# Thermal and electric field driven breakdown precursor formation on metal
surfaces

Ryo Shinohara  Department of Electrical and Computer Engineering, Michigan
State University, MI 48824, USA  Department of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  Theoretical Division, Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  Soumendu Bagchi
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  Evgenya Simakov  Accelerator Operations and Technology Division, Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  Sergey V. Baryshev
Department of Electrical and Computer Engineering, Michigan State University,
MI 48824, USA  Department of Chemical Engineering and Material Science,
Michigan State University, MI 48824, USA  Danny Perez  Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA

###### Abstract

The phenomenon of electric breakdown poses serious challenge to the design of
devices that operate under high gradient environments. Experimental evidence
often points towards breakdown events that are accompanied by elevated
temperatures and dark current spikes, presumably due to high-asperity nano-
structure formation that enhances the local electric field and triggers a
runaway process. However, the exact mechanistic origin of such nano-structures
under typical macroscopic operational conditions of electric gradient and
magnetic-field-mediated heating remains poorly understood. In this work, a
model is presented that describes the evolution of a typical copper surface
under the combined action of the electric fields and elevated temperatures.
Using a mesoscale curvature-driven growth model, we show how the copper
surface can undergo a type of dynamical instability that naturally leads to
the formation of sharp asperities in realistic experimental conditions.
Exploring the combined effect of fields and temperature rise, we identify
critical regimes that allow for the formation of breakdown precursors. These
regimes strongly resonate with previous experimental findings on breakdown of
copper electrodes, hence suggesting surface diffusion to be a crucial
breakdown precursor mechanism.

##  I Introduction

Application of high electric field are commonplace in modern vacuum
technologies ranging from space propulsion to powerful linear accelerators
Teel1  _et al._ (2017); Simakov  _et al._ (2018). In this latter use case,
next-generation accelerator technologies aim to operate in large field
gradient regimes, over 100 MV/m Limborg-Deprey  _et al._ (2016); Lucas  _et
al._ (2021). However, such high field are known to lead to vacuum arcs
launched by so-called breakdown (BD) events. These events not only perturb
normal operation, but also cause damages to the devices, ultimately limiting
the maximum achievable fields. In spite of the fact that BD has been a known
phenomenon for over a century, the detailed mechanism behind BD nucleation and
evolution remain poorly understood, in great part because BD events destroys
their precursors.

Previous experimental, theoretical, and computational efforts Han  _et al._
(2005); Descoeudres  _et al._ (2009a); Kyritsakis  _et al._ (2018); Eimre1
_et al._ (2015); Wang and Loew (1997) have proposed essential processes, later
categorizing them into two major groups: 1) processes that lead to the
formation of (i.e. nucleate) BD events and 2) processes that take place during
the BD event, including its terminal stage. It is hypothesized that a BD event
likely nucleates through the formation of geometrical asperities. These
asperities can locally enhance the applied gradient, triggering a runaway
process eventually leading to material evaporation and the terminal arc
formation. However, the exact mechanisms behind the formation of such nano-
structures on mirror-quality polished surface with nanoscale roughness remain
unknown.

What is known is that critical breakdown rates strongly depend on applied
electric and microstructural variations Simakov  _et al._ (2018). While
plastic deformation driven by electric fields Engelberg  _et al._ (2019);
Pohjonen  _et al._ (2011); Bagchi and Perez (2022) have been computationally
explored as candidate mechanisms to understand BD phenomenon, the applied
fields considered in these studies (e.g., 5-10 GV/m) have been mostly orders
of magnitude higher than those applied in experiments (e.g. 100-200 MV/m). On
the other hand, the breakdown rate is known to also be very sensitive to the
temperature Cahill  _et al._ (2018); Döbert (2004); Zha and Grudiev (2016).
While reducing temperature immediately decreases the BDR Cahill  _et al._
(2018), increasing the operating temperature above room temperature by only 50
K quickly increases rate of breakdown and essentially limits the practical
gradients Cahill  _et al._ (2018); Döbert (2004). When not actively cooled,
the operating temperature naturally grows due to a process called pulse
heating, where it is capable of producing temperature rise in the order of
several hundred degrees Pritzkau (2001). These results from the effects
associated with the magnetic field component of the radio-frequency (rf)
signal and the duty cycle (rf pulse length vs. repetition rate). The combined
pulsed heating and peak electric fields set an upper limits on the gradients
that can be used in normal operation if a sufficiently low breakdown rate is
to be maintained.

The synergy between the electric gradient and temperature, as well as the
effect of nominal surface geometries are relatively unexplored in previous
analyses Engelberg  _et al._ (2019). Here, we explore surface diffusion as a
possible mechanisms enabling BD nucleation under nominal experimental
$E$-field and heating conditions. By considering the competition between
surface tension and the external electric and thermo-mechanical driving
forces, we show that moderate initial surface perturbations are capable of
triggering a structural instability through surface diffusion, leading to the
formation of BD precursors at electric field and temperature gradients
comparable to experimental measurements reported in operating accelerators.

##  II Problem formulation

Figure 1: Example of FEM mesh used for the simulation where the vacuum medium
is shown as the green mesh and the bulk copper material is shown here as
orange. The red line shows the vacuum-copper boundary

Consider 2D domains that contain a vacuum and an elastic half-space that here
corresponds to bulk copper (Fig. 1). In this domain, surface evolution follows
from surface diffusion driven by electrostatic, thermo-elastic, and surface
energy contributions. In the following, we provide details of the multi-
physics workflow which accounts for diffusive evolution of surfaces under
combined effects of temperature-increase and electric fields.

###  II.1 Electrostatics analysis

In order to determine the local electric-field ($E$) along the materials
surface, we solve for the electrostatic potential ($\phi$) using Gauss’s law
in the domain shown in Fig. 1 under an applied electric field ($E_{a}$):

$\displaystyle\nabla\cdot(\nabla\phi)=0$ (1)

The material (copper) is idealized as a perfect conductor where the electric
potential vanishes, enforcing Dirichlet boundary condition at the
copper/vacuum interface. At a large enough distance away from the material
surface, we employ Neumann boundary conditions where the field is constrained
to the nominal applied field $E_{a}$. Once Eq. 1 is solved, the electric field
at any position can be obtained by taking the negative gradient of the
potential $-\nabla\phi=E$. Although the motivation of our model is to capture
surface evolution in the context of rf breakdown, the applied field is kept
constant during the simulation, since the time scale for surface morphology
evolution is much slower than that of an rf cycle (of roughly 100 ps in
X-band.) In this context, we can assume effective dc-field conditions at the
surface corresponding to the RMS of an rf cycle.

###  II.2 Thermo-elasticity analysis

To capture the effects of rf losses, we model the elastic deformation in the
copper domain resulting from a uniform temperature rise. Such uniform
temperature approximation stems from the fact that the high thermal
conductivity of copper quickly equilibrates temperature gradients over the
micron lengthscales simulated here. Indeed, experiments with rf pulse heating
show that temperature gradients tend to form on the scale of
$$\mathrm{mm}$\sim$\mathrm{cm}$$Laurent  _et al._ (2011), which is consistent
with our assumption.

The governing equations for small deformations on isotropic and homogeneous
materials can be written as in Refs. Anand and Govindjee (2020); Boresi and
Schmidt (2002):

$\displaystyle-\nabla\cdot\sigma(u)=0$ (2a) $\displaystyle\sigma(u)=\lambda
tr(\epsilon(u))\mathbb{1}+2\mu\epsilon(u)-\frac{\alpha Y}{1-2\nu}\Delta
T\mathbb{1}$ (2b) $\displaystyle\epsilon(u)=\frac{1}{2}(\nabla u+(\nabla
u)^{T})$ (2c)

where $\sigma$ is the stress tensor, $u$ is the displacement vector,
$\epsilon$ is the infinitesimal strain tensor, $\lambda$ and $\mu$ are the
Lamé’s elasticity parameters, $\alpha$ is the thermal expansion coefficient,
$Y$ is the Young’s modulus, $\nu$ is Poisson’s ratio, and $\Delta T$ is the
temperature change from the reference room temperature of 300 K.

We use fixed displacement boundary conditions at the bottom of the domain and
periodic boundary conditions along the lateral boundaries. The solution of Eq.
2a produces the displacement vector field $u$, describing deformation of the
material under a given $\Delta T$. With the displacement vector, the
corresponding strain and stress tensor can be obtained from Eq. 2b and Eq. 2c,
respectively.

It is important to note that thermo-elasticity only acts as a driving force if
the system is mechanically constrained in some directions (perpendicular to
the surface in this case). In an accelerator setting where heating from RF
losses mainly affects the first few microns from the surface, confinement is
provided by the rest of the material which remains colder and hence at a
shorter elastic constant. Thermo-elastic driving forces would however vanish
for uniformly heated systems that are allowed to freely thermally-expand in
all directions.

###  II.3 Surface Diffusion

We incorporate material transport through a surface diffusion process, where
the diffusion flux ($J$) is proportional to the surface gradient of the
chemical potential ($\mu_{c}$) as:

$J=-\frac{D_{s}}{k_{b}T}\frac{\partial\mu_{c}}{\partial s}$ (3)

where $D_{s}$ is surface diffusivity, $k_{b}$ is the Boltzmann constant, $T$
is the absolute temperature, and derivative with respect to $s$ is evaluated
along the surface.

From mass conservation, the normal velocity $V_{n}$ of the surface can be
expressed using the surface divergence of $-J$. $V_{n}$ can then be
reformulated in terms of $\partial h(x,t)/\partial t$, the vertical velocity
of the surface profile. The latter can be written as Yang and Srolovitz
(1994); Mullins (1957):

$\displaystyle V_{n}=\frac{D_{s}\Omega
v_{s}}{k_{b}T}\frac{\partial^{2}\mu_{c}}{\partial s^{2}}$ (4a)
$\displaystyle\frac{\partial h}{\partial
t}=\frac{D_{s}v_{s}\Omega}{k_{B}T}\frac{\partial}{\partial
x}[(1+h^{\prime}_{x})^{-1/2}\frac{\partial}{\partial x}(\mu_{c})]$ (4b)

Here $h^{\prime}_{x}$ indicates the derivative with respect to $x$, and
$v_{s}$ is the number of atoms per unit area.

Aggregating the contribution of electrostatics, thermo-elastic stresses, and
surface tension, the chemical potential on the surface can be written as Yang
and Srolovitz (1994):

$\displaystyle\mu_{c}=\Omega(\gamma\kappa+\omega_{T}-U_{E})$ (5)
$\displaystyle\kappa=-\frac{h^{\prime\prime}_{x}}{(1+h_{x}^{\prime 2})^{3/2}}$
$\displaystyle\omega_{T}=\frac{1}{2}(\sum_{j=1}^{2}\sum_{i=1}^{2}\sigma_{ij}\epsilon_{ij})$
$\displaystyle U_{E}=\frac{1}{2}\epsilon_{0}E_{local}^{2}$

Here, $\omega_{T}$ is the strain-energy density function for linear isotropic
materials, $U_{E}$ is the electric field-energy density on the material’s
surface, $\gamma$ is the isotropic surface free energy, and $\kappa$ is the
local surface curvature. The surface field-energy density and the strain
energy density are calculated simultaneously on their respective mesh, and Eq.
4b is solved to calculate the vertical velocity of the surface. The meshes for
the vacuum and material domains are then updated to conform to the new
interface. The field and thermal stress is recalculated again for each domain
after approximately 100 time steps, when the surface has evolved a sufficient
amount. This frequency is adjusted depending on the geometry and the surface
velocity. The open source PDE solver FEniCS Alnaes  _et al._ (2015) was used
to solve the governing equations using the finite-element method. Variational
forms are specified in the FEniCS workflow in the high-level Python-based
Unified Form Language (UFL) Kirby and Logg (2006). These forms are then
automatically compiled and executed through high-performance computational
kernels using the finite element library DOLFIN Logg and Wells (2010). This
computational approach was incorporated in a framework called SurFE-XD
(Surface curvature-driven Finite Elements model for Diffusion under eXtreme
conditions, previously introduced to investigate electric-field-driven surface
evolution Bagchi  _et al._ (2023)).

###  II.4 Initial surface morphology

We consider three different types of surface geometries: isolated
perturbations, sinusoidal perturbations, and random surfaces. An example of
isolated perturbation is shown in Fig. 1 where the simulated initial profile
is modeled by a Gaussian on an otherwise flat surface. For sinusoidal
surfaces, the material’s surface is perturbed by a perfect sine wave with
constant wavelength $\lambda$ commensurate with the size of the domain. For
both Gaussian and sinusoidal surfaces, the aspect ratio of the perturbation is
defined as the ratio of the height over either two lateral standard deviation
(for isolated perturbations) or the wavelength (for sinusoidal perturbations)
of the respective initial surface profile.

