# Local Hölder regularity for nonlocal parabolic $\displaystyle p$-Laplace
equations

Karthik Adimurthi111Supported by the Department of Atomic Energy, Government
of India, under project no. 12-R&D-TFR-5.01-0520 and SERB grant
SRG/2020/000081<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Harsh
Prasad222Supported by the Department of Atomic Energy, Government of India,
under project no. 12-R&D-TFR-5.01-0520<EMAIL_ADDRESS>Vivek
Tewary333Supported by the Department of Atomic Energy, Government of India,
under project no. 12-R&D-TFR-5.01-0520<EMAIL_ADDRESS>Tata Institute
of Fundamental Research, Centre for Applicable Mathematics, Bangalore,
Karnataka, 560065, India  School of Interwoven Arts and Sciences, Krea
University, Sri City, Andhra Pradesh, 517646, India

###### Abstract

We prove local Hölder regularity for nonlocal parabolic equations of the form

$\displaystyle\displaystyle\partial_{t}u+\text{P.V.}\int_{\mathbb{R}^{N}}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\,dy=0,$

for $\displaystyle p\in(1,\infty)$ and $\displaystyle s\in(0,1)$. Our
contribution extends the fundamental work of Caffarelli, Chan and Vasseur on
linear nonlocal parabolic equations to equations modeled on the fractional
$\displaystyle p$-Laplacian.

###### keywords:

Nonlocal operators; Weak Solutions; Hölder regularity

###### MSC:

[2010] 35K92, 35B65, 35K51, 35A01, 35A15, 35R11.

###### Contents

  1. 1 Introduction
    1. 1.1 A brief history of the problem
    2. 1.2 On historical development of intrinsic scaling
  2. 2 Notations and Auxiliary Results
    1. 2.1 Notations
    2. 2.2 Structure of the equation
    3. 2.3 Function spaces
    4. 2.4 Definition of weak solution
    5. 2.5 Auxiliary Results
    6. 2.6 Main results
  3. 3 Preliminary Results
    1. 3.1 Energy estimates
    2. 3.2 Shrinking Lemma
    3. 3.3 Tail Estimates
    4. 3.4 De Giorgi iteration Lemma
    5. 3.5 De Giorgi lemma: Forward in time version
  4. 4 Qualitative expansion of positivity in time
  5. 5 Expansion of positivity for Nonlocal Degenerate Equations
  6. 6 Expansion of positivity for Nonlocal Singular Equations
  7. 7 Hölder regularity for degenerate parabolic fractional $\displaystyle p$-Laplace equations
  8. 8 Hölder regularity for singular parabolic fractional $\displaystyle p$-Laplace equations

##  1 Introduction

In this article, we prove local Hölder regularity for a nonlocal parabolic
equation whose prototype equation is the parabolic fractional $\displaystyle
p$-Laplacian of the form

$\displaystyle\displaystyle\partial_{t}u+\text{P.V.}\int_{\mathbb{R}^{N}}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\,dy=0,$

with $\displaystyle p\in(1,\infty)$ and $\displaystyle s\in(0,1)$. The main
tools in our investigation are

  * •

the exponential change of variable in time discovered by E.DiBenedetto,
U.Gianazza and V.Vespri, which they used to prove Harnack’s inequality for the
parabolic $\displaystyle p$-Laplace equations. The standard references for
these works are [22, 23]. The exponential change of variable in time leads to
an expansion of positivity result which is suitable for proving Hölder
regularity results. N.Liao in [32] has effected a direct proof of Hölder
regularity for parabolic $\displaystyle p$-Laplace equation by using the
expansion of positivity. It has also been used to give the first proofs of
Hölder regularity for sign-changing solutions of doubly nonlinear equations of
porous medium type in [6, 7, 33]. A recent work that employs the exponential
change of variable to deduce regularity for elliptic anisotropic equations is
[34].

  * •

the energy estimate for fractional parabolic $\displaystyle p$-Laplace
equations containing the “good term” which substitutes the De Giorgi
isoperimetric inequality in the so-called “shrinking lemma”. The “good term”
has been used to great effect in the works [11, 12]. It has been used to
define a novel De Giorgi class by Cozzi in [14] to prove Hölder regularity for
fractional $\displaystyle p$-minimizers. Earlier, the absence of De Giorgi
isoperimetric inequality was tackled by Di Castro et. al. through a
logarithmic estimate [19]. See [15, 1] for discussions on fractional versions
of De Giorgi isoperimetric inequalities.

We demonstrate two purely nonlocal phenomenon - one at a technical level which
has implications for what the corresponding De Giorgi conjectures related to
the "best" Holder exponent in the nonlocal case should be (see Remark 3.6) and
another at the level of regularity which says that in certain cases, time
regularity can beat space regularity when we are working with nonlocal
equations (see Remark 8.3).

Previously, local boundedness for fractional parabolic $\displaystyle
p$-Laplace equations is proved in the papers [39, 24, 36]. Regarding Hölder
regularity for fractional parabolic $\displaystyle p$-Laplace equations in the
spirit of De Giorgi-Nash-Moser, the only result that we are aware of is for
the case $\displaystyle 2\leq p<\infty$ which was studied in [24], though we
were unable to verify some of the calculations, particularly (5.23),
pertaining to the logarithmic estimates in their paper. Explicit exponents for
Hölder regularity for fractional parabolic $\displaystyle p$-Laplace equations
with no coefficients appears in [10].

###  1.1 A brief history of the problem

Much of the early work on regularity of fractional elliptic equations in the
case $\displaystyle p=2$ was carried out by Silvestre [38], Caffarelli and
Vasseur [11], Caffarelli, Chan, Vasseur [12] and also Bass-Kassmann [4, 5,
31]. An early formulation of the fractional $\displaystyle p$-Laplace operator
was done by Ishii and Nakamura [30] and existence of viscosity solutions was
established. Di Castro, Kuusi and Palatucci extended the De Giorgi-Nash-Moser
framework to study the regularity of the fractional $\displaystyle p$-Laplace
equation in [19, 18]. The subsequent work of Cozzi [14] covered a stable (in
the limit $\displaystyle s\to 1$) proof of Hölder regularity by defining a
novel fractional De Giorgi class. Explicit exponents for Hölder regularity
were found in [9, 10]. Other works of interest are [29, 17, 13, 25, 16] and in
the parabolic context, some works of interest are [2, 3, 24, 37, 36].

###  1.2 On historical development of intrinsic scaling

The method of intrinsic scaling was developed by E.DiBenedetto in [20] to
prove Hölder regularity for degenerate quasilinear parabolic equations
modelled on the $\displaystyle p$-Laplace operator. A technical requirement of
the proof was a novel logarithmic estimate which aids in the expansion of
positivity. Subsequently, the proof of Hölder regularity for the singular case
was given in [40] by switching the scaling from time variable to the space
variable. These results were collected in E.DiBenedetto’s treatise [21]. After
a gap of several years, E.DiBenedetto, U.Gianazza and V.Vespri [22, 23] were
able to prove Harnack’s inequality for the parabolic $\displaystyle p$-Laplace
equations by a new technique involving an exponential change of variable in
time. This proof relies on expansion of positivity estimates and does not
involve logarithmic test functions. Then, a new proof of Hölder regularity was
given with a more geometric flavour in [26]. This theory was extended to
generalized parabolic $\displaystyle p$-Laplace equations with Orlicz growth
in [27, 28].

######  Remark 1.1.

In this paper, we assume that solutions to the nonlocal equation are locally
bounded. Local boundedness for the nonlocal elliptic equations is proved in
[19, Theorem 1.1] and [14, Theorem 6.2]. In the parabolic case, local
boundedness is proved in [39, 24, 36].

##  2 Notations and Auxiliary Results

In this section, we will fix the notation, provide definitions and state some
standard auxiliary results that will be used in subsequent sections.

###  2.1 Notations

We begin by collecting the standard notation that will be used throughout the
paper:

N1:

We shall denote $\displaystyle N$ to be the space dimension and by
$\displaystyle z=(x,t)$ to be a point in
$\displaystyle\mathbb{R}^{N}\times(0,T)$.

N2:

We shall alternately use $\displaystyle\dfrac{\partial f}{\partial
t},\partial_{t}f,f^{\prime}$ to denote the time derivative of f.

N3:

Let $\displaystyle\Omega$ be an open bounded domain in
$\displaystyle\mathbb{R}^{N}$ with boundary $\displaystyle\partial\Omega$ and
for $\displaystyle 0<T\leq\infty$, let
$\displaystyle\Omega_{T}\coloneqq\Omega\times(0,T)$.

N4:

We shall use the notation

$\begin{array}[]{ll}B_{\rho}(x_{0})=\\{x\in\mathbb{R}^{N}:|x-x_{0}|<\rho\\},&\overline{B}_{\rho}(x_{0})=\\{x\in\mathbb{R}^{N}:|x-x_{0}|\leq\rho\\},\\\
I_{\theta}(t_{0})=\\{t\in\mathbb{R}:t_{0}-\theta<t<t_{0}\\},&Q_{\rho,\theta}(z_{0})=B_{\rho}(x_{0})\times
I_{\theta}(t_{0}).\end{array}$

N5:

The maximum of two numbers $\displaystyle a$ and $\displaystyle b$ will be
denoted by $\displaystyle a\wedge b\coloneqq\max(a,b)$.

N6:

Integration with respect to either space or time only will be denoted by a
single integral $\displaystyle\int$ whereas integration on
$\displaystyle\Omega\times\Omega$ or
$\displaystyle\mathbb{R}^{N}\times\mathbb{R}^{N}$ will be denoted by a double
integral $\displaystyle\iint$.

N7:

We will use $\displaystyle\iiint$ to denote integral over
$\displaystyle\mathbb{R}^{N}\times\mathbb{R}^{N}\times(0,T)$. More
specifically, we will use the notation $\displaystyle\iiint_{Q}$ and
$\displaystyle\iiint_{B\times I}$ to denote the integral over
$\displaystyle\iiint_{B\times B\times I}$.

N8:

The notation $\displaystyle a\lesssim b$ is shorthand for $\displaystyle a\leq
Cb$ where $\displaystyle C$ is a universal constant which only depends on the
dimension $\displaystyle N$, exponents $\displaystyle p$, and the numbers
$\displaystyle\Lambda$ and $\displaystyle s$.

N9:

For a function $\displaystyle u$ defined on the cylinder $\displaystyle
Q_{\rho,\theta}(z_{0})$ and any level $\displaystyle k\in\mathbb{R}$ we write
$\displaystyle w_{\pm}=(u-k)_{\pm}$

N10:

For any fixed $\displaystyle t,k\in\mathbb{R}$ and set
$\displaystyle\Omega\subset\mathbb{R}^{N}$, we denote $\displaystyle
A_{\pm}(k,t)=\\{x\in\Omega:(u-k)_{\pm}(\cdot,t)>0\\}$; for any ball
$\displaystyle B_{r}$ we write $\displaystyle A_{\pm}(k,t)\cap
B_{r}=A_{\pm}(k,r,t)$.

N11:

For any set $\displaystyle\Omega\subset\mathbb{R}^{N}$, we denote
$\displaystyle
C_{\Omega}:=(\Omega^{c}\times\Omega^{c})^{c}=\left(\Omega\times\Omega\right)\cup\left(\Omega\times(\mathbb{R}^{N}\setminus\Omega)\right)\cup\left((\mathbb{R}^{N}\setminus\Omega)\times\Omega\right)$.

###  2.2 Structure of the equation

Let $\displaystyle s\in(0,1)$ and $\displaystyle p>1$ be fixed, and
$\displaystyle\Lambda\geq 1$ be a given constant. For almost every
$\displaystyle x,y\in\mathbb{R}^{N}$, let $\displaystyle
K:\mathbb{R}^{N}\times\mathbb{R}^{N}\times\mathbb{R}\to[0,\infty)$ be a
symmetric measurable function satisfying

$\frac{(1-s)}{\Lambda|x-y|^{N+ps}}\leq
K(x,y,t)\leq\frac{(1-s)\Lambda}{|x-y|^{N+ps}}.$ (2.1)

In this paper, we are interested in the regularity theory for the equation

$\displaystyle\displaystyle\partial_{t}u+\mathcal{L}u=0,$ (2.2)

where $\displaystyle\mathcal{L}$ is the operator formally defined by

$\mathcal{L}u=\text{P.V.}\int_{\mathbb{R}^{N}}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))\,dy,\,x\in\mathbb{R}^{N}.$

###  2.3 Function spaces

Let $\displaystyle 1<p<\infty$, we denote by $\displaystyle
p^{\prime}=p/(p-1)$ the conjugate exponent of $\displaystyle p$. Let
$\displaystyle\Omega$ be an open subset of $\displaystyle\mathbb{R}^{N}$, we
define the Sobolev-Slobodekiĭ space, which is the fractional analogue of
Sobolev spaces as follows:

$\displaystyle\displaystyle W^{s,p}(\Omega):=\left\\{\psi\in
L^{p}(\Omega):[\psi]_{W^{s,p}(\Omega)}<\infty\right\\},\qquad\text{for}\
s\in(0,1),$

where the seminorm $\displaystyle[\cdot]_{W^{s,p}(\Omega)}$ is defined by

$\displaystyle\displaystyle[\psi]_{W^{s,p}(\Omega)}:=\left(\iint_{\Omega\times\Omega}\frac{|\psi(x)-\psi(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy\right)^{\frac{1}{p}}.$

The space when endowed with the norm
$\displaystyle\left\lVert\psi\right\rVert_{W^{s,p}(\Omega)}=\left\lVert\psi\right\rVert_{L^{p}(\Omega)}+[\psi]_{W^{s,p}(\Omega)}$
becomes a Banach space. The space $\displaystyle W^{s,p}_{0}(\Omega)$ is the
completion of $\displaystyle C_{c}^{\infty}(\Omega)$ in the norm of
$\displaystyle W^{s,p}(\mathbb{R}^{N})$. We will use the notation
$\displaystyle W^{s,p}_{u_{0}}(\Omega)$ to denote the space of functions in
$\displaystyle W^{s,p}(\mathbb{R}^{N})$ such that $\displaystyle u-u_{0}\in
W^{s,p}_{0}(\Omega)$.

Let $\displaystyle I$ be an interval and let $\displaystyle V$ be a separable,
reflexive Banach space, endowed with a norm
$\displaystyle\left\lVert\cdot\right\rVert_{V}$. We denote by $\displaystyle
V^{*}$ to be its topological dual space. Let $\displaystyle v$ be a mapping
such that for a.e. $\displaystyle t\in I$, $\displaystyle v(t)\in V$. If the
function $\displaystyle t\mapsto\left\lVert v(t)\right\rVert_{V}$ is
measurable on $\displaystyle I$, then $\displaystyle v$ is said to belong to
the Banach space $\displaystyle L^{p}(I;V)$ provided
$\displaystyle\int_{I}\left\lVert v(t)\right\rVert_{V}^{p}\,dt<\infty$. It is
well known that the dual space $\displaystyle L^{p}(I;V)^{*}$ can be
characterized as $\displaystyle L^{p^{\prime}}(I;V^{*})$.

Since the boundedness result requires some finiteness condition on the
nonlocal tails, we define the tail space for some $\displaystyle m>0$ and
$\displaystyle s>0$ as follows:

$L^{m}_{s}(\mathbb{R}^{N}):=\left\\{v\in
L^{m}_{\text{loc}}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\frac{|v(x)|^{m}}{1+|x|^{N+s}}\,dx<+\infty\right\\}.$

Then a nonlocal tail is defined by

$\text{Tail}_{m,s,\infty}(v;x_{0},R,I):=\text{Tail}_{\infty}(v;x_{0},R,t_{0}-\theta,t_{0}):=\sup_{t\in(t_{0}-\theta,t_{0})}\left(R^{sm}\int_{\mathbb{R}^{N}\setminus
B_{R}(x_{0})}\frac{|v(x,t)|^{m-1}}{|x-x_{0}|^{N+sm}}\,dx\right)^{\frac{1}{m-1}},$

where $\displaystyle(x_{0},t_{0})\in\mathbb{R}^{N}\times(-T,T)$ and the
interval $\displaystyle I=(t_{0}-\theta,t_{0})\subseteq(-T,T)$. From this
definition, it follows that for any $\displaystyle v\in
L^{\infty}(-T,T;L^{m-1}_{sm}(\mathbb{R}^{N}))$, there holds
$\displaystyle\text{Tail}_{m,s,\infty}(v;x_{0},R,I)<\infty$.

###  2.4 Definition of weak solution

Now, we are ready to state the definition of a weak sub(super)-solution.

