capbtabboxtable[][]

11institutetext: Vrije Universiteit Amsterdam
11email<EMAIL_ADDRESS>
# Text classification of column headers with a controlled vocabulary:
leveraging LLMs for metadata enrichment

Margherita Martorana 11 0000-0001-8004-0464 Tobias Kuhn 11 0000-0002-1267-0234
Lise Stork 11 0000-0002-2146-4803 Jacco van Ossenbruggen 11
0000-0002-7748-4715

###### Abstract

Traditional dataset retrieval systems index on metadata information rather
than on the data values. Thus relying primarily on manual annotations and
high-quality metadata, processes known to be labour-intensive and challenging
to automate. We propose a method to support metadata enrichment with topic
annotations of column headers using three Large Language Models (LLMs):
ChatGPT-3.5, GoogleBard and GoogleGemini. We investigate the LLMs ability to
classify column headers based on domain-specific topics from a controlled
vocabulary. We evaluate our approach by assessing the internal consistency of
the LLMs, the inter-machine alignment, and the human-machine agreement for the
topic classification task. Additionally, we investigate the impact of
contextual information (i.e. dataset description) on the classification
outcomes. Our results suggest that ChatGPT and GoogleGemini outperform
GoogleBard for internal consistency as well as LLM-human-alignment.
Interestingly, we found that context had no impact on the LLMs performances.
This work proposes a novel approach that leverages LLMs for text
classification using a controlled topic vocabulary, which has the potential to
facilitate automated metadata enrichment, thereby enhancing dataset retrieval
and the Findability, Accessibility, Interoperability and Reusability (FAIR) of
research data on the Web.

###### Keywords:

Text classification Large Language Models Automatic metadata enrichment FAIR
Guiding Principles

##  1 Introduction

Traditional dataset retrieval systems index on metadata information rather
than on the underlying data values. Despite the pivotal role of high-quality
metadata for data retrieval, many datasets still lack informative metrics and
annotations to facilitate their discovery [16]. The enrichment of metadata
with high-quality information and annotations is a labour intensive and
challenging process to automate, and it often relies on knowledge from domain
experts. Enriching metadata with column-level information poses particular
difficulty due to sparse contextual information and reliance on domain-
specific code-books, often not available in digital structured format.
Nevertheless, in recent years the FAIR Guiding Principles [17] have emphasized
the importance of high quality metadata to facilitate the Findability,
Accessibility, Interoperability and Reusability (FAIR) of data on the Web. In
fact, several studies have found that by applying the FAIR Principles we can,
not only improve data management and stewardship [1, 9], but also facilitate
the transparency, reproducibility, discovery and reuse of data [18] and
resource citation [7].

The rise of advanced Large Language Models (LLMs) has presented several
opportunities and challenges in automating data annotation and metadata
creation [15]. Studies have shown some preliminary results regarding the
advantages - and disadvantages - of various LLMs, and overall variations in
performances [14, 12]. However, it is still not clear how different LLMs
perform in text classification tasks, particularly when dealing with short
texts such as column headers.

In this work we address the challenges of automated metadata enrichment, by
investigating how we can leverage LLMs to perform text classification tasks of
column headers, specifically by using a controlled vocabulary of topics. To
guide our investigation, we formulate the following research questions:

  1. 1.

How can we measure the LLMs consistency in the classification task?

  2. 2.

How can we measure classification differences between LLMs and humans?

  3. 3.

Do hierarchical and contextual information effect the classifications?

Our work contributes to the current knowledge on the applications of LLMs by
assessing the performance of three LLMs in the text classification task of
column headers with a controlled vocabulary, and comparing it with human-made
classifications. To the best of our knowledge, this is the first work that
investigate performances of various LLMs in this specific task.

##  2 Related Work

###  2.1 Semantic Metadata Enrichment

Semantic metadata enrichment is the processes by which metadata information is
enhanced with additional meanings and contexts, to make it more human- and
machine- readable. This process usually involves adding semantic annotations,
derived from ontologies, thesauri, or domain-specific controlled vocabularies,
that provide insights into the content and relationships within the data.
Semantic enrichment, unlike simpler annotations, offers richer context and
structure. Previous work shows that semantic metadata enrichment and the use
of controlled vocabularies facilitates data FAIRification on the web, and also
enables interdisciplinary collaborations among research groups and
institutions [16]. Further, the authors of this paper have previously explored
how the incorporation of column-level metadata can enhance the Findability,
Accessibility, Interoperability and Reusability of research data [8]. This is
particularly important for discovering restricted access data, which often
lacks informative details due to its sensitive nature and limited
accessibility. Semantic column-level annotations can enhance the discovery of
such data by providing additional context without compromising privacy-
preserving guidelines.

###  2.2 Text Classification and LLMs

Large Language Models (LLMs) have revolutionised the field of Natural Language
Processing (NLP), with advances in a variety of applications such as content
creation, text classification and question answering. LLMs are trained on very
large amount of text data (and more recently multi-modal data), allowing the
models not only to recognise patterns and relationships between words and
concepts, but also to handle a wide range of tasks, even those that the models
have not been specifically trained for and without any explicit supervision
[11, 6]. LLMs are, in general, more accurate than traditional NLP methods, but
they are often considered ‘black boxes’ as the mechanisms behind an LLM
decision making are challenging to understand, which makes debugging and bias
detection more difficult to investigate. Further, a recent study has tested
the performance of ChatGPT across a variety of NLP tasks, and it has been
found that the more complex and pragmatic the task (e.g. emotion recognition),
the more the LLM has a loss in performance [5]. Further, another work found
significant biases and inconsistent performance across ChatGPT and
GoogleGemini in sentiment analysis detection [2]. Nevertheless, there are
still open possibilities and challenges involving the prompting of LLMs for
automated data - and metadata - annotation [15]. Studies have shown evidences
that ChatGPT have outperformed crowd-workers in the text classification of
tweets [4], and other conventional baselines [3]. Further, another work
suggests that the text classification tasks could be improved with the
additional of semantic technologies and knowledge graphs [13]. Based on this
findings, our research investigates the performance of ChatGPT, GoogleBard and
GoogleGemini in the text classification task of column headers with a
controlled vocabulary of topics. Further, we compare classifications outcomes
between the LLMs and human participants, as well as the effect of topics
hierarchy and contextual information. In the sections that follow, we describe
the experimental settings and evaluation metrics employed in our
investigation.

##  3 Experimental Design and Evaluation

In this section, we describe the data collection, experimental design for both
human and machine topic classification tasks, and evaluation methods. Our
experimental design aims to assess the consistency in topic classifications of
column headers of three LLMs (ChatGPT, GoogleBard and GoogleGemini), comparing
it with human-made classifications and investigate the impact of contextual
information (e.g. dataset description). A key aspect of our methodology is
providing the same prompt for both human and LLM task, allowing evaluation
from a neutral standpoint. Further, our analysis also considers differences in
the classifications based on the hierarchical structure of topic controlled
vocabulary, which distinguish between ‘general’ and ‘specific’ topics (e.g.
‘Education’ versus ‘Higher and Further Education’).

###  3.1 Data Collection

This work explores how can LLMs be leveraged for text classification of column
headers using a Controlled Vocabulary (CV). Our analysis employs the Topic
Classification CV provided by the Consortium of European Social Science Data
(CESSDA)111https://www.cessda.eu. The input column headers were sourced from
the CBS Open Data Portal222https://opendata.cbs.nl/statline/portal. We opted
for a random dataset selection approach while ensuring diversity across
various topics. A total of 10 datasets were selected, containing between 4 to
69 columns. All input data, CV, code and results discussed in this work are
available on GitHub 333https://github.com/ritamargherita/LLMs-topic-
classification/.

###  3.2 LLMs Topic Classification

The prompt used for the LLMs task initiation includes the task specification,
input data (i.e. column headers), the CESSDA CV and some formatting
constraints. The same prompt was used to query all LLMs, and the same task was
executed 10 times for each LLM. Sessions were refreshed to prevent result
propagation. Additionally, to assess the impact of contextual information, we
repeated the process with the inclusion of `*Dataset Description:..` in the
prompt inputs. Here is important to note that the task with context (i.e.
dataset description) could not be performed with GoogleBard due to size
limitation in the allowed prompt. The analysis regarding the effect of
contextual information is, therefore, only performed with ChatGPT and
GoogleGemini. A summary of the prompt is provided below.

We then labelled the classifications of each column header as follows:
‘Specific’ for the CESSDA sub-concepts, ‘General’ for top-concepts, ‘Other’
for the specific classification of the topic ‘other’ in CESSDA, ‘Unassigned’
when the classification was not executed, and ‘Wrongly Assigned’ when the
topic classified was not included in CESSDA. As evaluation, we performed a
Tukey’s Honest Significant Difference (HSD) test to assess significant
differences between labels.

Task: Column Header Classification with Controlled Vocabulary

You are provided with two inputs, below: 1) the column headers of a dataset
(in a list format), and 2) a controlled vocabulary of topics (in a CSV
format). Your goal is to classify each column header with a relevant topic.
The controlled vocabulary has two columns: ’Topic Label’ and ’Topic
Description’. For each column header, assign a topic from the controlled
vocabulary based on semantic relevance and the definition provided for each
topic. The result should be structured in JSON format, where each column
header is paired with its assigned topic’s label.

**Constraints:
Use only topics provided in the controlled vocabulary, do not add any topics
that are not included.
Do not change the text of the column headers or topic’s label.
Only return the output in a JSON format, and no additional text.

**Inputs:

*Column Headers (List):
$[$h(1), h(2), ....., h(n)$]$

*Controlled Vocabulary (CSV Format):
Topic Label,Topic Description

Further, we measured the Internal Consistency of each LLM, implementing the
Needleman-Wunsch (NW) algorithm [10]. The NW algorithm is conventionally used
for aligning genomic sequences but was adapted here to assess the uniformity
of topic classification outcomes across the 10 repeated task executions for
each LLM. Pairs of classifications for each LLM and set of column headers were
aligned and scored using the algorithm. We analyzed these scores using a
Multi-Way ANOVA and post-hoc Tukey’s HSD tests to investigate differences in
outcomes within each LLM.

Similarly, we measured the similarities in classifications across LLMs -
Inter-LLMs alignment \- employing again the NW algorithm, to execute and score
pairs of classifications across pairs of LLMs. The alignment scores for each
LLM-pair were summed and averaged to obtain the Inter-LLM Alignment score.
ANOVA and Tukey’s HSD tests were performed to investigate differences in
alignment scores for each LLM pairs.

###  3.3 Human Topic Classification

The human topic classifications were performed by three participants: M.M and
T.K, both authors of this paper, and a social scientist specializing in CBS
data. Each participant performed the classification task twice: first without
contextual information, then with the dataset descriptions included. Given the
minimal difference (less than 5%) between the two classifications, we kept
only the second classification (with context) for simplicity.

To measure the agreement between the humans and LLM classifications - Human-
Computer Agreement \- we compute the joint probability distribution, which
measures the probability of two events happening at the same time. In our
case, the events are: 1) each topic classification of column header X by one
LLM, and 2) all human-made classifications for column header X. Further, we
can sum these joint probabilities to measure the probability of the
classifications to be in agreements between each LLM and the human
participants. Below, we report the equations 12 and their variables
descriptions in table 1.

$\footnotesize P(x_{H}=t,c_{m}=t|c_{H}\in C_{H},c_{m}\in
C_{m})=P(c_{H}=t|c_{H}\in C_{H})\cdot P(c_{m}=t|c_{m}\in C_{m})$ (1)
$\footnotesize P(c_{H}=c_{m}|c_{H}\in C_{H}|c_{m}\in C_{m})=\sum_{t\in
T}P(c_{H}=t|c_{H}\in C_{H})\cdot P(c_{m}=t|c_{m}\in C_{m})$ (2)

Variable |  Definition |  Variable |  Definition
---|---|---|---
Topic |  $t\in\\{1,..,95\\}=T$ |  Classification |  $c\in C$
Human |  $h\in\\{1,2,3\\}=H$ |  Humans classification |  $c_{H}\in T$
Machine |  $m\in\\{1,2,3\\}=M$ |  Machine classification |  $c_{m}\in T$
Table 1: Variable descriptions of the joint probability scoring to measure the
agreement between LLMs and humans topic classifications.

##  4 Results

In this study, we investigated the performance of ChatGPT, GoogleBard and
GoogleGemini in the task of topic classification of column headers. We also
compared human- and LLM-topic classifications, and explore the effect of
hierarchical and contextual information in the classification outcomes. Below
we report our findings.