Finally, random surfaces were used to capture the evolution of realistic
copper surfaces. To model random rough surfaces, the vacuum-copper boundary
was represented as a sum of Fourier modes Hoyt1  _et al._ (2001):

$h(x)=\sum_{n}^{N}A_{n}sin(n\omega x+\phi_{n})$ (6)

where the amplitude $A_{n}$ is obtained by fitting an exponential decay to an
experimentally measured spatial frequency spectrum of electro-polished copper
photocathodes Qian  _et al._ (2012), and the phases $\phi_{n}$ are sampled
randomly.

###  II.5 Linear Stability Analysis

Linear stability analysis can be utilized to complement fully nonlinear
numerical solutions by providing analytical solutions in the limit of small
perturbations. Hence, it serves as a baseline verification of the numerical
results.

By inserting Eq. 5 into Eq. 4b and assuming a small initial perturbation
$h(x)=h_{0}\sin(kx)$, the solution in the frequency domain takes the form of
$h(k,t)$=$h_{0}\exp(g(k)t)$ where $g(k)$ is the growth rate corresponding to
wavenumber $k$. In the small perturbation regime, Eq. 5 can be simplified as
$\kappa\approx\frac{\partial^{2}h}{\partial x^{2}}=-Ak^{2}sin(kx)$,
$U_{E}$=$\epsilon_{0}E_{0}^{2}(ksin(kx))$, $\omega_{T}=2kY\alpha^{2}\Delta
T^{2}$, resulting in the temporal surface evolution described as Srolovitz
(1989); Du and Srolovitz (2004a):

$\displaystyle\frac{\partial h}{\partial t}=C[-\gamma
h^{\prime\prime\prime\prime}_{x}-kh^{\prime\prime}_{x}(2\alpha^{2}\Delta
T_{0}^{2}Y-\epsilon_{0}E_{0}^{2})]$ (7a) $\displaystyle
h(t,k)=h_{0}\exp[C(2k^{3}\alpha^{2}\Delta
T_{0}^{2}Y+\epsilon_{0}E_{0}^{2}k^{3}-\gamma k^{4})t]$ (7b)

where $C=\frac{D_{s}v_{s}\Omega^{2}}{k_{B}T}$.

Figure 2: Growth rate $g(k)$ for surface wavenumber $k$. Perturbations with
wavenumber $k<k_{0}$ will grow while those corresponding to larger wavenumbers
will decay. The maximum growth rate occurs at $k=k_{m}$. See text for details.

Fig. 2 reports the growth rate $g(k)=C(2k^{3}\alpha^{2}\Delta
T_{0}^{2}Y+\epsilon_{0}E_{0}^{2}k^{3}-\gamma k^{4})$. The result shows that
perturbations corresponding to wavenumber below a critical value
$k_{0}=1/\gamma(2\alpha^{2}\Delta T_{0}^{2}Y+\epsilon_{0}E_{0}^{2})$ will
grow, while larger wavenumber modes will decay. The fastest growing mode can
be found by solving $\partial g(k)/\partial k=0$, which gives
$k_{m}=3k_{0}/4$. The result indicates that the maximally unstable mode
$k_{m}$ and critical mode $k_{0}$ both depend quadratically on the
operating/applied field and temperature rises. In general, higher temperatures
or applied fields allow for the spontaneous growth of perturbations at
increasingly large wavenumbers (increasingly small wavelengths). For typical
applied field of 100 MV/m and $\Delta T$=100 K, the critical wavelength,
$\lambda_{0}=2\pi/k_{0}$, is approximately 12 $\mu$m. If the gradient
increases to 300 MV/m, $\lambda_{0}$ decreases to 6 $\mu$m.

Eq. 7b can be used to approximate small amplitude surface evolution by taking
the Fourier transform of the equation. The critical temperature rise,
$T_{cr}$, and the critical nominal field, $E_{cr}$, can be defined using this
approximation by finding the value in which linear stability approximation
predicts a zero growth velocity. It is important to note that this
approximation is only expected to hold for small perturbations, as
interactions between modes are not taken into account by this approach.

##  III Results

###  III.1 Effect of heating

Figure 3: Simulated copper surface profile after applying a
$120\text{\,}\mathrm{K}$ uniform temperature rise to an initial sinusoidal
perturbation.

In the following, the temporal evolution of the modeled surface profiles are
reported in dimensionless time $\tau$. Conversion to physical time units will
be discussed in the Section IV.

We first consider the evolution of a copper surface subjected to a uniform
temperature rise in the absence of applied electric fields Yang and Srolovitz
(1994). The results of a uniform temperature rise of 120 K applied to a
sinusoidal surface is shown in Fig. 3. A temperature rise in the bulk material
results in compressive stresses due to the material being constrained in the
lateral direction. The resulting deformation causes the strain-energy density
to be concentrated at the trough of the sinusoidal profile, resulting in
sharpening of the troughs and simultaneous broadening of the peaks. These
concomitant sharpening and broadening occur due to conservation of mass during
the surface diffusion: the material removed from the trough is redistributed
to the surroundings of the original maximum. This is a key point – the
evolution of a small amplitude perturbation dominated by temperature rise
alone is unlikely to drive the formation of a type of precursor that would
lead to breakdown, as flat peaks and sharp troughs do not couple efficiently
with an electric field in a way that is expected to cause runaway field
emission. Electric fields are therefore expected to be a critical ingredient
in the formation of BD precursors.

###  III.2 Effect of electric field

In contrast, the application of sufficiently large electric fields has been
shown to lead to runaway tip growth and sharpening, due to the localized
enhancement of electric fields in regions of high negative curvatures Bagchi
_et al._ (2023). This mechanism was observed to lead to the formation of
breakdown precursors on copper surfaces at fields on the order of 250 to 500
MV/m, depending on the characteristics of the initial surface perturbations.
Grooving instabilities were not observed under electric field alone. Similar
mechanisms drive the formation of Taylor cones in liquid conductors Suvorov
and Zubarev (2003) as well as on metal electrode surfaces Du and Srolovitz
(2004b); Gill  _et al._ (2008). Therefore, field driven evolution of initially
small-amplitude perturbations initially present on the metal surface is a
likely candidate for the formation of sharp asperity breakdown precursors.

###  III.3 Critical $E$-field and heating regimes for instability

Figure 4: (a) Phase Diagram showing the normalized time when initial surface
profile grow a 10$\%$ of the original height or reach instability for given
combination of parameters. (b) Interpolated horizontal slice of the phase
diagram at various aspect ratio with analytically predicted boundary for low
aspect ratio.

In order to unveil the synergistic effect between the two driving forces, i.e.
a combined thermal and electric field effects, we conducted series of
simulations by applying electric field ranging from 140-275 MV/m and
temperature increase ranging from 0-100 K to act on a surface profile having
an aspect ratio ranging from 0.1-1.5. Here, the initial asperity has a
Gaussian profile with a 2 standard deviation width of 10 $\mu$m.

Fig. 4(a) reports a phase diagram of simulations with different combinations
of the aspect ratio, applied field and temperature augmentation. Here, the
color scale represents the time at which the surface grows by 10% of the
initial height or reaches instability at the peak $k_{m}$, i.e. moves in the
supercritical regime (Fig. 2). The instability is characterized by the
acceleration of the surface velocity at the peak of profile when the surface
reaches instability or when elapsed computational time $t$ is 1 s in SI units.
The black points represents the simulations in which called operating
temperature and electric field were not sufficient to reach instability
runaway, i.e. resulting in the initial protrusion to decay back towards flat
surface. The black dashed lines and the colored lines in Fig 4(a) separate
these two regimes, but should not be taken as the precise location of the
actual boundary, given the limited number of simulations.

Figure 5: (a) Surface evolution with $48\text{\,}\mathrm{K}$ temperature rise,
192 MV/m, and 0.1 aspect ratio. (c) Surface evolution with
$48\text{\,}\mathrm{K}$, 192 MV/m, and 1.0 aspect ratio. (b,d) Fourier
transform of the surface profile from the FEM simulations (b) with analytical
growth rate curve calculated from Eq. 7b

At small aspect ratios (0.1 (red) and, to a lesser extent 0.5 (green)),
temperature and applied electric field display synergistic effects, causing
tip growth in a regime where electric fields alone would be sub-critical. This
behavior is initially captured by the linear stability analysis introduced in
the Section II.E above (blue dotted line), within the uncertainty of the exact
location of the boundary. As the temperature rises above a certain value
(about $55\text{\,}\mathrm{K}$), the trend reverses and heating contributes
less and less to the formation of breakdown precursors. In contrast, at higher
aspect ratios ($\geq 1$), the propensity for BD precursor formation is
unaffected by heating (as indicated by the vertical lines).

Note that high amounts of heating alone do not lead to BD, as it instead leads
to grooving instabilities with blunt peaks, as described above. The resulting
reduction in peak (negative) curvature creates microstructures that couple
weakly with the electric field, and hence to the absence of tip growth.
Therefore, breakdown would not be expected in the high-$T$/low-$E$ regime, as
observed in Fig. 4.

The opposite regime, high-$E$/low-$T$ has been previously investigated Bagchi
_et al._ (2023). In this case, the critical electric field required for
instability was shown to decrease with aspect ratio, i.e., a consequence of
the fact that the electric field enhancement factor at the tip is linearly
proportional to the aspect ratio. This effect is observed in Fig. 4, as
expected.

On general grounds, one might also expect that heating would not affect tip
growth at high aspect ratio, as the elastic energy due to thermal loading can
be efficiently relaxed since the material is not laterally constrained at the
tip (in contrast to bulk materials or low aspect ratio asperities). In this
limit, BD propensity can be expected to be unaffected by heating. This is
again consistent with the results shown in Fig. 4.

The intermediate regime of $E$ and $T$ at low aspect ratio is more complex.
Fig. 5(a) illustrates a surface profile for a simulation with initial aspect
ratio of 0.1 under application of a field of 192 MV/m and a temperature
augmentation of 48 K. This corresponds to conditions where BD would not have
been expected under this electric field alone; heating is therefore enhancing
BD precursor formation in this case. The analytical solution of Eq. 7b under
this small surface perturbations is plotted against the simulation results.
The agreement is excellent, thereby showing that the emerging wavelength,
shown in Fig. 5(b), follows the predictions of linear stability analysis.
Additionally, the spectral analysis in Fig. 5(b) illustrates the strong
selection of a particular surface wavelength corresponding to the maximum of
the predicted growth rate $g(k)$ for the applied parameters combination. This
amplification process is efficient as the initial surface profile has a
significant amplitude at that frequency. Higher frequency modes are quickly
suppressed, leading to an efficient wavelength selection process. The mode
amplitude obtained computationally are in excellent agreement with the
analytically predicted growth rate. Analytically, the fastest growing mode
$k_{m}/(2\pi)$ is predicted for a wavelength of 0.039 $\mu$m-1, compared to
0.04 $\mu$m-1 observed in the simulations. When the aspect ratio is much
higher, (1.0, which is a case that would be expected to experience BD under
the corresponding electric field alone) Fig. 5(c,d) show that wave-like
surface evolution with a strong wavelength selection does not occur (at least
initially). Instead, high frequency modes are amplified by coupling with the
$E$-field at the tip, leading to rapid tip growth and sharpening, and
ultimately to BD. As shown by the comparison to the analytically predicted
growth rates (dashed line), the evolution at high aspect ratio cannot be
described by linear perturbation and is instead dominated by non-linear
effects caused by the large curvature at the tip, which lead to the growth of
high-frequency modes.

Figure 6: (a) Velocity components of the final surface profile shown in FIG.
5(a) for surface tension, applied electric field, and temperature rise. The
applied parameter for the simulations are 0.1 aspect ratio,
$48\text{\,}\mathrm{K}$ temperature rise, and 192 MV/m. (c) Velocity component
for simulation with 1.0 aspect ratio., $$48\text{\,}\mathrm{K}$\Delta T$,
192Mv/m. (b,d) Fourier Transform of the individual velocity components
respectively for the final surface in (a) and (b).

In order to elucidate the mechanics behind the surface evolution in a coupled
setting, it is useful to isolate the surface velocity contributions stemming
from surface tension, electric field, and heating. Such a summary is shown in
Fig. 6(a,d) for two aspect ratios, 0.1 and 1.0.