######  Definition 2.1.

A function $\displaystyle u\in
L^{p}_{\text{loc}}(I;W^{s,p}_{\text{loc}}(\Omega))\cap
C_{\text{loc}}(I;L^{2}_{\text{loc}}(\Omega))\cap
L^{\infty}_{\text{loc}}(I;L^{p-1}_{sp}(\mathbb{R}^{N}))$ is said to be a local
weak sub(super)-solution to Eq. 2.2 in $\displaystyle\Omega\times I$ if for
any closed interval $\displaystyle[t_{1},t_{2}]\subset I$ and any compact set
$\displaystyle\Omega^{\prime}\subset\Omega$, the following holds:

$\displaystyle\displaystyle\int_{\Omega^{\prime}}$ $\displaystyle\displaystyle
u(x,t_{2})\phi(x,t_{2})\,dx-\int_{\Omega^{\prime}}u(x,t_{1})\phi(x,t_{1})\,dx-\int_{t_{1}}^{t_{2}}\int_{\Omega^{\prime}}u(x,t)\partial_{t}\phi(x,t)\,dx\,dt$
$\displaystyle\displaystyle+\int_{t_{1}}^{t_{2}}\iint_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\,K(x,y,t)|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(\phi(x,t)-\phi(y,t))\,dy\,dx\,dt\leq(\geq)0,$

for all $\displaystyle\phi\in
L^{p}_{\text{loc}}(I,W^{s,p}_{0}(\Omega^{\prime}))\cap
W^{1,2}_{\text{loc}}(I,L^{2}(\Omega^{\prime}))$.

###  2.5 Auxiliary Results

We collect the following standard results which will be used in the course of
the paper. We begin with the Sobolev-type inequality [24, Lemma 2.3].

######  Theorem 2.2.

Let $\displaystyle t_{2}>t_{1}>0$ and suppose $\displaystyle s\in(0,1)$ and
$\displaystyle 1\leq p<\infty$. Then for any $\displaystyle f\in
L^{p}(t_{1},t_{2};W^{s,p}(B_{r}))\cap L^{\infty}(t_{1},t_{2};L^{2}(B_{r}))$,
we have

$\displaystyle\displaystyle\int_{t_{1}}^{t_{2}}\fint_{B_{r}}|f(x,t)|^{p\left(1+\frac{2s}{N}\right)}\,dx\,dt\leq
C(N,s,p)$
$\displaystyle\displaystyle\left(r^{sp}\int_{t_{1}}^{t_{2}}\int_{B_{r}}\fint_{B_{r}}\frac{|f(x,t)-f(y,t)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\,dt+\int_{t_{1}}^{t_{2}}\fint_{B_{r}}|f(x,t)|^{p}\,dx\,dt\right)$
$\displaystyle\displaystyle\times\left(\sup_{t_{1}<t<t_{2}}\fint_{B_{r}}|f(x,t)|^{2}\,dx\right)^{\frac{sp}{N}}.$

We also list a number of algebraic inequalities that are customary in
obtaining energy estimates for nonlinear nonlocal equations.

######  Lemma 2.3.

([14, Lemma 4.1]) Let $\displaystyle p\geq 1$ and $\displaystyle a,b\geq 0$,
then for any $\displaystyle\theta\in[0,1]$, the following holds:

$\displaystyle\displaystyle(a+b)^{p}-a^{p}\geq\theta
pa^{p-1}b+(1-\theta)b^{p}.$

######  Lemma 2.4.

([14, Lemma 4.3]) Let $\displaystyle p\geq 1$ and $\displaystyle a\geq b\geq
0$, then for any $\displaystyle\varepsilon>0$, the following holds:

$\displaystyle\displaystyle a^{p}-b^{p}\leq\varepsilon
a^{p}+\left(\frac{p-1}{\varepsilon}\right)^{p-1}(a-b)^{p}.$

Finally, we recall the following well known lemma concerning the geometric
convergence of sequence of numbers (see [21, Lemma 4.1 from Section I] for the
details):

######  Lemma 2.5.

Let $\displaystyle\\{Y_{n}\\}$, $\displaystyle n=0,1,2,\ldots$, be a sequence
of positive number, satisfying the recursive inequalities

$\displaystyle Y_{n+1}\leq Cb^{n}Y_{n}^{1+\alpha},$

where $\displaystyle C>1$, $\displaystyle b>1$, and $\displaystyle\alpha>0$
are given numbers. If

$\displaystyle Y_{0}\leq C^{-\frac{1}{\alpha}}b^{-\frac{1}{\alpha^{2}}},$

then $\displaystyle\\{Y_{n}\\}$ converges to zero as $\displaystyle
n\to\infty$.

###  2.6 Main results

We prove the following main theorems.

######  Theorem 2.6.

Let $\displaystyle p\in(2,\infty)$ and let $\displaystyle u$ be a locally
bounded, local weak solution to Eq. 2.2 in $\displaystyle\Omega_{T}$. Then
$\displaystyle u$ is locally Hölder continuous in $\displaystyle\Omega_{T}$,
i.e., there exist constants $\displaystyle C_{0}>1$, and
$\displaystyle\beta\in(0,1)$ depending only on the data, such that with
$\displaystyle Q_{0}:=B_{C_{0}R}\times(-(C_{0}R)^{2s},0)$ and $\displaystyle
L$, $\displaystyle R_{i}$, $\displaystyle d_{i}$ for $\displaystyle
i\in\\{0,1,\ldots\\}$ as defined in Eq. 7.4 and $\displaystyle\mathbf{C}_{4}$
as defined in Eq. 7.19, we have

$\displaystyle\displaystyle|u(x_{1},t_{1})-u(x_{2},t_{2})|\leq\frac{L}{R^{\beta}}\left(\max\\{|x_{1}-x_{2}|,L^{\frac{p-2}{sp}}\mathbf{C}_{4}|t_{1}-t_{2}|^{\frac{1}{sp}}\\}\right)^{\beta}\,$

for every pair of points $\displaystyle(x_{1},t_{1}),(x_{2},t_{2})\in
B_{R}(0)\times(-d_{1}R^{sp},0)$.

######  Theorem 2.7.

Let $\displaystyle p\in(1,2)$ and let $\displaystyle u$ be a locally bounded,
local weak solution to Eq. 2.2 in $\displaystyle\Omega_{T}$. Then
$\displaystyle u$ is locally Hölder continuous in $\displaystyle\Omega_{T}$,
i.e., there exist constants $\displaystyle C_{0}>1$, and
$\displaystyle\beta\in(0,1)$ depending only on the data, such that for a fixed
$\displaystyle\varepsilon>0$ with $\displaystyle
Q_{0}:=B_{(C_{0}R)^{1-\varepsilon_{o}}}\times(-(C_{0}R)^{sp},0)$ and
$\displaystyle L$, $\displaystyle R_{i}$, $\displaystyle d_{i}$ for
$\displaystyle i\in\\{0,1,\ldots\\}$ as defined in Eq. 8.4 and
$\displaystyle\mathbf{C}_{4}$ as defined in Eq. 8.19, we have

$\displaystyle\displaystyle|u(x_{1},t_{1})-u(x_{2},t_{2})|\leq\frac{L}{R^{\beta}}\left(\max\\{L^{\frac{2-p}{sp}}\mathbf{C}_{4}|x_{1}-x_{2}|,|t_{1}-t_{2}|^{\frac{1}{sp}}\\}\right)^{\beta}\,$

for every pair of points $\displaystyle(x_{1},t_{1}),(x_{2},t_{2})\in
B_{d_{1}R}(0)\times(-R^{sp},0)$.

##  3 Preliminary Results

###  3.1 Energy estimates

######  Theorem 3.1.

Let $\displaystyle u$ be a local, weak sub(super)-solution in $\displaystyle
B_{R}(x_{0})\times(t_{0}-\theta,t_{0})\Subset E_{T}$. Let
$\displaystyle\tau_{1},\tau_{2}\in(0,\infty)$ be given such that
$\displaystyle t_{0}-\theta<t_{0}-\tau_{2}<t_{0}-\tau_{1}\leq t_{0}$ and
$\displaystyle 0<r<R$. Then there is positive universal constant
$\displaystyle C(N,p,s,\Lambda)$ such that for every level $\displaystyle
k\in\mathbb{R}$ and every piecewise smooth cutoff function
$\displaystyle\zeta(x,t)=\zeta_{1}(x)\zeta_{2}(t)$ with
$\displaystyle\zeta_{1}\in C_{c}^{\infty}(B_{R})$, $\displaystyle\zeta_{2}\in
C^{0,1}(t_{0}-\theta,t_{0})$, $\displaystyle 0\leq\zeta(x,t)\leq 1$,
$\displaystyle\zeta_{1}\equiv 1$ on $\displaystyle B_{r}(x_{0})$, we have

$\begin{array}[]{l}\underset{t_{0}-\tau_{2}<t<t_{0}-\tau_{1}}{\operatorname*{ess\,sup}}\int_{B_{r}(x_{0})}w_{\pm}^{2}\zeta^{p}(x,t)\,dx+\int_{t_{0}-\tau_{2}}^{t_{0}-\tau_{1}}\int_{B_{R}(x_{0})}\zeta^{p}(x,t)w_{\pm}(x,t)\int_{{B_{R}(x_{0})}}\frac{w_{\mp}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\,dx\,dt\\\
\hskip
56.9055pt+\int_{t_{0}-\tau_{2}}^{t_{0}-\tau_{1}}\iint_{B_{r}(x_{0})\times
B_{r}(x_{0})}\frac{|w_{\pm}(x,t)-w_{\pm}(y,t)|^{p}}{|x-y|^{N+ps}}\max\\{\zeta_{1}(x),\zeta_{1}(y)\\}^{p}\zeta_{2}^{p}(t)\,dx\,dy\,dt\\\
\hskip
85.35826pt\leq\int_{{B_{R}(x_{0})}}w_{\pm}^{2}\zeta^{p}(x,t_{0}-\tau_{2})\,dx\\\
\hskip
113.81102pt+C\int_{t_{0}-\tau_{2}}^{t_{0}-\tau_{1}}\iint_{{B_{R}(x_{0})}\times{B_{R}(x_{0})}}\max\\{w_{\pm}(x,t),w_{\pm}(y,t)\\}^{p}\frac{|\zeta_{1}(x)-\zeta_{1}(y)|^{p}}{|x-y|^{N+sp}}\zeta_{2}^{p}(t)\,dx\,dy\,dt\\\
\hskip
113.81102pt+C\int_{t_{0}-\tau_{2}}^{t_{0}-\tau_{1}}\int_{{B_{R}(x_{0})}}w_{\pm}^{2}(x,t)|\delta_{t}\zeta(x,t)|\,dx\,dt\\\
\hskip
113.81102pt+C\int_{t_{0}-\tau_{2}}^{t_{0}-\tau_{1}}\int_{{B_{R}(x_{0})}}\zeta^{p}(x,t)w_{\pm}(x,t)\left(\operatorname*{ess\,sup}\limits_{t\in(t_{0}-\tau_{2},t_{0}-\tau_{1})}\int_{{\mathbb{R}^{N}\setminus
B_{R}(x_{0})}}\frac{w_{\pm}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right)\,dx\,dt,\end{array}$

where $\displaystyle w_{\pm}=(u-k)_{\pm}$.

###### Proof.

All the heavy-lifting pertaining to time regularization has already been
performed in [24, Lemma 3.3]. In fact, the proof of our theorem is the same as
their proof except for their estimate for the terms $\displaystyle I_{2}$ and
$\displaystyle I_{3}$. For this reason and in the interest of brevity, we only
present below the part of their proof with the subheadings “The estimate of
$\displaystyle I_{2}$” and “The estimate of $\displaystyle I_{3}$” with
different estimates that produce the “good term”. To this end, we follow [14,
Prop. 8.5]. Moreover, we only write the calculation for sub-solutions since
those for super-solutions are similar and we assume without loss of generality
that $\displaystyle(x_{0},t_{0})=(0,0)$. Let $\displaystyle\rho_{1},\rho_{2}$
be positive numbers such that $\displaystyle 0<r\leq\rho_{1}<\rho_{2}\leq R$.
We take $\displaystyle\zeta_{1}$ such that $\displaystyle 0\leq\zeta_{1}\leq
1$,
$\displaystyle\operatorname{spt}(\zeta_{1})=B_{\frac{\rho_{1}+\rho_{2}}{2}}:=B_{\frac{\rho_{1}+\rho_{2}}{2}}(0)$,
$\displaystyle\zeta_{1}\equiv 1$ on $\displaystyle B_{\rho_{1}}$ and
$\displaystyle|\nabla\zeta_{1}|\leq\frac{1}{\rho_{2}-\rho_{1}}$. Moreover, let
$\displaystyle\tau_{1},\tau_{2}\in(0,\infty)$ be given such that
$\displaystyle-\theta\leq-\tau_{2}<-\tau_{1}\leq 0$ and $\displaystyle
d\mu:=\frac{dx\,dy}{|x-y|^{N+sp}}$.

The estimate of $\displaystyle I_{2}$:

Recall that

$\displaystyle\displaystyle
I_{2}:=\frac{1}{2}\int_{-\tau_{2}}^{-\tau_{1}}\iint_{B_{\rho_{2}}\times
B_{\rho_{2}}}|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))\zeta_{2}^{p}(t)\,d\mu\,dt.$

To estimate the nonlocal terms, we consider the following cases pointwise for
$\displaystyle(x,t)$ and $\displaystyle(y,t)$ in
$\displaystyle{B_{\rho_{2}}}\times{B_{\rho_{2}}}\times I$ where $\displaystyle
I=(-\tau_{2},-\tau_{1})$. Recalling the notation from (N10), for any fixed
$\displaystyle t\in I$, we claim that

  * •

If $\displaystyle x\notin A_{+}(k,\rho_{2},t)$ and $\displaystyle y\notin
A_{+}(k,\rho_{2},t)$ then

$|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))=0.$
(3.1)

This estimate follows from the fact that
$\displaystyle(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))=0$ when
$\displaystyle x\notin A_{+}(k,t)$ and $\displaystyle y\notin A_{+}(k,t)$.

  * •

If $\displaystyle x\in A_{+}(k,\rho_{2},t)$ and $\displaystyle y\notin
A_{+}(k,\rho_{2},t)$ then

$|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))\\\
\geq\min\\{2^{p-2},1\\}\left[|w_{+}(x,t)-w_{+}(y,t)|^{p}+w_{-}(y,t)^{p-1}w_{+}(x,t)\right]\zeta_{1}(x)^{p}.$
(3.2)

To obtain estimate Eq. 3.2, we note that

$|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))\\\
=(w_{+}(x,t)+w_{-}(y,t))^{p-1}w_{+}(x,t)\zeta_{1}^{p}(x),$

when $\displaystyle x\in A_{+}(k,\rho_{2},t)$ and $\displaystyle y\notin
A_{+}(k,\rho_{2},t)$. The estimate follows by an application of Jensen’s
inequality when $\displaystyle p\leq 2$ and applying Lemma 2.3 with
$\displaystyle\theta=0$ in the case $\displaystyle p\geq 2$ .