####  4.0.1 LLMs Topic Classification Summary Statistics

We have analysed the LLMs topic classifications in settings with and without
context with a pairwise Tukey’s HSD test. We focused on the nature of
classified topics: 1) Specific, 2) General, 3) ’Other’, 4) Unassigned and 5)
Wrongly Assigned topics, as mentioned in Section 3.2. For the task without
context, Tukey’s HSD test revealed significant differences between the
ChatGPT-GoogleBard pair in all topics classifications except for ‘Specific’
topics ($p(2)=0.02,p(3)=0.01,p(4)=0.03,p(5)=0.03$). The ChatGPT-GoogleGemini
pair, instead, showed significant differences only in the ‘Other’ topic
classifications ($p(3)=0.01$). Lastly, the GoogleGemini-GoogleBard pair had
significant differences for ‘Unassigned’ ($p(4)=0.02$) and ‘Wrongly assigned’
($p(5)=0.01$) topic classifications. For the task with context, we found no
significant differences in the topics classification between the ChatGPT-
GoogleGemini pair, which is an interesting results as we anticipated more
‘Specific’ topics once added contextual information to the prompt.

We report a summary of the classifications in Figure 2. Overall, GoogleBard
showed less instances of the classification of ‘Specific’ and ‘General’
topics. It also assigned the ‘Other’ topic more frequently, particularly
compared to ChatGPT. Moreover, instances of ‘Wrongly assigned’ topics were
more prevalent compared to the two other LLMs. In summary, our findings and
the Tukey’s results suggest that both ChatGPT and GoogleGemini outperformed
GoogleBard in the classification task. They also show that GoogleBard is more
likely to assign the topic ‘Other’, indicating a tendency to abstain from
making a definite classification. Moreover, the results suggest that
contextual information had no significant impact on the types of topics
classified (e.g. specific/general topics).

####  4.0.2 LLM Internal Consistency

With this measure we assessed the LLMs internal consistency in the
classification task of each dataset across the 10 task executions. Using the
NW algorithm we measured the internal consistency, and we evaluated it using
Multi-Way ANOVA and Tukey’s HSD tests. For the task without context, the
overall consistency scores for each LLM were: $ChatGPT=0.52$,
$GoogleBard=0.11$ and $GoogleGemini=0.81$, where 1 is absolute consistency for
all several executions for each dataset. The Multi-Way ANOVA showed a
significant effect on consistency scores based on the dataset (p =
$1.87^{-7}$). Tukey’s test confirmed significant differences in consistency
between: the ChatGPT-GoogleBard pair($p=0.007$), and GoogleGemini-GoogleBard
pair ($p=0.0001$). No difference was found between the ChatGPT-GoogleGemini
pair ($p=0.3$), suggesting similar consistency scores across all task
executions and datasets. For the task with context, we found the overall
internal consistency scores of $ChatGPT=0.46$ and $GoogleGemini=0.51$. ANOVA
and Tukey’s test found no significant differences in consistency scores
between these LLMs, supporting the findings above.

These results indicate that, overall, GoogleGemini appears the LLM that is
more consistent in the classification of topics across the repeated task
executions. GoogleBard, instead, shows much lower scores for the internal
consistency measure. The ANOVA test also suggest that the dataset to which the
column headers are being classified can have an impact in the consistency
score. This result needs further investigation, to evaluate whether there is a
correlation between different aspects of the datasets (e.g. number of columns,
domain, expressivity of columns) and the internal consistency score. Further,
it appears that GoogleGemini and ChatGPT had similar internal consistency
scores across all datasets, and no significant differences were found between
these two LLMs even when the task was performed with context.

####  4.0.3 Inter-LLMs Agreement

To measure topic classifications agreement between LLMs, we calculated the
Inter-LLMs Alignment score using the NW algorithm and performed ANOVA and
Tukey’s tests. We computed the alignment scores for each LLM pair in both
tasks with and without context. In all cases, alignment scores were
approximately 0, suggesting different classified topics for each LLM. For the
task without context, ANOVA revealed that the datasets which the column
headers belonged to had significant effects for both the ChatGPT-GoogleBard
and GoogleGemini-GoogleBard pairs ($p=1.05^{-8}$ and $p=2.75^{-9}$
respectively). No significant effects by the dataset were found for the
GoogleGemini-ChatGPT pair. Further, for the task with context, ANOVA found no
significant effect of the dataset between the ChatGPT-GoogleGemini pair,
supporting previous findings. Similarly to the results from the Internal
Consistency scoring, the effect that the dataset might have on the LLM
performance needs further investigation. Nevertheless, no significant effect
was found for the ChatGPT-GoogleGemini pair for both the task with and without
context, indicating that these two LLMs might have comparable underlying
processes.

####  4.0.4 Human-Computer Agreement

We calculated the Human-Computer Agreement (HCA) scores based on the joint
probability metrics introduced in 3.3. The scores are reported in table 2,
where an alignment score of 1 indicates agreement between LLM classifications
and at least one human’s classification. The table reports the scores for the
tasks with and without context, as well as based on the hierarchy of topics in
CESSDA. In the table, the ’specific’ setting compares precisely classified
topics, while the ’general’ setting involves mapping topics to their general
category. This allows us to assess whether LLMs and human participants
classify column headers with the same general topics, even if specific topics
differ. We find that ChatGPT classifications aligns most closely with the
human-made classifications in the ‘general’ setting for both tasks with and
without context, and it also shows a slightly higher HCA for the task with
context compared to the one without. Interestingly, GoogleGemini shows lower
HCAs for the task with context compared to the one without. While we lack
sufficient data for statistical significance between HCAs for tasks with and
without context, these initial findings support, again, that contextual
information may not significantly impact the classification task.

Figure 1: Summary of the topic classification task by the three LLMs. We show
the distribution of the topics classified based on 5 groups: ‘Specific’
topics, ‘General’ topics, the ‘Other’ topic, ‘Unassigned’ topics and ‘Wrong’
topics, i.e. outside of the controlled vocabulary.

| no-context | context
---|---|---
| specific | general | specific | general
ChatGPT | 0.29 | 0.5 | 0.33 | 0.53
GoogleGemini | 0.28 | 0.46 | 0.15 | 0.37
GoogleBard | 0.24 | 0.31 | X | X

Figure 2: The table shows for each LLM and settings (context and no-context)
the alignment between machine and human classifications, where 1 is complete
alignment and 0 is no alignment at all.

##  5 Conclusion and Future Work

In this work we propose a novel approach that leverages LLMs for text
classification of column headers with a topic controlled vocabulary. Our
experimental design focuses on exploring the impact of contextual and
hierarchical information in the topic classification task. We have evaluated
the performance of the three LLMs (ChatGPT, GoogleBard and GoogleGemini)
through various metrics: 1) we investigated the nature of classified topics,
including the hierarchical structure of the controlled vocabulary; 2) we
measured the internal consistency of each LLM in the classification task; 3)
we assessed the alignment of classified topics across LLMs; and 4) we measured
the classifications agreement between the LLMs and human participants. Our
findings suggest that, overall, ChatGPT and GoogleGemini outperform.
Interestingly, contextual information appears to have no impact on the
consistency and agreement of classification tasks for the LLMs. We also didn’t
find any strong evidences suggesting that hierarchical information effect the
classification task. Future work could involve utilizing larger input datasets
to better support statistical analysis, and explore how LLMs capture
relationships between columns by classifying similar topics to related
columns.

####  5.0.1 Acknowledgements

We would like to thank Emma Beauxis-Aussalet for the help with the
mathematical notation of the Human-Machine Agreement scoring function.
Further, we acknowledge that ChatGPT was utilized to generate and debug part
of the python and latex code utilised in this work. This work is funded by the
Netherlands Organisation of Scientific Research (NWO), ODISSEI Roadmap
project: 184.035.014.

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††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Exploring the chiral and deconfinement phase transitions in a self-
consistent PNJL model

Xiaozhu Yu  Department of Physics, Jiangsu University, Zhenjiang 212013, P.R.
China  Liangkai Wu  Department of Physics, Jiangsu University, Zhenjiang
212013, P.R. China  Lang Yu  College of Physics, JiLin University, Changchun
130012, P.R. China  Xinyang Wang  Department of Physics, Jiangsu University,
Zhenjiang 212013, P.R. China  School of Fundamental Physics and Mathematical
Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, P.R.
China

###### Abstract

In this work, we study the chiral and deconfinement phase transitions in a
two-flavor Polyakov loop extended Nambu–Jona-Lasinio (PNJL) model. And note
that the self-consistent mean field approximation is employed by introducing
an arbitrary parameter $\alpha$ to measure the weights of the Fierz-
transformed interaction channels. By making use of this model, we
systematically investigate the chiral and deconfinement phase transition lines
(as well as the chiral ones in the NJL model for comparison) under different
values of $\alpha$. It is found that, the increasing of $\alpha$ helps to
enhance the chiral (pseudo)critical temperature at fixed chemical potential,
and also to enhance the chiral (pseudo)critical chemical potential at fixed
temperature. And the critical end point (CEP) vanishes when $\alpha$ becomes
large enough. Besides, we find that the incorporation of Polyakov loop
increases $T_{CEP}$ but does not change $\mu_{CEP}$ for small values of
$\alpha$.

##  I Introduction

Exploring the phase structure of Quantum Chromodynamics (QCD) matter is one of
the most important topics in the high energy nuclear physics. With the
increasing temperature or baryon density, the hadronic phase will transfer to
the deconfined phase with the approximate chiral symmetry restoration, which
is so-called quark-gluon plasma (QGP). It is well know now that the QGP should
exist in the early universe in nature. In recent years, the heavy-ion
collision experiments at the Relativistic Heavy Ion Collider (RHIC) in
Brookhaven and the Large Hadron Collider (LHC) at CERN have produced the QGP
at very high energies. On the other hand, the astrophysics observations could
help us to have a better understanding QCD matter at the large chemical
potential.

From the theoretical point of view, because the perturbation QCD method
becomes unavailable in the low energy regime where the strong coupling
constant is not small, we have to rely on the non-perturbation methods, such
as lattice QCD. As we know, lattice QCD calculations show that at small baryon
chemical potential the chiral phase transition as well as deconfinement phase
transition are crossover at finite temperature. Unfortunately, the study of
QCD using lattice simulations at large baryon chemical potential is not
possible at present due to the famous fermion sign problem. Therefore, the
effective theories and models of QCD in the low energy regime are needed, and
the QCD phase transitions have been explored by a variety of different
theories and models. For instance, it is generally believed that at high
baryon density the chiral phase transition is of the first order, and there
exists a QCD critical end point (CEP) in the temperature and baryon chemical
potential plane. The Nambu-Jona-Lasinio (NJL) model Nambu:1961tp ;
Nambu:1961fr  is one of the most useful tools to study the QCD matter. The NJL
model incorporates the chiral symmetry and its spontaneous breaking, the
gluonic degrees of freedom are replaced by a local four-point fermion
interactions. And this model has a wide range of applications for low-energy
QCD research, e.g.,the weak processes in hadrons, chiral phase transition,
spin structures, the form factors of the nucleons, and etc. (for reviews see
Refs. Klevansky:1992qe ; Hatsuda:1994pi ; Buballa:2003qv )

But since there is no gluonic degrees of freedom in the NJL model, it can not
describe the color confinement of QCD. Fortunately, the decofinement phase
transition could be investigated effectively in a slightly simple way that we
can introduce the contribution of the Polyakov loop into the NJL model
Fukushima:2003fw . This is so called the Polyakov loop extended Nambu–Jona-
Lasinio (PNJL) model. The essential feature of the PNJL model is that we could
study both chiral phase transition and deconfinement phase transition at the
same time.

As discussed in Ref. Klevansky:1992qe , the Fierz transformation of the NJL
interaction Lagrangian density $\mathcal{L}_{F}$ is actually of equal
importance with the original Lagrangian density $\mathcal{L}$. Therefore, by
inserting a new arbitrary parameter $\alpha$, a redefined Lagrangian density
could be constructed by linearly combining these two types of Lagrangian
densities i.e., $\mathcal{L}_{R}=(1-\alpha)\mathcal{L}+\alpha\mathcal{L}_{F}$.
Different from the conventional setting of $\alpha=0$ in the original NJL
model and the setting of $\alpha=1/2$ suggested in Ref. Klevansky:1992qe , the
authors of Ref. Wang:2019uwl  pointed out that there is no strict physical
requirement that the value of $\alpha$ must be 0 or $1/2$. The weighting
parameter $\alpha$ should determined by the heavy ion collision experiments or
astronomy observations. Theoretically, the QCD phase diagram with different
$\alpha$ have been studied by applying the self-consistent mean field
approximation in the two-flavor and three-flavor NJL model Wang:2019uwl ;
Yu:2020dnj . Therefore, in order to have a better understanding of the QCD
phase transitions, including both chiral phase transition and deconfinement
phase transition, the method of the self-consistent mean field approximation
is extended to the PNJL model. In this paper we will show how the weighting
parameter $\alpha$ affects the chiral and deconfinement phase transitions of
QCD matter.

