localization and concludes by outlining future trends associated with each
sensing technology.

We foresee numerous research directions leveraging ML tools in both indoor and
outdoor localization techniques to enhance their widespread deployability. ML
tools help to address the noisy environment in NLoS conditions and enable more
efficient data analysis. Floor detection for multi-storey buildings is a
crucial research direction in indoor localization, necessitating a transition
to 3D localization techniques. ML-based 3D location estimation employs neural
networks to extract relevant features and predict environmental data
efficiently and more accurately. Consistency and reliability are essential for
indoor localization applications, where ML methods notably impact human
detection, intruder identification, and other tasks. By utilizing ML tools,
data analysis becomes more reliable, even in complex environments. Traditional
approaches perform differently indoors and outdoors due to environmental
variations, necessitating adaptable solutions. Addressing cost concerns,
recent advancements enable high localization performance at reduced deployment
costs. The widespread deployment of these systems facilitates large-scale
location-based data mining, supporting applications like indoor landmark
discovery, queue detection, and geofencing.

Considering different data types, CSI is likely to replace RSSI due to its
higher robustness and performance in fingerprinting. However, smartphone
limitations hinder CSI data collection, necessitating specialized Wi-Fi-
enabled infrastructure. Additionally, Channel impulse response (CIR) and
Signal Eigenvector are also drawing attention for better indoor localization.
Integrating vision with fingerprinting offers promising avenues, while
combining various technologies like sound detection, inertial sensors, and
motion sensors can improve localization, particularly for assisting physically
disabled individuals [2].

Establishing a dependable indoor positioning system remains challenging due to
environmental factors. Integrating ML into the system enhances localization
reliability. Despite cost and complexity, indoor positioning systems are
widely adopted worldwide. Diverse applications necessitate innovative systems,
fostering competitiveness. As a comprehensive survey covering all the
subdomains is currently absent, this paper serves as a valuable resource for
referencing state-of-the-art approaches within this emerging realm. The
principal motivation driving this endeavor is to offer insights into this
intricate domain and to contribute to research challenges associated with it.
Its aim is to provide essential references for readers to gain knowledge and
contribute further in this field.

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|  Amala Sonny received the B.Tech. degree in electronics and communication
engineering from the TKM college of engineering, Kollam, India, in 2014, and
the M.Tech. degree from the National Institute of Technology Calicut, Calicut,
India, in 2017. She is pursuing the PhD degree with the Department of EE,
Indian Institute of Technology Hyderabad, Hyderabad, India. She was a visiting
researcher at the Department of ICT, University of Agder, Kristiansand,
Norway. Her research interests include Indoor and outdoor localization,
machine learning, and wireless communication.
---|---
|  Linga Reddy Cenkeramaddi (Senior Member, IEEE) received the master’s degree
in electrical engineering from the Indian Institute of Technology Delhi (IIT
Delhi), New Delhi, India, in 2004, and the Ph.D. degree in electrical
engineering from the Norwegian University of Science and Technology (NTNU),
Trondheim, Norway, in 2011. He worked at Texas Instruments on mixed-signal
circuit design before joining the Ph.D. Program at NTNU. After finishing his
Ph.D. degree, he worked on radiation imaging for an atmosphere–space
interaction monitor (ASIM mission to the International Space Station) at the
University of Bergen, Bergen, Norway, from 2010 to 2012. He is currently the
Leader of the Autonomous and Cyber-Physical Systems (ACPS) Research Group and
a Professor at the University of Agder, Grimstad, Norway. He has co-authored
over 150 research publications that have been published in prestigious
international journals and standard conferences in the research areas Internet
of Things (IoT), Cyber-Physical systems, Autonomous systems, Robotics and
Automation involving advanced sensor systems, Computer vision, Thermal
imaging, LiDAR imaging, Radar imaging, wireless sensor networks, smart
electronic systems, advanced machine learning techniques, connected autonomous
systems including drones/unmanned aerial vehicles (UAVs), unmanned ground
vehicles (UGVs), unmanned underwater systems (UUSs), 5G- (and beyond) enabled
autonomous vehicles, and Socio-Technical Systems like urban transportation
systems, Smart Agriculture and Smart Cities. He is also quite active in
medical imaging. Several of his master’s students won the best master thesis
award in information and communication technology (ICT).
---|---
|  Abhinav Kumar (Senior Member, IEEE)  received the B.Tech. degree in
electrical engineering, and the M.Tech. degree in information and
communication technology, and the Ph.D. degree in electrical engineering from
the Indian Institute of Technology (IIT) Delhi, in 2009, 2009, and 2013,
respectively. He was a Research Associate with IIT Delhi in 2013. From 2013 to
2014, he was a Postdoctoral Fellow at the University of Waterloo, Canada.
Since 2014, he has been with IIT Hyderabad, India, where he is currently an
Associate Professor. His research interests are in different aspects of
wireless communications and networking.
---|---


On $L^{\infty}$ estimates for Monge-Ampère and Hessian equations on nef
classes 111Work supported in part by the National Science Foundation under
grant DMS-1855947.

Bin Guo, Duong H. Phong, Freid Tong222F.T. is supported by Harvard’s Center
for Mathematical Sciences and Applications. , and Chuwen Wang

###### Abstract

The PDE approach developed earlier by the first three authors for $L^{\infty}$
estimates for fully non-linear equations on Kähler manifolds is shown to apply
as well to Monge-Ampère and Hessian equations on nef classes. In particular,
one obtains a new proof of the estimates of Boucksom-Eyssidieux-Guedj-Zeriahi
and Fu-Guo-Song for the Monge-Ampère equation, together with their
generalization to Hessian equations.