The surface tension contribution (blue line) is observed to counteract any
curvature: when negative curvature develops (at tips), the surface tension’s
contribution is negative, leading to blunting; in contrast, when the curvature
is positive (at grooves), the surface tension component is positive, again
leading to blunting. In contrast, the electric field contributions (orange
line), tend to amplify both tips and grooves, as the material diffuses along
the field gradient (from grooves where the applied field is partially shielded
towards tips where the field is amplified). Finally, heating (green lines)
tends to amplify grooves much more than tips, as discussed below. At low
aspect ratios, the velocity contributions from each driving forces are
similar. In contrast, at higher aspect ratios, corresponding to Fig. 6(d), the
relative contribution from heating is marginal compared to the surface and
electric field components, consistent with the above discussion that thermo-
elasticity couples weakly with tip-like structures, in contrast to electric
fields that are strongly amplified at such features Bagchi  _et al._ (2023).
In this case, the surface evolution is only weakly affected by heating,
consistent with the results shown in Fig. 4.

Electric fields therefore provide the main driving force leading to the
increase in negative curvature necessary for BD to occur. Nonetheless, Fig. 4
clearly shows that thermo-elastic driving forces can lead assist tip growth
when acting concurrently with an electric field. At low heating, the observed
enhancement follows the predictions of the analytic linear perturbation model.
This suggests that the main effect in this case is the amplification of the
growth rate of high frequency modes that are required for the eventual
creation of sharp features that efficiently couple with the electric field to
cause BD. This is confirmed in Fig. 7(a), where it can be seen that thermo-
elasticity contribute a relatively small but positive velocity at the tip
location at short times. As the aspect ratio of the asperity increases, this
velocity component correspondingly decreases as expected.

Figure 7: Velocity contribution from (a) temperature rise and (b) applied
field at various $t/\tau$ for the surfaces shown in Fig. 5(a)

As the temperature increases further, the trend reverses and the assistance
provided by heating becomes less significant. Departure from the linear-
stability predictions suggest that non-linear/large-deformation effects are
responsible for this reversal. The behavior observed for strong heating in
Fig. 3 shows that runaway groove growth (which form in the non-linear, large
amplitude, regime) leads to a blunting of the tip region. The fact that the
location of the initial tip moved up during surface evolution shows that
blunting resulted from the mass flowing out of the groove, and not primarily
from surface tension (which would have pulled the tip region down). The same
effect appears to be responsible for the decrease in assistance provided by a
large amounts of heating with respect to BD formation at small aspect ratio.
Our simulations show that if runaway groove growth sets in before tip growth,
the mass flowing out of the grooves contributes to a decrease in the curvature
in the tip region that hampers the sharpening of the tip, hence leading to a
relative decrease in the efficiency of the coupling with the electric field,
and a relative decrease of BD propensity. Note that even beyond the turnover
point, thermo-elasticity still contributes to a decrease in the critical
electric field for BD, although this enhancement is lessened for very large
temperature rises.

###  III.4 Random Surface Simulation

Figure 8: (a) Surface evolution of the random surface under application of
electric and thermal gradients of 460 MV/m and 130 K. The initial surface
structure was initialized following the frequency content observed
experimentally for polished copper surfaces Qian  _et al._ (2012). (b) Fourier
transforms of the surface profiles as compared to the growth rate $g(k)$. (c)
Log scale contour plot showing the fastest growing wavelength $\lambda_{m}$
for given combination of applied field and $\Delta T$.

The analysis above focused on isolated perturbations. In realistic surfaces,
roughness will initially be present on a wide range of scales Fig. 8. In the
following, realistic surface profiles were generated using the
amplitude/wavelength characteristics measured on actual copper photocathode
surfaces Qian  _et al._ (2012).

Field gradient of 460 MV/m and $\Delta K$ of 130K were applied on the copper
surface to simulate its morphology. The simulation result and growth behavior
of the surface remains consistent with the linear stability analysis and the
previous result for isolated geometry. The evolution of the frequency spectra
in the simulations show the preferential growth of the maximally unstable
frequency modes, while high frequency modes above the critical frequency
$k/2\pi>k_{0}/2\pi$, sharply decay. The combination of gradient and $\Delta T$
caused the surface to morph into a new more regular wave-like surface as
presented by Fig. 8(a,b).

This wavelength selection is in excellent agreement with the prediction of the
linear stability analysis. The peak in the spatial frequency spectrum forms at
$0.3~{}\mu\text{m}^{-1}$ which is very close to the maximal unstable frequency
$1/\lambda_{m}$ = $0.32~{}\mu\text{m}^{-1}$ as calculated analytically from
Eq. 7b.

As shown, the linear stability analysis gives an excellent prediction for the
growth behaviour of copper surfaces in the small amplitude (aspect ratio)
regime. To understand the relation between the maximally unstable mode $k_{m}$
and the applied parameters, Fig. 8(c) charts the evolution of maximally-
unstable wavelength $\lambda_{m}$ against given combination of applied
gradient and $\Delta T$. The results show that $\lambda_{m}$ rapidly decreases
from 1,000 to 10 $\mu$m as temperature increases by 200 K and the gradient
increases to 300 MV/m. The decrease in $\lambda_{m}$ becomes more gradual at
larger applied parameters. For instance, $\lambda_{m}$=$2~{}\mu m$ under 600
MV/m and 230 K condition. This analytical treatment provides useful guidelines
for the analysis of surfaces exposed to combinations of electrostatic and
thermo-elastic driving forces, as a robust wavelength selection would
constitute the strongest experimental signature of the effects discussed here.

##  IV Discussion

The simulation results presented above suggested that surface diffusion can be
an effective mechanism that contributes to breakdown precursor formation. To
predict the physical timescale over which this mechanism can be expected to
occur, the simulation time unit ($\tau$) are related to physical times as
follows:

$\tau=\frac{k_{b}T}{\gamma D_{s}v_{s}\Omega^{2}}.$ (8)

The main unknown in this conversion factor is the effective diffusivity of
copper atoms on surfaces, which, on clean surfaces is of the form
$D_{s}=3.615^{2}/2\times 10^{12}\exp(-E_{b}/k_{b}T)\>\AA^{2}/$s Butrymowicz
_et al._ (1973), where $E_{b}$ is the migration barriers, and a standard
prefactor of $10^{12}$/s was assumed. According to both experimental and
theoretical calculation, the migration barriers vary considerably for
different surface orientations: 0.1-0.15 eV for (111), 0.25-0.30 eV for (110),
and 0.38-0.69 eV for (100). Hansen  _et al._ (1991); Karimi  _et al._ (1995);
Schulze Icking-Konert  _et al._ (1998); Kallinteris  _et al._ (1996);
Evangelakis  _et al._ (1998); Evangelakis and Papanicolaou (1996) Using
$\gamma=1.5$ J/m2 Kumykov  _et al._ (2017), $\Omega=11.81\>\AA^{3}$,
$v_{s}=0.306\>\AA^{-2}$ Simon  _et al._ (1992), this wide range of possible
diffusivities translates into a broad range of physical timescales $\tau$
varying from 4.75 ms ($E_{b}=0.1$ eV) to 11 seconds ($E_{b}=0.3$ eV) with room
temperature. Due to the exponential dependence of the diffusivity with
temperature, a temperature rise can significantly decrease the timescale,
e.g., with $\Delta T$ of 100 K, the characteristic timescale vary from 2.4 ms
($E_{b}=0.1$ eV) to 0.81 seconds ($E_{b}=0.3$ eV). Further, the growth rates
of perturbations can vary by orders of magnitude depending on the applied
fields and temperature rises, as shown in Fig. 4. From the linear stability
analysis described above, one can indeed show that the growth rate of the
maximally unstable mode will increase as
$\mathcal{O}[\exp(-E_{b}/k_{b}(T+\Delta T_{0}))(E_{0}^{2}+\Delta
T_{0}^{2})^{4}]$. This is qualitatively consistent with the dramatic
acceleration of the breakdown rate with increasing gradient and temperature
observed in the literature Simakov  _et al._ (2018) (where it was observed to
scale with applied field to a large power Descoeudres  _et al._ (2009b) ).
Non-linearities due to high-aspect ratios are expected to further increase the
breakdown rates Bagchi  _et al._ (2023).

Our results are consistent with a number of experimental observations. First,
the peak pulse heating temperature is known to be strongly correlated with the
breakdown rate Simakov  _et al._ (2018). This is consistent with the
synergistic effect of electric fields and heating in increasing the breakdown
rate, and with the exponential increase of the surface diffusivity with
temperature, which is also expected to reduce to time required for precursors
to form, as discussed below. Second, the localization of breakdown events
(i.e., the propensity of breakdown craters to overlap with one another)
observed in pulsed DC experiments Profatilova  _et al._ (2020) strongly
suggests that debris and craters created by previous BD events are efficient
nucleation sites from subsequent BD events. This is consistent with the
results shown in Fig. 4, as preexisting large amplitude roughness leads to the
formation of further BD precursors at low fields and in short amounts of time.
Further, it could be expected that breakdown precursors would develop faster
on Cu(111) facets of polycrystalline surfaces. Indeed, it was found by
electron back-scattering diffraction measurements of conditioned Cu samples
Laurent  _et al._ (2011) exposed to cyclic pulsed heating (without a
significant electric field applied), that the resulting surface roughness was
heavily dependent the orientations of the surface. The surface orientation
with the largest damage/roughness development was observed to be (111),
followed by (110), and finally by (100). This experimentally distinct
difference in the extent of roughness and damage evolution is consistent with
the predictions of our model which predicts a very strong dependence of the
rate at which surface features would grow depending on the surface migration
energy. We however note that this specific experiment was carried out under
conditions of thermal cycling, so other mechanism (e.g., the formation of
shear bands) could have contributed to the development of surface roughening
in this case.

##  V Conclusion

In this work, we presented a numerical model describing how metallic surfaces
undergo tip-growth instabilities through surface diffusion under combined
applied electric fields and thermal stresses. Such instabilities can lead to
the formation of geometrically sharp features that can act as breakdown
precursor. The results show that thermal stressed induced by temperature rises
of a few tens of Kelvin (which are typical of conditions caused by Joule
heating in copper accelerator cavities) can significantly lower the electric
fields needed for breakdown precursors to form compared to situations where
heating is absent. Our results show that diffusion-driven surface evolution
can spontaneously create breakdown precursors in typically accessible regimes
of applied fields and temperature rises for accelerator applications. Surface
diffusion is therefore a viable candidate mechanism to explain the formation
of sharp breakdown precursors.

## Acknowledgments

Ryo Shinohara, Soumendu Bagchi, Evgenya Simakov, and Danny Perez were
supported by the Laboratory Directed Research and Development program of Los
Alamos National Laboratory under project number 20230011DR. The work by Sergey
Baryshev was supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics under Award No. DE-SC0020429. Los Alamos
National Laboratory is operated by Triad National Security, LLC, for the
National Nuclear Security Administration of U.S. Department of Energy
(Contract No. 89233218CNA000001).

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# Subelliptic Operations on Weighted Folland-Stein Spaces

Hung-Lin Chiu  Department of Mathematics, National Tsing Hua University,
Hsinchu, Taiwan 300, R.O.C<EMAIL_ADDRESS>
###### Abstract.

In this paper, we show that the sub-Laplacian of an asymptotically flat
pseudo-hermitian manifold defined on a suitable weighted Folland-Stein spaces
is an isomorphism. It turns out that the CR positive mass problem is resolved,
and hence CR Yamabe problem.

##  1\. Introduction

The CR positive mass problem plays an essential role in CR geometry. As well
known, we need a CR positive mass theorem to solve the CR Yamabe problem for
the cases which either the CR dimension $n=1$ or the CR manifold $M$ is
spherical with higher CR dimension. When $n=1$, this was shown by Cheng,
Malchiodi and Yang [3]. On the other hand, when $n\geq 2$ and $M$ is
spherical, this was finished by Cheng, Yang and the author [2] through showing
that the developing map is injective. However in the case $n=2$, we need an
extra condition on the growth rate of the Green’s function on universal cover
of $M$. So in the case $n=2$, the CR positive mass theorem is not really
completed. In the paper [1], Cheng and the author showed that for $n=2$, $M$
being spherical, if moreover $M$ has a spin structure, then we have the CR
positive mass theorem built up through a spinorial approach.