This paper is organized as follows: in the next section, we apply the self-
consistent mean field method on the 2-flavor PNJL model. The results of the
phase transitions are discussing in section III. The summary and discussion
are given at the end of this paper. And we assume the masses of u and d quarks
are the same in this work.

##  II Self-consistent mean field approximation in the PNJL model and the gap
equations

The combination of the original Lagrangian density $\mathcal{L}$ and its Fierz
transformation $\mathcal{L}_{F}$ in the 2-flavor NJL model is given by
Wang:2019uwl :

$\displaystyle\mathcal{L}_{R}=(1-\alpha)\mathcal{L}+\alpha\mathcal{L}_{F},$
(1)

where the quark masses $m=diag(m_{u},m_{d})$,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\partial-m)\psi+G\left[(\bar{\psi}\psi)^{2}+\left(\bar{\psi}i\gamma^{5}\cdot\tau\psi\right)^{2}\right],$
(2)

and

$\displaystyle\mathcal{L}_{F}$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\partial-m)\psi+\frac{G}{8N_{c}}\left[2(\bar{\psi}\psi)^{2}+2\left(\bar{\psi}i\gamma^{5}\tau\psi\right)^{2}-2(\psi\tau\psi)^{2}\right.$
(3)
$\displaystyle-2\left(\bar{\psi}i\gamma^{5}\psi\right)^{2}-4\left(\bar{\psi}\gamma^{\mu}\psi\right)^{2}-4\left(\bar{\psi}i\gamma^{\mu}\gamma^{5}\psi\right)^{2}+\left.\left(\bar{\psi}\sigma^{2m}\psi\right)^{2}-\left(\bar{\psi}\sigma^{\mu\nu}\tau\psi\right)^{2}\right].$

In order to study the deconfinement phase transition effectively, we add the
Polyakov-loop effective potential $\mathcal{U}(\Phi,\bar{\Phi},T)$ in the
following ansatz

$\displaystyle\frac{\mathcal{U}(\Phi,\bar{\Phi},T)}{T^{4}}$ $\displaystyle=$
$\displaystyle-\frac{a(T)}{2}\bar{\Phi}\Phi+b(T)\ln\left[1-6\bar{\Phi}\Phi+4\left(\bar{\Phi}^{3}+\Phi^{3}\right)-3(\bar{\Phi}\Phi)^{2}\right],$
(4)

where
$a(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}\text{
and }b(T)=b_{3}\left(\frac{T_{0}}{T}\right)^{3}$. In this manner, we obtain
the Lagrangian density of the self consistent PNJL model

$\displaystyle\mathcal{L}_{PNJL}$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\partial-m)\psi+(1-\alpha)G\left[(\bar{\psi}\psi)^{2}+\left(\bar{\psi}i\gamma^{5}\cdot\tau\psi\right)^{2}\right]+\alpha\frac{G}{8N_{c}}\left[2(\bar{\psi}\psi)^{2}+2\left(\bar{\psi}i\gamma^{5}\tau\psi\right)^{2}-2(\psi\tau\psi)^{2}\right.$
(5) $\displaystyle-$
$\displaystyle\left.2\left(\bar{\psi}i\gamma^{5}\psi\right)^{2}-4\left(\bar{\psi}\gamma^{\mu}\psi\right)^{2}-4\left(\bar{\psi}i\gamma^{\mu}\gamma^{5}\psi\right)^{2}+\left(\bar{\psi}\sigma^{2m}\psi\right)^{2}-\left(\bar{\psi}\sigma^{\mu\nu}\tau\psi\right)^{2}\right]-\mathcal{U}(\Phi,\bar{\Phi},T).$

By following the same procedure in the mean field approximation Kapusta:2006pm
, we could write down the corresponding thermodynamic potential
$\Omega_{PNJL}$,

$\displaystyle\Omega_{PNJL}$ $\displaystyle=$
$\displaystyle\mathcal{U}(\Phi,\bar{\Phi},T)+\frac{(M-m)^{2}}{4G(1-\alpha+\frac{\alpha}{4N_{c}})}-\frac{N_{c}(\mu-\tilde{\mu})^{2}}{2\alpha
G}-2N_{c}N_{f}\int_{0}^{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\left[E_{p}\right]$
(6) $\displaystyle-2N_{f}T\int\frac{d^{3}p}{(2\pi)^{3}}\left[\ln\left(1+3\Phi
e^{-\beta\left(E_{p}-\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}-\tilde{\mu}\right)}\right)\right]$
$\displaystyle-2N_{f}T\int\frac{d^{3}p}{(2\pi)^{3}}\left[\ln\left(1+3\bar{\Phi}e^{-\beta\left(E_{p}+\tilde{\mu}\right)}+3\Phi
e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}+\tilde{\mu}\right)}\right)\right].$

Here, we have $N_{c}=3$ and $N_{f}=2$. The dispersion relation of the quark is
defined by $E_{p}=\sqrt{\mathbf{p}^{2}+M^{2}}$, where $M$ is the constituent
quark mass.

And then, the gap equations are determined by the stationary point of the
themodynamic potential, i.e.,

$\displaystyle\frac{\partial\Omega_{P}}{\partial
M}=\frac{\partial\Omega_{P}}{\partial\Phi}=\frac{\partial\Omega_{P}}{\partial\bar{\Phi}}=\frac{\partial\Omega_{P}}{\partial\tilde{\mu}}=0.$
(7)

The above oequations can be written explicitly in the following,

$\displaystyle M$ $\displaystyle=$ $\displaystyle
m+4G(1-\alpha+\frac{\alpha}{4N_{c}})N_{c}N_{f}\int_{0}^{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\frac{M}{E_{p}}$
(8a)
$\displaystyle-4G(1-\alpha+\frac{\alpha}{4N_{c}})N_{f}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{M}{E_{p}}\frac{3\Phi
e^{-\beta\left(E_{p}-\tilde{\mu}\right)}+6\bar{\Phi}e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}+3e^{-3\beta\left(E_{p}-\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}-\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}-\tilde{\mu}\right)}\right)}$
$\displaystyle-4G(1-\alpha+\frac{\alpha}{4N_{c}})N_{f}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{M}{E_{p}}\frac{3\Phi
e^{-\beta\left(E_{p}+\tilde{\mu}\right)}+6\bar{\Phi}e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}+3e^{-3\beta\left(E_{p}+\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}+\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}+\tilde{\mu}\right)}\right)},$
$\displaystyle\tilde{\mu}$ $\displaystyle=$ $\displaystyle\mu-\frac{2\alpha
G}{N_{c}}N_{f}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{3\Phi
e^{-\beta\left(E_{p}-\tilde{\mu}\right)}+6\bar{\Phi}e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}+3e^{-3\beta\left(E_{p}-\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}-\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}-\tilde{\mu}\right)}\right)}$
(8b) $\displaystyle+\frac{2\alpha
G}{N_{c}}N_{f}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{3\Phi
e^{-\beta\left(E_{p}+\tilde{\mu}\right)}+6\bar{\Phi}e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}+3e^{-3\beta\left(E_{p}+\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}+\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}+\tilde{\mu}\right)}\right)},$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
T^{4}\left[-\frac{a(T)}{2}\bar{\Phi}+b(T)\frac{-6\bar{\Phi}+12\Phi^{2}-6\bar{\Phi}^{2}\Phi}{1-6\bar{\Phi}\Phi+4\left(\bar{\Phi}^{3}+\Phi^{3}\right)-3(\bar{\Phi}\Phi)^{2}}\right]$
(8c)
$\displaystyle-2N_{f}T\int\frac{d^{3}p}{(2\pi)^{3}}\frac{3e^{-\beta\left(E_{p}-\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}-\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}-\tilde{\mu}\right)}\right)}$
$\displaystyle-2N_{f}T\int\frac{d^{3}p}{(2\pi)^{3}}\frac{3e^{-\beta\left(E_{p}+\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}+\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}+\tilde{\mu}\right)}\right)}$
and $\displaystyle 0$ $\displaystyle=$ $\displaystyle
T^{4}\left[-\frac{a(T)}{2}\Phi+b(T)\frac{-6\Phi+12\bar{\Phi}^{2}-6\bar{\Phi}\Phi^{2}}{1-6\bar{\Phi}\Phi+4\left(\bar{\Phi}^{3}+\Phi^{3}\right)-3(\bar{\Phi}\Phi)^{2}}\right]$
(8d)
$\displaystyle-2N_{f}T\int\frac{d^{3}p}{(2\pi)^{3}}\frac{3e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}-\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}-\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}-\tilde{\mu}\right)}\right)}$
$\displaystyle-2N_{f}T\int\frac{d^{3}p}{(2\pi)^{3}}\frac{3e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}}{\left(1+3\Phi
e^{-\beta\left(E_{p}+\tilde{\mu}\right)}+3\bar{\Phi}e^{-2\beta\left(E_{p}+\tilde{\mu}\right)}+e^{-3\beta\left(E_{p}+\tilde{\mu}\right)}\right)}.$

By fitting the empirical values of the pion mass $m_{\pi}=138$ MeV, the pion
decay constant $f_{\pi}=93$ MeV and the quark condensate per flavour
$\left<\bar{\phi}\phi\right>=-(250\text{MeV})^{3}$, we first fix three
parameters, i.e., the current mass $m=5.5\text{MeV}$ Hatsuda:1994pi , the
coupling constant of the conventional NJL model $g=5.074\times
10^{-6}\text{MeV}^{-2}$ and the three dimensional cut-off
$\Lambda=631\text{MeV}$ . And then, we redefine the coupling constant G as
Ref. Yang:2019lyn  that,

$\displaystyle G=\frac{1+\frac{1}{4N_{c}}}{1-\alpha+\frac{\alpha}{4N_{c}}}g.$
(9)

Besides, we take $T_{0}=270{\rm MeV}$ with $a_{0}=3.51$, $a_{1}=-2.47$,
$a_{2}=15.22$, $b_{3}=-1.75$ in our model Li:2018ygx . Since the evaluation of
the chiral condensate is our main purpose in this paper, we use the cut-off on
the thermal part of the integrals in the gap equations Xue:2021ldz .

##  III chiral and deconfinement phase transitions

Figure 1: Chiral phase transition in NJL model(left panel) and PNJL
model(right panel) with $\alpha=0,0.5,0.925,1.009,1.044$. The dashed line are
represented the crossover transition and the bold line are represented the
first order transition. The CEP for $\alpha=0,0.5$ are shown as big dots.

Figure 2: The comparison of chiral phase transition in the NJL and PNJL models
with $\alpha=0,0.5,0.925,1.009,1.044$.

Figure 3: Left panel: Deconfinement phase transition with
$\alpha=0,0.5,0.925,1.009,1.044$. Right panel: Deconfinement phase transition
scaled by $T_{d,0}$ with $\alpha=0,0.5,0.925,1.009,1.044$.

The chiral phase transition lines of the NJL and PNJL models with different
values of parameter $\alpha$, which are obtained by solving Eqs. (8a)-(8d),
are present in Fig. 1. We find that, when $\alpha=0$, the CEPs are located at
$(\mu_{CEP},T_{CEP})=(327,115)$ MeV and $(\mu_{CEP},T_{CEP})=(328,47)$ MeV for
the PNJL and NJL models, respectively. And when $\alpha=0.5$, the CEPs are
located at $(\mu_{CEP},T_{CEP})=(347,78)$ MeV and
$(\mu_{CEP},T_{CEP})=(348,29)$ MeV, respectively. Whereas, when $\alpha$ is
large enough, e.g., $\alpha=0.925,1.009$ and $1.044$, there is no CEPs for
both the NJL and PNJL models, and it means that the chiral phase transitions
in these cases are crossover. With resort to the chiral susceptibility, we
find that the critical value $\alpha_{c}$ is approximately 0.71 in both
models, and there is first order phase transition only when
$\alpha<\alpha_{c}$. Moreover, the critical temperature $T_{c}$ is around 234
MeV at zero chemical potential in the PNJL model, which is much higher than
$T_{c}=178$ MeV at $\mu=0$ in the NJL model. The comparison of the chiral
phase transition lines between the PNJL model and the NJL model with different
values of the weighting parameter $\alpha$ is given in Fig. 2. The critical
chemical potential $\mu_{c}$ at zero temperature in the NJL and PNJL models is
coincided with each other at the same $\alpha$. When $\alpha$ is smaller than
1, the phase transition line in the PNJL model is always higher than the line
in the NJL model. But when $\alpha>1$, e.g., $\alpha=1.009$ and
$\alpha=1.044$, there is a intersection point between the two linesin the NJL
and PNJL models, as shown in Fig. 2.