  * •

If $\displaystyle x,y\in A_{+}(k,\rho_{2},t)$ then

$\displaystyle\displaystyle|u(x,t)-u(y,t)|^{p-2}$
$\displaystyle\displaystyle(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))$
$\displaystyle\displaystyle\geq\frac{1}{2}|w_{+}(x,t)-w_{+}(y,t)|^{p}\max\\{\zeta_{1}(x),\zeta_{1}(y)\\}^{p}$
$\displaystyle\displaystyle\qquad-C(p)\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}|\zeta_{1}(x)-\zeta(y)|^{p}.$
(3.3)

Since $\displaystyle x,y\in A_{+}(k,\rho_{2},t)$, we have

$|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))\\\
=\left\\{\begin{array}[]{ll}(w_{+}(x,t)-w_{+}(y,t))^{p-1}(\zeta_{1}^{p}(x)w_{+}(x,t)-\zeta_{1}^{p}(y)w_{+}(y,t))&\text{when}\,u(x,t)\geq
u(y,t),\\\
(w_{+}(y,t)-w_{+}(x,t))^{p-1}(\zeta_{1}^{p}(y)w_{+}(y,t)-\zeta_{1}^{p}(x)w_{+}(x,t))&\text{when}\,u(x,t)\leq
u(y,t).\end{array}\right.$

Thus, without loss of generality, we can assume $\displaystyle u(x,t)\geq
u(y,t)$. Thus, we get

$|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))\\\
=\left\\{\begin{array}[]{ll}(w_{+}(x,t)-w_{+}(y,t))^{p}\zeta_{1}^{p}(x)&\text{if}\,\zeta_{1}(x)\geq\zeta_{1}(y),\\\
(w_{+}(x,t)-w_{+}(y,t))^{p}(\zeta_{1}(y))^{p}-(w_{+}(x,t)-w_{+}(y,t))^{p-1}w_{+}(x,t)(\zeta_{1}^{p}(x)-\zeta_{1}^{p}(y))&\text{if}\,\zeta_{1}(x)\leq\zeta_{1}(y).\end{array}\right.$

In the second case, we apply Lemma 2.4 with $\displaystyle a=\zeta_{1}(y)$ and
$\displaystyle b=\zeta_{1}(x)$ and
$\displaystyle\varepsilon=\tfrac{1}{2}\tfrac{w_{+}(x,t)-w_{+}(y,t)}{w_{+}(x,t)}$
to obtain

$(w_{+}(x,t)-w_{+}(y,t))^{p-1}w_{+}(x,t)(\zeta_{1}^{p}(x)-\zeta_{1}^{p}(y))\\\
\leq\frac{1}{2}(w_{+}(x,t)-w_{+}(y,t))^{p}\zeta_{1}^{p}(y)+[2(p-1)]^{p-1}w_{+}^{p}(x,t)(\zeta_{1}(y)-\zeta(x))^{p}.$

Combining the previous three estimates, we obtain • ‣ The estimate of
$\displaystyle I_{2}$: when $\displaystyle u(x,t)\geq u(y,t)$. The case
$\displaystyle u(x,t)\leq u(y,t)$ can be handled analogously by interchanging
the role of $\displaystyle x$ and $\displaystyle y$

As a consequence of these cases and Eq. 2.1, we have

$\displaystyle\displaystyle I_{2}\geq$ $\displaystyle\displaystyle
C_{11}\Biggr{[}\int_{I}\iint_{B_{\rho_{2}}\times
B_{\rho_{2}}}\frac{|w_{+}(x,t)-w_{+}(y,t)|^{p}}{|x-y|^{N+ps}}\max\\{\zeta_{1}(x),\zeta_{1}(y)\\}^{p}\zeta_{2}^{p}(t)\,dx\,dy\,dt$
$\displaystyle\displaystyle\qquad+\int_{I}\int_{B_{\rho_{2}}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left(\int_{B_{\rho_{2}}}\frac{w_{-}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right)\,dx\,dt\Biggr{]}$
$\displaystyle\displaystyle\qquad\qquad-
C_{22}\int_{I}\iint_{B_{\rho_{2}}\times
B_{\rho_{2}}}\max\\{w_{+}(x,t),w_{+}(y,t)\\}^{p}\frac{|\zeta_{1}(x)-\zeta_{1}(y)|^{p}}{|x-y|^{N+sp}}\zeta_{2}^{p}(t)\,dx\,dy\,dt.$

The estimate of $\displaystyle I_{3}$:

Recall that

$\displaystyle\displaystyle I_{3}:$
$\displaystyle\displaystyle=\int_{-\tau_{2}}^{-\tau_{1}}\iint_{(\mathbb{R}^{N}\setminus
B_{\rho_{2}})\times
B_{\rho_{2}}}|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))(w_{+}(x,t)\zeta_{1}^{p}(x)-w_{+}(y,t)\zeta_{1}^{p}(y))\zeta_{2}^{p}(t)\,d\mu\,dt$
$\displaystyle\displaystyle=\int_{-\tau_{2}}^{-\tau_{1}}\int_{B_{\rho_{2}}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left[\int_{\mathbb{R}^{N}\setminus
B_{\rho_{2}}}{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}K(x,y,t)\,dy\right]\,dx\,dt$
$\displaystyle\displaystyle\geq
C_{33}\underbrace{\int_{-\tau_{2}}^{-\tau_{1}}\int_{A^{+}(k,\rho_{1},t)}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left[\int_{(B_{R}(x)\setminus
B_{\rho_{2}})\cap\\{u(x,t)\geq
u(y,t)\\}}\frac{(u(x,t)-u(y,t))^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt}_{T_{1}}$
$\displaystyle\displaystyle\qquad-
C_{44}\underbrace{\int_{-\tau_{2}}^{-\tau_{1}}\int_{A^{+}\left(k,\frac{\rho_{1}+\rho_{2}}{2},t\right)}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left[\int_{(\mathbb{R}^{N}\setminus
B_{\rho_{2}})\cap\\{u(y,t)>u(x,t)\\}}\frac{(u(y,t)-u(x,t))^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt}_{T_{2}}.$
(3.4)

We estimate $\displaystyle T_{1}$ as follows:

$\displaystyle\displaystyle T_{1}$
$\displaystyle\displaystyle\geq\int_{-\tau_{2}}^{-\tau_{1}}\int_{B_{\rho_{1}}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left[\int_{(B_{R}(x)\cap
A^{-}(k,t))\setminus
B_{\rho_{2}}}\frac{(w_{+}(x,t)+w_{-}(y,t))^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt$
$\displaystyle\displaystyle\geq\int_{-\tau_{2}}^{-\tau_{1}}\int_{B_{\rho_{1}}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left[\int_{B_{R}(x)\setminus
B_{\rho_{2}}}\frac{w_{-}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt.$ (3.5)

We estimate $\displaystyle T_{2}$ as follows:

$\displaystyle\displaystyle T_{2}$
$\displaystyle\displaystyle\leq\int_{-\tau_{2}}^{0}\int_{B_{\frac{\rho_{1}+\rho_{2}}{2}}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left[\int_{{\mathbb{R}^{N}\setminus
B_{\rho_{2}}}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt$
$\displaystyle\displaystyle\leq\iint_{I\times
B_{R}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left(\operatorname*{ess\,sup}\limits_{t\in(-\tau_{2},0)}\left[\int_{{\mathbb{R}^{N}\setminus
B_{\rho_{2}}}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\right)\,dx\,dt.$
(3.6)

Combining this observation with The estimate of $\displaystyle I_{3}$: and The
estimate of $\displaystyle I_{3}$: into The estimate of $\displaystyle
I_{3}$:, we obtain

$\displaystyle\displaystyle I_{3}\geq C_{33}$
$\displaystyle\displaystyle\underbrace{\int_{I}\int_{B_{\rho_{1}}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left[\int_{B_{R}(x)\setminus
B_{\rho_{2}}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right]\,dx\,dt}_{\text{ignore
this term which appears on the left hand side}}$ $\displaystyle\displaystyle-
C_{44}\iint_{I\times
B_{R}}\zeta_{2}^{p}(t)\zeta_{1}^{p}(x)w_{+}(x,t)\left(\operatorname*{ess\,sup}\limits_{t\in(-\tau_{2},0)}\int_{{\mathbb{R}^{N}\setminus
B_{\rho_{2}}}}\frac{w_{+}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right)\,dx\,dt.$

As mentioned earlier, the rest of the proof of Theorem 3.1 is the same as in
[24, Lemma 3.3]. ∎

######  Remark 3.2.

This energy estimate first appears in [36] under the assumption that
$\displaystyle\partial_{t}u\in L^{2}(E_{T})$ (see Lemma 3.1 and Remark 3.2 in
[36]). This assumption is generally dropped by working with regularizations in
time, such as Steklov averages [21]. In fact, the estimate in [36] is proved
for parabolic minimizers, however the same method works for solutions of
equations.

Henceforth, excepting the final section, we will let $\displaystyle u$ denote
a nonnegative, locally bounded supersolution.

###  3.2 Shrinking Lemma

One of the main difficulties we face when dealing with regularity issues for
nonlocal equations is the lack of a corresponding isoperimetric inequality for
$\displaystyle W^{s,p}$ functions. Indeed, since such functions can have
jumps, a generic isoperimetric inequality seems out of reach at this time (see
[15, 1]). One way around this issue is to note that since we are working with
solutions of an equation, which we expect to be continuous and hence have no
jumps, we could try and cook up an isoperimetric inequality for solutions.
Such a strategy turns out to be feasible due to the presence of the “good
term” or the isoperimetric term in the Caccioppoli inequality.

######  Lemma 3.3.

Let $\displaystyle k<l<m$ be arbitrary levels and $\displaystyle A\geq 1$.
Then,

$\displaystyle(l-k)(m-l)^{p-1}\left|[u>m]\cap B_{\rho}\right|\left|[u<k]\cap
B_{\rho}\right|\leq
C\rho^{N+sp}\int_{B_{\rho}}(u-l)_{-}(x)\int_{B_{A\rho}}\frac{(u-l)_{+}^{p-1}(y)}{|x-y|^{N+sp}}\,dy\,dx,$

where $\displaystyle C=C(N,s,p,A)>0$.

###### Proof.

We estimate from below

$\displaystyle\displaystyle\int_{B_{\rho}}(u-l)_{-}(x)\int_{B_{A\rho}}\frac{(u-l)_{+}^{p-1}(y)}{|x-y|^{N+sp}}\,dy\,dx$
$\displaystyle\displaystyle\geq\frac{C}{\rho^{N+sp}}\int_{B_{\rho}}(u-l)_{-}(x)\int_{B_{\rho}}(u-l)_{+}^{p-1}(y)\,dy\,dx$
$\displaystyle\displaystyle\geq\frac{C}{\rho^{N+sp}}\int_{B_{\rho}\cap\\{u<k\\}}(u-l)_{-}(x)\int_{B_{\rho}\cap\\{u>m\\}}(u-l)_{+}^{p-1}(y)\,dy\,dx$
$\displaystyle\displaystyle\geq\frac{C}{\rho^{N+sp}}\int_{B_{\rho}\cap\\{u<k\\}}(l-k)\int_{B_{\rho}\cap\\{u>m\\}}(m-l)^{p-1}\,dy\,dx.$

∎

We now prove the shrinking lemma using Lemma 3.3. It is usually proved in the
local case using the De Giorgi isoperimetric inequality.

######  Lemma 3.4.

Let $\displaystyle u$ be a super-solution of Eq. 2.2. Suppose that for some
level $\displaystyle m$, some constant $\displaystyle\nu\in(0,1)$ and all time
levels $\displaystyle\tau$ in some interval $\displaystyle J$ we have

$\displaystyle|[u(\cdot,\tau)>m]\cap B_{\rho}|\geq\nu|B_{\rho}|,$

and we can arrange that for some $\displaystyle A\geq 1$, the following is
also satisfied:

$\displaystyle\displaystyle\int_{J}\int_{B_{\rho}}(u-l)_{-}(x,t)\int_{B_{A\rho}(x)}\frac{(u-l)_{+}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\,dx\,dt\leq
C_{1}\frac{l^{p}}{\rho^{sp}}|B_{\rho}\times J|,$ (3.7)

where $\displaystyle l=\frac{m}{2^{j}}$ for some $\displaystyle j\geq 1$, then
we have the following conclusion:

$\displaystyle\left|\left[u<\frac{m}{2^{j+1}}\right]\cap B_{\rho}\times
J\right|\leq\left(\frac{C_{(C_{1},A,N,\nu)}}{2^{j}-1}\right)^{p-1}|B_{\rho}\times
J|.$

###### Proof.

In the conclusion of Lemma 3.3 we put $\displaystyle k=\frac{m}{2^{j+1}}$ and
$\displaystyle l=\frac{m}{2^{j}}$, use the hypothesis and then integrate over
the time interval $\displaystyle J$ to get

$\displaystyle\frac{m}{2^{j+1}}m^{p-1}\left(\frac{2^{j}-1}{2^{j}}\right)^{p-1}\left|\left[u<\frac{m}{2^{j+1}}\right]\cap
B_{\rho}\times
J\right|\leq\frac{C}{\nu}\left(\frac{m}{2^{j}}\right)^{p}|B_{\rho}\times J|,$

where $\displaystyle C$ depends on $\displaystyle C_{1},A$ and $\displaystyle
N$. The conclusion follows after a simple rearrangement. ∎

######  Remark 3.5.

In applying the shrinking lemma in proving the expansion of positivity lemmas,
in order to get the smallness condition Eq. 3.7, we shall need to impose the
smallness condition on the Tail term - this is one of the ways in which the
Tail alternatives enter the picture.

######  Remark 3.6.

An intriguing aspect of our lemma is the fact that the dependence between the
levels in the conclusion and how small we can make the corresponding level set
is not exponential; such a dependence is reflected in the best known lower
bound for the Hölder exponent for solutions to second order linear nonlocal
equations [35], whereas, in the local case, this lower bound is much worse
[8].

###  3.3 Tail Estimates

In this section, we shall outline how the estimate for the tail term is made
and we shall refer to this section whenever a similar calculation is required
in subsequent sections.

Given a function $\displaystyle\zeta\in C_{c}^{\infty}(B_{\rho})$ and any
level $\displaystyle k$, we want to estimate

$\displaystyle\underset{\stackrel{{\scriptstyle t\in
J}}{{x\in\operatorname{spt}\zeta}}}{\operatorname*{ess\,sup}}\int_{B_{\rho}^{c}(y_{0})}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy,$

where $\displaystyle J$ is some time interval. In order to do this, we will
typically choose $\displaystyle\zeta$ to be supported in $\displaystyle
B_{\vartheta\rho}$ for some $\displaystyle\vartheta\in(0,1)$. With such a
choice, we have

$\displaystyle|y-y_{0}|\leq|x-y|\left(1+\frac{|x-y_{0}|}{|x-y|}\right)\leq|x-y|\left(1+\frac{\vartheta}{(1-\vartheta)}\right),$

so we can make a first estimate

$\displaystyle\underset{\stackrel{{\scriptstyle t\in
J}}{{x\in\operatorname{spt}\zeta}}}{\operatorname*{ess\,sup}}\int_{B_{\rho}^{c}(y_{0})}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\leq\frac{1}{(1-\vartheta)^{N+sp}}\underset{t\in
J}{\operatorname*{ess\,sup}}\int_{B_{\rho}^{c}(y_{0})}\frac{(u-k)_{-}^{p-1}(y,t)}{|y-y_{0}|^{N+sp}}\,dy.$

In the local case, we could always estimate $\displaystyle(u-k)_{-}\leq k$
because we take $\displaystyle u\geq 0$ locally. However, in the Tail term we
have no information regarding the solution outside the ball and so unless we
make a global boundedness assumption (for e.g. $\displaystyle u\geq 0$ in full
space), the best we can do is

$\displaystyle(u-k)_{-}\leq u_{-}+k,$

which leads us to the next estimate

$\displaystyle\underset{t\in
J}{\operatorname*{ess\,sup}}\int_{B_{\rho}^{c}(y_{0})}\frac{(u-k)_{-}^{p-1}(y,t)}{|y-y_{0}|^{N+sp}}\,dy\leq
C(p)\frac{k^{p-1}}{\rho^{sp}}+C(p)\,\underset{t\in
J}{\operatorname*{ess\,sup}}\int_{B_{\rho}^{c}(y_{0})}\frac{u_{-}^{p-1}(y,t)}{|y-y_{0}|^{N+sp}}\,dy.$

Putting together the above estimates yield

$\displaystyle\underset{\stackrel{{\scriptstyle t\in
J}}{{x\in\operatorname{spt}\zeta}}}{\operatorname*{ess\,sup}}\int_{B_{\rho}^{c}(y_{0})}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\leq\frac{C(p)}{\rho^{sp}}\frac{1}{(1-\vartheta)^{N+sp}}\left[k^{p-1}+\operatorname{Tail}_{\infty}^{p-1}(u_{-};y_{0},\rho,J)\right].$

Finally we usually want an estimate of the form

$\displaystyle\underset{\stackrel{{\scriptstyle t\in
J}}{{x\in\operatorname{spt}\zeta}}}{\operatorname*{ess\,sup}}\int_{B_{\rho}^{c}(y_{0})}\frac{(u-k)_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\leq\frac{C(p)}{(1-\vartheta)^{N+sp}}\frac{k^{p-1}}{\rho^{sp}},$

and so we impose the condition that

$\displaystyle\operatorname{Tail}_{\infty}^{p-1}(u_{-};y_{0},\rho,J)\leq
k^{p-1}.$

This is the origin of the various Tail alternatives.

######  Remark 3.7.

We will see that it is the Tail that captures the nonlocality of our equation
in the sense that if it is small then the proofs become local. In fact, when
dealing with nonnegative supersolutions, we only need the Tail of the negative
part of the solution to be small and so if we assume that it is nonnegative in
full space then the Tail alternatives are automatically verified and the
proofs of the constituent lemmas become “local proofs”.