Recall that in 1982, E. Witten described a proof of the positive mass theorem
using spinors (see [7, 6]). The key to Witten’s method is to find a spinor
$\psi$ satisfying $D\psi=0$ and approaching a constant spinor at infinity.
Then applying the Weitzenbock-type formula to this spinor field $\psi$ and
integrating after taking the inner product with $\psi$, we then pick up the
$p$-mass from the boundary integral and obtain a Witten-type formula for
$p$-mass So the non-negativity of $p$-mass follows. Therefore, in this paper,
for the CR positive mass theorem, it suffices to show a kind of result about
the isomorphism of the square of the Dirac operator $D^{2}$ on some suitable
weighted Folland-Stein spaces. And this result is based on the main theorem
(see Theorem 3.2) of this paper.

##  2\. Operations on Weighted Folland-Stein Spaces

Let $H_{n}$ be the Heisenberg group and $Q=2n+2$ be the homogeneous dimension
of $H_{n}$. The summation convention applies and derivative along
$\mathring{e}_{a},1\leq a\leq 2n$, may be denoted by subscripts
$u_{a}=\mathring{e}_{a}u$. The sub-gradient
$\mathring{\nabla}_{b}u=\sum_{a=1}^{2n}u_{a}\mathring{e}_{a}$. The sub-
laplacian $\mathring{\Delta}_{b}u=\sum_{a=1}^{2n}(\mathring{e}_{a})^{2}u$. Set

(2.1) $\rho=(|z|^{4}+t^{2})^{\frac{1}{2}},\ \
\sigma=(1+\rho^{4})^{\frac{1}{4}},$

Denote $B_{R}=B_{R}(0)$ as the closed Heisenberg ball of radius $R$ and center
$0$. The annulus is $A_{R}=B_{2R}\setminus B_{R}$, and the exterior domain is
$E_{R}=H_{n}\setminus B_{R}$.

On $H_{n}$, we have the dilation operator defined by

(2.2) $R(z,t)=(Rz,R^{2}t)$

###  2.1. The weighted Folland-Stein spaces

For $\delta\in\operatorname{\mathbb{R}}$, we define the weighted Lebesgue
spaces $L^{p}_{\delta},L^{\prime p}_{\delta}$ with weight
$\delta\in\operatorname{\mathbb{R}}$ as the spaces of measurable functions in
$L^{p}_{loc}(H_{n}),L^{p}_{loc}(H_{n}\setminus\\{0\\})$, respectively, for
which the norms defined by

(2.3)
$\|u\|_{p,\delta}=\left\\{\begin{array}[]{ll}\left(\int_{H_{n}}|\sigma^{-\delta}u|^{p}\sigma^{-Q}dV_{(z,t)}\right)^{\frac{1}{p}},&\
\ p<\infty,\\\ \ \ \textrm{ess sup}_{H_{n}}(\sigma^{-\delta}|u|),&\ \
p=\infty,\end{array}\right.$

and

(2.4)
$\|u\|^{\prime}_{p,\delta}=\left\\{\begin{array}[]{ll}\left(\int_{H_{n}\setminus\\{0\\}}|\rho^{-\delta}u|^{p}\rho^{-Q}dV_{(z,t)}\right)^{\frac{1}{p}},&\
\ p<\infty,\\\ \ \ \textrm{ess
sup}_{H_{n}\setminus\\{0\\}}(\rho^{-\delta}|u|),&\ \
p=\infty,\end{array}\right.$

are finite, where $dV_{(z,t)}=\Theta\wedge(d\Theta)^{n}$.

For $k\in\operatorname{\mathbb{N}}\cup\\{0\\}$, the weighted Folland-Stein
spaces $S^{p}_{k,\delta},S^{\prime p}_{k,\delta}$ are the spaces of $u$ such
that

(2.5) $|\nabla^{j}u|\in L^{p}_{\delta-j},L^{\prime p}_{\delta-j},\ \
\textrm{for}\ 0\leq j\leq k,$

respectively, with norms

(2.6) $\|u\|_{k,p,\delta}=\sum_{j=0}^{k}\|\nabla^{j}u\|_{p,\delta-j},$ (2.7)
$\|u\|^{\prime}_{k,p,\delta}=\sum_{j=0}^{k}\|\nabla^{j}u\|^{\prime}_{p,\delta-j}.$

Observe that $C^{\infty}_{0}(H_{n}),C^{\infty}_{0}(H_{n}\setminus\\{0\\})$ are
dense in $S^{p}_{k,\delta},S^{\prime p}_{k,\delta}$, respectively, for
$\delta<0$ and $1\leq p<\infty$ and that $L^{p}_{\delta}=L^{p}(H_{n})$ for
$\delta=-Q/p$.

###  2.2. Global weighted inequalities

As R. Bartnik pointed out in [4] that the indexing chosen for the weights has
the advantage that it directly describes the growth at infinity (see (2.18)
below). In addition, the rescaled function, defined by

(2.8) $u_{R}(z,t)=u(R(z,t)),$

will satisfy the estimates (2.9) and (2.10) which follow by a simple change of
variables. We have

(2.9) $\|u_{R}\|^{\prime}_{k,p,\delta}=R^{\delta}\|u\|^{\prime}_{k,p,\delta}.$

Similarly, with an obvious notation for norms over subsets of $H_{n}$, we have

(2.10) $\|u\|_{k,p,\delta;A_{R}}\approx
R^{-\delta}\|u_{R}\|_{k,p,\delta;A_{1}},$

for $R\geq 1$, where ”$\approx$” means ”is comparable to”, independent of
$R\geq 1$. The estimate (2.10) is the key to proving global weighted
inequalities from local inequalities.

######  Proposition 2.1.

If $1\leq p\leq q\leq\infty,\ \delta_{2}<\delta_{1}$ and $u\in
L^{q}_{\delta_{2}}$, then

(2.11) $\|u\|_{p,\delta_{1}}\leq c\|u\|_{q,\delta_{2}},$

and hence $L^{q}_{\delta_{2}}\subset L^{p}_{\delta_{1}}$.

###### Proof.

The estimate (2.11) follows directly from the definition and Hölder
inequality. If $p=q<\infty$, we write

$\frac{|u|^{p}}{\sigma^{\delta_{1}p+Q}}=\frac{|u|^{q}}{\sigma^{\delta_{2}q+Q}}\
\frac{1}{\sigma^{(\delta_{1}-\delta_{2})q}}.$

Then (2.11) holds in this case. If $p<q<\infty$, we write

$\frac{|u|^{p}}{\sigma^{\delta_{1}p+Q}}=\frac{|u|^{p}}{\sigma^{\delta_{2}p+\frac{Qp}{q}}}\
\frac{1}{\sigma^{(\delta_{1}-\delta_{2})p}+\frac{Q(q-p)}{q}}.$

Then by Hölder inequality

$\begin{split}\int_{E_{R}}\frac{|u|^{p}}{\sigma^{\delta_{1}p+Q}}\
dV&\leq\left[\int_{E_{R}}\left(\frac{|u|^{p}}{\sigma^{\delta_{2}p+\frac{Qp}{q}}}\right)^{q/p}\right]^{p/q}\left[\int_{E_{R}}\left(\frac{1}{\sigma^{(\delta_{1}-\delta_{2})p}+\frac{Q(q-p)}{q}}\right)^{q/(q-p)}\right]^{(q-p)/q}\\\
&=\left[\int_{E_{R}}\left(\frac{|u|^{q}}{\sigma^{\delta_{2}q+Q}}\right)dV\right]^{p/q}\left[\int_{E_{R}}\left(\frac{1}{\sigma^{(\delta_{1}-\delta_{2})\frac{pq}{q-p}+Q}}\right)dV\right]^{(q-p)/q},\end{split}$

where the second term is integrable on $E_{R}$. Therefore (2.11) holds.
Finally, suppose that $p<q=\infty$, then

$\frac{|u|^{p}}{\sigma^{\delta_{1}p+Q}}=\frac{|u|^{p}}{\sigma^{\delta_{2}p}}\frac{1}{\sigma^{(\delta_{1}-\delta_{2})p+Q}}\leq\|u\|^{p}_{\infty,\delta_{2}}\
\frac{1}{\sigma^{(\delta_{1}-\delta_{2})p+Q}},\ a.e.$

Again that $\frac{1}{\sigma^{(\delta_{1}-\delta_{2})p+Q}}$ is integrable on
$H_{n}$, we get (2.11). Suppose $p=q=\infty$. We fix a point, since
$\sigma\geq 1$, we have that $(\sigma^{-1})^{x}$ is a decreasing function,
hence $(\sigma^{-1})^{\delta_{1}}\leq(\sigma^{-1})^{\delta_{2}}$. And it is
easy to see that if

$m\\{(z,t)\ |\ \sigma^{-\delta_{2}}|u(z,t)|>M\\}=0$

then

$m\\{(z,t)\ |\ \sigma^{-\delta_{1}}|u(z,t)|>M\\}=0.$

Therefore

$\begin{split}\|u\|_{\infty,\delta_{1}}&=ess\ sup(\sigma^{-\delta_{1}}|u|)\\\
&=\inf\\{M\ |\ m\\{(z,t)\ |\ \sigma^{-\delta_{1}}|u(z,t)|>M\\}=0\\}\\\
&\leq\inf\\{M\ |\ m\\{(z,t)\ |\ \sigma^{-\delta_{2}}|u(z,t)|>M\\}=0\\}\\\
&=ess\ sup(\sigma^{-\delta_{2}}|u|)=\|u\|_{\infty,\delta_{2}},\end{split}$

which gives (2.11). Finally, suppose that $p<q=\infty$, since

$\begin{split}|\sigma^{-\delta_{1}}u^{p}|\sigma^{-Q}&=|\sigma^{-\frac{\delta_{2}}{p}}u\sigma^{-\delta_{1}+\frac{\delta_{2}}{p}}|^{p}\sigma^{-Q}\\\
&=\sigma^{-\delta_{2}}|u|\sigma^{-Q-p\delta_{1}+\delta_{2}}\\\ &\leq
M\frac{1}{\sigma^{Q+(p\delta_{1}-\delta_{2})}},\ \ \textrm{almost
everywhere},\end{split}$

we have

$\begin{split}\int_{H_{n}}\frac{|u|^{p}}{\sigma^{\delta_{1}p+Q}}dV_{(z,t)}&\leq\int_{H_{n}}\frac{M}{\sigma^{Q+(p\delta_{1}-\delta_{2})}}dV_{(z,t)}\\\
&\leq C\ ess\ sup(\sigma^{-\delta_{2}}|u|).\end{split}$

We thus complete the proof. ∎

######  Proposition 2.2 (Hölder inequality).

If $u\in L^{q}_{\delta_{1}},v\in L^{r}_{\delta_{2}}$ and
$\delta=\delta_{1}+\delta_{2},\frac{1}{p}=\frac{1}{q}+\frac{1}{r},1\leq
p,q,r\leq\infty$, then

(2.12) $\|uv\|_{p,\delta}\leq\|u\|_{q,\delta_{1}}\|v\|_{r,\delta_{2}},$

###### Proof.

The estimate (2.12) follows directly from the definition and Hölder
inequality. If $1<p,q,r<\infty$, then we write

$\frac{|uv|^{p}}{\sigma^{\delta
p+Q}}=\frac{|u|^{p}}{\sigma^{\delta_{1}p+\frac{Qp}{q}}}\
\frac{|v|^{p}}{\sigma^{\delta_{2}p+\frac{Qp}{r}}}.$

By Hölder inequality, we have

$\int_{H_{n}}\frac{|uv|^{p}}{\sigma^{\delta
p+Q}}dV=\left(\int_{H_{n}}\left(\frac{|u|^{p}}{\sigma^{\delta_{1}p+\frac{Qp}{q}}}\right)^{q/p}dV\right)^{p/q}\left(\int_{H_{n}}\left(\frac{|v|^{p}}{\sigma^{\delta_{2}p+\frac{Qp}{r}}}\right)^{r/p}dV\right)^{p/r}.$

This shows (2.12). If $p=q<\infty,\ r=\infty$, then

$\frac{|uv|^{p}}{\sigma^{\delta p+Q}}=\frac{|u|^{p}}{\sigma^{\delta_{1}p+Q}}\
\frac{|v|^{p}}{\sigma^{\delta_{2}p}}\leq\frac{|u|^{p}}{\sigma^{\delta_{1}p+Q}}\
\|v\|^{p}_{\infty,\delta_{2}},\ a.e.$

This shows (2.12). The case $p=q<\infty,\ r=\infty$ is similar. In the end,
suppose that $p=q=r=\infty$. Let $M_{1},M_{2}$ be two numbers such that

$m\\{(z,t)\ |\ \sigma^{-\delta_{1}}|u(z,t)|>M_{1}\\}=0,\ m\\{(z,t)\ |\
\sigma^{-\delta_{2}}|v(z,t)|>M_{2}\\}=0.$

Then we have

$\sigma^{-\delta}|uv|=\sigma^{-\delta_{1}}|u|\cdot\sigma^{-\delta_{2}}|v|\leq
M_{1}M_{2},\ \ \textrm{almost everywhere},$

that is,

$m\\{(z,t)\ |\ \sigma^{-\delta}|uv|>M_{1}M_{2}\\}=0,$

which implies

(2.13) $\begin{split}\|uv\|_{\infty,\delta}&=ess\ sup(\sigma^{-\delta}|uv|)\\\
&=\inf\\{M\ |\ m\\{(z,t)\ |\ \sigma^{-\delta}|uv|>M\\}=0\\}\\\ &\leq
M_{1}M_{2},\ \ \textrm{for any such}\ M_{1},M_{2}.\end{split}$

Then (2.12) follows. We thus complete the proof. ∎

The following two estimates, Interpolation inequality and Sobolev inequality,
follow from the technique of rescaling and applying local estimate:

######  Proposition 2.3 (Interpolation inequality).

For any $\varepsilon>0$ and any non-negative integer $s$, there is a
$C=C(Q,p,\delta,s)$ such that

(2.14)
$\|u\|_{s+1,p,\delta}\leq\varepsilon\|u\|_{s+2,p,\delta}+\frac{C}{\varepsilon}\|u\|_{0,p,\delta},$

for all $u\in S^{p}_{s+2,\delta},1\leq p<\infty$.