The deconfinement phase transition lines in the PNJL model are plotted in
panel(a) of Fig. 3. The critical temperatures $T_{d}$ at zero chemical
potential for different values of $\alpha$ are the same, which are almost
coincided with the critical temperature of the chiral phase transition
$T_{c}$, i.e., $T_{d}\approx T_{c}\approx 234$ MeV. It is found that, when
$\alpha$ is small, the deconfinement phase transition lines decrease first and
then increase with the increasing of the chemical potential. And as $\alpha$
increases, the kink of the curve becomes smaller and smaller, and finally it
disappears. It shows that the transition line becomes smoother and smoother,
and when $\alpha$ is lager enough, the transition line decreases monotonically
with the increasing chemical potential. However, from panel(b) in Fig. 3 we
could find within a large range of chemical potential $0-700$ MeV, the change
of the critical temperature with the increasing of the chemical potential are
less than $10\%$ even for $\alpha=0$.

##  IV Summary and Discussion

In this paper, we studied the QCD phase transition in the framework of a self-
consistent PNJL model. By comparing the chiral phase transition lines between
the NJL model and the PNJL model, the Polyakov loop from the gluonic
contribution helps to increase the (pseudo)critical temperature at the same
chemical potential. When $\alpha>1$, there is a cross point between the phase
transition lines of these two models. As expect, the incorporation of the
Polyakov loop does not change the order of the chiral phase transition with
the same $\alpha$. And the deconfinement phase transition lines become
smoother when $\alpha$ increases. However, the lines are always relatively
flatten, the change of the critical temperature are smaller than $10\%$.

As discussed in Ref.Wang:2019uwl , the real weighting constant $\alpha$ should
be determined by experiments measurement at the large chemical potential
physical environments, e.g., the properties of neutron stars by observing the
binary neutron star mergers Zhao:2019xqy ; Zuo:2022rks , the meson properties
from the observation of the future collision experiments, and etc. Since the
gluons in the QCD matter do contribute to the equation of state, it is
valuable to introduce the Polyakov loop potential to investigate the equation
of state of neutron stars and the meson properties in the future.

###### Acknowledgements.

The authors thanks to Lin Zhang for the discussion of the numerical code. The
work of L.Y. is supported by the NSFC under Grant No. 11605072 and the Seeds
Funding of Jilin University. The work of X.W. was supported by the start-up
funding No. 4111190010 of Jiangsu University and NSFC under Grant No.
12147103.

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# Regularity for solutions of non-uniformly elliptic equations in non-
divergence form

Jongmyeong Kim  Institute of Mathematics, Academia Sinica, Taipei 106319,
Republic of China<EMAIL_ADDRESS>and  Se-Chan Lee  Research
Institute of Mathematics, Seoul National University, Seoul 08826, Republic of
Korea<EMAIL_ADDRESS>
###### Abstract.

We prove the Aleksandrov–Bakelman–Pucci estimate for non-uniformly elliptic
equations in non-divergence form. Moreover, we investigate local behaviors of
solutions of such equations by developing local boundedness and weak Harnack
inequality. Here we impose an integrability assumption on ellipticity
representing degeneracy or singularity, instead of specifying the particular
structure of ellipticity.

###### Key words and phrases:

Non-uniformly elliptic equations; Aleksandrov–Bakelman–Pucci maximum
principle; Interior estimates

###### 2020 Mathematics Subject Classification:

35B50, 35B65, 35D35, 35J70, 35J75

Se-Chan Lee is supported by Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (2022R1A6A3A01086546).

##  1\. Introduction

In this paper, we study regularity properties for solutions of non-uniformly
elliptic equations in non-divergence form. To illustrate the issues, let us
begin with the simplest example: a second-order, linear elliptic equation in
non-divergence form

(1.1) $a_{ij}D_{ij}u=f\quad\text{in $B_{1}$},$

where a coefficient matrix $a=(a_{ij})_{1\leq i,j\leq n}$ and an
nonhomogeneous term $f$ are measurable. In order to capture the ellipticity of
$a$, we introduce

(1.2) $\lambda(x):=\inf_{\xi\in\mathbb{R}^{n}}\frac{\xi\cdot
a(x)\xi}{|\xi|^{2}}\quad\text{and}\quad\Lambda(x):=\sup_{\xi\in\mathbb{R}^{n}}\frac{\xi\cdot
a(x)\xi}{|\xi|^{2}}$

In particular, we say $a=(a_{ij})$ is uniformly elliptic if there exist
ellipticity constants $0<\lambda_{0}\leq\Lambda_{0}<\infty$ such that

$\lambda_{0}\leq\lambda(x)\leq\Lambda(x)\leq\Lambda_{0}.$

The regularity theory of (possibly nonlinear) uniformly elliptic operators in
non-divergence form is by now classical; we refer to comprehensive books [11,
20] and references therein. In particular, Aleksandrov [1], Bakelman [4], and
Pucci [29] independently proved a maximum principle: if $u\in
C(\overline{B_{1}})\cap W^{2,n}_{\mathrm{loc}}(B_{1})$ is a strong subsolution
of (1.1), then there exists a constant $C=C(n,\lambda_{0},\Lambda_{0})>0$ such
that

(1.3) $\sup_{B_{1}}u\leq\sup_{\partial
B_{1}}u^{+}+C\|f^{-}\|_{L^{n}(\Gamma^{+}(u^{+}))},$

where $\Gamma^{+}(u^{+})$ is the upper contact set; see Section 2 for the
precise definition. The ABP maximum principle became a fundamental tool in
establishing local estimates for the associated equations, such as local
boundedness, weak Harnack inequality, and interior Hölder estimate.

The goal of this paper is to develop the ABP maximum principle and to derive
interior a priori estimates for solutions of non-uniformly, nonlinear elliptic
equations. In our framework, the ellipticity functions $1/\lambda$ and
$\Lambda$ are not necessarily bounded, but they satisfy some integrability
conditions. To be precise, we define two measurable functions
$\lambda,\Lambda:B_{1}\to[0,\infty]$ such that $\lambda\leq\Lambda$,

$\displaystyle 1/\lambda\in L^{p}(B_{1}),\quad\text{and}\quad\Lambda\in
L^{q}(B_{1}).$

It is noteworthy that the uniformly elliptic case corresponds to the choice
$p=q=\infty$. Moreover, we define a generalized version of Pucci extremal
operators $\mathcal{M}_{\lambda,\Lambda}^{\pm}$ by

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(M)(x)$
$\displaystyle:=\Lambda(x)\sum_{e_{i}\geq
0}e_{i}(M)+\lambda(x)\sum_{e_{i}<0}e_{i}(M),$
$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{-}(M)(x)$
$\displaystyle:=\lambda(x)\sum_{e_{i}\geq
0}e_{i}(M)+\Lambda(x)\sum_{e_{i}<0}e_{i}(M),$

where $x\in B_{1}$, $M\in\mathcal{S}^{n}:=\\{M:\text{$M$ is an $n\times n$
real symmetric matrix}\\}$, and $e_{i}(M)$’s are the eigenvalues of $M$. For
constant ellipticity $\lambda_{0}$ and $\Lambda_{0}$, it reduces to the
classical Pucci extremal operators; see [11, 12] for instance.

Throughout the paper, we assume that a pair $(p,q)$ satisfies

(1.4) $\frac{1}{p}+\frac{1}{q}\leq\frac{1}{n}.$

and we set the constants $\theta,\tau\in[n,\infty]$ to satisfy

(1.5)
$\frac{1}{\theta}=\frac{1}{n}-\frac{1}{p}-\frac{1}{q}\quad\text{and}\quad\frac{1}{\tau}=\frac{1}{n}-\frac{1}{p}.$

Then we are concerned with an $L^{\theta}$-strong solution $u$ of

$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)(x)\geq f(x)$

or

$\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)(x)\leq f(x)$

for an nonhomogeneous term $f\in L^{\tau}(B_{1})$; see Section 2 for details.

We begin with the Aleksandrov-Bakelman-Pucci estimates for $L^{\theta}$-strong
subsolutions. Several corollaries of Theorem 1.1 are discussed at the end of
Section 3.

######  Theorem 1.1 (ABP estimates).

Let $f\in L^{\tau}(B_{1})$. Suppose that $u\in W^{2,\theta}(B_{1})$ is an
$L^{\theta}$-strong solution of

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq f\quad\text{in
$B_{1}$}.$

Then there exists a universal constant $C=C(n)>0$ such that

$\displaystyle\sup_{B_{1}}u\leq\sup_{\partial
B_{1}}u^{+}+C\left(\int_{\Gamma^{+}(u^{+})}\left(\frac{f^{-}(x)}{\lambda(x)}\right)^{n}\,\mathrm{d}x\right)^{1/n}.$

After the pioneering works by Aleksandrov–Bakelman–Pucci, the ABP maximum
principle has been widely studied in different contexts. Just to name a few,
the ABP estimate, concerning uniformly elliptic/parabolic equations in non-
divergence form, was achieved for

  1. (i)

viscosity solutions of fully nonlinear elliptic equations [7, 10];

  2. (ii)

strong solutions of linear parabolic equations [26, 31];

  3. (iii)

viscosity solutions of fully nonlinear elliptic equations [32];

  4. (iv)

$L^{p}$-viscosity solutions of fully nonlinear elliptic/parabolic equations
[8, 13];

  5. (v)

viscosity solutions of fully nonlinear elliptic equations with gradient growth
terms [24, 25].

We refer to [6, 17] for improvements of the ABP estimates in other directions.
On the other hand, non-uniformly elliptic equations with particular structure
have been considered relatively recently by several authors in various
circumstances: [3, 16, 21] when an operator is given by
$|Du|^{\gamma}\mathcal{M}_{\lambda_{0},\Lambda_{0}}^{\pm}(D^{2}u)$ with
$\gamma>-1$, [2] for $p$-Laplace equations and the mean curvature flow, and
[22, 27] for elliptic equations that hold only where the gradient is large. In
this paper, we concentrate on analyzing non-uniformly elliptic equations whose
degeneracy and singularity are implicitly encoded in the integrability of
$1/\lambda$ and $\Lambda$.

We next move our attention to local estimates for solutions of Pucci extremal
operators. We first show the local boundedness result for strong subsolutions.

######  Theorem 1.2 (Local boundedness).

Let $f\in L^{\tau}(B_{1})$. Suppose that $u\in
W^{2,\theta}_{\mathrm{loc}}(B_{1})$ is an $L^{\theta}$-strong solution of

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq f\quad\text{in
$B_{1}$}.$

Then for $0<t\leq n$, we have

$\displaystyle\sup_{B_{1/2}}u\leq
C\left(\left\|(u^{+})^{t/n}\Lambda/\lambda\right\|_{L^{n}(B_{1})}^{n/t}+\|f^{-}/\lambda\|_{L^{n}(B_{1})}\right)$

for a universal constant $C=C(n,t)>0$.

In particular, for $t>0$, there exists
$C=C(n,t,\|1/\lambda\|_{L^{p}(B_{1})},\|\Lambda\|_{L^{q}(B_{1})})>0$ such that

$\displaystyle\sup_{B_{1/2}}u\leq C\left(\|u^{+}\|_{L^{\theta
t/n}(B_{1})}+\|f^{-}/\lambda\|_{L^{n}(B_{1})}\right).$

We also prove the weak Harnack inequality for viscosity supersolutions under a
stronger assumption on $(p,q)$.

######  Theorem 1.3 (Weak Harnack inequality).

Let $f\in L^{\tau}(B_{1})$ and assume that

$\displaystyle\frac{1}{p}+\frac{1}{q}<\frac{1}{2n}.$

Moreover, suppose that $u\in W^{2,\theta}_{\mathrm{loc}}(B_{1})$ is an
$L^{\theta}$-strong solution of

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\leq f\quad\text{in
$B_{1}$}.$

If $u$ is nonnegative in $B_{1}$, then we have

$\displaystyle\left\|u\right\|_{L^{t}(B_{1/2})}\leq
C\left(\inf_{B_{1/2}}u+\|f/\lambda\|_{L^{n}(B_{1})}\right)$

for some positive constants $t$ and $C$ which depend only on $n$,
$\|1/\lambda\|_{L^{p}(B_{1})}$, and $\|\Lambda\|_{L^{q}(B_{1})}$.

As immediate consequences of Theorem 1.2 and Theorem 1.3, we provide a Harnack
inequality and an interior Hölder regularity of strong solutions in Section 4.

We now describe two simple, but interesting observations regarding our main
theorems:

  1. (i)

For $n=1$ and $\gamma>0$, let us consider a linear operator
$Lu=|x|^{\gamma}u_{xx}$ in $B_{1}=(-1,1)$. We then claim that $u(x)=|x|$ is a
$C$-viscosity solution of $Lu=0$ in $B_{1}$; see Definition 2.4 for the
definition of $C$-viscosity solutions. Indeed, for $x_{0}\in
B_{1}\setminus\\{0\\}$, then $u_{xx}(x_{0})=0$ and so $Lu(x_{0})=0$ in the
classical sense. For $x_{0}=0$, if we let $\varphi\in C^{2}(B_{1})$ be a test
function such that $u-\varphi$ has a local maximum (or minimum) at $0$, then

$L\varphi(0)=|x|^{\gamma}\varphi_{xx}|_{x=0}=0.$

Therefore, we conclude that $u$ is a viscosity solution of $Lu=0$ in $B_{1}$.

