf{(}2^{n}-1)$.

######  Proposition A.3.

Given $x,y\in\\{0,1,\dots,2^{n}-1\\}$ two positive integer, one has that
$(\mathbf{x}-\mathbf{y})_{n+1}=(x<y)$.

###### Proof.

Assume that $x_{j}=y_{j}$ for any $k<j<n$ and the $k$-th bit is the most
significant one such that $x_{k}\neq y_{k}$. According to definition 1.5, one
has $d_{j}=b_{j}=b_{k+1}$ for any $k<j\leq n$. Therefore if $x>y$ one must
have $x_{k}=1$ and $y_{k}=0$, thence $d_{n+1}=b_{k+1}=0$. On the contrary, one
has $d_{n+1}=b_{k+1}=1$. ∎

######  Remark A.4 (Binary encoding of signed integer).

Any signed integer number can be encoded as a bit string via the following
bijective function

$\displaystyle\\{0,1\\}^{n+1}$
$\displaystyle\leftrightarrow\\{-2^{n},\dots,0,1,\dots,2^{n}-1\\}$
$\displaystyle\mathbf{x}$ $\displaystyle\mapsto-
x_{n}2^{n}+\sum_{k=0}^{n-1}x_{k}2^{k}$ $\displaystyle\mathbf{y|_{2}}$
$\displaystyle\mapsfrom y$

where $\mathbf{y|_{n+1}}$ denotes the string encoding the signed integer $y$
in the _2’s complement_ , i.e.

$\mathbf{y|_{n+1}}=\begin{cases}\mathbf{y}&y\geq 0\\\
\overline{\overline{\mathbf{|y|}}}&y<0~{},\end{cases}$ (10)

and $\mathbf{y}$ and $\mathbf{|y|}$ are the string encoding the positive
numbers $y$ and $|y|$ according to remark A.2. According to this convention,
the most significant bit of the string encodes $\text{sign}(y)$.

######  Proposition A.5.

For any two $n$ bits strings $\mathbf{x},\mathbf{y}$ with corresponding
unsigned integers $x$ and $y$, one has

$\mathbf{x}-\mathbf{y}=\mathbf{(x-y)|_{n+1}}~{},$ (11)

where the left-hand side is the $n$ strings subtraction and the right-hand
side is the string corresponding to the signed integer $(x-y)$ in the $2$’s
complement convention.

###### Proof.

One has to check two separate cases. First case, when $x\geq y$, one has that
$(x-y)$ is a positive number and $\mathbf{x}-\mathbf{y}$ correspond to the
string $\mathbf{(x-y)}$. In the other case, when $x<y$, one gets from the
definition of $n$ strings subtraction that

$\displaystyle\mathbf{x}-\mathbf{y}=$
$\displaystyle~{}\mathbf{x}+\overline{\overline{\mathbf{y}}}$ $\displaystyle=$
$\displaystyle~{}\overline{\mathbf{y}}+\mathbf{x}+\mathbf{1}$ $\displaystyle=$
$\displaystyle~{}\overline{\mathbf{(y-x)}}+1$ $\displaystyle=$
$\displaystyle~{}\overline{\overline{\mathbf{(y-x)}}}$ $\displaystyle=$
$\displaystyle~{}\overline{\overline{\mathbf{|x-y|}}}~{}.$

∎

The upshot is that given any classical circuit implementing $n$ strings
addition, one can implement signed integer arithmetic, encoding them in $2$’s
complement.

######  Proposition A.6.

For any two $n$ bits strings $\mathbf{x|_{n}},\mathbf{y|_{n}}$, corresponding
to the signed integers $x$ and $y$ in the $2$’s complement convention, one has
$\mathbf{x|_{n}}+\mathbf{y|_{n}}=\mathbf{(x+y)|_{n+1}}$, where the the left-
hand side is the $n$ strings addition and the right-hand side is the string
corresponding to the signed integer $(x+y)$ in the $2$’s complement
convention.

###### Proof.

One has to consider four separate cases. In the first case $x,y\geq 0$, one
gets the usual unsigned addition. When $x\geq 0$ and $y<0$ one has

$\displaystyle\text{lhs}=\mathbf{x}+\overline{\overline{\mathbf{|y|}}}=\mathbf{x}-\mathbf{|y|}=\mathbf{(x-|y|)|_{n+1}}$
(12)

employing proposition A.5 in the last equality. Similarly, when $x<0$ and
$y\geq 0$, one has

$\displaystyle\text{lhs}=\overline{\overline{\mathbf{|x|}}}+\mathbf{y}=\mathbf{|y|}-\mathbf{x}=\mathbf{(y-|x|)|_{n+1}}~{}.$
(13)

At last, when $x,y<0$, one notice that

$\displaystyle\text{lhs}=\overline{\overline{\mathbf{|x|}}}+\overline{\overline{\mathbf{|y|}}}={\overline{\mathbf{|x|}}}+\overline{\overline{\mathbf{|y|}}}+1=\overline{\overline{\mathbf{|x|}-\overline{\overline{|y|}}}}=\overline{\overline{\mathbf{|x|+|y|}}}~{}.$
(14)

∎


# Sharp bounds for a class of integral operators in weighted-type spaces on
Heisenberg group 00footnotetext: Key words and phrases: a class of integral
operators, Hardy operator ,Hardy Littlewood-Pólya operator, Hilbert operator,
weighted-type spaces, Heisenberg group.
2020 Mathematics Subject Classification: Primary 42B35; Secondary 26D15.

Xiang Li, Zhanpeng Gu,Dunyan Yan and Zhongci Hang111Corresponding author

Abstract: In this paper, we will use the conclusions and methods in [9] to
obtain the sharp bounds for a class of integral operators with the nonnegative
kernels in weighted-type spaces on Heisenberg group. As promotions, the sharp
bounds of Hardy operator , Hardy Littlewood-Pólya operator and Hilbert
operator are also obtained.

##  1 Introduction

In [10], Stein and Weiss gave a class of integral operators with the
nonnegative kernels which satisfy some homogeneity and the rotational
invariance. Beckner [1] represented it as Stein-Weiss lemma and obtained the
following theorem.