###  3.4 De Giorgi iteration Lemma

Let

$\displaystyle\mu_{-}\leq\underset{(x_{0},t_{0})+Q_{2\rho}^{-}(\theta)}{\operatorname*{ess\,inf}}u$

and $\displaystyle M>0$. Let $\displaystyle\xi\in(0,1]$ and $\displaystyle
a\in(0,1)$ be fixed numbers.

######  Lemma 3.8.

Let $\displaystyle u$ be a locally bounded, local, weak supersolution of Eq.
2.2 in $\displaystyle E_{T}$. Then there exists a number
$\displaystyle\nu_{-}$ depending only on the data and the parameters
$\displaystyle\theta,\xi,M$ and $\displaystyle a$ such that if the following
is satisfied

$\displaystyle|[u\leq\mu_{-}+\xi
M]\cap(x_{0},t_{0})+Q_{2\rho}^{-}(\theta)|\leq\nu_{-}|Q_{2\rho}^{-}(\theta)|,$

then one of the following two conclusion holds:

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}((u-\mu)_{-};x_{0},\rho,(t_{0}-\theta(2\rho)^{sp},t_{0}])>\xi
M,$

or

$\displaystyle u\geq\mu_{-}+a\xi M\text{ a.e. in
}[(x_{0},t_{0})+Q_{\rho}^{-}(\theta)].$

######  Remark 3.9.

An analogous result holds for subsolutions at the supremum.

###### Proof.

Without loss of generality, we will assume that
$\displaystyle(x_{0},t_{0})=(0,0)$ and for $\displaystyle n=0,1,2,\ldots$, we
set

$\rho_{n}=\rho+2^{-n}\rho,\qquad B_{n}=B_{\rho_{n}}\quad\text{and}\quad
Q_{n}=B_{n}\times(-\theta\rho_{n}^{sp},0].$

We apply the energy estimates over $\displaystyle B_{n}$ and $\displaystyle
Q_{n}$ to $\displaystyle(u-b_{n})_{-}$ for the levels

$b_{n}=\mu_{-}+\xi_{n}M\qquad\text{ where
}\qquad\xi_{n}=a\xi+\frac{1-a}{2^{n}}\xi.$

Let us define

$\displaystyle\hat{\rho_{n}}:=\frac{3\rho_{n}+\rho_{n+1}}{4},\quad\tilde{\rho_{n}}:=\frac{\rho_{n}+\rho_{n+1}}{2},\quad\text{and}\quad\bar{\rho_{n}}:=\frac{\rho_{n}+3\rho_{n+1}}{4},$

and denote
$\displaystyle\hat{Q_{n}}:=B_{\hat{\rho_{n}}}\times(-\theta\hat{\rho}_{n}^{sp},0]$,
$\displaystyle\tilde{Q_{n}}:=B_{\tilde{\rho_{n}}}\times(-\theta\tilde{\rho}_{n}^{sp},0]$
and
$\displaystyle\bar{Q_{n}}:=B_{\bar{\rho_{n}}}\times(-\theta\bar{\rho}_{n}^{sp},0]$.

We shall consider cut-off functions of the form $\displaystyle\bar{\zeta_{n}}$
and $\displaystyle\zeta_{n}$ satisfying

$\begin{array}[]{llll}\bar{\zeta_{n}}\equiv 1\text{ on
}B_{n+1},\quad\bar{\zeta_{n}}\in
C_{c}^{\infty}(B_{\bar{\rho_{n}}}),\quad|\nabla\bar{\zeta_{n}}|\apprle\frac{1}{\bar{\rho_{n}}-\rho_{n+1}}\approx\frac{2^{n}}{\rho}\quad\text{and}\quad|\delta_{t}\bar{\zeta_{n}}|\apprle\frac{1}{\theta(\bar{\rho}_{n}^{sp}-\rho_{n+1}^{sp})}\approx\frac{2^{nsp}}{\theta\rho^{sp}},\\\
{\zeta_{n}}\equiv 1\text{ on }B_{\tilde{\rho_{n}}},\quad{\zeta_{n}}\in
C_{c}^{\infty}(B_{\hat{\rho_{n}}}),\quad|\nabla{\zeta_{n}}|\apprle\frac{1}{\hat{\rho_{n}}-\tilde{\rho_{n}}}\approx\frac{2^{n}}{\rho}\quad\text{and}\quad|\delta_{t}{\zeta_{n}}|\apprle\frac{1}{\theta(\hat{\rho}_{n}^{sp}-\tilde{\rho_{n}}^{sp})}\approx\frac{2^{nsp}}{\theta\rho^{sp}},\end{array}$
(3.8)

Let us also denote $\displaystyle A_{n}:=[u<b_{n}]\cap Q_{n}$, then we can
apply Theorem 2.2 to get

$\begin{array}[]{rcl}\frac{(1-a)\xi
M}{2^{n}}|A_{n+1}|&\leq&\iint_{Q_{n+1}}(u-b_{n})_{-}\
dz\leq\iint_{\bar{Q_{n}}}(u-b_{n})_{-}\bar{\zeta_{n}}\ dz\\\
&\leq&\left(\iint_{\tilde{Q_{n}}}\left((u-b_{n})_{-}\bar{\zeta_{n}}\right)^{p\frac{N+2s}{N}}\
dz\right)^{\frac{N}{p(N+2s)}}|A_{n}|^{\frac{p(N+2s)-N}{p(N+2s)}}\\\
&\apprle&\bigg{[}\tilde{\rho}_{n}^{sp}\iiint_{\tilde{Q_{n}}\times
B_{\tilde{\rho_{n}}}}\frac{|(u-b_{n})_{-}\bar{\zeta_{n}}(x,t)-(u-b_{n})_{-}\bar{\zeta_{n}}(y,t)|^{p}}{|x-y|^{N+sp}}\\\
&&+\iint_{\tilde{Q_{n}}}((u-b_{n})_{-}\bar{\zeta_{n}})^{p}\bigg{]}^{\frac{N}{p(N+2s)}}\left(\sup_{-\theta\tilde{\rho}_{n}^{sp}<t<0}\fint_{B_{\tilde{\rho_{n}}}}((u-b_{n})_{-}\bar{\zeta_{n}})^{2}\
dz\right)^{\frac{s}{N+2s}}|A_{n}|^{\frac{p(N+2s)-N}{p(N+2s)}}\end{array}$
(3.9)

From Young’s inequality, we have

$|(u-b_{n})_{-}\bar{\zeta_{n}}(x,t)-(u-b_{n})_{-}\bar{\zeta_{n}}(y,t)|^{p}\leq
c|(u-b_{n})_{-}(x,t)-(u-b_{n})_{-}(y,t)|^{p}\bar{\zeta_{n}}^{p}(x,t)\\\
+c|(u-b_{n})_{-}(y,t)|^{p}|\bar{\zeta_{n}}(x,t)-\bar{\zeta_{n}}(y,t)|^{p}$
(3.10)

Combining Eq. 3.9 and Eq. 3.10, we get

$\begin{array}[]{rcl}\frac{(1-a)\xi
M}{2^{n}}|A_{n+1}|&\apprle&\bigg{[}\tilde{\rho}_{n}^{sp}\iiint_{\tilde{Q_{n}}\times
B_{\tilde{\rho_{n}}}}\frac{|(u-b_{n})_{-}(x,t)-(u-b_{n})_{-}(y,t)|^{p}}{|x-y|^{N+sp}}\\\
&&+\tilde{\rho}_{n}^{sp}\iiint_{\tilde{Q_{n}}\times
B_{\tilde{\rho_{n}}}}\frac{|(u-b_{n})_{-}(y,t)|^{p}|\bar{\zeta_{n}}(x,t)-\bar{\zeta_{n}}(y,t)|^{p}}{|x-y|^{N+sp}}\\\
&&+\iint_{\tilde{Q_{n}}}((u-b_{n})_{-})^{p}\bigg{]}^{\frac{N}{p(N+2s)}}\left(\sup_{-\theta\tilde{\rho}_{n}^{sp}<t<0}\fint_{B_{\tilde{\rho_{n}}}}(u-b_{n})_{-}^{2}\
dz\right)^{\frac{s}{N+2s}}|A_{n}|^{\frac{p(N+2s)-N}{p(N+2s)}}.\end{array}$
(3.11)

The energy inequality applied with the cutoff function
$\displaystyle\zeta_{n}$ over $\displaystyle Q_{n}$ from Theorem 3.1 gives

$\underset{-\theta\tilde{\rho}_{n}^{sp}<t<0}{\operatorname*{ess\,sup}}\int_{B_{\tilde{\rho_{n}}}}(u-b_{n})_{-}^{2}\,dx+{\iiint_{\tilde{Q_{n}}\times
B_{\tilde{\rho_{n}}}}}\frac{|(u-b_{n})_{-}(x,t)(x,t)-(u-b_{n})_{-}(y,t)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\,dt\\\
\leq\underbrace{\int_{-\theta\rho_{n}^{sp}}^{0}\iint_{B_{\rho_{n}}\times
B_{\rho_{n}}}\max\\{(u-b_{n})_{-}(x,t),(u-b_{n})_{-}(y,t)\\}\frac{|\zeta_{n}(x,t)-\zeta_{n}(y,t)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\,dt}_{:=I}\\\
+\underbrace{\int_{-\theta\rho_{n}^{sp}}^{0}\int_{B_{\rho_{n}}}(u-b_{n})_{-}^{2}|\partial_{t}\zeta_{n}(x,t)|\,dx\,dt}_{:=II}\\\
+\underbrace{\int_{-\theta\rho_{n}^{sp}}^{0}\int_{B_{\rho_{n}}}(u-b_{n})_{-}(x,t)\,dx\,dt\left(\operatorname*{ess\,sup}_{\stackrel{{\scriptstyle-\theta\rho_{n}^{sp}<t<0}}{{x\in\operatorname{spt}\zeta_{n}}}}\int_{\mathbb{R}^{N}\setminus
B_{\rho_{n}}}\frac{(u-b_{n})_{-}^{p-1}}{|x-y|^{N+sp}}\,dy\right)}_{:=III}.$
(3.12)

We now estimate each of the terms in the previous display as follows:

Estimate for $\displaystyle I$:

Using the fact that $\displaystyle(u-b_{n})_{-}\leq\xi M$ and Eq. 3.8, we have

$I\leq C\frac{2^{np}(\xi M)^{p}}{\rho^{sp}}|A_{n}|.$ (3.13)

Estimate for $\displaystyle II$:

Analogously, we also get

$II\leq C\frac{2^{nsp}(\xi M)^{2}}{\theta\rho^{sp}}|A_{n}|.$ (3.14)

Estimate for $\displaystyle III$:

To estimate this, we note that if $\displaystyle x\in B_{\tilde{\rho_{n}}}$
and $\displaystyle y\in\mathbb{R}^{N}\setminus B_{n}$, then we have

$\displaystyle|y|\leq|x-y|\left(1+\frac{|x|}{|x-y|}\right)\leq c2^{n}|x-y|\
\Longrightarrow\ \frac{1}{|x-y|^{N+sp}}\leq c2^{n(N+sp)}\frac{1}{|y|^{N+sp}}.$

Next, we note that the following

$\displaystyle(u-b_{n})_{-}\leq\xi_{n}M-(u-\mu_{-})\leq\xi M+(u-\mu_{-})_{-},$

holds globally. Therefore we have

$\begin{array}[]{l}\underset{\stackrel{{\scriptstyle-\theta\rho_{n}^{sp}<t<0;}}{{x\in\operatorname{spt}\zeta_{1}}}}{\operatorname*{ess\,sup}}\int_{\mathbb{R}^{N}\setminus
B_{\rho_{n}}}\frac{(u-b_{n})_{-}^{p-1}(y,t)}{|x-y|^{N+sp}}\,dy\\\ \hskip
56.9055pt\apprle
2^{n(N+sp)}\underset{-\theta\rho_{n}^{sp}<t<0}{\operatorname*{ess\,sup}}\int_{\mathbb{R}^{N}\setminus
B_{\rho_{n}}}\frac{(u-\mu_{-})_{-}^{p-1}(y,t)+(\xi
M)^{p-1}}{|y|^{N+sp}}\,dy\\\ \hskip 56.9055pt\apprle 2^{n(N+sp)}\frac{(\xi
M)^{p-1}}{\rho^{sp}}+2^{n(N+sp)}\underset{-\theta\rho_{n}^{sp}<t<0}{\operatorname*{ess\,sup}}\int_{\mathbb{R}^{N}\setminus
B_{\rho_{n}}}\frac{(u-\mu_{-})_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy\\\ \hskip
56.9055pt\leq c2^{n(N+sp)}\left(\frac{(\xi
M)^{p-1}}{\rho^{sp}}+\frac{1}{\rho^{sp}}\operatorname{Tail}_{\infty}^{p-1}((u-\mu)_{-};0,\rho,(-\theta(2\rho)^{sp},0])\right),\end{array}$
(3.15)

In particular, we get

$\begin{array}[]{rcl}III&\apprle&2^{n(N+sp)}\left(\frac{(\xi
M)^{p-1}}{\rho^{sp}}+\frac{1}{\rho^{sp}}\operatorname{Tail}_{\infty}^{p-1}((u-\mu)_{-};0,2\rho,(-\theta(2\rho)^{sp},0])\right)(\xi
M)|A_{n}|\\\ &\leq&C2^{n(N+sp)}\frac{(\xi
M)^{p}}{\rho^{sp}}|A_{n}|,\end{array}$ (3.16)

where to obtain the last estimate, we made use of the hypothesis

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}((u-\mu)_{-};x_{0},\rho,(-\theta(2\rho)^{sp},0])\leq\xi
M.$

Combining Eq. 3.13, Eq. 3.14 and Eq. 3.16 into Eq. 3.12, we get

$\underset{-\theta\tilde{\rho}_{n}^{sp}<t<0}{\operatorname*{ess\,sup}}\int_{B_{\tilde{\rho_{n}}}}(u-b_{n})_{-}^{2}\,dx+{\iiint_{\tilde{Q_{n}}\times
B_{\tilde{\rho_{n}}}}}\frac{|(u-b_{n})_{-}(x,t)(x,t)-(u-b_{n})_{-}(y,t)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\,dt\\\
\leq\underbrace{C\frac{2^{n(N+p)}}{\rho^{sp}}(\xi
M)^{p}\left(2+\frac{1}{\theta}(\xi M)^{2-p}\right)|A_{n}|}_{:=G},$ (3.17)

where we have denoted $\displaystyle G:=\frac{\Gamma(\xi
M)^{p}}{\rho^{sp}}|A_{n}|$ and
$\displaystyle\Gamma:=C2^{n(N+p)}\left(2+\frac{1}{\theta}(\xi
M)^{2-p}\right)$.

Now we can make use of Eq. 3.17 to further estimate the terms appearing on the
right hand side of Eq. 3.11 to get

$\begin{array}[]{rcl}\frac{(1-a)\xi M}{2^{n}}|A_{n+1}|&\leq&C\left(\Gamma(\xi
M)^{p}|A_{n}|\right)^{\frac{N}{p(N+2s)}}\left(\frac{\Gamma(\xi
M)^{p}}{\rho^{N+sp}}|A_{n}|\right)^{\frac{s}{N+2s}}|A_{n}|^{\frac{p(N+2s)-N}{p(N+2s)}}.\end{array}$
(3.18)

We now divide Eq. 3.18 by $\displaystyle|Q_{n+1}|$ noting that
$\displaystyle|Q_{n}|\approx|Q_{n+1}|$ and denote $\displaystyle
Y_{n}:=\frac{|A_{n}|}{|Q_{n}|}$ from which we get

$\displaystyle Y_{n+1}\leq\frac{Cd^{n}}{(1-a)}\left(\frac{\theta}{(\xi
M)^{2-p}}\right)^{\frac{s}{N+2s}}\left(2+\frac{(\xi
M)^{2-p}}{\theta}\right)^{\frac{N+sp}{p(N+2s)}}Y_{n}^{1+\frac{s}{N+2s}},$

where $\displaystyle d=d(N,s,p,\Lambda)>1$ and $\displaystyle
C=C(N,s,p,\Lambda)>0$ depends only on the data. By Lemma 2.5, we see that
$\displaystyle Y_{\infty}=0$ provided

$Y_{0}\leq\left(\frac{C}{1-a}\right)^{-\frac{N+2s}{s}}d^{-\left(\frac{N+2s}{s}\right)^{2}}\frac{\left(\frac{(\xi
M)^{2-p}}{\theta}\right)}{\left(2+\frac{(\xi
M)^{2-p}}{\theta}\right)^{\frac{N+sp}{sp}}}=:\nu_{-},$ (3.19)

which completes the proof of the lemma. ∎

######  Remark 3.10.

In the conclusion of Lemma 3.8, we can replace
$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}((u-\mu)_{-};x_{0},\rho,(t_{0}-\theta(2\rho)^{sp},t_{0}])>\xi
M$ with
$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}((u-\mu)_{-};x_{0},2\rho,(t_{0}-\theta(2\rho)^{sp},t_{0}])>\xi
M$.