###### Proof.

The estimates (2.14) follows from the technique of rescaling and applying
local estimates. By the usual interpolation inequality, we have, for $1\leq
p<\infty$,

(2.15) $\begin{split}\|u\|_{s+1,p,\delta;A_{R}}&\leq
CR^{-\delta}\|u\|_{s+1,p,\delta;A_{1}}\\\ &\leq
CR^{-\delta}\left(\varepsilon\|u_{R}\|_{s+2,p,\delta;A_{1}}+\frac{C}{\varepsilon}\|u_{R}\|_{0,p,\delta;A_{1}}\right)\\\
&\leq\varepsilon\|u\|_{s+2,p,\delta;A_{R}}+\frac{C}{\varepsilon}\|u\|_{0,p,\delta;A_{R}},\end{split}$

where $C$ is a constant independent of $A_{R}$, for $R\geq 1$. Therefore,
writing $u=\sum_{j=0}^{\infty}u_{j}$ with
$u_{0}=u|_{B_{1}},u_{j}=u|_{A_{2^{j-1}}}$, we see that

$\begin{split}\|u\|_{s+1,p,\delta}&=\sum_{k=0}^{s+1}\|\nabla^{k}u\|_{p,\delta-k}=\sum_{k=0}^{s+1}\left(\sum_{j=0}^{\infty}\|\nabla^{k}u_{j}\|^{p}_{p,\delta-k}\right)^{1/p}\\\
&\preceq\left(\sum_{k=0}^{s+1}\sum_{j=0}^{\infty}\|\nabla^{k}u_{j}\|^{p}_{p,\delta-k}\right)^{1/p}\preceq\left(\sum_{j=0}^{\infty}\|u_{j}\|^{p}_{s+1,p,\delta}\right)^{1/p}\\\
&\preceq\left(\varepsilon\sum_{j=0}^{\infty}\|u_{j}\|^{p}_{s+2,p,\delta}+\frac{C}{\varepsilon}\sum_{j=0}^{\infty}\|u_{j}\|^{p}_{0,p,\delta}\right)^{1/p},\
\ \textrm{by}\ \eqref{opw121},\\\
&\preceq\varepsilon\|u\|_{s+2,p,\delta}+\frac{C}{\varepsilon}\|u\|_{0,p,\delta}.\end{split}$

This completes the proof. We remark here that the Interpolation inequality
also holds when $p=\infty$. But we don’t need for it in this paper. ∎

######  Proposition 2.4 (Sobolev inequality).

If $u\in S^{q}_{k,\delta}$, then we have

(2.16) $\|u\|_{\frac{Qp}{Q-kp},\delta}\leq C\|u\|_{k,q,\delta},$

provided that $Q-kp>0$ and $1<p\leq q\leq\frac{Qp}{Q-kp}$.
We also have that

(2.17) $\|u\|_{\infty,\delta}\leq C\|u\|_{k,p,\delta},\ \ \textrm{if}\
Q-kp<0.$

and in fact

(2.18) $|u(z,t)|=o(\rho^{\delta})\ \ \textrm{as}\ \ \rho\rightarrow\infty.$

###### Proof.

Suppose that $p^{*}=Qp/(Q-kp)<\infty$, we have

(2.19)
$\begin{split}\|u\|_{p^{*},\delta;A_{R}}&=\left(\int_{A_{R}}(\sigma^{-\delta}u)^{p^{*}}\sigma^{-Q}dV_{(z,t)}\right)^{1/p^{*}}\\\
&\leq CR^{-\delta}\|u_{R}\|_{p^{*},\delta;A_{1}}\\\ &\leq
CR^{-\delta}\|u_{R}\|_{k,q,\delta;A_{1}},\end{split}$

where, for the last inequality, we have used the usual Sobolev inequality
applied to $A_{1}$, and by Hölder inequality. Rescaling gives

(2.20) $\|u\|_{p^{*},\delta;A_{R}}\leq C\|u\|_{k,q,\delta;A_{R}}$

Now using the same notation $u_{j}$ as above, we see that

(2.21)
$\begin{split}\|u\|_{p^{*},\delta}&=\left(\int_{H_{n}}\|u\|^{p^{*}}\sigma^{-\delta
p^{*}-Q}dV_{(z,t)}\right)^{1/p^{*}}\\\
&=\left(\sum_{j-0}^{\infty}\int_{A_{2^{j-1}}}|u|^{p^{*}}\sigma^{-\delta
p^{*}-Q}dV_{(z,t)}\right)^{1/p^{*}}=\left(\sum_{j-0}^{\infty}\|u\|^{p^{*}}_{p^{*},\delta;A_{2^{j-1}}}\right)^{1/p^{*}}\\\
&\leq
C\left(\sum_{j-0}^{\infty}\|u\|^{p^{*}}_{k,q,\delta;A_{2^{j-1}}}\right)^{1/p^{*}},\
\ \textrm{by}\ \eqref{opw131}\\\ &\leq
C\left(\sum_{j-0}^{\infty}\|u\|^{q}_{k,q,\delta;A_{2^{j-1}}}\right)^{1/q},\ \
q\leq p^{*}\\\ &=C\|u\|_{k,q,\delta}.\end{split}$

Therefore, we obtain (2.16). The same rescaling argument implies that

(2.22) $\sup_{A_{R}}|u|\sigma^{-\delta}=\|u\|_{\infty,\delta,A_{R}}\leq
C\|u\|_{k,p,\delta;A_{R}},$

which gives (2.17). Since $\|u\|_{k,p,\delta}<\infty$, we have

$\|u\|_{k,p,\delta;A_{R}}\rightarrow 0\ \ \ \textrm{as}\ R\rightarrow\infty,$

that is,

$\|u\|_{k,p,\delta;A_{R}}o(1)\ \ \ \textrm{as}\ R\rightarrow\infty,$

which gives (2.18), by means of (2.22). We therefore complete the proof.

∎

###  2.3. Weighted estimates for sub-elliptic operators

The scaling argument and the standard local estimates gives estimates in
weighted spaces for sub-elliptic operators whose coefficients are well behaved
at infinity.

######  Definition 2.5.

The operator $u\rightarrow Pu$ defined by

(2.23)
$Pu=a^{ij}(z,t)\mathring{e}_{i}\mathring{e}_{j}u+b^{i}(z,t)\mathring{e}_{i}u+c(z,t)u$

will be said to be asymptotic to $\mathring{\Delta}_{b}$ (at rate $\tau$) if
there exists $Q<q<\infty$ and $\tau\geq 0$ and constants $C_{1},\lambda$ such
that

(2.24) $\lambda|\zeta|^{2}\leq
a^{ij}(z,t)\zeta_{i}\zeta_{j}\leq\lambda^{-1}|\zeta|^{2},\ \ \textrm{for all}\
(z,t)\in H_{n},\ \zeta=\sum_{j=1}^{2n}\zeta_{j}\mathring{e}_{j}\in\xi,$

and

(2.25)
$\|a^{ij}-\delta_{ij}\|_{1,q,-\tau}+\|b^{i}\|_{0,q,-1-\tau}+\|c\|_{0,q/2,-2-\tau}\leq
C_{1},$

where $\delta_{ij}$ corresponds to the standard sub-laplacian
$\mathring{\Delta}_{b}$ on $H_{n}$.

It is clear that if $P$ is asymptotic to $\mathring{\Delta}_{b}$, then the map
$P:S^{p}_{2,\delta}\rightarrow S^{p}_{0,\delta-2}$ is bounded for $1\leq p\leq
q$ and $\delta\in\operatorname{\mathbb{R}}$. In fact the following weighted
estimate holds.

######  Proposition 2.6.

Suppose that $P$ is asymptotic to $\mathring{\Delta}_{b},1<p\leq q$ and
$\delta\in\operatorname{\mathbb{R}}$. There is a constant
$C=C(Q,p,q,s,\delta,C_{1},\lambda)$ such that if $u\in L^{p}_{\delta}$ and
$Pu\in S^{p}_{s,\delta-2}$, then $u\in S^{p}_{s+2,\delta}$. Moreover, we have

(2.26) $\|u\|_{s+2,p,\delta}\leq C(\|Pu\|_{s,p,\delta-2}+\|u\|_{p,\delta}).$

###### Proof.

Sub-elliptic regularity applies to show that $u\in S^{p}_{s+2,loc}$, and the
remaining conclusions follow from the usual interior $L^{p}$ estimates and the
scaling technique. ∎

Observe that the same argument gives the estimate (2.26) with the $S^{\prime
p}_{k,\delta}$ norms instead. We now investigate the Fredholm properties of
$P$. We need the following lemma:

######  Lemma 2.7.

Fix $p\in(1,\infty),p^{\prime}=p/(p-1)$, and let
$a,b\in\operatorname{\mathbb{R}}$ be such that $a+b>0$. Suppose that
$K^{\prime}(x,y)$ is the Kernel

$K(x,y)=|x|^{-a}|y^{-1}x|^{-Q+a+b}|y|^{-b},\ \ \textrm{for}\ x\neq y,$

For $u\in L^{p}(H_{n})$ define

$Ku(x)=\int_{H_{n}}K(x,y)u(y)dV_{y}.$

Then there is a constant $c=c(n,p,a,b)$ such that

$\|Ku\|_{L^{p}}\leq c\|u\|_{L^{p}}$

if and only if $a<Q/p$ and $b<Q/p^{\prime}$.