######  Theorem 1.

Suppose $K$ is a non-negative kernel defined on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ that satisfies continuous on any domain
that excludes the point $(0,0)$, homogeneous of degree $-n,K(\delta u,\delta
v)=\delta^{-n}K(u,v)$, and $K(Ru,Rv)=$ $K(u,v)$ for any $R\in SO(n)$. Then $T$
is a from $L^{p}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ bounded integral
operator defined by

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

with

$\|Tf\|_{L^{p}\left(\mathbb{R}^{n}\right)}\leq
A\|f\|_{L^{p}\left(\mathbb{R}^{n}\right)}$ (1)

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

$A=\int_{\mathbb{R}^{n}}K\left(x,\hat{e}_{1}\right)|x|^{-n/p^{\prime}}dx$ (2)

and $\hat{e}_{1}$ is a unit vector in the first coordinate direction.

The inequality (1) was given by Stein and Weiss [10]. In [1] and [2], Beckner
proved that the constant $A$ in (2) is the best constant. Inspired by Theorem
1, we investigate a class of integral operators on Heisenberg group and
obtained sharp bounds in weighted type spaces.

Let us first recall some basic knowledge about the Heisenberg group. The
Heisenberg group $\mathbb{H}^{n}$ is a non-commutative nilpotent Lie group,
with the underlying manifold $\mathbb{R}^{2n}\times\mathbb{R}$ with the group
law

$x\circ
y=\left(x_{1}+y_{1},\ldots,x_{2n}+y_{2n},x_{2n+1}+y_{2n+1}+2\sum_{j=1}^{n}\left(y_{j}x_{n+j}-x_{j}y_{n+j}\right)\right)$

and

$\delta_{r}\left(x_{1},x_{2},\ldots,x_{2n},x_{2n+1}\right)=\left(rx_{1},rx_{2},\ldots,rx_{2n},r^{2}x_{2n+1}\right),\quad
r>0,$

where $x=(x_{1},\cdots,x_{2n},x_{2n+1})$ and
$y=(y_{1},\cdots,y_{2n},y_{2n+1})$. The Haar measure on $\mathbb{H}^{n}$
coincides with the usual Lebesgue measure on $\mathbb{R}^{2n+1}$. We denote
the measure of any measurable set $E\subset\mathbb{H}^{n}$ by $|E|$.Then

$|\delta_{r}(E)|=r^{Q}|E|,d(\delta_{r}x)=r_{Q}dx,$

where $Q=2n+2$ is called the homogeneous dimension of $\mathbb{H}^{n}$.

The Heisenberg distance derived from the norm

$|x|_{h}=\left[\left(\sum_{i=1}^{2n}x_{i}^{2}\right)^{2}+x_{2n+1}^{2}\right]^{1/4},$

where $x=(x_{1},x_{2},\cdots,x_{2n},x_{2n+1})$,is given by

$d(p,q)=d(q^{-1}p,0)=|q^{-1}p|_{h}.$

This distance $d$ is left-invariant in the sense that $d(p,q)$ remains
unchanged when $p$ and $q$ are both left-translated by some fixed vector on
$\mathbb{H}^{n}$.Furthermore,$d$ satisfies the triangular inequality[7]

$d(p,q)\leq d(p,x)+d(x,q),\quad p,x,q\in\mathbb{H}^{n}.$

For $r>0$ and $x\in\mathbb{H}^{n}$,the ball and sphere with center $x$ and
radius $r$ on $\mathbb{H}^{n}$ are given by

$B(x,r)=\\{y\in\mathbb{H}^{n}:d(x,y)<r\\},$

and

$S(x,r)=\\{y\in\mathbb{H}^{n}:d(x,y)=r\\},$

respectively.And we have

$|B(x,r)|=|B(0,r)|=\Omega_{Q}r^{Q},$

where

$\Omega_{Q}=\frac{2\pi^{n+\frac{1}{2}}\Gamma(n/2)}{(n+1)\Gamma(n)\Gamma((n+1)/2)}$

is the volume of the unit ball $B(0,1)$ on $\mathbb{H}^{n}$,
$\omega_{Q}=Q\Omega_{Q}$ (see [8]). For more details about Heisenberg group
can be refer to [4] and [11].

The weighted-type space on Euclidean space
$H^{\infty}_{\alpha}(\mathbb{R}^{n})$ with $\alpha>0$ consists all measurable
functions $f$ satisfying

$\|f\|_{H^{\infty}_{\alpha}(\mathbb{R}^{n})}:=\text{ess}\sup_{x\in\mathbb{R}^{n}}|x|^{\alpha}|f(x)|<\infty.$

The function $\|\cdot\|_{H^{\infty}_{\alpha}(\mathbb{R}^{n})}$ is a norm on
the space which is $dx$ almost everywhere equal. In this paper, we will
consider the weighted-type space on Heisenberg group. Our definition is as
follow.

######  Definition 1.

The weighted-type space on Heisenberg group
$H^{\infty}_{\alpha}(\mathbb{H}^{n})$ with $\alpha>0$ consists of all
measurable functions $f$ satisfying

$\|f\|_{H^{\infty}_{\alpha}(\mathbb{H}^{n})}:=\text{ess}\sup_{x\in\mathbb{H}^{n}}|x|_{h}^{\alpha}|f(x)|<\infty,$

where $\|\cdot\|_{H_{\alpha}^{\infty}}$ is a norm on the space which also is
$dx$ almost everywhere equal.

In harmonic analysis, the study of multilinear integral operators and their
properties is often more desirable and interesting. In [12], Wu and Yan
extened Theorem 1 to the case of multilinearity, now we utilize their
promotional methods to give $m$-linear integral operator with the nonnegative
kernels on Heisenberg group.

######  Definition 2.

Suppose that $K$ is a nonnegative kernel defined on $\mathbb{H}^{nm}$ and
satisfies the homogenous degree $-mn$,

$K(|t|_{h}x,|t|_{h}y_{1},\ldots,|t|_{h}y_{m})=|t|_{h}^{-mn}K(x,y_{1},\ldots,y_{m}),$

and

$K(Rx,Ry_{1},\ldots,Ry_{m})=K(x,y_{1},\ldots,y_{m})$

for any $R\in SO(n)$, then $K$ denotes the kernel of

$T_{m}(f_{1},\ldots,f_{m})(x)=\int_{\mathbb{H}^{nm}}K(x,y_{1},\ldots,y_{m})f_{1}(y_{1})\cdots
f_{m}(y_{m})dy_{1}\cdots dy_{m}.$ (3)

Taking $e_{1}$ is a unit vector in the first coordinate direction in
$\mathbb{H}^{n}$, then we have

$T_{m}(f_{1},\ldots,f_{m})(x)=\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})f_{1}(\delta_{|x|_{h}}y_{1})\cdots
f_{m}(\delta_{|x|_{h}}y_{m})dy_{1}\cdots dy_{m}.$ (4)

Next, we will provide our main result of mulitilinear integral operator with
kernel $K$ in weighted type space on Heisenberg group.