This requires a finer refinement for the estimate of $\displaystyle III$
appearing in Eq. 3.12 which proceeds as follows:

$\displaystyle\displaystyle\underset{{-\theta\rho_{n}^{sp}<t<0}}{\operatorname*{ess\,sup}}\int_{\mathbb{R}^{N}\setminus
B_{\rho_{n}}}\frac{(u-b_{n})_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy=\underset{{-\theta\rho_{n}^{sp}<t<0}}{\operatorname*{ess\,sup}}\left(\int_{\mathbb{R}^{N}\setminus
B_{2\rho}}\frac{(u-b_{n})_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy+\int_{B_{2\rho}\setminus
B_{\rho_{n}}}\frac{(u-b_{n})_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy\right).$

The first term appearing on the right hand side of the previous display can be
estimated analogously to Eq. 3.15 in terms of
$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}((u-\mu)_{-};x_{0},2\rho,(t_{0}-\theta(2\rho)^{sp},t_{0}])$.

In order to estimate the second term, we note that
$\displaystyle(u-b_{n})_{-}\leq(\xi M)$ on $\displaystyle
B_{2\rho}\times(-\theta\rho_{n}^{sp},0]$ and
$\displaystyle|y|\geq\rho_{n}\geq\rho$ on $\displaystyle B_{2\rho}\setminus
B_{\rho_{n}}$ from which we get

$\displaystyle\underset{{-\theta\rho_{n}^{sp}<t<0}}{\operatorname*{ess\,sup}}\int_{B_{2\rho}\setminus
B_{\rho_{n}}}\frac{(u-b_{n})_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy\leq C\frac{(\xi
M)^{p-1}}{\rho^{N+sp}}|B_{2\rho}\setminus B_{\rho_{n}}|\leq C\frac{(\xi
M)^{p-1}}{\rho^{sp}},$

which implies Eq. 3.16 holds for this term.

###  3.5 De Giorgi lemma: Forward in time version

Let $\displaystyle u$ denote a nonnegative, local, weak supersolution of Eq.
2.2 in $\displaystyle E_{T}$ and let $\displaystyle B_{R}\times I\subset
E_{T}$ denote our reference cylinder. Suppose that we have the following
information at a time level $\displaystyle t_{0}$:

$u(x,t_{0})\geq\xi M\qquad\text{ for a.e. }x\in B_{2\rho}(y),$ (3.20)

for some $\displaystyle M>0$ and $\displaystyle\xi\in(0,1]$. Then in the
energy inequality from Theorem 3.1, for any level $\displaystyle k\leq\xi M$
over $\displaystyle[(x_{0},t_{0})+Q_{2\rho}^{+}(\theta)]$, the first term on
the right hand side of Theorem 3.1 over $\displaystyle
B_{2\rho}\times\\{t_{0}\\}$ vanishes.

Moreover, taking a test function independent of time also kills the integral
on the right involving the time derivative of the test function. Therefore we
may repeat the same arguments as in Lemma 3.8 for
$\displaystyle(u-\xi_{n}M)_{-}$ over the cylinders $\displaystyle Q_{n}^{+}$
where

$\displaystyle\xi_{n}=a\xi+\frac{1-a}{2^{n}}\xi\quad\text{and}\quad
Q_{n}^{+}=B_{n}\times(0,\theta(2\rho)^{sp}].$

Denoting $\displaystyle Y_{n}^{\prime}=\frac{|[u<\xi_{n}M]\cap
Q_{n}^{+}|}{|Q_{n}^{+}|}$, we see that if
$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};x_{0},2\rho,(t_{0},t_{0}+\theta(2\rho)^{sp}])\leq\xi
M$ is satisfied, then the following estimate holds:

$\displaystyle
Y_{n+1}^{\prime}\leq\frac{Cb^{n}}{(1-a)^{(N+2s)\frac{p}{N}}}\left(\frac{\theta}{(\xi
M)^{2-p}}\right)^{\frac{sp}{N}}Y_{n}^{\prime 1+\frac{sp}{N}}.$

Thus, by Lemma 2.5, we have $\displaystyle Y_{\infty}^{\prime}=0$ if

$Y_{0}^{\prime}\leq\nu_{0}\left(\frac{(\xi
M)^{2-p}}{\theta}\right)^{s}=:\tilde{\nu}.$ (3.21)

for a constant $\displaystyle\nu_{0}\in(0,1)$ depending only on $\displaystyle
a$ and data. The foregoing discussion leads to the following variant of the De
Giorgi lemma.

######  Lemma 3.11.

Let $\displaystyle u$ denote a nonnegative, local, weak supersolution to Eq.
2.2 in $\displaystyle E_{T}$. Let $\displaystyle M$ and $\displaystyle\xi$ be
positive numbers and suppose Eq. 3.20 holds at time $\displaystyle t=t_{0}$
and Eq. 3.21 is satisfied for
$\displaystyle\tilde{\nu}=\tilde{\nu}(\xi,M,\theta,a,N,\Lambda)$ as
determined. Then one of the following two alternatives hold:

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};x_{0},2\rho,(t_{0},t_{0}+\theta(2\rho)^{sp}])>\xi
M\qquad\text{OR}\qquad u\geq a\xi M\,\,\text{ a.e. in
}B_{\rho}(x_{0})\times(t_{0},t_{0}+\theta(2\rho)^{sp}].$

##  4 Qualitative expansion of positivity in time

In this section, we shall prove a general expansion of positivity estimate
that will be used in Section 5 and Section 6.

######  Lemma 4.1.

Let $\displaystyle u$ denote a nonnegative, local, weak supersolution to Eq.
2.2 in $\displaystyle E_{T}$. Assume that for some
$\displaystyle(x_{0},t_{0})\in E_{T}$ and some $\displaystyle\rho>0$,
$\displaystyle M>0$ and $\displaystyle\alpha\in(0,1)$, the following
hypothesis is satisfied:

$\displaystyle|[u(\cdot,t_{0})\geq M]\cap
B_{\rho}(x_{0})|\geq\alpha|B_{\rho}(x_{0})|,$

then there exists $\displaystyle\delta$ and $\displaystyle\epsilon$ in
$\displaystyle(0,1)$ depending on $\displaystyle\\{N,s,p,\Lambda,\alpha\\}$
such that either

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};x_{0},\rho,(t_{0},t_{0}+\delta\rho^{sp}M^{2-p}))>M,$

or

$\displaystyle|[u(\cdot,t)\geq\epsilon M]\cap
B_{\rho}(x_{0})|\geq\frac{1}{2}\alpha|B_{\rho}(x_{0})|,$

holds for all $\displaystyle t\in(t_{0},t_{0}+\delta\rho^{sp}M^{2-p}]$.

###### Proof.

Without loss of generality, we will assume that
$\displaystyle(x_{0},t_{0})=(0,0)$. For $\displaystyle k>0$ and $\displaystyle
t>0$, we set

$\displaystyle A_{k,\rho}(t):=[u(\cdot,t)<k]\cap B_{\rho},$

then the hypothesis of the lemma can be restated as:

$|A_{M,\rho}(0)|\leq(1-\alpha)|B_{\rho}|.$ (4.1)

We consider the energy estimate from Theorem 3.1 for $\displaystyle(u-M)_{-}$
over the cylinder $\displaystyle B_{\rho}\times(0,\theta\rho^{sp}]$ where
$\displaystyle\theta>0$ will be chosen later. Note that
$\displaystyle(u-M)_{-}\leq M$ in $\displaystyle B_{\rho}$ because
$\displaystyle u$ is nonnegative in $\displaystyle E_{T}$. For
$\displaystyle\sigma\in(0,1/8]$ to be chosen later, we take a cutoff function
$\displaystyle\zeta=\zeta(x)$, nonnegative such that it is supported in
$\displaystyle B_{(1-\tfrac{\sigma}{2})\rho}$, $\displaystyle\zeta=1$ on
$\displaystyle B_{(1-\sigma)\rho}$ and
$\displaystyle|\nabla\zeta|\leq\tfrac{2}{\sigma\rho}$ to get

$\begin{array}[]{l}\int_{B_{(1-\sigma)\rho}}(u-M)_{-}^{2}(x,\bar{t})\zeta^{p}(x)\,dx\\\
\hskip 85.35826pt\leq\int_{{B_{\rho}}}(u-M)_{-}^{2}(x,0)\zeta^{p}(x)\,dx\\\
\hskip
113.81102pt+C\int_{0}^{\bar{t}}\iint_{{B_{\rho}}\times{B_{\rho}}}\max\\{(u-M)_{-}(x,t),(u-M)_{-}(y,t)\\}^{p}\frac{|\zeta(x)-\zeta(y)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\,dt\\\
\hskip
113.81102pt+C\underbrace{\int_{0}^{\bar{t}}\int_{{B_{\rho}}}\zeta^{p}(x)(u-M)_{-}(x,t)\left(\operatorname*{ess\,sup}\limits_{\stackrel{{\scriptstyle
t\in(0,\bar{t})}}{{x\in\operatorname{spt}\zeta}}}\int_{{\mathbb{R}^{N}\setminus
B_{\rho}}}\frac{(u-M)_{-}(y,t)^{p-1}}{|x-y|^{N+sp}}\,dy\right)\,dx\,dt}_{:=III},\end{array}$
(4.2)

To estimate the tail term we first note that when $\displaystyle x\in
B_{(1-\tfrac{\sigma}{2})\rho}$ and $\displaystyle y\in B_{\rho}^{c}$ then

$\displaystyle|y|\leq|x-y|\left(1+\frac{|x|}{|x-y|}\right)\leq\frac{2}{\sigma}|x-y|\quad\Longrightarrow\quad\frac{1}{|x-y|^{N+sp}}\leq
c\sigma^{-(N+sp)}\frac{1}{|y|^{N+sp}}.$

Furthermore, noting that $\displaystyle(u-M)_{-}\leq M-u\leq M+u_{-}$, we get

$III\leq\frac{c}{{\sigma}^{N+sp}}\left(\frac{M^{p-1}}{\rho^{sp}}+\frac{1}{\rho^{sp}}\operatorname{Tail}_{\infty}^{p-1}(u_{-};0,\rho,(0,\bar{t}))\right)M\bar{t}|B_{\rho}|\overset{\text{hypothesis}}{\leq}\frac{c}{{\sigma}^{N+sp}}\frac{M^{p}}{\rho^{sp}}\bar{t}|B_{\rho}|.$
(4.3)

Substituting Eq. 4.3 into Eq. 4.2, we get

$\int_{B_{(1-\sigma)\rho}}(u-M)_{-}^{2}(x,\bar{t})\,dx\leq
M^{2}|A_{M,\rho}(0)|+\frac{CM^{p}}{\sigma^{p}\rho^{sp}}\bar{t}|B_{\rho}|+\frac{c}{{\sigma}^{N+sp}}\frac{M^{p}}{\rho^{sp}}\bar{t}|B_{\rho}|.$

Now restricting $\displaystyle\bar{t}\leq\delta\rho^{sp}M^{2-p}$ for some
$\displaystyle\delta\in(0,1)$ to be chosen, we get

$\begin{array}[]{rcl}\int_{B_{(1-\sigma)\rho}}(u-M)_{-}^{2}(x,\bar{t})\,dx&\leq&M^{2}|A_{M,\rho}(0)|+\delta\frac{CM^{2}}{\sigma^{p}}|B_{\rho}|+\frac{c}{{\sigma}^{N+sp}}\delta
M^{2}|B_{\rho}|\\\
&\overset{\lx@cref{creftype~refnum}{eq41.1}}{\leq}&M^{2}(1-\alpha)|B_{\rho}|+\delta\frac{CM^{2}}{\sigma^{p}}|B_{\rho}|+\frac{c}{{\sigma}^{N+sp}}\delta
M^{2}|B_{\rho}|\end{array}$

On the other hand, we have

$\displaystyle\displaystyle\int_{B_{(1-\sigma)\rho}}(u-M)_{-}^{2}(x,t)\geq\int_{B_{(1-\sigma)\rho}\cap[u<\epsilon
M]}(u-M)_{-}^{2}(x,t)\geq M^{2}(1-\epsilon)^{2}|A_{\epsilon
M,(1-\sigma)\rho}(t)|,$

where $\displaystyle\epsilon\in(0,1)$ will be chosen below depending only on
$\displaystyle\alpha$. Next, we note that

$\displaystyle|A_{\epsilon M,\rho}(t)|\leq|A_{\epsilon
M,(1-\sigma)\rho}(t)|+|B_{\rho}-B_{(1-\sigma)\rho}|\leq|A_{\epsilon
M,(1-\sigma)\rho}(t)|+N\sigma|B_{\rho}|.$

Thus, combining everything, we have

$\displaystyle|A_{\epsilon
M,\rho}(t)|\leq\frac{1}{(1-\epsilon)^{2}}\left((1-\alpha)+\delta\frac{\bar{C}}{\sigma^{N+sp}}+(1-\epsilon)^{2}N\sigma\right)|B_{\rho}|.$

Now we choose

$\sigma=\frac{\alpha}{8N},\qquad\epsilon\leq
1-\frac{\sqrt{1-\frac{3}{4}\alpha}}{\sqrt{1-\frac{1}{2}\alpha}}\Longrightarrow\frac{1-\alpha}{(1-\epsilon)^{2}}\leq
1-\frac{3\alpha}{4},\quad\text{and}\quad\delta=\frac{\alpha\sigma^{N+sp}(1-\epsilon)^{2}}{8\bar{C}},$

to get the desired conclusion. ∎

##  5 Expansion of positivity for Nonlocal Degenerate Equations

We will assume that $\displaystyle u$ is a nonnegative, local, weak
supersolution to Eq. 2.2 in $\displaystyle E_{T}$ and $\displaystyle p>2$. For
$\displaystyle(x_{0},t_{0})\in E_{T}$ and some given positive number
$\displaystyle M$ we consider the cylinder

$\displaystyle B_{8\rho}(x_{0})\times\left(t_{0},t_{0}+\frac{b^{p-2}}{(\eta
M)^{p-2}}\delta\rho^{sp}\right]\subset
B_{R}(x_{0})\times[t_{0}-R^{sp},t_{0}+R^{sp}]\subset E_{T},$

where $\displaystyle
B_{R}(x_{0})\times[t_{0}-R^{sp},t_{0}+R^{sp}]=B_{R}(x_{0})\times I$ is our
reference cylinder. The constants $\displaystyle b,\eta,\delta$ are constants
given by Proposition 5.1 and $\displaystyle\rho>0$ is chosen small enough.