##  1 Introduction

The goal of this short note is to show that the PDE approach introduced in
[13, 14] for $L^{\infty}$ and Trudinger-type estimates for general classes
fully non-linear equations on a compact Kähler manifold applies as well to
Monge-Ampère and Hessian equations on nef classes.

The key to the approach in [13, 14] is an estimate of Trudinger-type, obtained
by comparing the solution $\varphi$ of the given equation to the solution of
an auxiliary Monge-Ampère equation with the energy of the sublevel set
function $-\varphi+s$ on the right hand side. We shall see that, in the
present case of nef classes, the argument can still be made to work by
replacing $\varphi$ by $\varphi-V$, where $V$ is the envelope of the nef
class. Applied to the Monge-Ampère equation, this gives a PDE proof of the
estimates obtained earlier for nef classes by Boucksom-Eyssidieux-Guedj-
Zeriahi [3] and Fu-Guo-Song [10]. The estimates which we obtain with this
method applied to Hessian equations seem new.

We note that the use of an auxiliary Monge-Ampère equation had been
instrumental in the recent progress of Chen and Cheng [4] on the constant
scalar curvature Kähler metrics problem. There the auxiliary equation involved
the entropy, and not the energy of sublevel set functions as in our case. More
generally, auxiliary equations have often been used in the theory of partial
differential equations, notably by De Giorgi [6], and more recently by Dinew
and Kolodziej [8, 7] in their approach to Hölder estmates for the complex
Monge-Ampère equation.

##  2 The Monge-Ampère equation

We begin with the Monge-Ampère equation. Let $(X,\omega)$ be a compact Kähler
manifold and $\chi$ be a closed $(1,1)$-form on $X$. We assume the cohomology
class $[\chi]$ is nef and let $\nu\in\\{0,1,\ldots,n\\}$ be the numerical
dimension of $[\chi]$, i.e.

$\nu=\max\\{k~{}|~{}[\chi]^{k}\neq 0\mbox{ in }H^{k,k}(X,\bf C)\\}.$

When $\nu=n$ we say the class $[\chi]$ is big.

Let $\hat{\omega}_{t}=\chi+t\omega$ for $t\in(0,1]$. The form
$\hat{\omega}_{t}$ may not be positive but its class is Kähler. We consider
the following family of complex Monge-Ampère equations

$(\hat{\omega}_{t}+i\partial\bar{\partial}\varphi_{t})^{n}=c_{t}e^{F}\omega^{n},\quad\sup_{X}\varphi_{t}=0$
(2.1)

where $c_{t}=[\hat{\omega}_{t}]^{n}=O(t^{n-\nu})$ is a normalizing constant
and $F\in C^{\infty}(X)$ satisfies
$\int_{X}e^{F}\omega^{n}=\int_{X}\omega^{n}$. This equation admits a unique
smooth solution $\varphi_{t}$ by Yau’s theorem [18].

The form $\chi$ is not assumed to be semipositive, so the usual $L^{\infty}$
estimate of $\varphi_{t}$ may not hold [16]. As in [3, 10], we need to modify
the solution $\varphi_{t}$ by an envelope $V_{t}$ of the class
$[\hat{\omega}_{t}]$, defined as follows,

$V_{t}=\sup\\{v~{}|~{}v\in PSH(X,\hat{\omega}_{t}),\,v\leq 0\\}.$

Then we have:

###### Theorem 1

Consider the equation (2.1), and assume that the cohomology class of $\chi$ is
nef. For any $s>0$, let $\Omega_{s}=\\{\varphi_{t}-V_{t}\leq-s\\}$ be the sub-
level set of $\varphi_{t}-V_{t}$.

(a) Then there are constants $C=C(n,\omega,\chi)>0$ and
$\alpha_{0}=\alpha_{0}(n,\omega,\chi)>0$ such that

$\displaystyle\int_{\Omega_{s}}\,{\rm
exp}\,\big{\\{}\alpha_{0}\Big{(}\frac{-(\varphi_{t}-V_{t}+s)}{A_{s}^{1/(1+n)}}\Big{)}^{\frac{n+1}{n}}\big{\\}}\omega^{n}\leq
C\,{\rm exp}\,{(CE_{t})},$ (2.2)

where $A_{s}=\int_{\Omega_{s}}(-\varphi_{t}+V_{t}-s)e^{F}\omega^{n}$ and
$E_{t}=\int_{X}(-\varphi_{t}+V_{t})e^{F}\omega^{n}$.

(b) Fix $p>n$. There is a constant $C(n,p,\omega,\chi,\|e^{F}\|_{L^{1}(\,{\rm
log}\,L)^{p}})$ so that for all $t\in(0,1]$, we have

$\displaystyle 0\leq-\varphi_{t}+V_{t}\leq
C(n,p,\omega,\chi,\|e^{F}\|_{L^{1}(\,{\rm log}\,L)^{p}}).$ (2.3)

We remark that the estimates in Theorem 1 continue to hold for a family of
Kähler metrics (maybe with distinct complex structures) which satisfy a
uniform $\alpha$-invariant type estimate.