###### Proof.

To show that the conditions $a<Q/p$ and $b<Q/p^{\prime}$ are necessary,
suppose that $K$ is a bounded operator on $L^{p}(H_{n})$. We first define the
function

$v(x)=\int_{|y|\leq 1}K(x,y)dV_{y}.$

Then we have

$v(x)=\int_{|y|\leq 1}K(x,y)dV_{y}=\int_{H_{n}}K(x,y)u(y)dV_{y},$

where

$u(y)=\left\\{\begin{array}[]{cc}1&|y|\leq 1\\\ 0&|y|>1\end{array}\right.$

Since $u(y)\in L^{p}(H_{n})$ and $K$ is a bounded operator on $L^{p}(H_{n})$,
we see that $v(x)$ is in $L^{p}(H_{n})$. Next, we study the behavior of $v(x)$
at infinity. By the triangle inequality (see [5]), there exists a constant
$\gamma\geq 1$ such that

$\frac{1}{\gamma}-\frac{|y|}{|x|}\leq\frac{|y^{-1}x|}{|x|}\leq\gamma(1+\frac{|y|}{|x|}).$

Therefore, for $|y|\leq 1$, if $|x|$ is large enough, then
$K(x,y)\approx\frac{1}{|x|^{Q-b}|y|^{b}}$, and hence $v(x)$ behaves like a
constant multiple of $|x|^{-Q+b}$. Then $v(x)$ belongs to $L^{p}(H_{n})$ only
if $p(Q-b)>Q$, i.e. only if $b<Q/p^{\prime}$. On the other hand, we define the
function $w(y)$ by

$w(y)=\int_{|x|\leq 1}K(x,y)dV_{x}.$

Then, by Fubini Theorem and Hölder inequality, for $u(y)\in L^{p}(H_{n})$, we
have

$\begin{split}\int_{H_{n}}w(y)u(y)dV_{y}&=\int_{H_{n}}\left(\int_{|x|\leq
1}K(x,y)u(y)dV_{x}\right)dV_{y}\\\ &=\int_{|x|\leq
1}\left(\int_{H_{n}}K(x,y)u(y)dV_{y}\right)dV_{x}\\\
&\leq\|Ku(x)\|_{L^{p}(B_{1})}\left(\int_{|x|<1}1\
dV_{x}\right)^{1/p^{\prime}}\\\
&\leq\omega_{n}^{1/p^{\prime}}\|Ku(x)\|_{L^{p}(H_{n})}\leq c\
\omega_{n}^{1/p^{\prime}}\|u\|_{L^{p}(H_{n})},\end{split}$

for some constant $c>0$, where $\omega_{n}$ is the volume of the unit ball
$B_{1}$. That is, $w(y)$ is a bounded linear functional on $L^{p}(H_{n})$, and
hence belongs to $L^{p}(H_{n})$. As we discuss the behavior of $v(x)$ above,
it is easy to see that, for large $|y|$, $w(y)$ behaves like a constant
multiple of $|y|^{-Q+a}$. Then $w(y)$ belongs to $L^{p}(H_{n})$ only if
$a<Q/p$.

To show that the conditions $a<Q/p$ and $b<Q/p^{\prime}$ are sufficient, note
first that it may be assumed without loss of generality that both $a$ and $b$
are non-negative. (If, say, $a$ is negative, then $b$ is positive, and from
the triangle inequality $|x|\leq\gamma(|y|+|y^{-1}x|)$ for some $\gamma\geq
1$, we have

$\frac{|x|}{|y^{-1}x|}\leq\gamma\left(1+\frac{|y|}{|y^{-1}x|}\right),$

which implies that

$\left(\frac{|x|}{|y^{-1}x|}\right)^{-a}\leq
C\left(1+\frac{|y|}{|y^{-1}x|}\right)^{-a},$

for some $C>0$. It is seen that

$K(x,y)\leq\frac{C}{|y^{-1}x|^{Q-b}|y|^{b}}+\frac{C}{|y^{-1}x|^{Q-a-b}|y|^{a+b}}.$

The boundedness of $K$ in this case then follows immediately from the result
in the case in which both $a$ and $b$ are non-nagative.) And it suffices to
show that

(2.27) $\int_{H_{n}}\int_{H_{n}}K(x,y)u(y)v(x)dV_{y}dV_{x}\leq
C\left(\int_{H_{n}}u(y)^{p}dV_{y}\right)^{1/p}\left(\int_{H_{n}}v(x)^{p^{\prime}}dV_{x}\right)^{1/p^{\prime}},$

hence

$Ku(x)=\int_{H_{n}}K(x,y)u(y)dV_{y}\in(L^{p^{\prime}})^{*}(H_{n})=L^{p}(H_{n})$

with operator norm less than $C\left(\int_{H_{n}}u(y)^{p}dV_{y}\right)^{1/p}$.
By Hölder inequailty, and notice that $K(x,y)$ is homogeneous of degree $-Q$,
we have

(2.28) $\begin{split}&\ \ \ \
\int_{H_{n}}\int_{H_{n}}K(x,y)u(y)v(x)dV_{y}dV_{x}\\\
&=\int_{H_{n}}\int_{H_{n}}K^{1/p}u(y)\left(\frac{|y|}{|x|}\right)^{Q/pp^{\prime}}K^{1/p^{\prime}}v(x)\left(\frac{|x|}{|y|}\right)^{Q/pp^{\prime}}dV_{y}dV_{x}\\\
&\leq\left[\int_{H_{n}}\int_{H_{n}}\left(K^{1/p}u(y)\left(\frac{|y|}{|x|}\right)^{Q/pp^{\prime}}\right)^{p}\right]^{1/p}\left[\int_{H_{n}}\int_{H_{n}}\left(K^{1/p^{\prime}}v(x)\left(\frac{|x|}{|y|}\right)^{Q/pp^{\prime}}\right)^{p^{\prime}}\right]^{1/p^{\prime}}\\\
&=P^{1/p}Q^{1/p^{\prime}},\end{split}$

where

(2.29)
$\begin{split}P&=\int_{H_{n}}\int_{H_{n}}\left(K(x,y)u^{p}(y)\left(\frac{|y|}{|x|}\right)^{Q/p^{\prime}}\right)dV_{y}dV_{x}\\\
&=\int_{H_{n}}u^{p}(y)\left(\int_{H_{n}}K(x,y)\left(\frac{|y|}{|x|}\right)^{Q/p^{\prime}}dV_{x}\right)dV_{y}\\\
\end{split}$

and

(2.30)
$\begin{split}Q&=\int_{H_{n}}\int_{H_{n}}\left(K(x,y)v^{p^{\prime}}(x)\left(\frac{|x|}{|y|}\right)^{Q/p}\right)dV_{y}dV_{x}\\\
&=\int_{H_{n}}v^{p^{\prime}}(x)\left(\int_{H_{n}}K(x,y)\left(\frac{|x|}{|y|}\right)^{Q/p}dV_{x}\right)dV_{x}\\\
\end{split}$

From (2.28),(2.29),(2.30), we obtain (2.27), provided that the integral
$\int_{H_{n}}K(x,y)\left(\frac{|y|}{|x|}\right)^{Q/p^{\prime}}dV_{x}$ has an
upper bound which is independent of $y$ and
$\int_{H_{n}}K(x,y)\left(\frac{|x|}{|y|}\right)^{Q/p}dV_{y}$ have an upper
bound which is independent of $x$. Actually, fix $y$ and by change of
variables, we have

(2.31)
$\begin{split}\int_{H_{n}}K(x,y)\left(\frac{|y|}{|x|}\right)^{Q/p^{\prime}}dV_{x}&=\int_{H_{n}}|y|^{-Q}K(\frac{x}{|y|},\frac{y}{|y|})\left(\frac{|y|}{|x|}\right)^{Q/p^{\prime}}dV_{x}\\\
&=\int_{H_{n}}K(z,y^{*})|z|^{-Q/p^{\prime}}dV_{z},\ z=\frac{x}{|y|},\
\textrm{and}\ y^{*}=\frac{y}{|y|}\\\
&=\int_{H_{n}}\frac{1}{|z|^{a+\frac{Q}{p^{\prime}}}|(y^{*})^{-1}z|^{Q-(a+b)}}dV_{z},\end{split}$

where

(2.32)
$\frac{1}{|z|^{a+\frac{Q}{p^{\prime}}}|(y^{*})^{-1}z|^{Q-(a+b)}}\approx\left\\{\begin{array}[]{ll}\frac{1}{|z|^{a+\frac{Q}{p^{\prime}}}}&,x\approx
0\\\ \frac{1}{|(y^{*})^{-1}z|^{Q-(a+b)}}&,x\approx 1\\\
\frac{1}{|z|^{Q+\frac{Q}{p^{\prime}}-b}}&,x\approx\infty\end{array}\right.$

Since $a<\frac{Q}{p}$ and $b<\frac{Q}{p^{\prime}}$, we have that
$0<a+\frac{Q}{p^{\prime}}<Q,\ 0<Q-(a+b)<Q$ and $Q+\frac{Q}{p^{\prime}}-b>Q$,
thus the integral
$\int_{H_{n}}K(x,y)\left(\frac{|y|}{|x|}\right)^{Q/p^{\prime}}dV_{x}$
converges, depending on $y^{*}$ which belongs to the unit sphere. Therefore,
it has an upper bound which is independent of $y$. Similarly,
$\int_{H_{n}}K(x,y)\left(\frac{|x|}{|y|}\right)^{Q/p}dV_{y}$ have an upper
bound which is independent of $x$. Therefore, we complete the proof. ∎

######  Theorem 2.8.

Suppose that $\delta<0$, $1<p<\infty$, and $s$ is a non-negative integer. Then
the map

(2.33) $\mathring{\Delta}_{b}:S^{\prime p}_{s+2,\delta}\rightarrow S^{\prime
p}_{s,\delta-2}$

is an isomorphism and there is a constant $C=C(Q,p,\delta,s)$ such that

(2.34) $\|u\|^{\prime}_{s+2,p,\delta}\leq
C\|\mathring{\Delta}_{b}u\|^{\prime}_{s,p,\delta-2}.$

###### Proof.

It will suffice to prove (2.34) for $s=0$. We first assume that $-2n<\delta<0$
and show that the distribution inverse has convolution kernel $K_{0}(x,y)$:

(2.35) $c_{0}K_{0}(x,y)=c_{0}\Phi_{0}(y^{-1}x)=|y^{-1}x|^{-2n}.$

Let us first show that (2.35) defines a bounded operator from $S^{\prime
p}_{0,\delta-2}$ to $S^{\prime p}_{0,\delta}$. Lemma 2.7 shows that the kernel

$K(x,y)=|x|^{-\delta-\frac{2n+2}{p}}K_{0}(x,y)|y|^{\delta-2+\frac{2n+2}{p}}$

defines a bounded operator $L^{p}\rightarrow L^{p}$ when $-2n<\delta<0$. Then
we have

(2.36)
$\begin{split}\|K_{0}u\|^{\prime}_{p,\delta}&=\left(\int_{H_{n}\setminus\\{0\\}}|K_{0}u(x)|^{p}|x|^{-\delta
p-Q}dV_{x}\right)^{1/p}\\\
&=\left(\int_{H_{n}\setminus\\{0\\}}\left|\int_{H_{n}\setminus\\{0\\}}|x|^{-\delta-\frac{Q}{p}}K_{0}(x,y)u(y)dV_{y}\right|^{p}dV_{x}\right)^{1/p}\\\
&=\left(\int_{H_{n}\setminus\\{0\\}}\left|(KU)(x)\right|^{p}dV_{x}\right)^{1/p},\
\ \textrm{where}\ U(y)=u(y)|y|^{-(\delta-2)-\frac{Q}{p}}\\\
&=\|KU\|_{L^{p}}\leq c\|U\|_{L^{p}},\ \ (\textrm{by Lemma}\ \ref{conbon}),\\\
&=c\|u\|^{\prime}_{p,\delta-2}.\end{split}$

Thus we have shown that $K_{0}:S^{\prime p}_{0,\delta-2}\rightarrow S^{\prime
p}_{0,\delta}$ is bounded when $-2n<\delta<0$. Recall that we have
$K_{0}\mathring{\Delta}_{b}u=u$, for all $u\in
C^{\infty}_{0}(H_{n}\setminus\\{0\\})$, so the boundness of $K_{0}$ gives

$\|u\|^{\prime}_{p,\delta}\leq C\|\mathring{\Delta}u\|^{\prime}_{p,\delta-2},\
\ \ \textrm{for all}\ u\in S^{\prime p}_{0,\delta},$

since $C^{\infty}_{0}(H_{n}\setminus\\{0\\})$ is dense. This and scaling
estimate (2.26) yield (2.34).