######  Proposition 5.1.

Assume that for some $\displaystyle(x_{0},t_{0})\in E_{T}$,
$\displaystyle\rho>0$, $\displaystyle M>0$ and some
$\displaystyle\alpha\in(0,1)$ the following assumption is satisfied:

$|[u(\cdot,t_{0})\geq M]\cap B_{\rho}(x_{0})|\geq\alpha|B_{\rho}(x_{0})|.$
(5.1)

Then there exist constants $\displaystyle\eta,\delta,\sigma^{\prime}\in(0,1)$
and $\displaystyle b>0$ depending only on the data and $\displaystyle\alpha$
such that either

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}\left(u_{-};x_{0},\rho,\left(t_{0},t_{0}+\frac{b^{p-2}}{(\eta
M)^{p-2}}\delta\rho^{sp}\right)\right)>\eta M\quad\text{OR}\quad
u(\cdot,t)\geq\eta M\,\,\text{a.e. in }B_{2\rho}(x_{0}),$

holds for all times

$t_{0}+\frac{b^{p-2}}{(\eta M)^{p-2}}\sigma^{\prime}\delta\rho^{sp}\leq t\leq
t_{0}+\frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp}.$

###### Proof.

Without loss of generality, we will assume that
$\displaystyle(x_{0},t_{0})=(0,0)$. The proof of the proposition is split into
the following steps:

Step 1:

Changing the time variable. For all $\displaystyle\sigma\leq 1$, it is easy to
see that Eq. 5.1 implies

$|[u(\cdot,0)\geq\sigma M]\cap B_{\rho}|\geq\alpha|B_{\rho}|\qquad\forall\
\sigma\leq 1.$

For $\displaystyle\tau\geq 0$, let us also set

$\sigma_{\tau}:=\exp\left(-\frac{\tau}{p-2}\right)\leq 1,$

Assuming

$\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,\rho,J_{1})\leq\sigma_{\tau}M\qquad\text{where}\quad
J_{1}:=(0,\delta\rho^{sp}(\sigma_{\tau}M)^{2-p}),$ (5.2)

then we can apply Lemma 4.1 to get

$\left|\left[u\left(\cdot,\frac{\delta\rho^{sp}}{(\sigma_{\tau}M)^{p-2}}\right)\geq\epsilon\sigma_{\tau}M\right]\cap
B_{\rho}\right|\geq\frac{1}{2}\alpha|B_{\rho}|,$ (5.3)

for universal constants $\displaystyle\epsilon,\delta$ (depending only on
$\displaystyle p,s,N,\Lambda,\alpha$) in $\displaystyle(0,1)$. In particular,
we can rewrite Eq. 5.3 as

$\left|\left[u\left(\cdot,\frac{e^{\tau}}{M^{p-2}}\delta\rho^{sp}\right)\geq\epsilon
M\sigma_{\tau}\right]\cap B_{\rho}\right|\geq\frac{1}{2}\alpha|B_{\rho}|.$
(5.4)

As in [23, Section 4.2], let us perform the change of variable

$w(x,\tau):=\frac{1}{\sigma_{\tau}M}(\delta\rho^{sp})^{\frac{1}{p-2}}u\left(x,\frac{e^{\tau}}{M^{p-2}}\delta\rho^{sp}\right),$
(5.5)

then Eq. 5.4 written in terms of $\displaystyle w$ translates to

$|[w(\cdot,\tau)\geq b_{0}]\cap
B_{\rho}|\geq\frac{1}{2}\alpha|B_{\rho}|\quad\text{for all}\ \tau>0,$ (5.6)

where

$b_{0}:=\epsilon(\delta\rho^{sp})^{\frac{1}{p-2}}.$ (5.7)

We can further rewrite Eq. 5.6 as

$|B_{4\rho}\setminus[w(\cdot,\tau)<b_{0}]|\geq\frac{1}{2}\alpha
4^{-N}|B_{4\rho}|\qquad\text{for all}\ \tau>0.$ (5.8)

Step 2:

Relating $\displaystyle w$ to the evolution equation. Since $\displaystyle
u\geq 0$ in $\displaystyle E_{T}$, by formal calculations, we have

$\displaystyle\displaystyle w_{\tau}$
$\displaystyle\displaystyle=\left(\frac{1}{\sigma_{\tau}M}(\delta\rho^{sp})^{\frac{1}{p-2}}\right)^{p-1}u_{t}+\frac{1}{p-2}\frac{1}{M\sigma_{\tau}}(\delta\rho^{sp})^{\frac{1}{p-2}}u$
$\displaystyle\displaystyle\geq-\left(\frac{1}{\sigma_{\tau}M}(\delta\rho^{sp})^{\frac{1}{p-2}}\right)^{p-1}Lu$
$\displaystyle\displaystyle\geq-L_{1}w,$

in $\displaystyle E\times\mathbb{R}_{+}$ where

$\displaystyle\left\\{\begin{array}[]{l}L_{1}\varphi(x,\tau)=P.V.\int_{\mathbb{R}^{N}}B_{1}(x,y,t)J_{p}(\varphi(x,\tau)-\varphi(y,\tau))dy,\\\
K_{1}(x,y,\tau)=K\left(x,y,\frac{e^{\tau}}{M^{p-2}}\delta\rho^{sp}\right),\\\
\frac{\Lambda^{-1}}{|x-y|^{N+sp}}\leq
K_{1}(x,y,\tau)\leq\frac{\Lambda}{|x-y|^{N+sp}}.\end{array}\right.$

The formal calculation can be made rigorous by appealing to the weak
formulation and the energy estimates for $\displaystyle u$ can be transferred
to energy estimates for $\displaystyle w$ by change of variable.

For any level $\displaystyle k\in\mathbb{R}$, the energy estimate from Theorem
3.1 for $\displaystyle(w-k)_{-}$ in $\displaystyle Q_{8\rho}^{+}(\theta)$
yields

$\int_{\theta(4\rho)^{sp}}^{\theta(8\rho)^{sp}}\int_{B_{4\rho}}(w-k)_{-}(x,\tau)\int_{B_{4\rho}}\frac{|(w-k)_{+}(y,\tau)|^{p-1}}{|x-y|^{N+sp}}\,dx\,dy\,d\tau\\\
\leq
C\int_{0}^{\theta(8\rho)^{sp}}\int_{B_{8\rho}}\int_{B_{8\rho}}\max\\{(w-k)_{-}(x,\tau),(w-k)_{-}(y,\tau)\\}^{p}\frac{|\zeta(x,\tau)-\zeta(y,\tau)|^{p}}{|x-y|^{N+sp}}\,dx\,dy\,d\tau\\\
+C\int_{0}^{\theta(8\rho)^{sp}}\int_{B_{8\rho}}(w-k)_{-}^{2}(x,\tau)|\delta_{\tau}\zeta(x,\tau)|\,dx\,d\tau\\\
+C\int_{0}^{\theta(8\rho)^{sp}}\int_{B_{8\rho}}(w-k)_{-}(x,\tau)\zeta(x,\tau)\left(\underset{\tau\in(0,\theta(8\rho)^{sp})}{\operatorname*{ess\,sup}}\int_{B_{8\rho}^{c}}\frac{(w-k)_{-}^{p-1}(y,\tau)}{|x-y|^{N+sp}}\,dy\right)\,dx\,d\tau,$
(5.9)

for any non-negative piecewise smooth cutoff vanishing on the parabolic
boundary of $\displaystyle Q_{8\rho}^{+}(\theta)$. In particular, we choose
$\displaystyle\zeta\equiv 1$ on
$\displaystyle{Q}_{4\rho}(\theta)=B_{4\rho}\times((4\rho)^{sp}\theta,(8\rho)^{sp}\theta]$
with $\displaystyle\operatorname{spt}(\zeta)\subset
B_{6\rho}\times(0,(8\rho)^{sp}\theta]$ and satisfying

$\displaystyle|\nabla\zeta|\leq\frac{1}{2\rho}\qquad\text{and}\qquad|\zeta_{\tau}|\leq\frac{1}{\theta(4\rho)^{sp}}.$

Step 3:

Shrinking lemma for $\displaystyle w$. We claim that for every
$\displaystyle\nu>0$ there exist $\displaystyle\epsilon_{\nu}\in(0,1)$
depending only on the data and $\displaystyle\alpha$, and
$\displaystyle\theta=\theta(b_{0},\epsilon_{\nu})>0$ depending only on
$\displaystyle b_{0}$ and $\displaystyle\epsilon_{\nu}$ such that if

$\left\\{\begin{array}[]{l}\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,8\rho,J_{2})\leq
2\epsilon_{\nu}\epsilon
M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),\\\
J_{2}:=\left(M^{2-p}\delta\rho^{sp},\exp\left(\frac{8^{sp}}{(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right)M^{2-p}\delta\rho^{sp}\right],\end{array}\right.$
(5.10)

then the following conclusion holds:

$\displaystyle|[w<\epsilon_{\nu}B_{0}]\cap\mathcal{Q}_{4\rho}(\theta)|\leq\nu|\mathcal{Q}_{4\rho}(\theta)|.$

###### Proof of Claim:.

In Eq. 5.9 we work with the levels $\displaystyle b_{j}$ and the parameter
$\displaystyle\theta$ as follows

$\displaystyle b_{j}=\frac{1}{2^{j}}b_{0}\quad\text{ for }\quad 1\leq j\leq
j_{\ast}\qquad\text{ and }\qquad\theta=b_{j_{\ast}}^{2-p},$

where $\displaystyle b_{0}$ is given by Eq. 5.7 and $\displaystyle{j_{\ast}}$
is to be chosen in Eq. 5.12. We first note that since $\displaystyle u\geq 0$
in $\displaystyle Q_{8\rho}^{+}(\theta)$, all the local integrals on the right
in Eq. 5.9 can be estimated using

$\displaystyle(w-b_{j})_{-}\leq b_{j}\qquad\text{on}\ Q_{8\rho}^{+}(\theta).$

Thus, by our choice of the test function and Eq. 5.9, we get

$\displaystyle\displaystyle\int_{\theta(4\rho)^{sp}}^{\theta(8\rho)^{sp}}\int_{B_{4\rho}}(w-b_{j})_{-}(x,\tau)$
$\displaystyle\displaystyle\int_{B_{4\rho}}\frac{|(w-b_{j})_{+}(y,\tau)|^{p-1}}{|x-y|^{N+sp}}\,dy\,dx\,d\tau\
$ $\displaystyle\displaystyle\leq
C\left(\frac{b_{j}^{p}}{\rho^{sp}}+C\frac{b_{j}^{2}}{\theta\rho^{sp}}+b_{j}\underset{\tau\in(0,\theta(8\rho)^{sp})}{\operatorname*{ess\,sup}}\int_{B_{8\rho}^{c}}\frac{(w-b_{j})_{-}^{p-1}(y,\tau)}{|y|^{N+sp}}\,dy\right)|{Q}_{4\rho}(\theta)|.$

From Subsection 3.3, suppose the following holds:

$\operatorname{Tail}_{\infty}^{p-1}(w_{-};0,8\rho,(0,\theta(8\rho)^{sp}))\leq
b_{j_{*}}^{p-1},$ (5.11)

then we have

$\displaystyle\underset{\tau\in(0,\theta(8\rho)^{sp})}{\operatorname*{ess\,sup}}\int_{B_{8\rho}^{c}}\frac{(w-b_{j})_{-}^{p-1}(y,\tau)}{|y|^{N+sp}}\,dy\leq
C\frac{b_{j}^{p}}{\rho^{sp}}|{Q}_{4\rho}(\theta)|.$

In particular, recalling the choice of $\displaystyle\theta$, for any
$\displaystyle 1\leq j\leq j_{\ast}$, we have

$\displaystyle\displaystyle\int_{\theta(4\rho)^{sp}}^{\theta(8\rho)^{sp}}\int_{B_{4\rho}}(w-b_{j})_{-}(x,\tau)\int_{B_{4\rho}}\frac{|(w-b_{j})_{+}(y,\tau)|^{p-1}}{|x-y|^{N+sp}}\,dy\,dx\,d\tau\leq
C\frac{b_{j}^{p}}{\rho^{sp}}|{Q}_{4\rho}(\theta)|$

The estimate from Eq. 5.8 satisfies hypothesis of Lemma 3.4, thus we get

$\displaystyle|[w<b_{j+1}]\cap{Q}_{4\rho}(\theta)|\leq\left(\frac{C}{2^{j}-1}\right)^{p-1}|{Q}_{4\rho}(\theta)|,$

for a constant $\displaystyle C=C(N,p,s,\Lambda,\alpha)>0$. Thus for a given
$\displaystyle\nu\in(0,1)$, we choose $\displaystyle j_{*}$ such that

$\left(\frac{C}{2^{j_{*}}-1}\right)^{p-1}\leq\nu\qquad\text{and}\quad\epsilon_{\nu}:=\frac{1}{2^{j_{*}+1}}.$
(5.12)

To conclude, we rewrite the Tail alternative in terms of $\displaystyle u_{-}$
as follows.

$\begin{array}[]{rcl}\underset{\tau\in(0,\theta(8\rho)^{sp})}{\operatorname*{ess\,sup}}\int_{B_{8\rho}^{c}}\frac{w_{-}^{p-1}(y,\tau)}{|y|^{N+sp}}\,dy&=&(\delta\rho^{sp})^{\frac{p-1}{p-2}}\frac{1}{M^{p-1}}\underset{\tau\in(0,\theta(8\rho)^{sp})}{\operatorname*{ess\,sup}}\int_{B_{8\rho}^{c}}\frac{u_{-}^{p-1}(y,e^{\tau}M^{2-p}\delta\rho^{sp})}{|y|^{N+sp}}e^{\tau\frac{p-1}{p-2}}\,dy\\\
&\leq&(\delta\rho^{sp})^{\frac{p-1}{p-2}}\frac{\exp\left(\theta(8\rho)^{sp}\frac{p-1}{p-2}\right)}{M^{p-1}}\underset{t\in
J_{2}}{\operatorname*{ess\,sup}}\int_{B_{8\rho}^{c}}\frac{u_{-}^{p-1}(y,t)}{|y|^{N+sp}}\,dy,\end{array}$

where

$\displaystyle
J_{2}=\left(M^{2-p}\delta\rho^{sp},\exp((8\rho)^{sp}\theta)M^{2-p}\delta\rho^{sp}\right].$

Thus, Eq. 5.11 is verified provided

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,8\rho,J_{2})\leq\frac{b_{j_{*}}}{(\delta\rho^{sp})^{\frac{1}{p-2}}}\frac{M}{\exp\left(\theta(8\rho)^{sp}\frac{1}{p-2}\right)}.$

Recalling that

$\displaystyle
b_{j_{*}}=\frac{b_{0}}{2^{j_{*}}}=\frac{\epsilon(\delta\rho^{sp})^{\frac{1}{p-2}}}{2^{j_{*}}}=2\epsilon_{\nu}\epsilon(\delta\rho^{sp})^{\frac{1}{p-2}}=\theta^{\frac{1}{2-p}},$

we get

$\displaystyle\exp\left(\frac{\theta(8\rho)^{sp}}{p-2}\right)=\exp\left(\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),$

and

$\displaystyle\frac{b_{j_{*}}}{(\delta\rho^{sp})^{\frac{1}{p-2}}}\frac{M}{\exp\left(\theta(8\rho)^{sp}\frac{1}{p-2}\right)}=2\epsilon_{\nu}\epsilon
M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),$

which completes the proof of the claim. ∎

Step 4:

Expansion of positivity for $\displaystyle w$. We claim that there exists a
$\displaystyle\nu=\nu(N,s,p,\Lambda,\alpha)\in(0,1)$ such that either

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,4\rho,J_{2})>2\epsilon_{\nu}\epsilon
M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right)$

or

$\displaystyle w(\cdot,\tau)\geq\frac{1}{2}\epsilon_{\nu}b_{0}\qquad\text{
a.e. in
}B_{2\rho}\times\left(\frac{(8\rho)^{sp}-(2\rho)^{sp}}{(2\epsilon_{\nu}b_{0})^{p-2}},\frac{(8\rho)^{sp}}{(2\epsilon_{\nu}b_{0})^{p-2}}\right]$

holds, where $\displaystyle\epsilon_{\nu}$ is the number corresponding to
$\displaystyle\nu$ in (Step 3) (see Eq. 5.12) and $\displaystyle J_{2}$ is as
defined in Eq. 5.10.