Proof of Theorem 1

We would like to find an auxiliary equation with smooth coefficients, so that
its solvability can be guaranteed by Yau’s theorem. For this, we need a lemma
due to Berman [2] on a smooth approximation for $V_{t}$ (see also Lemma 2
below). Fix a time $t\in(0,1]$.

###### Lemma 1

Let $u_{\beta}$ be the smooth solution to the complex Monge-Ampère equation

$(\hat{\omega}_{t}+i\partial\bar{\partial}u_{\beta})^{n}=e^{\beta
u_{\beta}}\omega^{n}.$

Then $u_{\beta}$ converges uniformly to $V_{t}$ as $\beta\to\infty$.

We remark that by [5], $V_{t}$ is a $C^{1,1}$ function on $X$, although this
fact is not used in this note. We now return to the proof of Part (a) of
Theorem 1.

We choose a sequence of positive functions $\tau_{k}:\bf R\to\bf R_{+}$ such
that $\tau_{k}(x)$ decreases to $x\cdot\chi_{\bf R_{+}}(x)$ as $k\to\infty$.
Fix a smooth function $u_{\beta}$ as in Lemma 1. The $u_{\beta}$ depends on
$t$, but for simplicity we omit the subscript $t$. We solve the following
auxiliary Monge-Ampère equation on $X$

$(\hat{\omega}_{t}+i\partial\bar{\partial}\psi_{t,k})^{n}=c_{t}\frac{\tau_{k}(-\varphi_{t}+u_{\beta}-s)}{A_{s,k,\beta}}e^{F}\omega^{n},\quad\sup_{X}\psi_{t,k}=0,$
(2.4)

where
$A_{s,k,\beta}=\int_{X}\tau_{k}(-\varphi_{t}+u_{\beta}-s)e^{F}\omega^{n}.$
Since $\psi_{t,k}\leq V_{t}$ and $u_{\beta}$ converges uniformly to $V_{t}$.
By taking $\beta$ large enough, we may assume $\psi_{t,k}<u_{\beta}+1$.

Define a function

$\Phi=-\varepsilon(-\psi_{t,k}+u_{\beta}+1+\Lambda)^{n/(n+1)}-(\varphi_{t}-u_{\beta}+s),$

with the constants

$\varepsilon^{n+1}=A_{s,k,\beta}n^{-n}(n+1)^{n},\quad\Lambda=n^{n+1}(n+1)^{-n-1}\varepsilon^{n+1}.$
(2.5)

As a smooth function on the compact manifold $X$, $\Phi$ must achieve its
maximum at some $x_{0}\in X$. If $x_{0}\in X\backslash\Omega_{s}^{\circ}$,
then

$\Phi(x_{0})\leq-(\varphi_{t}-u_{\beta}+s)\leq-
V_{t}+u_{\beta}\leq\epsilon_{\beta}$

where $\epsilon_{\beta}\to 0$ as $\beta\to\infty$. On the other hand, if
$x_{0}\in\Omega_{s}^{\circ}$ we calculate ($\Delta_{t}$ denotes the Laplacian
with respect to the metric
$\omega_{t}=\hat{\omega}_{t}+i\partial\bar{\partial}\varphi_{t}$)

$\displaystyle 0$ $\displaystyle\geq$ $\displaystyle\Delta_{t}\Phi(x_{0})$
$\displaystyle=$
$\displaystyle-\varepsilon\frac{n}{n+1}(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{-\frac{1}{n+1}}{\rm
tr}_{\omega_{t}}(-i\partial\bar{\partial}\psi_{t,k}+i\partial\bar{\partial}u_{\beta})-{\rm
tr}_{\omega_{t}}(i\partial\bar{\partial}\varphi_{t}-i\partial\bar{\partial}u_{\beta})$
$\displaystyle+\frac{n\varepsilon}{(n+1)^{2}}(-\psi_{t,k}+u_{\beta}+1+\Lambda)^{-\frac{n+2}{n+1}}{\rm
tr}_{\omega_{t}}i\partial(\psi_{t,k}-u_{\beta})\wedge\bar{\partial}(\psi_{t,k}-u_{\beta})$
$\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{-\frac{1}{n+1}}{\rm
tr}_{\omega_{t}}(\hat{\omega}_{t,\psi_{t,k}}-\hat{\omega}_{t,u_{\beta}})-n+{\rm
tr}_{\omega_{t}}\hat{\omega}_{t,u_{\beta}}$ $\displaystyle\geq$
$\displaystyle\frac{n\varepsilon}{n+1}(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{-\frac{1}{n+1}}n\Big{(}\frac{\hat{\omega}_{t,\psi_{t,k}}^{n}}{\omega_{t}^{n}}\Big{)}^{1/n}-n+(1-\frac{n\varepsilon}{n+1}(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{-\frac{1}{n+1}}){\rm
tr}_{\omega_{t}}\hat{\omega}_{t,u_{\beta}}$ $\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{-\frac{1}{n+1}}\Big{(}\tau_{k}(-\varphi_{t}+u_{\beta}-s)A_{s,k,\beta}^{-1}\Big{)}^{1/n}-n+(1-\frac{n\varepsilon}{n+1}\Lambda^{-\frac{1}{n+1}}){\rm
tr}_{\omega_{t}}\hat{\omega}_{t,u_{\beta}}$ $\displaystyle\geq$
$\displaystyle\frac{n^{2}\varepsilon}{n+1}(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{-\frac{1}{n+1}}(-\varphi_{t}+u_{\beta}-s)^{1/n}A_{s,k,\beta}^{-1/n}-n.$