Now suppose that $\\{u_{j}\\}\subset S^{\prime
p}_{2,\delta},\\{f_{j}\\}\subset S^{\prime p}_{0,\delta-2}$ are sequences such
that $f_{j}=\mathring{\Delta}u_{j}$ and $f_{j}\rightarrow f$. By (2.34),
$\\{u_{j}\\}$ is a Cauchy sequence and hence is convergent to $u\in S^{\prime
p}_{2,\delta}$ with $\mathring{\Delta}u=f$, since $\mathring{\Delta}:S^{\prime
p}_{2,\delta}\rightarrow S^{\prime p}_{0,\delta-2}$ is bounded. This map thus
has closed range. On the other hand, by (2.34), the kernel of this map is
trivial. Also, by (2.34), we have $K_{0}f\in S^{\prime p}_{2,\delta}$, and
hence $\mathring{\Delta}(K_{0}f)(x)=f$, for all $f\in
C_{0}^{\infty}(H_{n}\setminus\\{0\\})$, so it is also surjective and this
establishes the isomorphism for the cases $-2n<\delta<0$. For the other cases
$\delta\leq-2n$, we need the following commutative diagram

$\begin{array}[]{cccc}\mathring{\Delta}_{b}:&S^{\prime
p}_{s+2,\delta_{1}}&\longrightarrow&S^{\prime p}_{s,\delta_{1}-2}\\\
&\uparrow&&\uparrow\\\ \mathring{\Delta}_{b}:&S^{\prime
p}_{s+2,\delta_{2}}&\longrightarrow&S^{\prime p}_{s,\delta_{2}-2}\end{array}$

where $\delta_{2}<\delta_{1}$ and the vertical maps are the inclusions, which,
by proposition 2.1, are continuous. Suppose that
$\mathring{\Delta}_{b}:S^{\prime p}_{s+2,\delta_{1}}\longrightarrow S^{\prime
p}_{s,\delta_{1}-2}$ is isomorphic, then from the above diagram, it is easy to
see that the map $\mathring{\Delta}_{b}:S^{\prime
p}_{s+2,\delta_{2}}\longrightarrow S^{\prime p}_{s,\delta_{2}-2}$ is one-to-
one. Since $\mathring{\Delta}_{b}$ is self-adjoint, the map
$\mathring{\Delta}_{b}:S^{\prime p}_{s+2,\delta_{2}}\longrightarrow S^{\prime
p}_{s,\delta_{2}-2}$ is onto, and hence an isomorphic. We claim that (2.34)
holds for $\delta_{2}$. For a contradiction, we assume that (2.34) doesn’t not
hold for $\delta_{2}$. Then there exists a sequence of $u_{j}\in S^{\prime
p}_{s+2,\delta_{2}}$ such that $\|u_{j}\|^{\prime}_{s+2,p,\delta_{2}}=1$ and

(2.37)
$\|\mathring{\Delta}_{b}u_{j}\|^{\prime}_{s,p,\delta_{2}-2}<\frac{1}{j}.$

Thus there exists a subsequence $\\{u_{j}\\}$ converges weakly to $u_{\infty}$
such that $\|u_{\infty}\|^{\prime}_{s+2,p,\delta_{2}}=1$ and
$\|\mathring{\Delta}_{b}u_{\infty}\|^{\prime}_{s,p,\delta_{2}-2}=0$, which
implies that $u_{\infty}=0$, due to that $\mathring{\Delta}_{b}$ is one-to-
one. We therefore get a contradiction, and thus have (2.34) for all
$\delta<0$. ∎

The scale-broken estimate (2.38) which follows is the key to proving Fredholm
properties.

######  Theorem 2.9.

Suppose that $P$ is asymptotic to $\mathring{\Delta}_{b}$ at rate $\tau>0$,
and $\delta<0$, $s$ is a non-negative integer. Then, for $1<p\leq q$, the map

$P:S^{p}_{s+2,\delta}\rightarrow S^{p}_{s,\delta-2}$

has finite-dimensional kernel and closed range. In addition, for any $u\in
S^{p}_{s+2,\delta}$, we have

(2.38) $\|u\|_{s+2,p,\delta}\leq
C(\|Pu\|_{s,p,\delta-2}+\|u\|_{L^{p}(B_{R})}),$

where $C,R$ are constants depending on $P,\delta,Q,p,q$.

###### Proof.

It will suffice to prove (2.38) for $s=0$. Define the operator norm

$\|P-\mathring{\Delta}_{b}\|_{op}=\sup\\{\|(P-\mathring{\Delta}_{b})u\|_{p,\delta-2}\
:\ u\in S^{p}_{2,\delta},\ \|u\|_{2,p,\delta}=1\\}$

and let $\|\cdot\|_{op,R}$ denote the same norm restricted to function with
support in $E_{R}=H_{n}\setminus B_{R}$. For (2.38), first, we claim that

(2.39) $\|P-\mathring{\Delta}_{b}\|_{op,R}=o(1),\ \ \textrm{as}\
R\rightarrow\infty.$

If $\operatorname{supp}{u}\subset E_{R}$, since $Q<q$,

(2.40) $\begin{split}&\|(P-\mathring{\Delta}_{b})u\|_{p,\delta-2}\\\
\leq&\|(a^{ij}(z,t)-\delta_{ij})\partial_{ij}u+b^{i}(z,t)\partial_{i}u+c(z,t)u\|_{p,\delta-2}\\\
\leq&\|(a^{ij}(z,t)-\delta_{ij})\partial_{ij}u\|_{p,\delta-2;E_{R}}+\|b^{i}(z,t)\partial_{i}u\|_{p,\delta-2;E_{R}}+\|c(z,t)u\|_{p,\delta-2;E_{R}},\end{split}$

where

(2.41)
$\|(a^{ij}(z,t)-\delta_{ij})\partial_{ij}u\|_{p,\delta-2;E_{R}}\leq\sup_{|(z,t)|>R}\left\\{|a^{ij}(z,t)-\delta_{ij}|\right\\}\|\partial_{ij}u\|_{p,\delta-2;E_{R}};$

(2.42)
$\begin{split}\|b^{i}(z,t)\partial_{i}u\|_{p,\delta-2;E_{R}}&\leq\|b^{i}\|_{Q,-1;E_{R}}\|\partial_{i}u\|_{\frac{Qp}{Q-p},\delta-1},\
\ \textrm{by}\ \eqref{opw11}\\\ &\leq
C\|b^{i}\|_{Q,-1;E_{R}}\|\partial_{i}u\|_{1,p,\delta-1},\ \ \textrm{by}\
\eqref{opw13}\\\ &\leq
C\|b^{i}\|_{q,-1-\tau;E_{R}}\|\partial_{i}u\|_{1,p,\delta-1},\ \ \textrm{by}\
\eqref{opw10};\end{split}$

and, similarly, we have

(2.43) $\|c(z,t)u\|_{p,\delta-2;E_{R}}\leq
C\|c\|_{q/2,-2-\tau;E_{R}}\|u\|_{2,p,\delta}.$

Substituting (2.41),(2.42) and (2.43) into (2.40), we obtain

$\begin{split}\|(P-\mathring{\Delta}_{b})u\|_{p,\delta-2}&\leq\sup_{|(z,t)|>R}\left\\{|a^{ij}(z,t)-\delta_{ij}|\right\\}\|\partial_{ij}u\|_{p,\delta-2;E_{R}}\\\
&\leq+C\|b^{i}\|_{q,-1-\tau;E_{R}}\|\partial_{i}u\|_{1,p,\delta-1}+C\|c\|_{q/2,-2-\tau;E_{R}}\|u\|_{2,p,\delta},\end{split}$

and hence (2.39) holds.

Let $\chi\in C_{0}^{\infty}(B_{2})$ be a patch function, $0\leq\chi\leq
1,\chi=1$ in $B_{1}$, and set $\chi_{R}(x)=\chi(R^{-1}x)$. Writting
$u=u_{0}+u_{\infty},\ u_{0}=\chi_{R}u,u_{\infty}=(1-\chi_{R})u$ with $R$ a
constant to be determined, the sharp estimate (2.34) yields

$\begin{split}\|u_{\infty}\|_{2,p,\delta}&\leq
C\|\mathring{\Delta}_{b}u_{\infty}\|_{0,p,\delta-2}\\\ &\leq
C(\|Pu_{\infty}\|_{0,p,\delta-2}+\|(P-\mathring{\Delta}_{b})u_{\infty}\|_{0,p,\delta-2})\\\
&\leq
C(\|Pu_{\infty}\|_{0,p,\delta-2}+\|P-\mathring{\Delta}_{b}\|_{op,R}\|u_{\infty}\|_{2,p,\delta}),\end{split}$

in which the second term of the right hand side can be absorbed into the left
hand side, for $R$ sufficiently large, and we estimate

$\begin{split}\|Pu_{\infty}\|_{0,p,\delta-2}&=\|Pu-Pu_{0}\|_{0,p,\delta-2}\\\
&\leq\|Pu\|_{0,p,\delta-2}+\|2a^{ij}(\partial_{i}u)(\partial_{j}\chi_{R})+(a^{ij}\partial_{ij}\chi_{R}+b^{i}\partial_{i}\chi_{R})u\|_{0,p,\delta-2;A_{R}}\\\
&\leq\|Pu\|_{0,p,\delta-2}+C\|u\|_{1,p,\delta;A_{R}}.\end{split}$

We therefore get

$\|u_{\infty}\|_{2,p,\delta}\leq
C(\|Pu\|_{0,p,\delta-2}+\|u\|_{1,p,\delta;A_{R}},$

for $R$ large enough. Using (2.26) and the interpolation inequality (2.14)
gives (2.38). Now suppose that $\\{u_{j}\\}$ is a sequence in Ker$P$
satisfying $\|u_{j}\|_{2,p,\delta}=1$, so that by the Rellich lemma we may
assume, without loss of generality, that $\\{u_{j}\\}$ converges strongly in
$L^{p}(B_{R})$. Estimate (2.38) now shows that $\\{u_{j}\\}$ is a Cauchy
sequence and hence convergent in $S^{p}_{2,\delta}$ which implies that Ker$P$
is finite-dimensional. To show that $P$ has closed range, as the argument in
Theorem 2.8, it suffices to show that there is a constant $C$ such that

$\|u\|_{2,p,\delta}\leq C\|Pu\|_{0,p,\delta-2},\ \ \ \textrm{for all}\ u\in
Z,$

where $Z$ is a closed subspace such that $S^{p}_{2,\delta}=Z+\textrm{Ker}P$.
For if this were not the case, there would be a sequence $\\{u_{j}\\}\subset
Z$ such that $\|u_{j}\|_{2,p,\delta}=1$ and $\|Pu\|_{0,p,\delta-2}\rightarrow
0$. The usual Rellich lemma applied to (2.38) shows that $u_{j}\\}$ has a
subsequence which is Cauchy in $Z$ and whose limit is a non-zero element of
Ker$P\cap Z$, which is a contradiction. Therefore, $P$ has closed range. ∎

We are interested in the dimension of the kernel of $P$, which will be denoted
by

(2.44) $N(P,\delta)=\textrm{dim\ ker}(P:S^{p}_{2,\delta}\rightarrow
S^{p}_{0,\delta-2})$

with $1<p\leq q$.

######  Proposition 2.10.

Suppose that $P$ and its formal adjoint $P^{*}$ both satisfy conditions (2.24)
and (2.25), and $-2n<\delta<0,\ \frac{q}{q-1}<p\leq q$. Then
$P:S^{p}_{2,\delta}\rightarrow S^{p}_{0,\delta-2}$ is a Fredholm operator with
Fredholm index $\iota(P,\delta)=N(P,\delta)-N(P^{*},2-Q-\delta)$.

###### Proof.

Suppose that the formal adjoint

$P^{*}:(S^{p}_{0,\delta-2})^{*}\rightarrow(S^{p}_{2,\delta})^{*}$

of $P$ is also asymptotic to $\mathring{\Delta}_{b}$ which satisfies the
condition (2.24) and (2.25). Here $(S^{p}_{2,\delta})^{*}$ is the subspace of
$D^{\prime}(H_{n})$ consisting of those distributions which extend to give
bounded linear functionals on $S^{p}_{2,\delta}$, endowed with the dual norm.
From the Hölder inequality (2.12), we have

$\|uv\|_{1,-Q}\leq\|u\|_{p,\delta-2}\|v\|_{p^{\prime},2-Q-\delta},$

which implies that $:(S^{p}_{0,\delta-2})^{*}=S^{p^{\prime}}_{2-Q-\delta}$,
where $p^{\prime}=p/(p-1)$. If $-2n<\delta<0$, which implies that
$-2n<2-Q-\delta<0$, and moreover $\frac{q}{q-1}<p\leq q$ means that
$1<p^{\prime}\leq q$, then Proposition 2.6 shows that $Ker(P^{*})\subset
S^{p^{\prime}}_{2,2-Q-\delta}$ and hence

$\textrm{dim\ coker}P=\textrm{dim\ ker}P^{*}=N(P^{*},2-Q-\delta).$

This is finite by Theorem 2.9; hence $P$ is Fredholm with Fredhom index

(2.45) $\iota(P,\delta)=N(P,\delta)-N(P^{*},2-Q-\delta).$

∎

Let $\xi_{0}=\ker{\Theta}$ be the standard contact form on the Heisenberg
group. Instead of the standard pseudo-hermitian structure $(J_{0},\Theta)$, we
consider another pseudo-hermitian structure $(J,\theta)$ with
$\ker{\theta}=\xi_{0}$. Let $P=\Delta_{b}$ be the corresponding sub-Laplacian,
then $P^{*}=P$. We write

$\Delta_{b}=g_{ij}(z,t)\mathring{e}_{i}\mathring{e}_{j}+\cdots.$

We have the following corollary.