##  2 Sharp bound for m-linear integral operator

######  Theorem 2.

Let $\alpha>0$ and $\alpha=\alpha_{1}+\cdots+\alpha_{m}$ with $\alpha_{j}>0$
for any $j=1,\ldots,m$, if

$H_{m}:=\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{i=1}^{m}|y_{i}|_{h}^{-\alpha_{i}}dy_{1},\ldots
dy_{m}<\infty,$ (5)

then $T(f_{1},\ldots f_{m})(x)$ is bounded from
$H_{a_{1}}^{\infty}(\mathbb{H}^{n})\times\cdots\times
H_{\alpha_{m}}^{\infty}(\mathbb{H}^{n})$ to
$H_{\alpha}^{\infty}(\mathbb{H}^{n})$.

Moreover, if condition (4) is satisfied, then the following formula for the
norm of the operator hold

$\|T_{m}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}=H_{m}.$ (6)

###### Proof of Theorem 2.

For convenience, we take

$T_{m}(f_{1},\ldots,f_{m})(x)=\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{i=1}^{m}f_{i}(\delta_{|x|_{h}}y_{j})dy_{1}\cdots
dy_{m}.$

Using the above identity and the definition of weighted-type space on
Heisenberg group $H^{\infty}_{\alpha}(\mathbb{H}^{n})$, we can infer that

$\displaystyle\|T_{m}\|_{H^{\infty}_{\alpha}(\mathbb{H}^{n})}$
$\displaystyle=$
$\displaystyle\text{ess}\sup_{x\in\mathbb{H}^{n}}|x|_{h}^{\alpha}\left|\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{i=1}^{m}f_{i}(\delta_{|x|_{h}}y_{i})dy_{1}\cdots
dy_{m}\right|$ $\displaystyle\leq$
$\displaystyle\prod_{i=1}^{m}\|f_{i}\|_{H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})}\sup_{x\in\mathbb{H}^{n}\backslash\\{0\\}}|x|_{h}^{\alpha}\left|\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{i=1}^{m}|\delta_{|x|_{h}}y_{i}|_{h}^{-\alpha_{i}}dy_{1}\cdots
y_{m}\right|$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{m}\|f_{i}\|_{H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})}\sup_{x\in\mathbb{H}^{n}\backslash\\{0\\}}|x|_{h}^{\alpha}\left|\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{i=1}^{m}|x|_{h}^{-\alpha_{i}}|y_{i}|_{h}^{-\alpha_{i}}dy_{1}\cdots
y_{m}\right|$ $\displaystyle=$ $\displaystyle
H_{m}\prod_{i=1}^{m}\|f_{i}\|_{H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})}$

for every
$(f_{1},\ldots,f_{m})\in\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})$,
from which by taking the supremum in
$H_{\alpha_{i}}^{\infty}(\mathbb{H}^{n})$, it follows that

$\|T_{m}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}\leq H_{m},$ (7)

and we obtain the boundedness of $T_{m}$ on
$H^{\infty}_{\alpha}(\mathbb{H}^{n})$.

Then by taking

$f_{\alpha_{j}}(x)=\begin{cases}1/|x|^{\alpha_{j}}_{h},&{x\neq 0},\\\
0,&{x=0},\end{cases}$ (8)

it is clear that for any $j=1,\ldots,y_{m}$, we have

$\|f_{\alpha_{j}}\|_{H^{\infty}_{\alpha_{j}}(\mathbb{H}^{n})}=1.$ (9)

Bring (8) into identity (4) and through simple calculations, we have

$\displaystyle|x|_{h}^{\alpha}\left|\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{j=1}^{m}f_{\alpha_{j}}(\delta_{|x|_{h}}y_{j})dy_{1}\cdots
dy_{m}\right|$ (10) $\displaystyle=$
$\displaystyle|x|_{h}^{\alpha}\left|\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{j=1}^{m}|\delta_{|x|_{h}}y_{j}|_{h}^{-\alpha_{j}}dy_{1}\cdots
dy_{m}\right|$ $\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{j=1}^{m}|y_{j}|_{h}^{-\alpha_{j}}dy_{1}\cdots
y_{m}=H_{m}$

for $x\neq 0$, where we used the condition
$\alpha=\alpha_{1}+\cdots+\alpha_{m}$. From (10) and by using the definition
of norm in the weighted-type space on Heisenberg group
$H^{\infty}_{\alpha}(\mathbb{H}^{n})$, we can infer that

$\displaystyle\|T_{m}\|_{H^{\infty}_{\alpha}(\mathbb{H}^{n})}$ (11)
$\displaystyle=$
$\displaystyle\text{ess}\sup_{x\in\mathbb{H}^{n}}|x|_{h}^{\alpha}\left|\int_{\mathbb{H}^{nm}}K(e_{1},y_{1},\ldots,y_{m})\prod_{j=1}^{m}f_{\alpha_{j}}(\delta_{\alpha_{j}}y_{j})dy_{1}\cdots
dy_{m}\right|=H_{m}.$

Combining (9) with (11) it follows that

$\|T_{m}\|_{\prod_{j=1}^{m}H^{\infty}_{\alpha_{j}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}\geq H_{m}.$ (12)

Through (7) and (12), it implies that (6) holds. We have finished the proof of
Theorem 2. ∎

##  3 Sharp bound for Hardy operator

In this section, we will study $m$-linear $n$-dimensional Hardy operator on
Heisenberg group and obtain its sharp bound in weighted-type space on
Heisenberg group.