Therefore, at $x_{0}\in\Omega_{s}^{\circ}$

$-(\varphi_{t}-u_{\beta}+s)\leq(\frac{n\varepsilon}{n+1})^{n}A_{s,k,\beta}(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{\frac{n}{n+1}}=\varepsilon(-\psi_{t,k}+u_{\beta}+\Lambda+1)^{\frac{n}{n+1}},$

i.e. $\Phi(x_{0})\leq 0$. Combining the two cases, we conclude that
$\sup_{X}\Phi\leq\epsilon_{\beta}\to 0$ as $\beta\to\infty.$ It then follows
that on $\Omega_{s}$

$(-\varphi_{t}+u_{\beta}-s)^{\frac{n+1}{n}}\leq
C_{n}A_{s,k,\beta}^{1/n}(-\psi_{t,k}+u_{\beta}+1+A_{s,k,\beta})+\epsilon_{\beta}^{(n+1)/n}$

Letting $\beta\to\infty$ we have

$(-\varphi_{t}+V_{t}-s)^{\frac{n+1}{n}}\leq
C_{n}A_{s,k}^{1/n}(-\psi_{t,k}+V_{t}+1+A_{s,k}),$

where $A_{s,k}=\int_{X}\tau_{k}(-\varphi_{t}+V_{t}+s)e^{F}\omega^{n}$. Observe
that by definition $V_{t}\leq 0$ and by the $\alpha$-invariant estimate [15,
17], there exists an $\alpha_{0}(n,\omega,\chi)$ such that

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{(}\alpha_{0}\frac{(-\varphi_{t}+V_{t}-s)^{\frac{n+1}{n}}}{A_{s,k}^{1/n}}\Big{)}\omega^{n}\leq\int_{\Omega_{s}}\,{\rm
exp}\,\Big{(}\alpha_{0}C_{n}(-\psi_{t,k}+1+A_{s,k})\Big{)}\omega^{n}\leq
Ce^{CA_{s,k}}.$ (2.6)

Letting $k\to\infty$, we obtain

$\int_{\Omega_{s}}\,{\rm
exp}\,\Big{(}\alpha_{0}\frac{(-\varphi_{t}+V_{t}-s)^{\frac{n+1}{n}}}{A_{s}^{1/n}}\Big{)}\omega^{n}\
\leq Ce^{CA_{s}}.$

Part (a) of Theorem 1 is proved by noting that $A_{s}\leq E_{t}$ for any
$s>0$.

Once Part (a) of Theorem 1 has been proved, Part (b) can be proved by
following closely the arguments in [13].

Fix $p>n$, and define $\eta:{\bf R}_{+}\to{\bf R}_{+}$ by $\eta(x)=(\,{\rm
log}\,(1+x))^{p}$. Note that $\eta$ is a strictly increasing function with
$\eta(0)=0$, and let $\eta^{-1}$ be its inverse function. Denote

$\displaystyle
v:=\frac{\alpha_{0}}{2}\big{(}\frac{-\varphi_{t}+V_{t}-s}{A_{s}^{1/(n+1)}}\big{)}^{(n+1)/n}$
(2.7)

then by the generalized Young’s inequality with respect to $\eta$, for any
$z\in\Omega_{s}$,

$\displaystyle v(z)^{p}e^{F(z)}$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\,{\rm
exp}\,({F(z)})}\eta(x)dx+\int_{0}^{v(z)^{p}}\eta^{-1}(y)dy$
$\displaystyle\leq$ $\displaystyle\,{\rm
exp}\,({F(z)})(1+|F(z)|)^{p}+C(p)\,{\rm exp}\,({2v(z)})$

We integrate both sides in the inequality above over $z\in\Omega_{s}$, and get
by Part (a), Theorem 1 that

$\displaystyle\int_{\Omega_{s}}v(z)^{p}e^{F(z)}\omega^{n}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega_{s}}e^{F}(1+|F(z)|)^{p}\omega^{n}+\int_{\Omega_{s}}e^{2v(z)}\omega^{n}$
$\displaystyle\leq$ $\displaystyle\|e^{F}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{E}_{t}},$

where the constant $C>0$ depends only on $n,\omega_{X},\chi$. In view of the
definition of $v$, this implies

$\int_{\Omega_{s}}(-\varphi_{t}+V_{t}-s)^{\frac{(n+1)p}{n}}e^{F(z)}\omega^{n}\leq
2^{p}\alpha_{0}^{-p}A_{s}^{\frac{p}{n}}\big{(}\|e^{F}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{E}_{t}}\big{)}.$ (2.8)

From the definition of $A_{s}$, it follows from Hölder inequality that

$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\int_{\Omega_{s}}(-\varphi_{t}+V_{t}-s)e^{F}\omega^{n}$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{\Omega_{s}}(-\varphi_{t}+V_{t}-s)^{\frac{(n+1)p}{n}}e^{F}\omega^{n}\Big{)}^{\frac{n}{(n+1)p}}\cdot\Big{(}\int_{\Omega_{s}}e^{F}\omega^{n}\Big{)}^{1/q}$
$\displaystyle\leq$ $\displaystyle
A_{s}^{\frac{1}{n+1}}\Big{(}2^{p}\alpha_{0}^{-p}\big{(}\|e^{F}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{E}_{t}}\big{)}\Big{)}^{\frac{n}{(n+1)p}}\cdot\Big{(}\int_{\Omega_{s}}e^{F}\omega^{n}\Big{)}^{1/q}$

where $q>1$ satisfies $\frac{n}{p(n+1)}+\frac{1}{q}=1$, i.e.
$q=\frac{p(n+1)}{p(n+1)-n}$. The inequality above yields