On the other hand, if we choose ellipticity functions
$\lambda(x)=|x|^{\gamma}$ and $\Lambda(x)=1$, then it immediately follows that
a viscosity solution $u$ of $Lu=0$ in $B_{1}$ satisfies

$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq
0\quad\text{and}\quad\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\leq
0\quad\text{in $B_{1}$}.$

Moreover, it is easy to see that $\Lambda\in L^{\infty}(B_{1})$ and
$1/\lambda\in L^{p}(B_{1})$ for any $p<1/\gamma$, while $u$ does not enjoy the
(weak) minimum principle in $B_{1}$. Hence, even though we impose stronger
integrability condition on $1/\lambda$ and $\Lambda$, Theorem 1.1 does not
hold for general “viscosity solution” $u$. In other words, this example shows
that the “strong solution” condition on $u$ is essential in our framework.

  2. (ii)

For $n=2$, we consider a linear operator $Lu=2u_{xx}+y^{2}u_{yy}$ in
$B_{1}=\\{(x,y):x^{2}+y^{2}<1\\}$. Then ellipticity functions are given by
$\lambda(x,y)=y^{2}$ and $\Lambda(x,y)=2$, where $1/\lambda=|y|^{-2}\notin
L^{1}(B_{1})$. It follows from a direct calculation that $u(x,y)=y^{2}\cos x$
is a classical (or strong) solution of $Lu=0$ in $B_{1}$. Since
$u(0,0)=0=\min_{\partial B_{1}}u$, $u$ does not satisfy the strong maximum
principle and the weak Harnack inequality. In short, this example guarantees
the necessity of (a version of) integrability criterion on $1/\lambda$ and
$\Lambda$ in Theorem 1.3. Nevertheless, the optimality of our assumption on
$(p,q)$ is not verified by this example, and it remains an interesting open
problem.

Let us finally discuss similar consequences for linear non-uniformly elliptic
equations in divergence form. In particular, as a variational counterpart of
(1.1), the authors of [5, 30] considered a weak solution $u$ of

$-D_{j}(a_{ij}D_{i}u)=0\quad\text{in $B_{1}$},$

where the ellipticity of $a$ is measured by $\lambda$ and $\Lambda$ defined in
(1.2). In [30], Trudinger established interior estimates such as local
boundedness, Harnack inequality, and Hölder regularity for weak solutions,
provided that $1/\lambda\in L^{p}(B_{1})$, $\Lambda\in L^{q}(B_{1})$ with

$\displaystyle\frac{1}{p}+\frac{1}{q}<\frac{2}{n}.$

Recently, Bella and Schäffner [5] improved the result by replacing the
condition with

$\displaystyle\frac{1}{p}+\frac{1}{q}<\frac{2}{n-1},$

and proved that this integrability condition is indeed sharp. The strategy of
both papers mainly relied on a modification of the Moser iteration method,
which is not available for operators in non-divergence form. We also refer to
[14, 28] for related results.

The paper is organized as follows. In Section 2, we summarize several
notations which will be used throughout the paper. Section 3 is devoted to the
proof of Theorem 1.1 by adopting sequential approximation techniques. Finally,
we investigate local behaviors of strong solutions: local boundedness for
subsolutions and weak Harnack inequality for supersolutions.

##  2\. Preliminaries

We first introduce a concept of $L^{\theta}$-strong solutions for Pucci
extremal operators $\mathcal{M}_{\lambda,\Lambda}^{\pm}$.

######  Definition 2.1 ($L^{\theta}$-strong solutions).

Let $f\in L^{\tau}_{\text{loc}}(B_{1})$. A function $u\in
W^{2,\theta}_{\mathrm{loc}}(B_{1})$ is an $L^{\theta}$-strong solution of
$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq f$ in $B_{1}$, if

$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u):=\Lambda(x)\sum_{e_{i}\geq
0}e_{i}(D^{2}u(x))+\lambda(x)\sum_{e_{i}<0}e_{i}(D^{2}u(x))\geq
f(x)\quad\text{a.e. in $B_{1}$},$

where $e_{i}(M)$’s are the eigenvalues of $M$.

In a similar way, a function $u\in W^{2,\theta}_{\mathrm{loc}}(B_{1})$ is an
$L^{\theta}$-strong solution of $\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\leq
f$ in $B_{1}$, if

$\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u):=\lambda(x)\sum_{e_{i}\geq
0}e_{i}(D^{2}u(x))+\Lambda(x)\sum_{e_{i}<0}e_{i}(D^{2}u(x))\leq
f(x)\quad\text{a.e. in $B_{1}$}.$

######  Remark 2.2.

The constants $\theta$ and $\tau$ are chosen to verify that
$(\Lambda/\lambda)D^{2}u$ and $f/\lambda$ are contained in $L^{n}$-space. If
$1/\lambda$ and $\Lambda$ further belong to $L^{\infty}$-space, then it
corresponds to the uniformly elliptic setting with $p=q=\infty$ and
$\theta=\tau=n$. In this case, Definition 2.1 coincides with the definition of
$L^{n}$-strong solutions given in [8].

We provide simple properties of
$\mathcal{M}^{\pm}=\mathcal{M}_{\lambda,\Lambda}^{\pm}$ as follows.

######  Lemma 2.3.

Let $M,N\in\mathcal{S}^{n}$. Then the following hold a.e.:

  1. (i)

$\mathcal{M}^{-}(M)\leq\mathcal{M}^{+}(M)$.

  2. (ii)

$\mathcal{M}^{-}(M)=-\mathcal{M}^{+}(-M)$.

  3. (iii)

$\mathcal{M}^{\pm}(tM)=t\mathcal{M}^{\pm}(M)$ if $t\geq 0$.

  4. (iv)

$\mathcal{M}^{+}(M)+\mathcal{M}^{-}(N)\leq\mathcal{M}^{+}(M+N)\leq\mathcal{M}^{+}(M)+\mathcal{M}^{+}(N)$.

For later uses, we also define $C$-viscosity solutions when $\lambda$,
$\Lambda$, and $f$ are continuous; see [11, 12] for instance.

######  Definition 2.4 ($C$-viscosity solutions).

Let $\lambda,\Lambda,f\in C(B_{1})$ with $0\leq\lambda(x)\leq\Lambda(x)$ for
$x\in B_{1}$. A function $u\in C(B_{1})$ is a $C$-viscosity solution of
$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq f$ in $B_{1}$, if for all
$\varphi\in C^{2}(B_{1})$ and point $x_{0}\in B_{1}$ at which $u-\varphi$ has
a local maximum, one has

$\mathcal{M}_{\lambda(x_{0}),\Lambda(x_{0})}^{+}(D^{2}\varphi(x_{0}))\geq
f(x_{0}).$

In a similar way, a function $u\in C(B_{1})$ is a $C$-viscosity solution of
$\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\geq f$ in $B_{1}$, if for all
$\varphi\in C^{2}(B_{1})$ and point $x_{0}\in B_{1}$ at which $u-\varphi$ has
a local minimum, one has

$\mathcal{M}_{\lambda(x_{0}),\Lambda(x_{0})}^{-}(D^{2}\varphi(x_{0}))\leq
f(x_{0}).$

The following contact set $\Gamma^{+}$ will be used for the proof of ABP
estimates.

######  Definition 2.5.

For a function $u:\Omega\to\mathbb{R}$ and $r>0$, the upper contact set is
defined by

$\displaystyle\Gamma^{+}(u)$
$\displaystyle=\Gamma^{+}(u,\Omega)=\\{x\in\Omega\text{ :
$\exists\,p\in\mathbb{R}^{n}$ such that $u(y)\leq u(x)+\langle p,y-x\rangle$,
$\forall y\in\Omega$}\\},$ $\displaystyle\Gamma_{r}^{+}(u)$
$\displaystyle=\Gamma_{r}^{+}(u,\Omega)=\\{x\in\Omega\text{ :
$\exists\,p\in\overline{B_{r}(0)}$ such that $u(y)\leq u(x)+\langle
p,y-x\rangle$, $\forall y\in\Omega$}\\}.$

For sets $A_{1}$, $A_{2}$, $\cdots$, we define

$\displaystyle\limsup_{j\to\infty}A_{j}:=\bigcap_{n=1}^{\infty}\bigcup_{k>n}A_{k}.$

######  Lemma 2.6 ([8, Lemma A.1]).

Let $u_{j}$, $j=1,2,\cdots$ be functions defined on sets $\Omega_{j}$, where
$\Omega_{j}$ are open and increase to $\Omega$; that is
$\Omega_{j}\subset\Omega_{j+1}$ and $\bigcup\Omega_{j}=\Omega$. Let $u_{j}$
converge uniformly to a continuous function $u$ on each $\Omega_{m}$. Then

  1. (i)

$\limsup_{j\to\infty}\Gamma^{+}(u_{j},\Omega_{j})\subset\Gamma^{+}(u,\Omega)$.

  2. (ii)

$\limsup_{j\to\infty}|\Gamma^{+}(u_{j},\Omega_{j})|\leq|\Gamma^{+}(u,\Omega)|$.

  3. (iii)

$\limsup_{j\to\infty}\Gamma^{+}_{r}(u_{j},\Omega_{j})\subset\Gamma^{+}_{r}(u,\Omega)$.

We finally state a version of the cube decomposition lemma, which is suitable
for our purpose in Section 4.

######  Lemma 2.7 ([20, Lemma 9.23]).

Let $K_{0}$ be a cube in $\mathbb{R}^{n}$, $w\in L^{1}(K_{0})$, and set

$D_{k}=\\{x\in K_{0}\,|\,w(x)\leq k\\}\quad\text{for $k\in\mathbb{R}$}.$

Suppose that there exist constants $\delta\in(0,1)$ and $C>0$ such that

$\sup_{K_{0}\cap K_{3r}(z)}\,(w-k)\leq C,$

whenever $k$ and $K=K_{r}(z)\subset K_{0}$ satisfy

$|D_{k}\cap K|\geq\delta|K|.$

Then it follows that, for all $k$,

$\sup_{K_{0}}\,(w-k)\leq
C\left(1+\frac{\log(|D_{k}|/|K_{0}|)}{\log\delta}\right).$

###  2.1. Applications

In this section, we present concrete examples of degenerate or singular
equations in non-divergence form, which are contained in our framework.

  1. (i)

(Issacs equations) Issaces equations, which naturally arise in probability
theory [19] (stochastic control and differential games), are given by

$\displaystyle\inf_{\alpha}\sup_{\beta}\left(A_{\alpha\beta}(x)D^{2}u(x)\right)=f\quad\text{in
$B_{1}$},$

where $A_{\alpha\beta}(\cdot)$ (for any $\alpha$, $\beta$ contained in index
sets) are matrices satisfying

$\displaystyle\lambda(x)I_{n}\leq A_{\alpha\beta}(x)\leq\Lambda(x)I_{n}$

with $1/\lambda\in L^{p}(B_{1})$ and $\Lambda\in L^{q}(B_{1})$. We note that
linear elliptic operators with ellipticity $\lambda$ and $\Lambda$, and the
Pucci extremal operators $\mathcal{M}_{\lambda,\Lambda}^{\pm}$ can be
understood as special cases of Issacs operators.

  2. (ii)

(Monge-Ampére equations) The Monge-Ampére equation, which appears from the
prescribed Gaussian curvature equation [18] (or “Minkowski problem”), is a
fully nonlinear, degenerate elliptic equation given by

$\displaystyle\mathrm{det}\,D^{2}u=f\quad\text{in $B_{1}$}.$

It has important applications in convex geometry and optimal transportation.
For simplicity, we consider an equation

(2.1) $G(D^{2}u):=\log\mathrm{det}\,D^{2}u=\log f.$

Then we have $G_{ij}=u^{ij}$, where $u^{ij}$ denote the inverse of the Hessian
matrix $D^{2}u$. Thus, if we denote $\lambda$ and $\Lambda$ are ellipticity
functions defined in (1.2) for $u^{ij}$, then we observe that $1/\Lambda$ and
$1/\lambda$ are ellipticity function for $D^{2}u$. Since

$u$ is convex if and only if (2.1) is degenerate elliptic and
$\displaystyle\text{$u$ is uniformly convex if and only if \eqref{eq-
mongeampere} is uniformly elliptic},$

the integrability assumption on $1/\lambda$ and $\Lambda$ corresponds to some
“intermediate” convexity on $u$.