The $n$-dimensional Hardy operator on Heisenberg group is defined by Wu and Fu
[13]

$Hf(x):=\frac{1}{|B(0,|x|_{h})|}\int_{B(0,|x|_{h})}f(y)dy,x\in\mathbb{H}^{n}\backslash\\{0\\},$
(13)

where $f$ be a locally integrable function on $\mathbb{H}^{n}$. And they
proved $H$ is bounded from $L^{p}(\mathbb{H}^{n})$ to $L^{p}(\mathbb{H}^{n})$,
$1<p\leq\infty$. Moreover,

$\|H\|_{L^{p}(\mathbb{H}^{n})\rightarrow
L^{p}(\mathbb{H}^{n})}=\frac{p}{p-1},1<p\leq\infty.$

In [5], Fu and Wu introduced the $m$-linear $n$-dimensional Hardy operator,
which is defined by

$\mathfrak{H}_{m}(f_{1},\ldots,f_{m})(x)=\frac{1}{\Omega_{n}^{m}|x|^{mn}}\int_{|(y_{1},\ldots,y_{m})|<|x|}f_{1}(y_{1})\cdots
f_{m}(y_{m}),$

where $x\in\mathbb{R}^{n}\backslash\\{0\\}$ and $f_{1},\ldots,f_{m}$ are
nonnegative locally integrable functions on $\mathbb{R}^{n}$. Inspired by [5],
we define the m-liner n-dimensional Hardy operator on Heisenberg group is as
follow.

######  Definition 3.

Let $m$ be a positive integer and $f_{1},\ldots,f_{m}$ be nonnegative locally
integrable functions on $\mathbb{H}^{n}$, then

$\mathfrak{H}_{m}^{h}(f_{1},\ldots,f_{m})(x)=\frac{1}{\Omega_{Q}^{m}|x|_{h}^{mQ}}\int_{|(y_{1},\ldots,y_{m})|_{h}<|x|_{h}}f_{1}(y_{1})\cdots
f_{m}(y_{m})dy_{1}\cdots dy_{m},$ (14)

where $x\in\mathbb{H}^{n}\backslash\\{0\\}$.

Now, we formulate our main results as follows.

######  Theorem 3.

Let $m\in\mathbb{N}$, $f_{i}\in H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})$,
$\alpha>0$ and $\alpha=\alpha_{1}+\cdots+\alpha_{m}$ with $\alpha_{j}>0$ for
any $j=1,\ldots,m.$ If

$\mathcal{A}^{h}_{m}=\frac{1}{\Omega_{Q}^{m}}\int_{|(y_{1},\ldots,y_{m})|_{h}<1}\prod_{i=1}^{m}|y_{i}|_{h}^{-\alpha_{i}}dy_{1}\cdots
dy_{m}<\infty,$ (15)

then $\mathfrak{H}_{m}^{h}$ is bounded from
$H_{a_{1}}^{\infty}(\mathbb{H}^{n})\times\cdots\times
H_{\alpha_{m}}^{\infty}(\mathbb{H}^{n})$ to
$H_{\alpha}^{\infty}(\mathbb{H}^{n})$, and the following formula for the norm
of the operator hold

$\|\mathfrak{H}_{m}^{h}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}=\mathcal{A}^{h}_{m}.$

According to Theorem 3, we have the following result holds.

######  Corollary 1.

Let $m\in\mathbb{N},\alpha>0$ and $\alpha=\alpha_{1}+\cdots+\alpha_{m}$ with
$\alpha_{i}>0$ for any $i=1,\ldots,m$.Then the operator $\mathfrak{H}_{m}^{h}$
is bounded from $H_{a_{1}}^{\infty}(\mathbb{H}^{n})\times\cdots\times
H_{\alpha_{m}}^{\infty}(\mathbb{H}^{n})$ to
$H_{\alpha}^{\infty}(\mathbb{H}^{n})$ if and only if (3) hold. Moreover, if
(15) hold, the following formula holds

$\mathcal{A}^{h}_{m}=\frac{2Q^{m}}{2^{m}(mQ-\alpha)}\frac{\prod_{i=1}^{m}\Gamma(\frac{Q}{2}-\frac{\alpha_{i}}{2})}{\Gamma(\frac{mQ}{2}-\frac{\alpha}{2})}.$

Now, we begin to prove our result.

###### Proof of Theorem 3.

The proof process in this section is similar to that of Theorem 2, so we only
provide a brief proof and focus on the calculation of the optimal constant.

Combining (14) with the definition of weighted-type space on Heisenberg group
$H^{\infty}_{\alpha}(\mathbb{H}^{n})$, we have

$\displaystyle\|\mathfrak{H}_{m}^{h}\|_{H^{\infty}_{\alpha}(\mathbb{H}^{n})}$
$\displaystyle=\text{ess}\sup_{x\in\mathbb{H}^{n}}|x|_{h}^{\alpha}\left|\frac{1}{\Omega_{Q}^{m}|x|_{h}^{mQ}}\int_{|(y_{1},\ldots,y_{m})|_{h}<|x|_{h}}f_{1}(y_{1})\cdots
f_{m}(y_{m})dy_{1}\cdots dy_{m}\right|$
$\displaystyle=\text{ess}\sup_{x\in\mathbb{H}^{n}}|x|_{h}^{\alpha}\left|\frac{1}{\Omega_{Q}^{m}}\int_{|(z_{1},\ldots,z_{m})|_{h}<1}f_{1}(\delta_{|x|_{h}}z_{1})\cdots
f_{m}(\delta_{|x|_{h}}z_{m})dz_{1}\cdots dz_{m}\right|$
$\displaystyle\leq\prod_{i=1}^{m}\|f_{i}\|_{H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})}\sup_{x\in\mathbb{H}^{n}\backslash\\{0\\}}|x|_{h}^{\alpha}\left|\frac{1}{\Omega_{Q}^{m}}\int_{|(y_{1},\ldots,y_{m})|_{h}<1}\prod_{i=1}^{m}|x|_{h}^{-\alpha}|y_{i}|_{h}^{-\alpha_{i}}dy_{1}\cdots
dy_{m}\right|$
$\displaystyle=\mathcal{A}_{m}^{h}\prod_{i=1}^{m}\|f_{i}\|_{H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})}.$

This indicates for every
$f_{i}\in\prod_{i=1}^{m}(H^{\infty}_{\alpha_{i}}),i=1,\ldots,m$, by taking the
supremum in $H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})$ that

$\|\mathfrak{H}_{m}^{h}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}\leq\mathcal{A}_{m}^{h}.$ (16)

Next, we take $f_{\alpha_{j}}(x)$ defined by (8), which satisfies (9). Using
(14), we can obtain

$|x|_{h}^{\alpha}\left|\frac{1}{\Omega_{Q}^{m}|x|_{h}^{mQ}}\int_{|(y_{1},\ldots,y_{m})|_{h}<|x|_{h}}f_{\alpha_{1}}(y_{1})\cdots
f_{\alpha_{m}}(y_{m})dy_{1}\cdots dy_{m}\right|=\mathcal{A}_{m}^{h}.$ (17)

Combining (9) with (17), it follows that

$\|\mathfrak{H}_{m}^{h}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}\geq\mathcal{A}_{m}^{h}.$

Thus, we have finished the proof of Theorem 3. ∎

In order to obtain the sharp constant, we learned and borrowed the method in
[5]. For convenience, we take several special cases for separate calculations,
and the general situation is a summary of these special cases.
Case 1 when $m=1$.