######  Corollary 2.11.

Suppose that $g_{ij}$ is uniformly sub-elliptic in
$(\operatorname{\mathbb{R}}^{2n+1},\xi_{0})$ and $(g_{ij}-\delta_{ij})\in
S^{q}_{1,-\tau}$ for some $Q<q<\infty$, and that $\delta<0$ and
$\frac{q}{q-1}<p\leq q$. Then $\Delta_{b}:S^{p}_{s+2,\delta}\rightarrow
S^{p}_{s,\delta-2}$ is an isomorphism.

###### Proof.

From Proposition 2.10, the sub-Laplacian
$\Delta_{b}:S^{p}_{2,\delta}\rightarrow S^{p}_{0,\delta-2}$ is Fredholm. And
since $-2n<\delta<0$ if and only if $-2n<\delta^{*}<0$, where
$\delta^{*}=2-Q-\delta$. It will suffice to show that
$N(\Delta_{b},\delta)=0$. But if $\Delta_{b}u=0$ and $u\in S^{p}_{2,\delta}$,
then $u=o(1)$ at infinity and the strong maximum principle for weak solutions
shows that $u=0$. ∎

##  3\. Asymptotically Flat Pseudo-hermitian Manifolds

######  Definition 3.1.

A $(2n+1)$-dimensional pseudo-hermitian manifold $(N,J,\theta)$ is said to be
asymptotically flat pseudo-hermitian if $N=N_{0}\cup N_{\infty}$, with $N_{0}$
compact and $N_{\infty}$ diffemorphic to $H_{n}\setminus B_{\rho_{0}}$ in
which $(J,\theta)$ is close to $(\mathring{J},\mathring{\theta})$ in the sense
that

(3.1)
$\begin{split}\theta&=(1+c_{n}A\rho^{-2n}+O(\rho^{-2n-1}))\mathring{\theta}+O(\rho^{-2n-1})_{\beta}dz^{\beta}+O(\rho^{-2n-1})_{\bar{\beta}}dz^{\bar{\beta}};\\\
\theta^{\alpha}&=O(\rho^{-2n-1})\mathring{\theta}+O(\rho^{-2n-2})_{\bar{\beta}}{}^{\alpha}dz^{\bar{\beta}}+(1+\tilde{c}_{n}A\rho^{-2n}+O(\rho^{-2n-1}))\sqrt{2}dz^{\alpha},\end{split}$

for some $A\in\operatorname{\mathbb{R}}$ and a unitary co-frame
$\theta^{\alpha}$ in coordinates $(z^{\beta},z^{\bar{\beta}},t)$ (called
asymptotic coordinates) for $N_{\infty}$ on which
$\rho=((\sum_{\beta=1}^{n}|z^{\beta}|^{2})^{2}+t^{4})^{1/4}$ is defined. We
also require the Tanaka-Webster scalar curvature $W\in L^{1}(N)$.

Suppose that $(M,J)$ is a spherical in a neighborhood of a point $p\in M$. If
we blow up at $p$ through the Green’s function of the CR invariant sub-
Laplacian

$G_{p}=\frac{a_{n}}{2\pi}\rho^{-2n}+A_{p}+O(\rho),$

which is expressed in terms of the CR normal coordinates, then we obtain an
asymptotically flat pseudo-hermitian. We denote such a manifold by $N(p)$. In
such a case, we have

(3.2)
$\begin{split}\theta&=(1+c_{n}A_{p}\rho^{-2n}+O(\rho^{-2n-1}))\mathring{\theta};\\\
\theta^{\alpha}&=O(\rho^{-2n-1})\mathring{\theta}+(1+\tilde{c}_{n}A_{p}\rho^{-2n}+O(\rho^{-2n-1}))\sqrt{2}dz^{\alpha},\end{split}$

where $c_{n}=\frac{4\pi}{na_{n}}$ and $\tilde{c}_{n}=\frac{2\pi}{na_{n}}$. Let
$\\{Z_{\alpha},Z_{\bar{\alpha}},T\\}$ be the frame dual to
$\\{\theta^{\alpha},\theta^{\bar{\alpha}},\theta\\}$. It is easy to see that

(3.3)
$Z_{\alpha}=(1-\tilde{c}_{n}A_{p}\rho^{-2n}+O(\rho^{-2n-1}))\mathring{Z}_{\alpha}+O(\rho^{-2n-2})_{\alpha}{}^{\bar{\beta}}\mathring{Z}_{\bar{\beta}}.$

On the other hand, from the structure equations of pseudo-hermitian manifolds,
we have the asymptotic behavior of the pseudo-hermitian connection forms

(3.4)
$\theta_{\alpha}{}^{\beta}=O(\rho^{-2n-2})\mathring{\theta}+B_{\alpha}{}^{\beta}{}_{\gamma}dz^{\gamma}+C_{\alpha}{}^{\beta}{}_{\bar{\gamma}}dz^{\bar{\gamma}},$

where for any fixed $\alpha$, we have

(3.5) $\begin{split}C_{\alpha}{}^{\beta}{}_{\bar{\gamma}}&=O(\rho^{-2n-2}),\
\textrm{for}\ \beta\neq\alpha,\gamma\neq\alpha\\\
C_{\alpha}{}^{\beta}{}_{\bar{\alpha}}&=\frac{-inc_{n}A_{p}z^{\beta}\omega}{\rho^{2n+4}}+O(\rho^{-2n-2}),\
\textrm{for}\ \beta\neq\alpha\\\
C_{\alpha}{}^{\alpha}{}_{\bar{\gamma}}&=\frac{-in\tilde{c}_{n}A_{p}z^{\gamma}\omega}{\rho^{2n+4}}+O(\rho^{-2n-2}),\
\textrm{for}\ \gamma\neq\alpha\\\
C_{\alpha}{}^{\alpha}{}_{\bar{\alpha}}&=\frac{-in(c_{n}+\tilde{c}_{n})A_{p}z^{\alpha}\omega}{\rho^{2n+4}}+O(\rho^{-2n-2}),\\\
\end{split}$

and
$B_{\alpha}{}^{\beta}{}_{\gamma}=-\overline{C_{\beta}{}^{\alpha}{}_{\bar{\gamma}}}$.
Therefore, we have

(3.6)
$\Delta_{b}=\left(1+\frac{4\tilde{c}_{n}A_{p}}{\rho^{2n}}\right)\mathring{\Delta}_{b}+\textrm{higher
decay order},$

which implies that $\Delta_{b}$ is asymptotic to $\mathring{\Delta}_{b}$. We
remark that if the manifold $(M,J)$ is not spherical, then, in general,
formula (3.6) does not hold.

As used in Corollary 2.11, the strong maximum principle implies the following
theorem.

######  Theorem 3.2.

Suppose that $(M,J)$ is a CR manifold which is spherical in a neighborhood of
a point $p\in M$. We consider the blow up manifold $N(p)$ at this point $p$.
Suppose that $\frac{q}{q-1}<p\leq q$ and $\delta<0$. Then

(3.7) $\Delta_{b}:S^{p}_{s+2,\delta}(N(p))\rightarrow
S^{p}_{s,\delta-2}(N(p))$

is an isomorphism.

###  3.1. The Weitzenbock formula for $n=2$

In this subsection, we will give an isomorphism theorem for Dirac operator on
a Spinor bundle. More details and notations on the geometry of context of
spinors here may be found in [1]. Suppose $(M,J,\theta)$ is a spherical CR
manifold with a spin structure $Spin(\xi)$ on the contact bundle
$\xi=\ker{\theta}$, together with the Levi metric $L_{\theta}$. Let
$\Lambda_{R}^{k}(n)$ and $\Lambda_{C}^{k}(n)$ denote the real and complex
vector spaces respectively, spanned by
$\\{\omega^{j_{1}}\wedge\cdots\wedge\omega^{j_{k}}\ |\ 1\leq
j_{1}<\cdots<j_{k}\leq n\\}$ (view symbols $\omega^{1},\cdots,\omega^{n}$ as
independent vectors). Recall, from [1], that there is a canonical
representation $\mathcal{N}$ of the Clifford algebra $C_{2n}(-1)$ such that
$\Lambda_{C}^{*}(n)$ is an irreducible $C_{2n}(-1)$-modules, and
$\Lambda_{C}^{even}(n),\Lambda_{C}^{odd}(n)$ are two irreducible
$Spin(2n)$-modules. We hence have the following three associated vector
bundles:

$S=Spin(\xi)\times_{\mathcal{N}}\Lambda_{C}^{*}(n),\
S^{+}=Spin(\xi)\times_{\mathcal{N}}\Lambda_{C}^{even}(n),\
S^{-}=Spin(\xi)\times_{\mathcal{N}}\Lambda_{C}^{odd}(n),$

which are called spinor bundles. On each spinor bundle there is associated a
unique spin connection $\nabla$, constructed from the pseudo-hermitian
connection, and hence a Dirac operator, which is denoted by $D_{\xi}$. We have
the Weitzenbock-type formula as follows:

$D_{\xi}^{2}=\nabla^{*}\nabla+W-2\sum_{\beta=1}^{n}e_{\beta}e_{n+\beta}\nabla_{T},$

where $\\{e_{1},\cdots,e_{2n}\\}$ is a local orthonormal frame field such that
$e_{n+\beta}=Je_{\beta}$ for $1\leq\beta\leq n$. Fortunately, due to some
algebra property in the can $n=2$, we reduce the formula to

(3.8) $D_{\xi}^{2}=\nabla^{*}\nabla+W,\ \ \ \textrm{on}\ S^{+}$

It is this formula make Witten’s approach work on this geometry. Actually we
have the following isomorphism theorem for Dirac operator.

######  Theorem 3.3.

Suppose $n=2$ and suppose that $(M,J)$ is a CR manifold which is spherical in
a neighborhood of a point $p\in M$. We consider the blow up manifold $N(p)$ at
this point $p$. Suppose that $\frac{q}{q-1}<p\leq q$ and $\delta<0$. Then

(3.9) $D^{2}_{\xi}:S^{p}_{s+2,\delta}(S^{+})\rightarrow
S^{p}_{s,\delta-2}(S^{+})$

is an isomorphism.

###### Proof.

Let $\varphi=\varphi^{\lambda}s_{\lambda}$ be a spinor spanned by a frame
field of spinors $\\{s_{\lambda}\\}$. We compute

(3.10)
$\begin{split}\nabla^{*}\nabla\varphi&=-\sum_{a=1}^{2n}\nabla^{2}_{e_{a},e_{a}}\varphi\\\
&=-\sum_{a=1}^{2n}\left(\nabla_{e_{a}}\nabla_{e_{a}}\varphi-\nabla_{\nabla_{e_{a}}e_{a}}\varphi\right)\\\
&=-\sum_{a=1}^{2n}(e_{a}e_{a}\varphi^{\lambda}-(\nabla_{e_{a}}e_{a})\varphi^{\lambda})s_{\lambda}+2(e_{a}\varphi^{\lambda})\nabla_{e_{a}}s_{\lambda}+\varphi^{\lambda}\left(\nabla_{e_{a}}\nabla_{e_{a}}s_{\lambda}-\nabla_{\nabla_{e_{a}}e_{a}}s_{\lambda}\right)\\\
&=-(\Delta_{b}\varphi^{\lambda})s_{\lambda}+\ \textrm{l.o.t.},\end{split}$

where

$\begin{split}\Delta_{b}f&=tr(\nabla df)=e_{a}e_{a}f-(\nabla_{e_{a}}e_{a})f,\
\ \textrm{for a function}\ f\\\
\textrm{l.o.t.}&=2(e_{a}\varphi^{\lambda})\nabla_{e_{a}}s_{\lambda}+\varphi^{\lambda}\left(\nabla_{e_{a}}\nabla_{e_{a}}s_{\lambda}-\nabla_{\nabla_{e_{a}}e_{a}}s_{\lambda}\right).\end{split}$

Therefore the principal part of each component of $D_{\xi}^{2}$ is the sub-
Laplacian $\Delta_{b}$, which is asymptotic to $\mathring{\Delta}_{b}$ at rate
$\tau>0$. By Theorem 3.2, $D_{\xi}^{2}$ is an isomorphism. ∎

## References

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