$A_{s}\leq\Big{(}2^{p}\beta_{0}^{-p}\big{(}\|e^{F}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{E}_{t}}\big{)}\Big{)}^{1/p}\cdot\Big{(}\int_{\Omega_{s}}e^{F}\omega^{n}\Big{)}^{\frac{1+n}{qn}}.$
(2.9)

Observe that the exponent of the integral on the right hand of (2.9) satisfies

$\frac{1+n}{qn}=\frac{pn+p-n}{pn}=1+\delta_{0}>1,$

for $\delta_{0}:=\frac{p-n}{pn}>0$. For notation convenience, set

$B_{0}:=\Big{(}2^{p}\alpha_{0}^{-p}\big{(}\|e^{F}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C+Ce^{C{E}_{t}}\big{)}\Big{)}^{1/p}.$ (2.10)

From ((2.9)) we then get

$A_{s}\leq
B_{0}\Big{(}\int_{\Omega_{s}}e^{F}\omega^{n}\Big{)}^{1+\delta_{0}}.$ (2.11)

If we define $\phi:{\bf R}_{+}\to{\bf R}_{+}$ by
$\phi(s):=\int_{\Omega_{s}}e^{F}\omega^{n}$ then ((2.11)) and the definition
of $A_{s}$ imply that

$r\phi(s+r)\leq B_{0}\phi(s)^{1+\delta_{0}},\quad\forall r\in[0,1]\mbox{ and
}s\geq 0.$ (2.12)

$\phi$ is clearly nonincreasing and continuous, so a De Giorgi type iteration
argument shows that there is some $S_{\infty}$ such that $\phi(s)=0$ for any
$s\geq S_{\infty}$. This finishes the proof of the $L^{\infty}$ estimate of
$\varphi_{t}-V_{t}$, combining with a bound on $E_{t}$ by
$\|e^{F}\|_{L^{1}(\,{\rm log}\,L)^{1}}$ which follows from Jensen’s inequality
(c.f. Lemma 6 in [13]). The proof of Theorem 1 is complete.

Finally, we note the recent advances in the theory of envelopes in [11] and
[12], which can provide an approach to $L^{\infty}$ estimates for Monge-Ampère
equations on Hermitian manifolds.

##  3 Complex Hessian equations

We explain in this section how the proof of Theorem 1 can be modified to give
a similar result for a degenerate family of complex Hessian equations. With
the same notations as above, we consider the $\sigma_{k}$-equations

$(\hat{\omega}_{t}+i\partial\bar{\partial}\varphi_{t})^{k}\wedge\omega^{n-k}=c_{t}e^{F}\omega^{n},\quad\sup_{X}\varphi_{t}=0.$
(3.1)

Define the envelope corresponding to the $\Gamma_{k}$-cone

$\tilde{V}_{t,k}=\sup\\{v|~{}v\in SH_{k}(X,\omega,\hat{\omega}_{t})\cap
C^{2},\,v\leq 0\\}$

where $v\in SH_{k}(X,\omega,\hat{\omega}_{t})\cap C^{2}$ means that the
eigenvalue vector of the linear transformation
$\omega^{-1}\cdot(\hat{\omega}_{t}+i\partial\bar{\partial}v)$ lies in the
$\Gamma_{k}$-cone.

Let

$E_{t}(\varphi_{t})=\int_{X}(-\varphi_{t}+\tilde{V}_{t,k})e^{nF/k}\omega^{n}$

be the entropy associated to the equation (3.1) as in [13] and let
$\bar{E}_{t}$ be an upper bound of $E_{t}(\varphi_{t})$. Then the following
$L^{\infty}$-estimate holds for the solution $\varphi_{t}$ to (3.1).

###### Theorem 2

Let $\varphi_{t}$ be the solution to (3.1), then there exists a constant
depending on $\bar{E}_{t}$, $\|e^{\frac{n}{k}F}\|_{L^{1}(\,{\rm
log}\,L)^{p}}$, $\frac{c_{t}}{[\hat{\omega}_{t}]^{k}[\omega]^{n-k}}$, $p>n$
such that

$0\leq-\varphi_{t}+\tilde{V}_{t,k}\leq C.$

This theorem can be derived using a similar argument as in Section 2 with
suitable modifications for $\sigma_{k}$ equations, c.f. [13], so we omit the
details. The only novel ingredient is the smooth approximation of
$\tilde{V}_{t,k}$, as in Lemma 1. One can adapt the method in [2] to derive
this required approximation. For the convenience of the reader, we present a
sketch of the proof.

###### Lemma 2

Fix $t\in(0,1]$. There exists a sequence of smooth functions $u_{\beta}\in
SH_{k}(X,\omega,\hat{\omega}_{t})$ converging uniformly to $\tilde{V}_{t,k}$
as $\beta\to\infty$.