This situation returns to equation (13), here we provide another form of
definition, then its norm of sharp bound is

$\mathcal{A}^{h}_{1}=\frac{1}{\Omega_{Q}}\int_{|y|_{h}<1}|y|_{h}^{-\alpha}dy<\infty.$

Using the polar coordinate transformation formula, we have

$\displaystyle\mathcal{A}^{h}_{1}$
$\displaystyle=\frac{1}{\Omega_{Q}}\int_{|y|_{h}<1}|y|_{h}^{-\alpha}dy$
$\displaystyle=\frac{1}{\Omega_{Q}}\int_{0}^{1}\int_{S(0,1)}r^{-\alpha+Q-1}drdy^{{}^{\prime}}$
$\displaystyle=\frac{\omega_{Q}}{\Omega_{Q}}\int_{0}^{1}r^{-\alpha+Q-1}dr$
$\displaystyle=\frac{Q}{Q-\alpha}.$

At this point, we have obtained the sharp constant when $m=1$, which is

$\|\mathfrak{H}_{1}^{h}\|_{H^{\infty}_{\alpha}(\mathbb{H}^{n})}=\frac{Q}{Q-\alpha}.$
(18)

Case 2 when $m=2$.

According to Theorem 3 and the polar coordinate transformation formula, we
have

$\displaystyle\mathcal{A}^{h}_{2}$
$\displaystyle=\frac{1}{\Omega_{Q}^{2}}\int_{|(y_{1},y_{2})|_{h}<1}|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{-\alpha_{2}}dy_{1}dy_{2}$
$\displaystyle=\frac{1}{\Omega_{Q}^{2}}\int_{0}^{1}\int_{0}^{1}\int_{S(0,1)}\int_{S(0,1)}r_{1}^{-\alpha_{1}+Q-1}r_{2}^{-\alpha_{2}+Q-1}dr_{1}dr_{2}dy_{1}^{{}^{\prime}}dy_{2}^{{}^{\prime}}$
$\displaystyle=\frac{\omega^{2}_{Q}}{\Omega^{2}_{Q}}\frac{1}{2Q-\alpha}\int_{0}^{1}(1-t^{2})^{\frac{1}{2}(-\alpha_{1}+Q-1)}t^{-\alpha_{2}+Q-1}(1-t^{2})^{-\frac{1}{2}}dt$
$\displaystyle=\frac{\omega^{2}_{Q}}{\Omega^{2}_{Q}}\frac{1}{4Q-2\alpha}\int_{0}^{1}(1-x)^{\frac{1}{2}(-\alpha_{1}+Q)-1}x^{\frac{1}{2}(-\alpha_{2}+Q)-1}dx$
$\displaystyle=\frac{Q^{2}}{2(2Q-\alpha)}B(\frac{Q}{2}-\frac{\alpha_{1}}{2},\frac{Q}{2}-\frac{\alpha_{2}}{2})$
$\displaystyle=\frac{Q^{2}}{2(2Q-\alpha)}\frac{\Gamma(\frac{Q}{2}-\frac{\alpha_{1}}{2})\Gamma(\frac{Q}{2}-\frac{\alpha_{2}}{2})}{\Gamma(Q-\frac{\alpha}{2})}.$

This indicates that when $m=2$, the sharp constant is

$\mathcal{A}^{h}_{2}=\frac{Q^{2}}{2(2Q-\alpha)}\frac{\Gamma(\frac{Q}{2}-\frac{\alpha_{1}}{2})\Gamma(\frac{Q}{2}-\frac{\alpha_{2}}{2})}{\Gamma(Q-\frac{\alpha}{2})}$
(19)

Case 3 when $m\geq 3$.

The proof of case when $m\geq 3$ is similar to that of case 2, we can continue
to use the same method.

$\displaystyle\mathcal{A}^{h}_{m}=$
$\displaystyle\frac{1}{\Omega_{Q}^{m}}\int_{|(y_{1},\ldots,y_{m})|_{h}<1}\prod_{i=1}^{m}|y_{i}|_{h}^{-\alpha_{i}}dy_{1}\cdots
dy_{m}$ (20) $\displaystyle=$
$\displaystyle\frac{1}{\Omega_{Q}^{m}}\int_{0}^{1}\cdots\int_{0}^{1}\int_{(S(0,1))^{m}}\prod_{i=1}^{m}r_{i}^{-\alpha_{i}+Q-1}dr_{1}\cdots
dr_{m}dy_{1}^{{}^{\prime}}\cdots dy_{m}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\omega_{Q}^{m}}{\Omega_{Q}^{m}}\frac{2}{2^{m}(mQ-\alpha)}\int_{0}^{1}\cdots\int_{0}^{1}(1-x)^{\frac{1}{2}(-\alpha_{1}+Q)-1}x^{\frac{1}{2}(-\alpha_{2}+Q)-1}$
$\displaystyle\times\cdots\times(1-x)^{\frac{1}{2}(-\alpha_{m-1}+Q)-1}x^{\frac{1}{2}(-\alpha_{m}+Q)-1}dx$
$\displaystyle=$
$\displaystyle\frac{2Q^{m}}{2^{m}(mQ-\alpha)}\frac{\prod_{i=1}^{m}\Gamma(\frac{Q}{2}-\frac{\alpha_{i}}{2})}{\Gamma(\frac{mQ}{2}-\frac{\alpha}{2})}.$

When we take $m=1$ and $m=2$ in (20), the result is the same as (18) and (19).
So we obtained the sharp constant for $\mathcal{A}_{m}^{h}$, which is

$\mathcal{A}^{h}_{m}=\frac{2Q^{m}}{2^{m}(mQ-\alpha)}\frac{\prod_{i=1}^{m}\Gamma(\frac{Q}{2}-\frac{\alpha_{i}}{2})}{\Gamma(\frac{mQ}{2}-\frac{\alpha}{2})}.$

##  4 Sharp bounds for Hardy-Littlewood-Pólya and Hilbert operators

In this section, we study $m$-linear $n$-dimensional Hardy-Littlewood-Pólya
and Hilbert operators in weighted-type space on Heisenberg group. First of
all, let’s recall their definitions and properties.