  3. (iii)

(Linear equations with particular degeneracy/singularity) In [15], the authors
employed the partial Legendre transform to convert the two-dimensional Monge-
Ampére equation

$\mathrm{det}\,D^{2}u=|x|^{\alpha}\quad\text{for $\alpha>0$}$

into the linear equation

$v_{xx}+|x|^{\alpha}v_{yy}=0\quad\text{in $B_{1}$}.$

Then the pair $(p,q)$ corresponding to the ellipticity functions given by
$\lambda(x,y)=|x|^{\alpha}$ and $\Lambda(x,y)=1$ satisfy the stuctural
condition (1.4) when $\alpha<1/2$.

Moreover, a similar type of equation can be found in an extension problem
related to the fractional Laplacian [9]. To be precise, the solution $u$ of
the degenerate/singular equations

$\left\\{\begin{aligned} \Delta_{x}u+z^{\frac{2s-1}{s}}u_{zz}&=0&&\text{in
$\mathbb{R}^{n}\times[0,\infty)$}\\\ u&=f&&\text{on
$\mathbb{R}^{n}\times\\{0\\}$}\end{aligned}\right.$

satisfies

$(-\Delta)^{s}f(x)=-C(n,s)u_{z}(x,0)$

for $s\in(0,1)$. It is easy to check that this example lies in our setting
when $(n+1)/(2n+3)<s<(n+1)/(2n+1)$. We refer to [23] for related examples.

##  3\. ABP estimates

In order to prove Theorem 1.1, we are going to provide a version of [8,
Proposition 2.12] (ABP estimates for continuous coefficients and $C$-viscosity
solutions) and [8, Lemma 3.1] (existence of $L^{n}$-strong
super/subsolutions). It is noteworthy that an additional approximation
technique is required to control ellipticity functions $\lambda$ and
$\Lambda$, which are not necessarily bounded in $L^{\infty}$.

######  Lemma 3.1.

Let $f\in C(B_{1})$. Assume $\lambda,\Lambda:B_{1}\to(0,\infty)$ such that
$1/\lambda,\Lambda\in C(\overline{B_{1}})$. Moreover, suppose that $u\in
C(\overline{B_{1}})$ is a $C$-viscosity solution of

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq f\quad\text{in
$B_{1}$}.$

Then there exists a universal constant $C=C(n)>0$ such that

$\displaystyle\sup_{B_{1}}u\leq\sup_{\partial
B_{1}}u^{+}+C\left(\int_{\Gamma^{+}(u^{+})}\left(\frac{f^{-}(x)}{\lambda(x)}\right)^{n}\,\mathrm{d}x\right)^{1/n}.$

###### Proof.

We will follow the proof provided in [8, Appendix A]. We begin by assuming
that $u\in C^{2}(B_{1})\cap C(\overline{B_{1}})$ and later remove this
assumption via approximations. We set

(3.1) $r_{0}=\frac{\sup_{B_{1}}u-\sup_{\partial B_{1}}u^{+}}{2}.$

For $r<r_{0}$, let $p\in B_{r}$ and $\hat{x}\in\overline{B_{1}}$ be a maximum
point of $u(\cdot)-\langle p,\cdot\rangle$ so that

$u(\hat{x})-\langle p,\hat{x}\rangle\geq u(x)-\langle p,x\rangle\text{ or
}u(x)-u(\hat{x})\leq\langle p,x-\hat{x}\rangle$

for any $x\in\overline{B_{1}}$. It follows that

$\displaystyle\sup_{B_{1}}u-u(\hat{x})\leq 2|p|\leq
2r<2r_{0}=\sup_{B_{1}}u-\sup_{\partial B_{1}}u^{+}$

and then $2(r_{0}-r)+\sup_{\partial B_{1}}u^{+}<u(\hat{x}).$ In particular,
$\hat{x}\in B_{1}$ and $u(\hat{x})>0$. Since $Du(\hat{x})=p$ and
$D^{2}u(\hat{x})\leq 0$, we conclude that for $0<r<r_{0}$,
$\Gamma^{+}_{r}(u^{+})$ is a compact subset of $B_{1}$ and

(3.2) $B_{r}=B_{r}(0)=Du(\Gamma^{+}_{r}(u^{+}))\text{ and }D^{2}u(x)\geq
0\text{ on }\Gamma^{+}_{r}(u^{+})\subset\\{u>0\\}.$

We now employ the change of variables $p=Du(x)$ to have

(3.3) $\int_{B_{r}}\,\mathrm{d}p\leq\int_{\Gamma^{+}_{r}(u^{+})}|\det
D^{2}u|\,\mathrm{d}x\leq\int_{\Gamma^{+}_{r}(u^{+})}\left(\frac{-\operatorname*{tr}D^{2}u}{n}\right)^{n}\,\mathrm{d}x.$

Since $\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)(x)\geq f(x)$ and $D^{2}u\leq
0$ on $\Gamma^{+}(u^{+})$, we have

$\lambda\operatorname*{tr}(D^{2}u)\geq f(x)\text{ on }\Gamma^{+}(u^{+})$

and (3.3) implies

$r^{n}|B_{1}|=\int_{B_{r}}\,\mathrm{d}p\leq\frac{1}{n^{n}}\int_{\Gamma^{+}(u^{+})}\left(\frac{f^{-}(x)}{\lambda(x)}\right)^{n}\,\mathrm{d}x.$

Since $\lambda$, $\Lambda$, and $f$ are continuous, the general case follows
from the standard approximation argument as in [8, Appendix A]. ∎

######  Lemma 3.2.

Let $f\in L^{n}(B_{1})$, $\psi\in C(\partial B_{1})$. Assume
$1/\lambda,\Lambda\in C(\overline{B_{1}})$. Then there exist $L^{n}$-strong
solutions $u,v\in C(\overline{B_{1}})\cap W^{2,n}_{\mathrm{loc}}(B_{1})$ of

$\displaystyle\mathcal{M}^{+}_{\lambda,\Lambda}(D^{2}u)(x)\leq
f(x)\quad\text{in $B_{1}$}$

and

$\displaystyle\mathcal{M}^{-}_{\lambda,\Lambda}(D^{2}v)(x)\geq
f(x)\quad\text{in $B_{1}$}$

such that $u=v=\psi$ on $B_{1}$. Moreover, $u$, $v$ satisfy the estimate

(3.4)
$\displaystyle\|u\|_{L^{\infty}(B_{1})},\|v\|_{L^{\infty}(B_{1})}\leq\|\psi\|_{L^{\infty}(\partial
B_{1})}+C\|f/\lambda\|_{L^{n}(B_{1})}.$

###### Proof.

The existence of a strong solution follows from [8, Lemma 3.1]. The
$L^{\infty}$-estimate is a consequence of Lemma 3.1 together with the
approximation. ∎

We are now ready to prove the first main theorem [Theorem 1.1].

###### Proof of Theorem 1.1.

We employ several regularization techniques; more precisely, we approximate
the ellipticity $\lambda$, $\Lambda$, and then the forcing term $f$. For
simplicity, we may omit “a.e.” if no confusion occurs.

  1. (i)

(Approximation of $\lambda$, $\Lambda$) We first let

$\displaystyle\widetilde{\lambda}_{j}:=\left(\lambda\lor j\right)\land
j^{-1}\quad\text{and}\quad\widetilde{\Lambda}_{j}:=\left(\Lambda\lor
j\right)\land j^{-1}$

so that

$\displaystyle\|1/\lambda_{j}-1/\widetilde{\lambda}_{j}\|_{p}\to
0,\quad\|\Lambda_{j}-\widetilde{\Lambda}_{j}\|_{q}\to 0,\quad\text{and}\quad
j^{-1}\leq\widetilde{\lambda}_{j}\leq\widetilde{\Lambda}_{j}\leq j.$

Since $C_{c}(B_{1})$ is a dense subspace of $L^{p}(B_{1})$ for any
$p\in[1,\infty)$, there exist two sequences
$\\{\lambda_{j}\\}_{j=1}^{\infty}\subset C_{c}(B_{1})$ and
$\\{\Lambda_{j}\\}_{j=1}^{\infty}\subset C_{c}(B_{1})$ such that

$\displaystyle\|1/\lambda_{j}-1/\widetilde{\lambda}_{j}\|_{p}<j^{-1}\quad\text{and}\quad\|\Lambda_{j}-\widetilde{\Lambda}_{j}\|_{q}<j^{-1}.$

Here we may assume $\lambda_{j}\in[j^{-1},j]$ and
$\lambda_{j}\leq\widetilde{\lambda}_{j}$ by replacing $\lambda_{j}$ with
$\left(\lambda_{j}\lor j\right)\land j^{-1}$ and
$\lambda_{j}\land\widetilde{\lambda}_{j}$, if necessary. A similar argument
also holds for $\Lambda_{j}$. Hence, it follows that
$\lambda_{j},\Lambda_{j}\in C(\overline{B_{1}})$,
$j^{-1}\leq\lambda_{j}\leq\Lambda_{j}\leq j$,

(3.5) $\displaystyle\|1/\lambda_{j}-1/\lambda\|_{p}\to
0,\quad\text{and}\quad\|\Lambda_{j}-\Lambda\|_{q}\to 0.$

We now would like to find the inequality satisfied by $u$, in terms of Pucci
extremal operators with ‘good’ ellipticity $\lambda_{j}$ and $\Lambda_{j}$.
Indeed, since $u\in W^{2,\theta}(B_{1})$ satisfies

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)=\Lambda(x)\sum_{e_{i}>0}e_{i}(D^{2}u(x))+\lambda(x)\sum_{e_{i}<0}e_{i}(D^{2}u(x))\geq
f(x),$

we observe that

$\displaystyle\mathcal{M}_{\lambda_{j},\Lambda_{j}}^{+}(D^{2}u)$
$\displaystyle=\Lambda_{j}\sum_{e_{i}>0}e_{i}(D^{2}u)+\lambda_{j}\sum_{e_{i}<0}e_{i}(D^{2}u)$
$\displaystyle=\Lambda\sum_{e_{i}>0}e_{i}(D^{2}u)+(\Lambda_{j}-\Lambda)\sum_{e_{i}>0}e_{i}(D^{2}u)$
$\displaystyle+\lambda\sum_{e_{i}<0}e_{i}(D^{2}u)+(\lambda_{j}-\lambda)\sum_{e_{i}<0}e_{i}(D^{2}u)=:f_{j}.$

By recalling that $f\in L^{\tau}(B_{1})$, $\Lambda\in L^{q}(B_{1})$, and
$D^{2}u\in L^{\theta}(B_{1})$, it turns out that

$f_{j}=f+(\Lambda_{j}-\Lambda)\sum_{e_{i}>0}e_{i}(D^{2}u)+(\lambda_{j}-\lambda)\sum_{e_{i}<0}e_{i}(D^{2}u)\in
L^{n}(B_{1}).$

  2. (ii)

(Approximation of $f_{j}$) For fixed $j\in\mathbb{N}$, let
$\\{f_{j,k}\\}_{k=1}^{\infty}\subset C^{\infty}(B_{1})$ be a sequence of
smooth functions such that

(3.6) $\displaystyle\|f_{j,k}-f_{j}\|_{L^{n}(B_{1})}\to 0\quad\text{as
$k\to\infty$}.$

Then we let $\psi_{j,k}\in W^{2,n}_{\text{loc}}(B_{1})\cap
C(\overline{B_{1}})$ solve

$\left\\{\begin{aligned}
\mathcal{M}_{\lambda_{j},\Lambda_{j}}^{-}(D^{2}\psi_{j,k})&\geq
f_{j,k}-f_{j}&&\text{in $B_{1}$}\\\ \psi_{j,k}&=0&&\text{on $\partial
B_{1}$},\end{aligned}\right.$

whose existence is guaranteed by Lemma 3.2. From the estimate (3.4),

$\displaystyle\|\psi_{j,k}\|_{L^{\infty}(B_{1})}\leq
C\|(f_{j,k}-f_{j})/\lambda_{j}\|_{L^{n}(B_{1})},$

where the constant $C>0$ is independent of $k\in\mathbb{N}$. Therefore, it
immediately follows that

(3.7) $\displaystyle\|\psi_{j,k}\|_{L^{\infty}(B_{1})}\to 0\quad\text{as
$k\to\infty$}.$

  3. (iii)

(Conclusion; ABP estimates) If we set
$w:=u+\psi_{j,k}-\|\psi_{j,k}\|_{L^{\infty}(B_{1})}$, then we observe that

$\displaystyle\mathcal{M}_{\lambda_{j},\Lambda_{j}}^{+}(D^{2}w)$
$\displaystyle\geq\mathcal{M}_{\lambda_{j},\Lambda_{j}}^{+}(D^{2}u)+\mathcal{M}_{\lambda_{j},\Lambda_{j}}^{-}(D^{2}\psi_{j,k})$
$\displaystyle\geq f_{j}+(f_{j,k}-f_{j})=f_{j,k}.$