Proof. Let $u_{\beta}\in SH_{k}(X,\omega,\hat{\omega}_{t})$ be the solution to
the $\sigma_{k}$-equations

$(\hat{\omega}_{t}+i\partial\bar{\partial}u_{\beta})^{k}\wedge\omega^{n-k}=c_{t}e^{\beta
u_{\beta}}\omega^{n},$ (3.2)

which admits a unique smooth solution by [9]. We claim that there is a
constant $C_{t}>0$ such that

$\sup_{X}|u_{\beta}-\tilde{V}_{t,k}|\leq\frac{C_{t}\,{\rm
log}\,\beta}{\beta},$

from which the lemma follows.

By the maximum principle, at a maximum point of $u_{\beta}$,
$i\partial\bar{\partial}u_{\beta}\leq 0$, so $\beta u_{\beta}\leq\,{\rm
log}\,\frac{\hat{\omega}_{t}^{k}\wedge\omega^{n-k}}{c_{t}\omega^{n}}\leq
C_{t}$, that is $u_{\beta}-\frac{C_{t}}{\beta}\leq 0$. By the definition of
$\tilde{V}_{t,k}$, it follows that

$u_{\beta}-\frac{C_{t}}{\beta}\leq\tilde{V}_{t,k}.$ (3.3)

On the other hand, we fix a smooth $u\leq 0$ such that
$\hat{\omega}_{t}+i\partial\bar{\partial}u>0$. Such a $u$ exists because
$[\hat{\omega}_{t}]$ is a Kähler class by assumption. For any $v\in
SH_{k}(X,\omega,\hat{\omega}_{t})\cap C^{2}$ with $v\leq 0$, we consider the
barrier function

$\tilde{u}=\frac{1}{\beta}u+(1-\frac{1}{\beta})v-\frac{C_{t}^{\prime}\,{\rm
log}\,\beta}{\beta}$

where $C_{t}^{\prime}>0$ is a large constant to be determined. By direct
calculation, we have

$(\hat{\omega}_{t}+i\partial\bar{\partial}\tilde{u})^{k}\wedge\omega^{n-k}\geq\frac{1}{\beta^{n}}(\hat{\omega}_{t}+i\partial\bar{\partial}u)^{k}\wedge\omega^{n-k}\geq
e^{\beta\tilde{u}}\omega^{n}$

where the last inequality holds if we choose $C_{t}^{\prime}$ large enough so
that

$e^{-C_{t}^{\prime}\,{\rm
log}\,\beta}\leq\frac{1}{\beta^{k}}\min_{X}\frac{(\hat{\omega}_{t}+i\partial\bar{\partial}u)^{k}\wedge\omega^{n-k}}{\omega^{n}}.$

Therefore we get

$(\hat{\omega}_{t}+i\partial\bar{\partial}\tilde{u})^{k}\wedge\omega^{n-k}\geq
e^{\beta(\tilde{u}-u_{\beta})}(\hat{\omega}_{t}+i\partial\bar{\partial}u_{\beta})^{k}\wedge\omega^{n-k}.$

At the maximum point of $\tilde{u}-u_{\beta}$,
$(\hat{\omega}_{t}+i\partial\bar{\partial}\tilde{u})^{k}\wedge\omega^{n-k}\leq(\hat{\omega}_{t}+i\partial\bar{\partial}u_{\beta})^{k}\wedge\omega^{n-k}$.
This shows that $\tilde{u}-u_{\beta}\leq 0$ on $X$. Taking supremum over all
such $v$’s in $\tilde{u}$, it follows that

$(1-\frac{1}{\beta})\tilde{V}_{t,k}\leq u_{\beta}+\frac{C_{t}\,{\rm
log}\,\beta}{\beta}.$

The lemma follows from this and (3.3).

## References

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  * [19]

Department of Mathematics & Computer Science, Rutgers University, Newark, NJ
07102

<EMAIL_ADDRESS>
Department of Mathematics, Columbia University, New York, NY 10027

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Center for Mathematical Sciences and Applications, Harvard University,
Cambridge, MA 02138

<EMAIL_ADDRESS>


is the parallel transport along the path $\{ \exp(v,(1-t)\xi)\}_{0\leq t \leq 1}$, and
$$\pi_j: W^{k,p}_\delta (\dot F, \wedge^{1}T^{*}F)\to W^{k,p}_\delta (\dot F, \wedge^{0,1}_jT^{*}F)$$

is the projection with respect to $g_{(\dot F,j)}$.

Let $\widetilde f$ be the section of $\mathcal E|_{\mathcal{N}(K)} \to \mathcal N(K)$ with

$$\widetilde f (j',u)(s,t)=\rho_{s_{+}}(s)f(t)\otimes \pi_{j'} (ds-idt),$$

where $(s,t)$ are viewed as coordinates of $\dot F$ via $\phi_u$ as usual.   We show that $\widetilde f$ is a $C^1$-section of $\mathcal E|_{\mathcal N(K)}\to \mathcal N(K)$. In the next several paragraphs we calculate the derivative $d\widetilde f(j',u)$.