The $m$-linear $n$-dimensional Hardy-Littlewood-Pólya is defined by

$P_{m}(f_{1},\ldots,f_{m})(x):=\int_{\mathbb{R}^{nm}}\frac{f_{1}(y_{1})\cdots
f_{m}(y_{m})}{[\text{max}(|x|^{n},|y_{1}|^{n},\ldots,|y_{m}|^{n})]^{m}}dy_{1}\cdots
dy_{m},x\in\mathbb{R}^{n}\backslash\\{0\\}.$

When $n=m=1$, the linear Hardy-Littlewood-Pólya operator $P_{1}$ is considered
in [6], Hardy obtained that the norm of Hardy-Littlewood-Pólya operator on
$L^{p}(\mathbb{R}^{+})(1<q<\infty)$, that is

$\|P_{1}\|_{L^{p}(\mathbb{R}^{+}\rightarrow
L^{p}(\mathbb{R}^{+}))}=\frac{p^{2}}{p-1}.$

The Hilbert operator is the essential extension of the classical Hilbert’s
inequality , the $m$-linear $n$-dimensional Hilbert operator is defined by

$P^{*}_{m}(f_{1},\ldots,y_{m})(x):=\int_{\mathbb{R}^{nm}}\frac{f_{1}(y_{1})\cdots
f_{m}(y_{m})}{(|x|^{n}+|y_{1}|^{n}+\cdots+|y_{m}|^{n})^{m}}dy_{1}\cdots
dy_{m},x\in\mathbb{R}^{n}\backslash\\{0\\}.$

For $n=1$, the following is a know sharp estimate

$\int_{0}^{\infty}T_{1}^{*}f(x)g(x)dx\leq\frac{\pi}{\sin(\pi/p)}\|f\|_{L^{p}(0,\infty)}\|g\|_{L^{p^{\prime}(0,\infty)}}.$

Now, we give the definitions of $m$-linear $n$-dimensional Hardy-Littlewood-
Pólya and Hilbert operators on Heisenberg group.

######  Definition 4.

Suppose that $f_{1},\ldots,f_{m}$ be nonnegative locally integrable functions
on $\mathbb{H}^{n}$. The $m$-linear $n$-dimensional Hardy-Littlewood-Pólya
operator is defined by

$P^{h}_{m}(f_{1},\ldots,f_{m})(x)=\int_{\mathbb{H}^{nm}}\frac{f_{1}(y_{1})\cdots
f_{m}(y_{m})}{[\text{max}(|x|_{h}^{Q},|y_{1}|_{h}^{Q},\ldots,|y_{m}|_{h}^{Q})]^{m}}dy_{1}\cdots
dy_{m},x\in\mathbb{H}^{n}\backslash\\{0\\}.$ (21)

######  Definition 5.

Suppose that $f_{1},\ldots,f_{m}$ be nonnegative locally integrable functions
on $\mathbb{H}^{n}$. The $m$-linear $n$-dimensional Hilbert operator is
defined by

$P^{h*}_{m}(f_{1},\ldots,f_{m})(x)=\int_{\mathbb{H}^{nm}}\frac{f_{1}(y_{1})\cdots
f_{m}(y_{m})}{(|x|^{Q}_{h}+|y_{1}|_{h}^{Q}+\cdots+|y_{m}|_{h}^{Q})^{m}}dy_{1}\cdots
dy_{m},x\in\mathbb{H}^{n}\backslash\\{0\\}.$ (22)

Now, we formulate our main results as follows.

######  Theorem 4.

Let $m\in\mathbb{N},f_{i}\in H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n}),\alpha>0$
and $\alpha=\alpha_{1}+\cdots+\alpha_{m}$ with $\alpha_{i}>0$ for any
$i=1,\ldots,m$. If

$\mathcal{B}^{h}_{m}=\int_{\mathbb{H}^{nm}}\frac{\prod_{i=1}^{m}|y_{i}|_{h}^{-\alpha_{i}}}{[\max(1,|y_{1}|_{h}^{Q},\ldots,|y_{m}|_{h}^{Q})]^{m}}dy_{1}\cdots
dy_{m}<\infty,$ (23)

then $P^{h}_{m}$ is bounded from
$H_{a_{1}}^{\infty}(\mathbb{H}^{n})\times\cdots\times
H_{\alpha_{m}}^{\infty}(\mathbb{H}^{n})$ to
$H_{\alpha}^{\infty}(\mathbb{H}^{n})$ if and only if the following formula for
the norm of the operator hold

$\|P_{m}^{h}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}=\frac{mQ\omega_{Q}^{m}}{\alpha\prod_{j=1}^{m}(Q-\alpha_{j})}.$

######  Theorem 5.

Let $m\in\mathbb{N}$,$f_{i}\in
H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})$,$\alpha>0$ and
$\alpha=\alpha_{1}+\cdots+\alpha_{m}$ with $\alpha_{i}>0$ for any
$i=1,\ldots,m$. If

$\mathcal{B}^{h*}_{m}=\int_{\mathbb{H}^{nm}}\frac{\prod_{i=1}^{m}|y_{i}|_{h}^{-\alpha_{i}}}{[\max(1+|y_{1}|_{h}^{Q}+\cdots+|y_{m}|_{h}^{Q})]^{m}}dy_{1}\cdots
dy_{m}<\infty,$ (24)

then $P^{h}_{m}$ is bounded from
$H_{a_{1}}^{\infty}(\mathbb{H}^{n})\times\cdots\times
H_{\alpha_{m}}^{\infty}(\mathbb{H}^{n})$ to
$H_{\alpha}^{\infty}(\mathbb{H}^{n})$ if and only if the following formula for
the norm of the operator hold

$\|P_{m}^{h*}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}=\left(\frac{2\pi^{n+\frac{1}{2}}\Gamma(n/2)}{(n+1)\Gamma(n)\Gamma((n+1)/2)}\right)^{m}\frac{\prod^{m}_{i=1}\Gamma(1-\frac{\alpha_{i}}{Q})\Gamma(\frac{\alpha}{Q})}{\Gamma(m)}.$

In the above proof,we have obtained that $\mathcal{B}^{h}_{m}$ and
$\mathcal{B}^{h*}_{m}$ are clearly valid, so we omit their proof and will
focus on proving their sharp constants.