###### Proof of the claim.

We apply Lemma 3.8 along with Remark 3.10 to $\displaystyle w$ over the
cylinder

$\displaystyle{Q}_{4\rho}(\theta)=(0,\tau^{*})+Q_{4\rho}^{-}(\theta)\qquad\text{
for }\tau_{*}=\theta(8\rho)^{sp}.$

We work with $\displaystyle\epsilon_{\nu}b_{0}$ instead of
$\displaystyle\xi\omega$, $\displaystyle a=1/2$ and $\displaystyle\mu_{-}\geq
0$ is ignored (since we assume $\displaystyle u$ is non-negative subsolution)
to get that either

$\operatorname*{\operatorname{Tail}_{\infty}}((w)_{-};0,4\rho,(\tau_{*}-\theta(4\rho)^{sp},\tau_{*}])>\epsilon_{\nu}b_{0},$
(5.13)

holds or

$\displaystyle w(x,\tau)\geq\frac{1}{2}\epsilon_{\nu}b_{0}\qquad\text{ for
a.e. }\qquad(x,\tau)\in[(0,\tau^{*})+Q_{2\rho}^{-}(\theta)],$

holds provided

$\displaystyle\frac{\left|[w<\epsilon_{\nu}b_{0}]\cap{Q}_{4\rho}(\theta)\right|}{|{Q}_{4\rho}(\theta)|}\leq\frac{1}{C}\left(\frac{1}{2}\right)^{N+2s}\frac{[\theta(\epsilon_{\nu}b_{0})^{p-2}]^{\frac{N}{sp}}}{[1+2\theta(\epsilon_{\nu}b_{0})^{p-2}]^{\frac{N+sp}{sp}}}=\frac{1}{C}\left(\frac{1}{2}\right)^{N+2s}\frac{1}{3^{\frac{N+sp}{sp}}}=\nu,$

for a universal constant $\displaystyle C>1$. With this choice of
$\displaystyle\nu$, we find $\displaystyle\epsilon_{\nu}$ from (Step 3) and
therefore
$\displaystyle\theta=\frac{(2\epsilon_{\nu}\epsilon)^{2-p}}{\delta\rho^{sp}}$
quantitatively. The tail alternative Eq. 5.13 can be reformulated in terms of
$\displaystyle u$ as in (Step 3) to get

$\left\\{\begin{array}[]{l}\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,4\rho,J_{3})\leq
2\epsilon_{\nu}\epsilon
M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),\\\
J_{3}:=(\exp((8\rho)^{sp}\theta-(4\rho)^{sp}\theta)M^{2-p}\delta\rho^{sp},\exp((8\rho)^{sp}\theta)M^{2-p}\delta\rho^{sp}].\end{array}\right.$
(5.14)

Since $\displaystyle J_{2}\supset J_{3}$, the claim follows. ∎

Step 5:

Expanding the positivity for $\displaystyle u$. Assume that the Tail
alternatives from Eq. 5.2,Eq. 5.10 and Eq. 5.14 hold . As $\displaystyle\tau$
ranges over the interval in (Step 4), we see that $\displaystyle
e^{\tfrac{\tau}{p-2}}$ ranges over the interval

$\displaystyle
k_{1}:=\exp\left(\frac{8^{sp}-2^{sp}}{(p-2)[2\epsilon_{\nu}\epsilon\delta^{\frac{1}{p-2}}]^{p-2}}\right)\leq
f(\tau)\leq\exp\left(\frac{8^{sp}}{(p-2)[2\epsilon_{\nu}\epsilon\delta^{\frac{1}{p-2}}]^{p-2}}\right)=:k_{2}.$

Rewriting the conclusion of (Step 4) in terms of $\displaystyle u$, we get
that

$u(x,t)\geq\frac{\epsilon_{\nu}\epsilon M}{2k_{2}}=:\eta M\qquad\text{ for
a.e. }\quad x\in B_{2\rho},$ (5.15)

holds for all times

$\displaystyle\frac{b^{p-2}}{(\eta M)^{p-2}}\sigma\delta\rho^{sp}\leq
t\leq\frac{b^{p-2}}{(\eta M)^{p-2}}\delta\rho^{sp},$

where $\displaystyle b=b(N,s,p,\Lambda,\alpha)>0$ and
$\displaystyle\sigma^{\prime}=\sigma(N,s,p,\Lambda,\alpha)\in(0,1)$. In fact,
we have

$\displaystyle
b=\frac{\epsilon_{\nu}\epsilon}{2}\quad\text{and}\qquad\text{and}\quad\sigma^{\prime}=\left(\frac{k_{1}}{k_{2}}\right)^{p-2}.$

Step 6:

Pulling the Tail alternatives together. We need the following Tail
alternatives going from (Step 1) and (Step 3) noting that Eq. 5.10 implies Eq.
5.14. In particular, we recall

$\displaystyle\begin{array}[]{rcl}\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,\rho,J_{1})&\leq&\sigma_{\tau}M,\\\
\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,4\rho,J_{2})&\leq&2\epsilon_{\nu}\epsilon
M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),\\\
\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,8\rho,J_{2})&\leq&2\epsilon_{\nu}\epsilon
M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right),\end{array}$

where we have taken

$\displaystyle\begin{array}[]{rcl}J_{1}&=&(0,\delta\rho^{sp}(\sigma_{\tau}M)^{2-p}),\\\
J_{2}&=&\left(M^{2-p}\delta\rho^{sp},\exp\left(\frac{8^{sp}}{(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right)M^{2-p}\delta\rho^{sp}\right].\end{array}$

Since we have $\displaystyle\tau\leq\theta(8\rho)^{sp}$ which implies the
following inclusion holds:

$\displaystyle
J_{1}\subset(0,\delta\rho^{sp}M^{2-p}\exp(\theta(8\rho)^{sp}))=(0,\delta\rho^{sp}M^{2-p}\exp(8^{sp}\delta^{-1}(2\epsilon_{\nu}\epsilon)^{2-p})),$

and
$\displaystyle\sigma_{\tau}\leq\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right)$.
Furthermore, we choose $\displaystyle\eta$ such that

$\displaystyle 2\epsilon_{\nu}\epsilon
M\exp\left(-\frac{8^{sp}}{(p-2)(2\epsilon_{\nu}\epsilon)^{p-2}\delta}\right)=\frac{2\epsilon_{\nu}\epsilon
M}{k_{2}}\overset{\lx@cref{creftype~refnum}{def_eta}}{=}4\eta M.$

Further choosing $\displaystyle\eta\in(0,1)$ small such that by an abuse of
notation, denoting $\displaystyle\eta=\frac{\eta}{8^{sp}}$, b the final Tail
alternative which subsumes all others is

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,\rho,J)\leq\eta
M\quad\text{where}\quad J:=\left(0,\frac{b^{p-2}}{(\eta
M)^{p-2}}\delta\rho^{sp}\right],$

where we recall $\displaystyle b=\frac{\epsilon_{\nu}\epsilon}{2}$.

∎

######  Remark 5.2.

The conclusion of Proposition 5.1 can also be written without Tail
alternatives as

$\displaystyle u(\cdot,t)\geq\eta
M-\operatorname*{\operatorname{Tail}_{\infty}}\left(u_{-};x_{0},\rho,\left(t_{0},t_{0}+\frac{b^{p-2}}{(\eta
M)^{p-2}}\delta\rho^{sp}\right)\right)\qquad\text{a.e. in }B_{2\rho}(x_{0}),$

holds for all times

$\displaystyle t_{0}+\frac{b^{p-2}}{(\eta
M)^{p-2}}\sigma^{\prime}\delta\rho^{sp}\leq t\leq t_{0}+\frac{b^{p-2}}{(\eta
M)^{p-2}}\delta\rho^{sp},$

because we are working with nonnegative solutions.

######  Remark 5.3.

In the proof of Hölder regularity we use Proposition 5.1 with
$\displaystyle\alpha=1/2$ so that
$\displaystyle\eta,\delta,\sigma^{\prime}\in(0,1)$ and $\displaystyle b>0$
depend only on the data in the proof.

##  6 Expansion of positivity for Nonlocal Singular Equations

In this section, we will consider $\displaystyle 1<p<2$ and assume that
$\displaystyle u$ is a nonnegative, local, weak supersolution to Eq. 2.2 in
$\displaystyle E_{T}$. For $\displaystyle(x_{0},t_{0})\in E_{T}$ and some
given positive numbers $\displaystyle M>0$ and $\displaystyle\delta\in(0,1)$,
we consider the cylinder

$\displaystyle(x_{0},t_{0})+Q_{16\rho}(\delta
M^{2-p}):=B_{16\rho}(x_{0})\times(t_{0},t_{0}+\delta M^{2-p}\rho^{sp}]\subset
B_{R}(x_{0})\times[t_{0}-R^{sp},t_{0}+R^{sp}]\subset E_{T},$

where $\displaystyle
B_{R}(x_{0})\times[t_{0}-R^{sp},t_{0}+R^{sp}]=B_{R}(x_{0})\times I$ is our
reference cylinder and $\displaystyle\rho>0$ is chosen small enough.

######  Proposition 6.1.

Assume that for some $\displaystyle(x_{0},t_{0})\in E_{T}$ and some
$\displaystyle\rho>0$, $\displaystyle M>0$ and $\displaystyle\alpha\in(0,1)$,
the following hypothesis holds

$|[u(\cdot,t_{0})\geq M]\cap B_{\rho}(x_{0})|\geq\alpha|B_{\rho}(x_{0})|.$

Then there exist constants $\displaystyle\eta,\delta$ and
$\displaystyle\varepsilon$ in $\displaystyle(0,1)$ depending only on the data
and $\displaystyle\alpha$ such that either

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};x_{0},\rho,(t_{0},t_{0}+\delta\rho^{sp}M^{2-p}))>\eta
M\qquad\text{OR}\qquad u(\cdot,t)\geq\eta M\qquad\text{a.e. in
}B_{2\rho}(x_{0}),$

holds for all times

$t_{0}+(1-\varepsilon)\delta M^{2-p}\rho^{sp}\leq t\leq t_{0}+\delta
M^{2-p}\rho^{sp}.$

###### Proof.

Without loss of generality, we will assume that
$\displaystyle(x_{0},t_{0})=(0,0)$.

Step 1:

Changing variables. From Section 4 we get that there exist
$\displaystyle\delta$ and $\displaystyle\epsilon$ in $\displaystyle(0,1)$
depending only on the data and $\displaystyle\alpha$ such that one of the
following two possibilities must hold:

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,\rho,(0,\delta\rho^{sp}M^{2-p}))>M,$

or

$|[u(\cdot,t)>\epsilon M]\cap
B_{\rho}|\geq\frac{1}{2}\alpha|B_{\rho}|\qquad\text{ for all }\qquad
t\in(0,\delta M^{2-p}\rho^{sp}].$ (6.1)

Let us now define the change of variables

$z=\frac{x}{\rho},\qquad-e^{-\tau}=\frac{t-\delta M^{2-p}\rho^{sp}}{\delta
M^{2-p}\rho^{sp}}\qquad\text{and}\quad
v(z,\tau)=\frac{1}{M}u(x,t)e^{\frac{\tau}{2-p}}.$ (6.2)

Then the cylinder $\displaystyle Q_{16\rho}(\delta M^{2-p})$ in
$\displaystyle(x,t)$ coordinates is mapped into $\displaystyle
B_{16}\times(0,\infty)$ in the $\displaystyle(z,\tau)$ coordinates. By formal
calculations (which can be made rigorous using the weak formulation), we have

$\displaystyle\displaystyle
v_{\tau}=\frac{1}{2-p}v+\frac{e^{\frac{p-1}{2-p}\tau}\delta\rho^{sp}}{M^{p-1}}u_{t}(z\rho,\delta
M^{2-p}\rho^{sp}(1-e^{-\tau}))=\frac{1}{2-p}v+\delta L_{1}v,$ (6.3)

in $\displaystyle B_{16}\times(0,\infty)$, where

$\displaystyle\begin{array}[]{rcl}L_{1}\varphi(z,\tau)&=&P.V.\int_{\mathbb{R}^{N}}K_{1}(z,z^{\prime},\tau)J_{p}(\varphi(z,\tau)-\varphi(z^{\prime},\tau))\,dz^{\prime},\\\
K_{1}(z,z^{\prime},\tau)&=&\rho^{N+sp}K\left(z\rho,z^{\prime}\rho,\delta
M^{2-p}\rho^{sp}(1-e^{-\tau})\right),\end{array}$

satisfying

$\displaystyle\frac{\Lambda^{-1}}{|z-z^{\prime}|^{N+sp}}\leq
K_{1}(z,z^{\prime},\tau)\leq\frac{\Lambda}{|z-z^{\prime}|^{N+sp}}.$

We rewrite Eq. 6.1 as

$|[v(\cdot,\tau)\geq\epsilon e^{\frac{\tau}{2-p}}]\cap
B_{1}|\geq\frac{1}{2}\alpha|B_{1}|\qquad\text{ for all }\tau\in(0,\infty).$
(6.4)

For constants $\displaystyle\tau_{0}>0$ (chosen in Eq. 6.15) and
$\displaystyle j_{*}\in\mathbb{N}$ (chosen in (Step 3)), we take

$b_{0}=\epsilon e^{\frac{\tau_{0}}{2-p}}\qquad\text{and}\quad
b_{j}=b_{0}2^{-j}\ \ \text{ for }1\leq j\leq j_{*}.$ (6.5)

Then from Eq. 6.4, we see that

$|[v(\cdot,\tau)\geq b_{j}]\cap B_{8}|\geq\frac{1}{2}\alpha
8^{-N}|B_{8}|\qquad\text{ holds for all
}\tau\in(\tau_{0},\infty)\quad\text{and}\quad\forall\,j\in\mathbb{N}.$ (6.6)

Consider the cylinders

$Q_{\tau_{0}}=B_{8}\times(\tau_{0}+b_{0}^{2-p},\tau_{0}+2b_{0}^{2-p})\quad\text{and}\quad
Q_{\tau_{0}^{\prime}}=B_{16}\times(\tau_{0},\tau_{0}+2b_{0}^{2-p}),$ (6.7)

and a nonnegative, piecewise smooth cutoff in $\displaystyle
Q_{\tau_{0}^{\prime}}$ of the form
$\displaystyle\zeta(z,\tau)=\zeta_{1}(z)\zeta_{2}(\tau)$ such that

$\displaystyle\begin{array}[]{ll}\zeta_{1}=\left\\{\begin{array}[]{lcl}1&\text{in}&B_{8}\\\
0&\text{on}&\mathbb{R}^{N}\setminus
B_{12}\end{array}\right.&|\delta\zeta_{1}|\leq\frac{1}{4},\\\
\zeta_{2}=\left\\{\begin{array}[]{lcl}0&\text{for}&\tau<\tau_{0}\\\
1&\text{for}&\tau\geq\tau_{0}+b_{0}^{2-p}\end{array}\right.&|\delta_{t}\zeta_{2}|\leq\frac{1}{b_{0}^{2-p}}.\end{array}$

From Theorem 3.1 applied for $\displaystyle(v-b_{j})_{-}$ over $\displaystyle
Q_{\tau_{0}^{\prime}}$ with the cutoff function $\displaystyle\zeta$, we get

$\displaystyle\displaystyle\int_{\tau_{0}}^{\tau_{0}+2b_{0}^{2-p}}$
$\displaystyle\displaystyle\int_{B_{16}}(v-b_{j})_{-}(z,\tau)\zeta^{p}(z,\tau)\int_{B_{16}}\frac{(v-b_{j})_{+}^{p-1}(z^{\prime},\tau)}{|z-z^{\prime}|^{N+sp}}\,dz^{\prime}\,dz\,d\tau\
$ $\displaystyle\displaystyle\leq
C\int_{\tau_{0}}^{\tau_{0}+2b_{0}^{2-p}}\int_{B_{16}}\int_{B_{16}}\max\\{(v-b_{j})_{-}(z,\tau),(v-b_{j})_{-}(z^{\prime},\tau)\\}^{p}\frac{|\zeta(z,\tau)-\zeta(z^{\prime},\tau)|^{p}}{|z-z^{\prime}|^{N+sp}}\,dz\,dz^{\prime}\,d\tau$
$\displaystyle\displaystyle+C\int_{\tau_{0}}^{\tau_{0}+2b_{0}^{2-p}}\int_{B_{16}}(v-b_{j})_{-}^{2}(z,\tau)|\delta_{\tau}\zeta(z,\tau)|\,dz\,d\tau$
$\displaystyle\displaystyle+C\int_{\tau_{0}}^{\tau_{0}+2b_{0}^{2-p}}\int_{B_{16}}(v-b_{j})_{-}(z,\tau)\zeta^{p}(z,\tau)\left(\underset{\tau\in(\tau_{0},\tau_{0}+2b_{0}^{2-p})}{\operatorname*{ess\,sup}}\int_{B_{16}^{c}}\frac{(v-b_{j})_{-}^{p-1}(z^{\prime},\tau)}{|z-z^{\prime}|^{N+sp}}\,dz^{\prime}\right)\,dz\,d\tau,$
(6.8)

where $\displaystyle C=\frac{C}{\delta}$ (note the $\displaystyle\delta$ in
Eq. 6.3) is a constant depending only on the data and $\displaystyle\delta$.
To obtain the above estimate, we tested Eq. 6.3 with
$\displaystyle-(v-b_{j})_{-}\zeta^{p}$ and discarded the nonpositive
contribution on the right-hand side from the nonnegative term
$\displaystyle\tfrac{1}{2-p}v$.

Step 2:

Shrinking lemma for $\displaystyle v$. For any $\displaystyle j\geq 2$, we
claim that either of the following two possibilities hold:

$\left\\{\begin{array}[]{l}\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,16\rho,J_{2}^{\prime})>\exp\left(\tfrac{-2b_{0}^{2-p}}{2-p}\right)\frac{\epsilon
M}{2^{j}},\\\
J_{2}^{\prime}=\left(M^{2-p}\delta\rho^{sp}(1-e^{-\tau_{0}}),M^{2-p}\delta\rho^{sp}(1-e^{-\tau_{0}-2b_{0}^{2-p}})\right),\end{array}\right.$
(6.9)

or

$\displaystyle|[v<b_{j}]\cap
Q_{\tau_{0}}|\leq\nu|Q_{\tau_{0}}|\qquad\text{where}\quad\nu=\left(\frac{C}{2^{j-1}-1}\right)^{p-1},$

for a universal constant $\displaystyle C>0$ depending only on
$\displaystyle\alpha$ and data.