We first calculate $d\widetilde f (j',u)(0,\xi)$, where $\{u_{l_1}\}_{-\epsilon\leq l_1 \leq \epsilon}$ is a smooth path in $\mathcal W$ with $u_0=u$ and $\xi= \frac{d}{dl_1}|_{l_1=0}u_{l_1}$:

\begin{align} \label{eqn: part 1}
d\widetilde f (j',u)(0,\xi)=&\tfrac{d}{dl}|_{l_1=0} \widetilde f (j',u_{l_1})\\
\label{eqn: part 1A}=& \tfrac{d}{dl_1} \rho_{s_{+}(l_1)}(s_{l_1})|_{l_1=0} f(t) \otimes \pi_{j'}(ds-idt)\\
\label{eqn: part 1B} &+\rho_{s_+(0)}(s) \tfrac{d}{dl_1} f(t_{l_1})|_{l_1=0}\otimes \pi_{j'}(ds-idt)\\
\label{eqn: part 1C} &+\rho_{s_+(0)}(s)f(t)\otimes \pi_{j'}\tfrac{d}{dl_1}(ds_{l_1}-idt_{l_1})|_{l_1=0},
\end{align}

where $(s_{l_1}, t_{l_1})=\phi_{u_{l_1}}(\sigma,\tau)$.

Until the end of the Appendix, $W^{k,p}_\delta$ means $W^{k,p}_\delta$ with domain $\dot D$.  We claim that $d\widetilde f(j',u)(0,\xi)\in W^{k,p}_\delta$. Note that the weight $\delta$ can be ignored since $d\widetilde f(j',u)(0,\xi)$ has compact support. Since $\xi \in W^{k+1,p}_\delta$,

* $\tfrac{d}{dl_1}s_{l_1}|_{l_1=0},\tfrac{d}{dl_1}t_{l_1}|_{l_1=0},$ $\tfrac{d}{dl_1} f(t_{l_1})|_{l_1=0}\in W^{k+1,p}_\delta$; and

* $\tfrac{d}{dl_1}(ds_{l_1}-idt_{l_1})|_{l_1=0}\in W^{k,p}_\delta$.

Next we consider the term $\tfrac{d}{dl_1} \rho_{s_{+}(l_1)}(s_{l_1})|_{l_1=0}$.  If we write $\rho_{s_+}(s)$ as $\rho(s_+,s)$, then $\rho(\cdot, \cdot)$ is smooth in both variables. Observe that

$$\tfrac{d}{dl_1} \rho(s_{+}(l_1),s_{l_1})|_{l_1=0}=\tfrac{\partial \rho}{\partial s_+}(s_+(0),s)\tfrac{d}{dl_1} s_+(l_1)|_{l_1=0}+\tfrac{\partial \rho}{\partial s}(s_+(0),s)\tfrac{d}{dl_1} s_{l_1} |_{l_1=0}.$$

$s_+:\mathcal N(K)\to \mathbb R$ is a $C^1$-map.

The proof of this lemma will be postponed until the very end. By Lemma <ref>, $\tfrac{d}{dl_1} \rho(s_{+}(l_1),s_{l_1})|_{l_1=0}\in W^{k+1,p}_\delta$.  It follows that $d\widetilde f (j',u)(0,\xi)\in W^{k,p}_\delta$.

Next we calculate $d\widetilde f (j',u)(\frak j,0)$ with $\mathfrak j\in T_{j'}\widetilde{ \mathcal U}$. Let $\{j_{l_2}\}_{-\epsilon\leq l_2\leq \epsilon}$ be a smooth path in $\widetilde{\mathcal U}$ with $j_0=j'$ and $\frac{d}{dl}|_{l_2=0}j_{l_2}=\mathfrak j$. Then

\begin{align} \label{eqn: part 2}
d\widetilde f (j',u)(\frak j,0)=& \tfrac{d}{dl}|_{l_2=0} \widetilde f (j_{l_2},u)\\
\label{eqn: part 2A} =&\tfrac{d}{dl_2} \rho_{s_{+}(l_2)}(s)|_{l_2=0} f(t) \otimes \pi_{j'}(ds-idt)\\
\label{eqn: part 2B} &+\rho_{s_{+}(0)}(s) f(t) \otimes \tfrac{d}{dl_2} \pi_{j_{l_2}}(ds-idt)|_{l_2=0}.
\end{align}

Since $\pi_j$ depends smoothly on $j$, we have $\tfrac{d}{dl_2}\pi_{j_{l_2}}(ds-idt)|_{l_2=0}\in W^{k,p}_{\op{loc}}$. Again we write $\rho_{s_{+}}(s)$ as $\rho(s_+,s)$ and obtain

$$\tfrac{d}{dl_2} \rho(s_{+}(l_2),s)|_{l_2=0}=\tfrac{\partial \rho}{\partial s_+}(s_+(0), s)\tfrac{d}{dl_2} s_+(l_2)|_{l_2=0}.$$

By Lemma <ref>, $\tfrac{d}{dl_2} \rho(s_{+}(l_2),s)|_{l_2=0}\in W^{k+1,p}_\delta$. Hence $d\widetilde f (j',u)(\frak j,0)\in W^{k,p}_\delta$.

We now bound the $W^{k,p}_\delta$-norm of each term in Equations (<ref>) and (<ref>):

* (<ref>) is bounded above by $C\|\xi\|_{W^{k+1,p}}\cdot C(j')\cdot \|gu\|_{W^{k+1,p}}$;

* (<ref>) by $C\|\xi\|_{W^{k+1,p}}\cdot C(j')\cdot \|gu\|_{W^{k+1,p}}$;

* (<ref>) by $C \cdot C(j') \cdot \|\xi\|_{W^{k+1,p}}$;

* (<ref>) by $C\|\frak j\|\cdot C(j')\cdot \|gu\|_{W^{k+1,p}}$; and

* (<ref>) by $C\|\frak j\|\cdot \|gu\|_{W^{k+1,p}}$.