Since $\lambda_{j}$, $\Lambda_{j}$, $f_{j,k}$ are regularized enough so that
Lemma 3.1 is applicable, we have

$\displaystyle\sup_{B_{1}}w\leq\sup_{\partial
B_{1}}w^{+}+C\left(\int_{\Gamma^{+}(w^{+})}\left(\frac{f_{j,k}^{-}(x)}{\lambda_{j}(x)}\right)^{n}\,\mathrm{d}x\right)^{1/n}.$

By letting $k\to\infty$ together with (3.6), (3.7), and Lemma 2.6, we deduce

$\displaystyle\sup_{B_{1}}u\leq\sup_{\partial
B_{1}}u^{+}+C\left(\int_{\Gamma^{+}(u^{+})}\left(\frac{f_{j}^{-}(x)}{\lambda_{j}(x)}\right)^{n}\,\mathrm{d}x\right)^{1/n}.$

Moreover, by applying Hölder inequality, we obtain

$\displaystyle\left\|\frac{f_{j}^{-}}{\lambda_{j}}-\frac{f^{-}}{\lambda}\right\|_{n}$
$\displaystyle\leq\left\|\frac{f_{j}}{\lambda_{j}}-\frac{f}{\lambda}\right\|_{n}$
$\displaystyle\leq\left\|\left(\frac{1}{\lambda}-\frac{1}{\lambda_{j}}\right)\lambda\sum_{e_{i}<0}e_{i}(D^{2}u)\right\|_{n}+\left\|\frac{1}{\lambda_{j}}(\Lambda_{j}-\Lambda)\sum_{e_{i}>0}e_{i}(D^{2}u)\right\|_{n}+\left\|\frac{f}{\lambda_{j}}-\frac{f}{\lambda}\right\|_{n}$
$\displaystyle\leq\|1/\lambda-1/\lambda_{j}\|_{p}\|\Lambda\|_{q}\|D^{2}u\|_{\theta}+\|1/\lambda_{j}\|_{p}\|\Lambda_{j}-\Lambda\|_{q}\|D^{2}u\|_{\theta}+\|1/\lambda_{j}-1/\lambda\|_{p}\|f\|_{\tau}.$

Therefore, by passing the limit $j\to\infty$ together with (3.5), we finally
conclude that

$\displaystyle\sup_{B_{1}}u\leq\sup_{\partial
B_{1}}u^{+}+C\left(\int_{\Gamma^{+}(u^{+})}\left(\frac{f^{-}(x)}{\lambda(x)}\right)^{n}\,\mathrm{d}x\right)^{1/n}$

as desired.

∎

######  Corollary 3.3.

Let $f\in L^{\tau}(B_{1})$. Suppose that $u\in W^{2,\theta}(B_{1})$ is an
$L^{\theta}$-strong solution of

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\leq f\quad\text{in
$B_{1}$}.$

Then there exists a universal constant $C=C(n)>0$ such that

$\displaystyle\sup_{B_{1}}u^{-}\leq\sup_{\partial
B_{1}}u^{-}+C\left(\int_{\Gamma^{+}(u^{-})}\left(\frac{f^{+}(x)}{\lambda(x)}\right)^{n}\,\mathrm{d}x\right)^{1/n}.$

###### Proof.

It immediately follows by considering $-u$ instead of $u$ in the proof of
Theorem 1.1. ∎

######  Corollary 3.4.

Let $\lambda,\Lambda,f\in C(B_{1})$ with $0\leq\lambda(x)\leq\Lambda(x)$ for
$x\in B_{1}$. Suppose that $u\in W^{2,\theta}_{\mathrm{loc}}(B_{1})$ is an
$L^{\theta}$-strong solution of $\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq
f$ in $B_{1}$. Then $u$ is also a $C$-viscosity solution of
$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq f$ in $B_{1}$.

###### Proof.

Since $\theta\geq n$, we have $u\in C(B_{1})$. We assume by contradiction that
for some $\varphi\in C^{2}(B_{1})$, $u-\varphi$ has a (strict) local maximum
at $x_{0}\in B_{1}$ and

$\mathcal{M}_{\lambda(x_{0}),\Lambda(x_{0})}^{+}(D^{2}\varphi(x_{0}))<f(x_{0}).$

By the continuity of $\lambda,\Lambda,f$, we have

$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}\varphi)<f$

near $x_{0}$. On the other hand, we observe from Lemma 2.3 that

$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}(u-\varphi))\geq\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)-\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}\varphi)>0\quad\text{a.e.
in $B_{\eta}(x_{0})$ for some $\eta>0$}.$

We now apply Theorem 1.1 in $B_{\eta}(x_{0})$ to conclude that

$(u-\varphi)(x_{0})\leq\sup_{\partial B_{\eta}(x_{0})}(u-\varphi),$

which leads to a contradiction. ∎

We say a measurable function $F:\mathcal{S}^{n}\times B_{1}\to\mathbb{R}$ is
$(\lambda(\cdot),\Lambda(\cdot))$-elliptic if

$\mathcal{M}_{\lambda,\Lambda}^{-}(N)(x)\leq
F(M+N,x)-F(M,x)\leq\mathcal{M}_{\lambda,\Lambda}^{+}(N)(x)$

for any $M,N\in\mathcal{S}^{n}$ and $x\in B_{1}$ a.e.. We note that the Pucci
extremal operators $\mathcal{M}_{\lambda,\Lambda}^{\pm}$ are
$(\lambda(\cdot),\Lambda(\cdot))$-elliptic. The notion of $L^{\theta}$-strong
solution defined in Definition 2.1 can be easily extended to such fully
nonlinear operators $F$.

######  Corollary 3.5 (Comparison principle).

Let $f\in L^{\tau}(B_{1})$ and $F$ be
$(\lambda(\cdot),\Lambda(\cdot))$-elliptic. Suppose that $u,v\in
W^{2,\theta}(B_{1})$ are $L^{\theta}$-strong subsolution and supersolution of
$F(D^{2}w,x)=f(x)$ in $B_{1}$, respectively. If $u\leq v$ on $\partial B_{1}$,
then $u\leq v$ in $B_{1}$.

###### Proof.

By the definition of $(\lambda(\cdot),\Lambda(\cdot))$-ellipticity, we have

$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}(u-v))(x)\geq
F(D^{2}u,x)-F(D^{2}v,x)\geq 0.$

The desired result follows from Theorem 1.1. ∎

##  4\. Local estimates

In this section, we develop interior a priori estimates of $L^{\theta}$-strong
solutions of non-uniformly elliptic Pucci extremal operators. The proof
utilizes the ABP maximum principle [Theorem 1.1].

We begin with the local boundedness for $L^{\theta}$-strong subsolutions.

###### Proof of Theorem 1.2.

For simplicity, we omit “a.e.” if no confusion occurs. For $\beta\geq 2$ to be
determined, we define a cut-off function $\eta$ by

(4.1) $\displaystyle\eta(x)=(1-|x|^{2})^{\beta}.$

Then we immediately observe

$\displaystyle D_{i}\eta$ $\displaystyle=-2\beta x_{i}(1-|x|^{2})^{\beta-1}.$
$\displaystyle D_{ij}\eta$
$\displaystyle=-2\beta\delta_{ij}(1-|x|^{2})^{\beta-1}+4\beta(\beta-1)x_{i}x_{j}(1-|x|^{2})^{\beta-2}.$

By setting $v=\eta u$, we have

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}v)$
$\displaystyle=\mathcal{M}_{\lambda,\Lambda}^{+}(\eta D^{2}u+Du\otimes
D\eta+D\eta\otimes Du+uD^{2}\eta)$
$\displaystyle\geq\mathcal{M}_{\lambda,\Lambda}^{+}(\eta
D^{2}u)+\mathcal{M}_{\lambda,\Lambda}^{-}(Du\otimes D\eta+D\eta\otimes
Du+uD^{2}\eta)$ $\displaystyle=:I_{1}+I_{2},$

where we write $(x\otimes y)_{ij}=x_{i}y_{j}$ for $x,y\in\mathbb{R}^{n}$. For
$I_{1}$, we have

$\displaystyle I_{1}=\eta\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq\eta
f\geq-f^{-}.$

For $I_{2}$, we let $\Gamma^{+}_{v}$ be the upper contact set of $v^{+}$ in
$B_{1}$; then we have $u\geq 0$ and $v$ is concave on $\Gamma_{v}^{+}$.
Moreover, on $\Gamma_{v}^{+}$, we observe that

$\displaystyle|Du|$ $\displaystyle=\frac{1}{\eta}|Dv-
uD\eta|\leq\frac{1}{\eta}\left(\frac{v}{1-|x|}+u|D\eta|\right)\leq
2(1+\beta)\eta^{-1/\beta}u.$

It follows from the estimates

$\displaystyle|Du||D\eta|$ $\displaystyle\leq
4\beta(1+\beta)\eta^{-2/\beta}v\leq 8\beta^{2}\eta^{-2/\beta}v,$
$\displaystyle u|D^{2}\eta|$
$\displaystyle\leq(2\beta\eta^{1/\beta}+4\beta(\beta-1))\eta^{-2/\beta}v\leq
4\beta^{2}\eta^{-2/\beta}v,$

that

$\displaystyle I_{2}\geq-20\Lambda\beta^{2}\eta^{-2/\beta}v.$

We now apply the ABP estimates [Corollary 3.3] to derive

$\displaystyle\sup_{B_{1}}v$ $\displaystyle\leq
C\left(\int_{\Gamma_{v}^{+}}\left[\left(\frac{\beta^{2}\Lambda(x)\eta^{-2/\beta}(x)v^{+}(x)}{\lambda(x)}\right)^{n}+\left(\frac{f^{-}(x)}{\lambda(x)}\right)^{n}\right]\,\mathrm{d}x\right)^{1/n}$
$\displaystyle\leq
C\left(\left(\sup_{B_{1}}v^{+}\right)^{1-2/\beta}\left\|(u^{+})^{2/\beta}\Lambda/\lambda\right\|_{L^{n}(B_{1})}+\|f^{-}/\lambda\|_{L^{n}(B_{1})}\right)$

By choosing $\beta=2n/t$ (provided that $t\leq n$) and using Young’s
inequality,

$\displaystyle\left(\sup_{B_{1}}v^{+}\right)^{1-t/n}\leq\varepsilon\sup_{B_{1}}v^{+}+\varepsilon^{1-n/t}\quad\text{for
any $\varepsilon>0$}.$

In particular, the choice

$\displaystyle\varepsilon=\frac{1}{2C}\left\|(u^{+})^{t/n}\Lambda/\lambda\right\|_{L^{n}(B_{1})}^{-1}$

yields that

$\displaystyle\sup_{B_{1/2}}u\leq
C\left(\left\|(u^{+})^{t/n}\Lambda/\lambda\right\|_{L^{n}(B_{1})}^{n/t}+\|f^{-}/\lambda\|_{L^{n}(B_{1})}\right).$

Finally, an application of Hölder’s inequality concludes that

$\displaystyle\sup_{B_{1/2}}u\leq
C\left(\|1/\lambda\|_{p}^{n/t}\|\Lambda\|_{q}^{n/t}\|u^{+}\|_{\theta
t/n}+\|f^{-}/\lambda\|_{n}\right).$

∎

We now move our attention to the weak Harnack inequality for
$L^{\theta}$-strong supersolutions.