###### Proof of Theorem 4 and 5.

In order to get the sharp constant of $m$-linear $n$-dimensional Hardy-
Littlewood-Pólya operator, we will consider the following two scenarios and
combine them to obtain our result.

Case 1 when $m=2$.

In this case , we have

$\mathcal{B}^{h}_{m}=\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{\alpha_{2}}}{[\text{max}(1,|y_{1}|_{h}^{Q},|y_{2}|_{h}^{Q})]^{2}}dy_{1}dy_{2}.$

By calculation,we have

$\displaystyle\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{\alpha_{2}}}{[\text{max}(1,|y_{1}|_{h}^{Q},|y_{2}|_{h}^{Q})]^{2}}dy_{1}dy_{2}=$
$\displaystyle\int_{|y_{1}|_{h}<1}\int_{|y_{2}|_{h}<1}|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{-\alpha_{2}}dy_{1}dy_{2}$
$\displaystyle+\int_{|y_{1}|_{h}>1}\int_{|y_{2}|_{h}\leq|y_{1}|_{h}}|y_{1}|_{h}^{-\alpha_{1}-2Q}|y_{2}|_{h}^{-\alpha_{2}}dy_{1}dy_{2}$
$\displaystyle+\int_{|y_{2}|_{h}>1}\int_{|y_{1}|_{h}<|y_{2}|_{h}}|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{-\alpha_{2}-2Q}dy_{1}dy_{2}$
$\displaystyle:=$ $\displaystyle I_{0}+I_{1}+I_{2}.$ $\displaystyle I_{0}=$
$\displaystyle\int_{|y_{1}|_{h}<1}\int_{|y_{2}|_{h}<1}|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{-\alpha_{2}}dy_{1}dy_{2}$
$\displaystyle=$
$\displaystyle\frac{\omega_{Q}^{2}}{(Q-\alpha_{1})(Q-\alpha_{2})},$
$\displaystyle I_{1}=$
$\displaystyle\int_{|y_{1}|_{h}>1}\int_{|y_{2}|_{h}\leq|y_{1}|_{h}}|y_{1}|_{h}^{-\alpha_{1}-2Q}|y_{2}|_{h}^{-\alpha_{2}}dy_{1}dy_{2}$
$\displaystyle=$
$\displaystyle\frac{\omega_{Q}}{Q-\alpha_{2}}\int_{|y_{1}|_{h}>1}|y_{1}|_{h}^{-\alpha-Q}dy_{1}$
$\displaystyle=$ $\displaystyle\frac{\omega_{Q}^{2}}{\alpha(Q-\alpha_{2})}.$

Similarly,

$\displaystyle I_{2}$
$\displaystyle=\int_{|y_{2}|_{h}>1}\int_{|y_{1}|_{h}<|y_{2}|_{h}}|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{-\alpha_{2}-2Q}dy_{1}dy_{2}$
$\displaystyle=\frac{\omega_{Q}}{Q-\alpha_{1}}\int_{|y_{2}|_{h}>1}|y_{2}|_{h}^{-\alpha-Q}dy_{2}$
$\displaystyle=\frac{\omega_{Q}^{2}}{\alpha(Q-\alpha_{1})}.$

Thus,we have

$\displaystyle\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{|y_{1}|_{h}^{-\alpha_{1}}|y_{2}|_{h}^{\alpha_{2}}}{[\text{max}(1,|y_{1}|_{h}^{Q},|y_{2}|_{h}^{Q})]^{2}}dy_{1}dy_{2}$
$\displaystyle=$ $\displaystyle I_{0}+I_{1}+I_{2}$ $\displaystyle=$
$\displaystyle\frac{2Q\omega_{Q}^{2}}{\alpha(Q-\alpha_{1})(q-\alpha_{2})}.$

Case 2 when $m\geq 3$.

Let

$\displaystyle
E_{0}=\\{(y_{1},\ldots,y_{m})\in\mathbb{H}^{n}\times\cdots\times\mathbb{H}^{n}:|y_{k}|_{h}\leq
1,1\leq k\leq m\\};$ $\displaystyle
E_{1}=\\{(y_{1},\ldots,y_{m})\in\mathbb{H}^{n}\times\cdots\times\mathbb{H}^{n}:|y_{1}|_{h}>1,|y_{k}|_{h}\leq|y_{1}|_{h},2\leq
k\leq m\\};$ $\displaystyle
E_{i}=\\{(y_{1},\ldots,y_{m})\in\mathbb{H}^{n}\times\cdots\times\mathbb{H}^{n}:|y_{i}|_{h}>1,|y_{j}|_{h}<|y_{k}|_{h},|y_{k}|_{h}\leq|y_{i}|_{h},1\leq
j<i<k\leq m\\};$ $\displaystyle
E_{m}=\\{(y_{1},\ldots,y_{m})\in\mathbb{H}^{n}\times\cdots\times\mathbb{H}^{n}:|y_{m}|_{h}>1,|y_{j}|_{h}<|y_{m}|_{h},1<j<m\\}.$

Obviously, we have

$\bigcup_{j=0}^{m}E_{j}=\mathbb{H}^{n}\times\cdots\times\mathbb{H}^{n},E_{i}\cap
E_{j}=\emptyset.$

Taking

$K_{j}=\int_{\mathbb{H}^{nm}}\frac{\prod_{i=1}^{m}|y_{i}|_{h}^{-\alpha_{i}}}{[\text{max}(1,|y_{1}|_{h}^{Q},\ldots,|y_{m}|_{h}^{Q})]^{m}}dy_{1}\cdots
dy_{m},$

then we begin to calculate $K_{j}$ with $j=0,1,\ldots,m$.