Here $C(j')$ is a positive continuous function of $j'$ and $g$ is a smooth cutoff function which equals one on $s_+(v)-c\leq s\leq s_+(v)+c$ for some $c>0$ and has compact support on $\dot D$.

$$\|d \widetilde f (j',u)(\frak j,\xi)\|_{W^{k,p}_\delta} \leq C(C(j') + C(j') \cdot \|gu\|_{W^{k+1,p}}+\|gu\|_{W^{k+1,p}}) (\| \frak j\| + \|\xi\|_{W^{k+1,p}_\delta}),$$

which implies that $\widetilde f$ is a differentiable section of $\mathcal E|_{\mathcal{N}(K)}\to \mathcal N(K)$. Almost the same calculation shows that $\widetilde f$ is in $C^1$.

Let $r_{j'}:\dot F\to \mathbb R$ be the injectivity radius map, i.e, $r_{j'}$ maps $x\in \dot F$ to the injectivity radius $\op{inj}(x)$ with respect to $g_{(\dot F,j)}$. The map $r_{j}$ is smooth on the ends of $\dot F$ and also smoothly depends on $j'$.  For any $(j',u)\in \mathcal D$, consider the map

\begin{equation}
R_{(j',u)}: [R,\infty) \to \mathbb R^+, \quad s\mapsto r_{j'}(\phi^{-1}_u(s,0)),
\end{equation}

where $R\gg 0$ and depends on $(j',u)$.
Note that $R_{(j',u)}$ is invertible near the positive end.

When $\chi(\dot F)<0$, we have

\begin{align} \label{s_+ explicit}
\end{align}

Differentiating the equation

\begin{equation}
\end{equation}

with respect to $l_1$ and rearranging, we obtain:

\begin{align}
ds_+(j',u)(0,\xi)=\tfrac{d}{dl_1}s_+(j',u_{l_1})|_{l_1=0}= -\frac{d r_{j'}(z)\circ (d\phi_u (z))^{-1} (\xi^{s,t}(z)) }{dr_{j'}(z)\circ (d\phi_u(z))^{-1} (\partial_s) },
\end{align}

where $\partial_s$ is a vector field on $\mathbb R^+ \times S^1$, $\xi^{s,t}(z)$ is the $(s,t)$-component of $\xi$ at $z$,  and $z=\phi_u^{-1}(s_+(j',u),0).$  Here ${d\over dl_1} \phi^{-1}_{u_{l_1}}(s_+(j',u),0))|_{l_1=0}=(d\phi_u (z))^{-1} (\xi^{s,t}(z))$.  Note that all terms are evaluated at the point $z$ and $z$ continuously depends on $(j',u)$, so $d r_{j'}(z)\circ (d\phi_u (z))^{-1}$ and $dr_{j'}(z)\circ (d\phi_u(z))^{-1} (\partial_s)$ depend continuously on $(j',u)$.  Hence $ds_+(j',u)(0,\cdot)$ is a bounded linear map from $W_\delta^{k+1,p}(\dot F, v^{*}T\widehat W)$ to $\mathbb R$ which depends continuously on $(j',u)$.

Next, differentiating the equation

\begin{equation}
\end{equation}

with respect to $l_2$, we obtain

\begin{align}
ds_+(j',u)(\frak j,0)=\tfrac{d}{dl_2}s_+(j_{l_2},u)|_{l_2=0}= -\frac{{d\over dl_2} r_{j_{l_2}}(z)|_{l_2=0}}{dr_{j'}(z)\circ(d\phi_u(z))^{-1}(\bdry_s)}.
\end{align}

Since ${d\over dl_2} r_{j_{l_2}}(z)|_{l_2=0}$ depends continuously on $(j',u)$,  $ds_+(j',u)(\cdot,0)$ is a bounded linear map from $T_{j'}\mathcal V$ to $\mathbb R$ which depends continuously on $(j',u)$. This proves that $s_+$ is a $C^1$-map when $\chi(\dot F)<0$.

When $\chi(\dot F)=0$ or $1$, $s_+$ is still given by Equation (<ref>), but the metric $g_{(\ddot F,j')}$ is now a function of $u$. More precisely, for any $u=(s,t,\eta)$, define

$$S_u:[R,\infty)\to \mathbb R^+,\quad s\mapsto \int_s^{+\infty}\hspace{-0.3cm}\int_0^{\mathcal A_\alpha(\gamma)} |\eta(s',t')|^2 dt'ds'=\varepsilon'$$

with $R\gg 0$ and then

Recall that the set $\mathbf{q}=(q_1,\dots,q_{2m(\gamma_{+})})$ of additional punctures that we put on $\dot F$ satisfies

$$q_k=\phi_u^{-1}\big(s_+', \tfrac{k}{2m(\gamma_{+})}\mathcal A_\alpha(\gamma_{+})\big)\in \dot F.$$

Then $g_{(\ddot F,j')}$ is smooth function of $\mathbf q$.  An argument similar to that of the $\chi(\dot F)<0$ case shows that $s_+$ is a $C^1$-map.

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