###### Proof of Theorem 1.3.

For $\varepsilon>0$, we set

$\displaystyle\overline{u}$
$\displaystyle=u+\varepsilon+\|f/\lambda\|_{L^{n}(B_{1})},$ $\displaystyle w$
$\displaystyle=-\log\overline{u},\quad v=\eta w,\quad\text{and}\quad
g=f/\overline{u},$

where $\eta$ is the cut-off function defined by (4.1) with $\beta\geq 2$ to be
determined. It is easily checked that

$\displaystyle D_{i}w$ $\displaystyle=-\overline{u}^{-1}D_{i}\overline{u},$
$\displaystyle D_{ij}w$
$\displaystyle=\overline{u}^{-2}D_{i}\overline{u}D_{j}\overline{u}-\overline{u}^{-1}D_{ij}\overline{u}=D_{i}wD_{j}w-\overline{u}^{-1}D_{ij}{u}.$

Then a direct calculation yields that

(4.2) $\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}v)$
$\displaystyle=\mathcal{M}_{\lambda,\Lambda}^{+}(\eta D^{2}w+2Dw\otimes
D\eta+wD^{2}\eta)$
$\displaystyle=\mathcal{M}_{\lambda,\Lambda}^{+}(-\eta\overline{u}^{-1}D^{2}{u}+\eta
Dw\otimes Dw+Dw\otimes D\eta+D\eta\otimes Dw+wD^{2}\eta)$
$\displaystyle\geq\mathcal{M}_{\lambda,\Lambda}^{+}(-\eta\overline{u}^{-1}D^{2}{u})+\mathcal{M}_{\lambda,\Lambda}^{-}(\eta
Dw\otimes Dw+Dw\otimes D\eta+D\eta\otimes
Dw)+\mathcal{M}_{\lambda,\Lambda}^{-}(wD^{2}\eta)$
$\displaystyle\geq-g\eta+\mathcal{M}_{\lambda,\Lambda}^{-}(\eta Dw\otimes
Dw+Dw\otimes D\eta+D\eta\otimes
Dw)+\mathcal{M}_{\lambda,\Lambda}^{-}(wD^{2}\eta).$

  1. (i)

We first prove the following Cauchy-Schwarz inequality for matrices:

$\displaystyle\pm(Dw\otimes D\eta+D\eta\otimes Dw)\leq\eta^{-1}D\eta\otimes
D\eta+\eta Dw\otimes Dw.$

It can be written in an equivalent form:

$\displaystyle\left|\langle(Dw\otimes D\eta+D\eta\otimes
Dw)a,a\rangle\right|\leq\langle(\eta^{-1}D\eta\otimes D\eta+\eta Dw\otimes
Dw)a,a\rangle\quad\text{for any $a\in\mathbb{R}^{n}$}.$

Indeed, this inequality follows from a simple observation

$\displaystyle\langle(b\otimes c)a,a\rangle=[(b\otimes c)a]_{i}a_{i}=(b\otimes
c)_{ij}a_{j}a_{i}=a_{i}b_{i}a_{j}c_{j}=\langle a,b\rangle\langle a,c\rangle$

for any $a,b,c\in\mathbb{R}^{n}$.

  2. (ii)

We control the term $\eta^{-1}|D\eta|^{2}$ as

$\displaystyle\eta^{-1}|D\eta|^{2}\leq 4\beta^{2}\eta^{1-2/\beta}.$

  3. (iii)

The eigenvalues of $D^{2}\eta$ are

$\displaystyle
4\beta(\beta-1)(1-|x|^{2})^{\beta-2}|x|^{2}-2\beta(1-|x|^{2})^{\beta-1}\quad$
$\displaystyle\text{with multiplicity $1$},$
$\displaystyle-2\beta(1-|x|^{2})^{\beta-1}\quad$ $\displaystyle\text{with
multiplicity $n-1$}.$

We note that the first eigenvalue is nonnegative if $\alpha\leq|x|\leq 1$ and
$\beta\geq 1+1/(2\alpha^{2})$ for $\alpha:=1/(3n)$. Therefore, for
$\alpha\leq|x|\leq 1$ and $\beta\geq 1+1/(2\alpha^{2})$, we obtain

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}\eta)$
$\displaystyle=\lambda[4\beta(\beta-1)(1-|x|^{2})^{\beta-2}|x|^{2}-2\beta(1-|x|^{2})^{\beta-1}]-\Lambda(n-1)[2\beta(1-|x|^{2})^{\beta-1}]$
$\displaystyle=\lambda[4\beta(\beta-1)(1-|x|^{2})^{\beta-2}|x|^{2}]-(\lambda+(n-1)\Lambda)[2\beta(1-|x|^{2})^{\beta-1}]$
$\displaystyle=2\beta(1-|x|^{2})^{\beta-2}\left[2\lambda(\beta-1)|x|^{2}-(\lambda+(n-1)\Lambda)(1-|x|^{2})\right].$

On the other hand, if $|x|\leq\alpha$, then

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}\eta)\geq-\Lambda
n[2\beta(1-|x|^{2})^{\beta-1}].$

By taking the previous estimates obtained in (i), (ii), and (iii) account into
(4.2), we have

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}v)$
$\displaystyle\geq-g\eta-\eta^{-1}\mathcal{M}_{\lambda,\Lambda}^{+}(D\eta\otimes
D\eta)+\mathcal{M}_{\lambda,\Lambda}^{-}(wD^{2}\eta)$
$\displaystyle=-g\eta-\eta^{-1}\Lambda|D\eta|^{2}+\mathcal{M}_{\lambda,\Lambda}^{-}(wD^{2}\eta)$
$\displaystyle\geq-|g|-4\beta^{2}\Lambda-\frac{2\Lambda
n\beta}{1-\alpha^{2}}v^{+}\mathbf{1}_{\\{|x|\leq\alpha\\}}$
$\displaystyle\quad+2\beta(1-|x|^{2})^{-2}\left[2\lambda(\beta-1)|x|^{2}-(\lambda+(n-1)\Lambda)(1-|x|^{2})\right]v^{+}\mathbf{1}_{\\{|x|\geq\alpha\\}}$
$\displaystyle=:\widetilde{f}$

on $\Gamma_{v}^{+}$. We now apply the ABP estimates [Corollary 3.3] to derive

$\displaystyle\sup_{B_{1}}v$ $\displaystyle\leq
C\left(\int_{\Gamma_{v}^{+}}\left(\frac{\widetilde{f}^{-}(x)}{\lambda(x)}\right)^{n}\,\mathrm{d}x\right)^{1/n}.$

Therefore, by recalling that $\|g/\lambda\|_{n}\leq 1$, we obtain

$\displaystyle\sup_{B_{1}}v$ $\displaystyle\leq
C+C\beta^{2}\|\Lambda/\lambda\|_{n}+C\beta\|\Lambda/\lambda\|_{n}\sup_{B_{1}}v\cdot|\\{|x|\leq\alpha\\}\cap\\{v>0\\}|^{1/n}$
$\displaystyle\quad+C\sup_{B_{1}}v\left(\int_{\alpha\leq|x|\leq
1}\left[\left(\frac{\Lambda}{\lambda}-\frac{\beta}{1-|x|^{2}}\right)_{+}\frac{\beta}{1-|x|^{2}}\right]^{n}\,\mathrm{d}x\right)^{1/n}.$

Since $1/p+1/q<1/(2n)$, an application of Hölder inequality yields that

$\displaystyle\int_{\alpha\leq|x|\leq
1}\left[\left(\frac{\Lambda}{\lambda}-\frac{\beta}{1-|x|^{2}}\right)_{+}\frac{\beta}{1-|x|^{2}}\right]^{n}\,\mathrm{d}x$
$\displaystyle\leq\int_{\\{\alpha\leq|x|\leq 1\\}\cap
U_{\beta}}\left(\frac{\Lambda}{\lambda}\right)^{2n}\,\mathrm{d}x$
$\displaystyle\leq\|1/\lambda\|_{p}^{2n}\|\Lambda\|_{q}^{2n}\left|U_{\beta}\right|^{1-(2n)/p-(2n)/q}$

where

$U_{\beta}:=\left\\{|x|\leq
1:\frac{\Lambda(x)}{\lambda(x)}\geq\frac{\beta}{1-|x|^{2}}\right\\}.$

We also have the following inequality:

$\displaystyle|U_{\beta}|\leq\left|\left\\{\frac{\Lambda}{\lambda}\geq\beta\right\\}\right|\leq\beta^{-2n}\int|\Lambda/\lambda|^{2n}.$

Hence, there exists a constant $\beta>0$ which depends only on
$\left\|1/\lambda\right\|_{p}$, $\|\Lambda\|_{L^{q}}$, and $n$ such that

$\displaystyle\int_{\alpha\leq|x|\leq
1}\left[\left(\frac{\Lambda}{\lambda}-\frac{\beta}{1-|x|^{2}}\right)_{+}\frac{\beta}{1-|x|^{2}}\right]^{n}\,\mathrm{d}x\leq\frac{1}{(2C)^{n}}.$

By combining all estimates above, we conclude that

(4.3) $\displaystyle\sup_{B_{1}}v\leq
C+C\sup_{B_{1}}v\cdot|\\{|x|\leq\alpha\\}\cap\\{v>0\\}|^{1/n}.$

In order to finish the proof, we would like to exploit the cube decomposition
lemma [Lemma 2.7]. For this purpose, let us define $K_{R}(z)$ be the open
cube, parallel to the coordinate axes, with center $z$ and the side length
$2R$. Since $B_{\alpha}\subset K_{\alpha}(0)\subset\subset B_{1}$ for
$\alpha=1/(3n)$, we have

$\displaystyle\sup_{B_{1}}v\leq
C(1+\sup_{B_{1}}v^{+}|K_{\alpha}^{+}|^{\frac{1}{n}}),$

where $K_{\alpha}^{+}:=\\{x\in K_{\alpha}\,|\,v>0\\}$. Hence, whenever

$\displaystyle\frac{|K_{\alpha}^{+}|}{|K_{\alpha}|}\leq\theta:=[2(2\alpha)^{n}C]^{-1},$

we obtain

$\displaystyle\sup_{B_{1}}v\leq 2C.$

We point out that

  1. (i)

this procedure is stable under the transformation $x\to\alpha(x-z)/r$ for
$B_{r/\alpha}(z)\subset B_{1}(0)$;

  2. (ii)

we can repeat this argument for $w-k$ instead of $w$ for arbitrary
$k\in\mathbb{R}$.

Thus, by applying Lemma 2.7 with $\delta=1-\theta,\;K_{0}=K_{\alpha}(0)$, and
$\alpha=1/(3n)$, we obtain

$\displaystyle\sup_{K_{0}}\,(w-k)\leq
C\left(1+\frac{\log(|D_{k}|/|K_{0}|)}{\log\delta}\right),$

where $D_{k}=\left\\{x\in K_{0}\,|\,w(x)\leq k\right\\}$. In other words, if
we write

$\displaystyle\mu_{t}=|\\{x\in K_{0}\,|\,\bar{u}>t\\}|\quad\text{with
$t=e^{-k}$},$

then

$\displaystyle\mu_{t}\leq C\left(\inf_{K_{0}}\bar{u}/t\right)^{\kappa},$

where $C$ and $\kappa$ are positive universal constants. By recalling [20,
Lemma 9.7], we obtain

$\displaystyle\int_{B_{\alpha}}\bar{u}^{t}\leq
C\left(\inf_{B_{\alpha}}\bar{u}\right)^{t},$

for $t=\kappa/2$. The desired weak Harnack inequality follows by letting
$\varepsilon\to 0$ together with the covering and scaling argument. ∎

We remark that if $u$ is a strong solution of $F(D^{2}u,x)=f(x)$ for a
$(\lambda(\cdot),\Lambda(\cdot))$-elliptic operator $F$ with $F(0,x)=0$, then
$u$ is contained in the (extended) Pucci class, i.e., $u$ satisfies

$\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq-|f|\quad\text{and}\quad\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\leq|f|.$

Indeed, the following corollaries hold for a wide class of functions: not only
solutions of degenerate/singular fully nonlinear equations, but also functions
in the (extended) Pucci class.

######  Corollary 4.1 (Harnack inequality).

Let $f\in L^{\tau}(B_{1})$. Assume that

$\displaystyle\frac{1}{p}+\frac{1}{q}<\frac{1}{2n}.$

Moreover, suppose that $u\in W^{2,\theta}_{\mathrm{loc}}(B_{1})$ be an
nonnegative $L^{\theta}$-strong solution of

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq-|f|\quad\text{and}\quad\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\leq|f|\quad\text{in
$B_{1}$}.$

Then there exists a constant $C>0$ depending only on $\|1/\lambda\|_{p}$,
$\|\Lambda\|_{q}$, and $n$ such that

$\displaystyle\sup_{B_{1/2}}u\leq
C\left(\inf_{B_{1/2}}u+\|f/\lambda\|_{L^{n}(B_{1})}\right).$

######  Corollary 4.2 (A priori Hölder estimates).

Let $f\in L^{\tau}(B_{1})$. Assume that

$\displaystyle\frac{1}{p}+\frac{1}{q}<\frac{1}{2n}.$

Moreover, suppose that $u\in W^{2,\theta}_{\mathrm{loc}}(B_{1})$ be an
$L^{\theta}$-strong solution of

$\displaystyle\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)\geq-|f|\quad\text{and}\quad\mathcal{M}_{\lambda,\Lambda}^{-}(D^{2}u)\leq|f|\quad\text{in
$B_{1}$}.$

Then there exist constants $C,\alpha>0$ depending only on $\|1/\lambda\|_{p}$,
$\|\Lambda\|_{q}$, and $n$ such that the following estimate holds:

$\displaystyle\|u\|_{C^{\alpha}(\overline{B_{1/2}})}\leq
C\left(\|u\|_{L^{\infty}(B_{1})}+\|f/\lambda\|_{L^{n}(B_{1})}\right).$

###### Proof of Corollary 4.1 and Corollary 4.2.

Once we obtain the local boundedness [Theorem 1.2] and the weak Harnack
inequality [Theorem 1.3], both corollaries easily follow; we refer to [11,
Section 4] for instance. ∎

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