$K_{0}=\prod_{j=1}^{m}\int_{|y_{j}|\leq
1}|y_{j}|_{h}^{-\alpha_{j}}dy_{j}=\frac{\omega_{Q}^{m}}{\prod_{j=1}^{m}(Q-\alpha_{j})},$
$\displaystyle K_{1}$
$\displaystyle=\int_{|y_{1}|>1}|y_{1}|_{h}^{-\alpha_{1}-Qm}dy_{1}\prod_{j=2}^{m}\int_{|y_{j}|_{h}\leq|y_{1}|_{h}}|y_{j}|_{h}^{-\alpha_{j}}dy_{j}$
$\displaystyle=\frac{\omega_{Q}^{m-1}}{\prod_{j=2}^{m}(Q-\alpha_{j})}\int_{|y_{1}|_{h}>1}|y_{1}|_{h}^{-\alpha-Q}dy_{1}$
$\displaystyle=\frac{\omega_{Q}^{m}}{\alpha\prod_{j=2}^{m}(Q-\alpha_{j})}.$

So we can deduce that

$K_{j}=\frac{\omega_{Q}^{m}}{\alpha\prod_{1\leq i\leq m,i\neq
j}(Q-\alpha_{j})}.$

Then we can obtain that

$K_{m}=\frac{mQ\omega_{Q}^{m}}{\alpha\prod_{j=1}^{m}(Q-\alpha_{j})}.$

Combining the above two cases, we have completed the Theorem 4. Next, we
follow the method given by Bényi and Oh [3] to calculate the sharp constant of
$m$-linear $n$-dimensional Hilbert operator. Using the polar coordinates and
making change of variables, we have

$\displaystyle\|P_{m}^{h*}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}$
$\displaystyle=\int_{\mathbb{H}^{nm}}\frac{|y_{1}|_{h}^{-\alpha_{1}}\cdots|y_{m}|_{h}^{-\alpha_{m}}}{(1+|y_{1}|_{h}^{Q}+\cdots+|y_{m}|_{h}^{Q})}dy_{1}\cdots
dy_{m}$
$\displaystyle=\omega_{Q}^{m}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{r_{1}^{Q-\alpha_{1}-1}\cdots
r_{m}^{Q-\alpha_{m}-1}}{(1+r_{1}^{Q}+\cdots+r_{m}^{Q})^{m}}dr_{1}\cdots
dr_{m}$
$\displaystyle=\frac{\omega_{Q}^{m}}{Q^{m}}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{t_{1}^{\frac{1}{Q}(Q-\alpha_{1})-1}\cdots
t_{m}^{\frac{1}{Q}(Q-\alpha_{m})-1}}{(1+t_{1}+\cdots+t_{m})^{m}}dt_{1}\cdots
dt_{m}$
$\displaystyle=\frac{\omega_{Q}^{m}}{Q^{m}}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{t_{1}^{-\frac{\alpha_{1}}{Q}}\cdots
t_{m}^{-\frac{\alpha_{m}}{Q}}}{(1+t_{1}+\cdots+t_{m})^{m}}dt_{1}\cdots dt_{m}$
$\displaystyle=\frac{\omega_{Q}^{m}}{Q^{m}}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{t_{1}^{-\beta_{1}}\cdots
t_{m}^{-\beta_{m}}}{(1+t_{1}+\cdots+t_{m})^{\alpha}}dt_{1}\cdots dt_{m}.$

Set

$I_{m}(\alpha,\beta_{1},\ldots,\beta_{m}):=\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{t_{1}^{-\beta_{1}}\cdots
t_{m}^{-\beta_{m}}}{(1+t_{1}+\cdots+t_{m})^{\alpha}}dt_{1}\cdots dt_{m}.$

By a simple calculate, we have

$\displaystyle\int_{0}^{\infty}\frac{1}{(1+t)^{\alpha}t^{\beta}}dt$
$\displaystyle=\int_{0}^{\infty}(1-t)^{-\beta}t^{\alpha+\beta-2}dt$
$\displaystyle=B(1-\beta,\alpha+\beta-1).$

Using the variable substitution $t_{m}=(1+t_{1}+\cdots+t_{m}-1)q_{m}$, we have

$\displaystyle I_{m}(\alpha,\beta_{1},\ldots,\beta_{m})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{t_{1}^{-\beta_{1}}\cdots
t_{m-1}^{-\beta_{m-1}}}{(1+t_{1}+\cdots+t_{m-1})^{\alpha-1+\beta_{m}}}dt_{1}\cdots
dt_{m-1}\int_{0}^{\infty}\frac{1}{(1+q_{m})^{\alpha}q_{m}^{\beta_{m}}}dq_{m}$
$\displaystyle=$ $\displaystyle
B(1-\beta_{m},\alpha+\beta_{m}-1)I_{m}(\alpha-1+\beta_{m},\beta_{1},\ldots,\beta_{m-1}).$

Using the inductive method combine with the properties of Gamma function, we
have

$I_{m}(\alpha,\beta_{1},\ldots,\beta_{m})=\frac{\prod_{i=1}^{m}\Gamma(1-\beta_{i})\Gamma(\alpha-k+\prod_{i=1}^{m}\beta_{i})}{\Gamma(\alpha)}.$

Thus, we can obtain

$\displaystyle\|P_{m}^{h*}\|_{\prod_{i=1}^{m}H^{\infty}_{\alpha_{i}}(\mathbb{H}^{n})\rightarrow
H^{\infty}_{\alpha}(\mathbb{H}^{n})}$
$\displaystyle=\frac{\omega_{Q}^{m}}{Q^{m}}\frac{\prod^{m}_{i=1}\Gamma(1-\frac{\alpha_{i}}{Q})\Gamma(\frac{\alpha}{Q})}{\Gamma(m)}$
$\displaystyle=\left(\frac{2\pi^{n+\frac{1}{2}}\Gamma(n/2)}{(n+1)\Gamma(n)\Gamma((n+1)/2)}\right)^{m}\frac{\prod^{m}_{i=1}\Gamma(1-\frac{\alpha_{i}}{Q})\Gamma(\frac{\alpha}{Q})}{\Gamma(m)}.$

This finishes the proof of Theorem 4 and 5. ∎

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant
No. 12271232) and Shandong Jianzhu University Foundation (Grant No.
X20075Z0101).

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

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

_E-mail address_<EMAIL_ADDRESS>
Dunyan Yan
School of Mathematical Sciences
University of Chinese Academy of Sciences
Beijing, 100049
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>