###### Proof of Claim.

In Step 1:, we work with levels $\displaystyle b_{j}$ defined by

$\displaystyle b_{j}=\frac{1}{2^{j}}b_{0}\qquad\text{ for }\qquad j\geq 1,$

where $\displaystyle b_{0}$ is given by Eq. 6.5. We first note that since
$\displaystyle v\geq 0$ in $\displaystyle Q_{\tau_{0}^{\prime}}$, we have
$\displaystyle(v-b_{j})_{-}\leq b_{j}$ on $\displaystyle
Q_{\tau_{0}^{\prime}}$. Hence from our choice of the test function, we get

$\int_{\tau_{0}+b_{0}^{2-p}}^{\tau_{0}+2b_{0}^{2-p}}\int_{B_{8}}(v-b_{j})_{-}(z,\tau)\int_{B_{8}}\frac{|(v-b_{j})_{+}(z^{\prime},\tau)|^{p-1}}{|z-z^{\prime}|^{N+sp}}\,dz^{\prime}\,dz\,d\tau\\\
\leq
C|Q_{\tau_{0}^{\prime}}|\left(b_{j}^{p}+b_{j}\underset{\tau\in(\tau_{0},\tau_{0}+2b_{0}^{2-p})}{\operatorname*{ess\,sup}}\int_{B_{16}^{c}}\frac{(v-b_{j})_{-}^{p-1}(z^{\prime},\tau)}{|z^{\prime}|^{N+sp}}\,dz^{\prime}\right)$

where $\displaystyle C>0$ is a universal constant. We now make use of the Tail
estimates from Subsection 3.3 to get

$\displaystyle\int_{\tau_{0}+b_{0}^{2-p}}^{\tau_{0}+2b_{0}^{2-p}}\int_{B_{8}}(v-b_{j})_{-}(z,\tau)\int_{B_{8}}\frac{|(v-b_{j})_{+}(z^{\prime},\tau)|^{p-1}}{|z-z^{\prime}|^{N+sp}}\,dz^{\prime}\,dz\,d\tau\
\leq Cb_{j}^{p}|Q_{\tau_{0}^{\prime}}|,$

holds provided the following is satisfied

$\operatorname{Tail}_{\infty}^{p-1}(v_{-};0,16,(\tau_{0},\tau_{0}+2b_{0}^{2-p}))\leq
b_{j}^{p-1}.$ (6.10)

We now invoke Lemma 3.4 and Eq. 6.4 to conclude that

$\displaystyle|[v<b_{j+1}]\cap
Q_{\tau_{0}}|\leq\left(\frac{C}{2^{j}-1}\right)^{p-1}|Q_{\tau_{0}^{\prime}}|\overset{\lx@cref{creftype~refnum}{6.7:eq}}{\leq}C\left(\frac{C}{2^{j}-1}\right)^{p-1}|Q_{\tau_{0}}|,$

for a constant $\displaystyle C>0$ depending only on $\displaystyle\alpha$ and
data. This proves the claim with the choice
$\displaystyle\nu=\left(\frac{C}{2^{j}-1}\right)^{p-1}.$

We now rewrite the Tail alternative in terms of $\displaystyle u_{-}$ as
follows.

$\begin{array}[]{rcl}\underset{\tau\in(\tau_{0},\tau_{0}+2b_{0}^{2-p})}{\operatorname*{ess\,sup}}\int_{B_{16}^{c}}\frac{v_{-}^{p-1}(z^{\prime},\tau)}{|z^{\prime}|^{N+sp}}\,dz^{\prime}&\overset{\lx@cref{creftype~refnum}{eq:covsing}}{\leq}&\frac{\exp((\tau_{0}+2b_{0}^{2-p})\frac{p-1}{2-p})}{M^{p-1}}\rho^{sp}\left(\underset{t\in
J_{2}^{\prime}}{\operatorname*{ess\,sup}}\int_{B_{16\rho}^{c}}\frac{u(y,t)^{p-1}_{-}}{|y|^{N+sp}}\,dy\right)\\\
&\leq&\frac{\exp((\tau_{0}+2b_{0}^{2-p})\frac{p-1}{2-p})}{M^{p-1}}\operatorname{Tail}_{\infty}^{p-1}(u_{-};0,16\rho,J_{2}^{\prime}),\end{array}$
(6.11)

where

$\displaystyle
J_{2}^{\prime}:=\left(M^{2-p}\delta\rho^{sp}(1-e^{-\tau_{0}}),M^{2-p}\delta\rho^{sp}(1-e^{-\tau_{0}-2b_{0}^{2-p}})\right).$

Thus, making use of Eq. 6.11, we see that Eq. 6.10 is satisfied provided

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,16\rho,J_{2}^{\prime})\leq\exp\left(\frac{-\tau_{0}-2b_{0}^{2-p}}{2-p}\right)Mb_{j}.$

Recalling Eq. 6.5, we get

$\displaystyle\exp\left(\frac{-\tau_{0}-2b_{0}^{2-p}}{2-p}\right)Mb_{j}=\exp\left(\frac{-2b_{0}^{2-p}}{2-p}\right)\frac{\epsilon
M}{2^{j}},$

which completes the proof of the claim. ∎

Step 3:

Obtaining a good time slice for $\displaystyle v$. We claim that there exist
$\displaystyle\sigma_{0}\in(0,1)$ and $\displaystyle j_{*}\in(1,\infty)$
depending only on the data such that either

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,8\rho,J_{2}^{\prime})>\exp\left(\frac{-2b_{0}^{2-p}}{2-p}\right)\frac{\epsilon
M}{2^{j_{*}}},$

holds (recall Eq. 6.9) or there exists a time level
$\displaystyle\tau_{1}\in(\tau_{0}+b_{0}^{2-p},\tau_{0}+2b_{0}^{2-p})$ such
that the following holds:

$v(z,\tau_{1})\geq\sigma_{0}e^{\frac{\tau_{0}}{2-p}}.$ (6.12)

###### Proof of the claim.

Assume momentarily that $\displaystyle j_{*}$ is fixed and hence
$\displaystyle\nu$ has been determined according to (Step 2). By increasing
$\displaystyle j_{*}$ if required to not necessarily be an integer, we can
further assume without loss of generality that $\displaystyle 2^{j_{*}(2-p)}$
is an integer. Next, we subdivide $\displaystyle Q_{\tau_{0}}$ into
$\displaystyle 2^{j_{*}(2-p)}$ cylinders each of length $\displaystyle
b_{j_{*}}^{2-p}$ given by

$\displaystyle
Q_{n}=B_{8}\times(\tau_{0}+b_{0}^{2-p}+nb_{j_{*}}^{2-p},\tau_{0}+b_{0}^{2-p}+(n+1)b_{j_{*}}^{2-p})\qquad\mbox{
for }n=0,1,\ldots,2^{j_{*}(2-p)}-1.$

Then from (Step 2), either Eq. 6.9 holds or for at least one of the sub-
cylinders, we must have

$\displaystyle|[v<b_{j_{*}}\cap Q_{n}]|\leq\nu|Q_{n}|.$

We now apply Lemma 3.8 to $\displaystyle v$ over $\displaystyle Q_{n}$ with

$\displaystyle\mu_{-}=0,\qquad\xi M=b_{j_{*}},\qquad
a=\frac{1}{2},\qquad\text{and}\quad\theta=b_{j_{*}}^{2-p},$

to obtain

$v(z,\tau_{0}+b_{0}^{2-p}+(n+1)b_{j_{*}}^{2-p})\geq\frac{1}{2}b_{j_{*}}\qquad\text{
a.e. }z\in B_{4}.$ (6.13)

provided the following two estimates hold

$\begin{array}[]{c}\operatorname*{\operatorname{Tail}_{\infty}}(v_{-};0,8,(\tau_{0}+b_{0}^{2-p}+nb_{j_{*}}^{2-p},\tau_{0}+b_{0}^{2-p}+(n+1)b_{j_{*}}^{2-p}))\leq
b_{j_{*}},\\\ \frac{|[v<b_{j_{*}}]\cap
Q_{n}|}{|Q_{n}|}\overset{\lx@cref{creftype~refnum}{eq3.18}}{\leq}3^{-\frac{N+sp}{sp}}\overline{\gamma_{0}}(\text{data})=:\nu.\end{array}$
(6.14)

With the choice of $\displaystyle\nu$ from Eq. 6.14, we can now choose
$\displaystyle j_{*}$ according to (Step 2) such that Eq. 6.13 holds.
Recalling the definition of $\displaystyle b_{0}$ from Eq. 6.5, if we set
$\displaystyle\sigma_{0}=\epsilon 2^{-(j_{*}+1)}$, then we see that Eq. 6.12
holds with $\displaystyle\tau_{1}=\tau_{0}+b_{0}^{2-p}+(n+1)b_{j_{*}}^{2-p}$.

Let us now rewrite the Tail alternative in Eq. 6.14 in terms of $\displaystyle
u$ as in (Step 2) to get

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,8\rho,J_{2}^{\prime})\leq\exp\left(\frac{-2b_{0}^{2-p}}{2-p}\right)\frac{\epsilon
M}{2^{j_{*}}}.$

In particular, the claim holds provided the Tail alternative from Eq. 6.14 is
satisfied. ∎

Step 4:

Expanding the positivity for $\displaystyle u$. Recalling the change of
variables from Eq. 6.2, we see that if the Tail alternatives in (Step 2) and
(Step 3) are satisfied, then the following conclusion holds

$\displaystyle
u(\cdot,t_{1})\geq\sigma_{0}Me^{-\frac{\tau_{1}-\tau_{0}}{2-p}}=:M_{0}\qquad\text{
in }B_{4\rho},$

where $\displaystyle t_{1}=\delta M^{2-p}\rho^{sp}(1-e^{-\tau_{1}})$.

With $\displaystyle\nu_{0}$ as obtained in Eq. 3.21, let us first choose
$\displaystyle\theta=\nu_{0}^{\frac{1}{s}}M_{0}^{2-p}$. We now apply Lemma
3.11 (noting that $\displaystyle\tilde{\nu}=1$ in Eq. 3.21 so that the
smallness assumption is trivially satisfied) with $\displaystyle M_{0}$ in
place of $\displaystyle M$ and $\displaystyle\xi=1$ over the cylinder
$\displaystyle B_{4\rho}\times(t_{1},t_{1}+\theta(4\rho)^{sp}]$ to see that
the following holds

$u(x,t)\geq\frac{1}{2}M_{0}\geq\frac{1}{2}\sigma_{0}\exp\left(-\frac{2}{2-p}e^{\tau_{0}}\right)M\qquad\text{
for all }(x,t)\in
B_{2\rho}\times(t_{1},t_{1}+\nu_{0}^{\frac{1}{s}}M_{0}^{2-p}(4\rho)^{sp}),$

provided
$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,4\rho,(t_{1},t_{1}+\theta(4\rho)^{sp}])\leq
M_{0}$ is satisfied.

We now choose $\displaystyle\tau_{0}$ satisfying

$\delta M^{2-p}\rho^{sp}e^{-\tau_{1}}=\delta
M^{2-p}\rho^{sp}-t_{1}=\nu_{0}^{\frac{1}{s}}\sigma_{0}^{2-p}M^{2-p}(4\rho)^{sp}e^{-(\tau_{1}-\tau_{0})}\iff\tau_{0}=\ln\left(\frac{\delta}{4^{sp}\nu_{0}^{\frac{1}{s}}\sigma_{0}^{2-p}}\right).$
(6.15)

This determines $\displaystyle\tau_{0}$ in terms of $\displaystyle\alpha$ and
data (note that by increasing $\displaystyle j_{*}$ if necessary we can always
ensure that $\displaystyle\tau_{0}>0$). Finally, setting
$\displaystyle\eta:=\frac{1}{2}\sigma_{0}\exp\left(-\frac{2}{2-p}e^{\tau_{0}}\right)\in(0,1)$
which depends only on $\displaystyle\alpha$ and data, we get that

$\displaystyle u(\cdot,t)\geq\eta M\qquad\text{ a.e. in }B_{2\rho},$

holds for all times

$(1-\varepsilon)\delta M^{2-p}\rho^{sp}\leq t\leq
M^{2-p}\delta\rho^{sp}\qquad\text{ where
}\varepsilon=\exp(-\tau_{0}-2e^{\tau_{0}}).$ (6.16)

Step 5:

Pulling the Tail alternatives together. We needed the following Tail
assumptions going from (Step 1) to (Step 4) (recall $\displaystyle j\leq
j_{*})$:

$\displaystyle\begin{array}[]{rcl}\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,\rho,(0,\delta\rho^{sp}M^{2-p}))&\leq&M,\\\
\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,16\rho,J_{2}^{\prime})&\leq&\exp\left(\tfrac{-2b_{0}^{2-p}}{2-p}\right)\frac{\epsilon
M}{2^{j}},\\\
\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,8\rho,J_{2}^{\prime})&\leq&\exp\left(\tfrac{-2b_{0}^{2-p}}{2-p}\right)\frac{\epsilon
M}{2^{j_{*}}},\\\
\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,4\rho,(t_{1},t_{1}+\theta(4\rho)^{sp}])&\leq&M_{0}.\end{array}$

We note that in all the Tail alternatives, $\displaystyle\rho$ is the smallest
radius and $\displaystyle(0,\delta\rho^{sp}M^{2-p})$ is the largest time
interval. We estimate

$\displaystyle\displaystyle\exp\left(\frac{-2b_{0}^{2-p}}{2-p}\right)\frac{\epsilon
M}{2^{j}}=\exp\left(-\frac{2\epsilon^{2-p}e^{\tau_{0}}}{2-p}\right)\frac{\epsilon
M}{2^{j}}\geq\exp\left(-\frac{2e^{\tau_{0}}}{2-p}\right)\frac{\epsilon
M}{2^{j}}=\frac{2\eta}{\sigma_{0}}\frac{\epsilon M}{2^{j}}\geq 4\eta M,$
$\displaystyle\displaystyle
M_{0}\geq\sigma_{0}\exp\left(-\frac{2}{2-p}e^{\tau_{0}}\right)M=2\eta M,$

where we recall $\displaystyle\epsilon$ is from Eq. 6.16 and
$\displaystyle\sigma_{0}=\frac{\epsilon}{2^{j_{*}+1}}$.

Since $\displaystyle\eta\in(0,1)$, we see that all the Tail assumptions are
satisfied if we require

$\displaystyle\operatorname*{\operatorname{Tail}_{\infty}}(u_{-};0,\rho,(0,\delta\rho^{sp}M^{2-p}))\leq\eta
M.$

This completes the proof of the proposition. ∎

######  Remark 6.2.

The conclusion of Proposition 6.1 can also be written without Tail
alternatives as

$\displaystyle u(\cdot,t)\geq\eta
M-\textup{Tail}_{\infty}(u_{-};x_{0},\rho,(t_{0},t_{0}+\delta\rho^{sp}M^{2-p}))\qquad\text{a.e.
in }B_{2\rho}(x_{0}),$

for all times

$\displaystyle t_{0}+(1-\varepsilon)\delta M^{2-p}\rho^{sp}\leq t\leq
t_{0}+M^{2-p}\delta\rho^{sp},$

because we are working with $\displaystyle u$ nonnegative.

######  Remark 6.3.

In the proof of Hölder regularity we use Proposition 6.1 with
$\displaystyle\alpha=1/2$ so that $\displaystyle\eta,\delta$ and
$\displaystyle\epsilon\in(0,1)$ depend only on the data in the proof.

######  Remark 6.4.

In the degenerate case the final time level depends on the final lower level
$\displaystyle\eta M$ that is achieved during the expansion whereas in the
singular case, the final time level depends on the starting level
$\displaystyle M$.

##  7 Hölder regularity for degenerate parabolic fractional $\displaystyle
p$-Laplace equations

In this section, we present the proof of Theorem 2.6 in the case
$\displaystyle p>2$. The induction argument is similar to the one in [14] and
the covering argument is taken from [27, 28]. Since the complete details are
not easily available in the literature, we present all the calculations for
the sake of completeness. We assume without loss of generality that
$\displaystyle(x_{0},t_{0})=(0,0)$ and choose $\displaystyle\beta\in(0,1)$
satisfying

$0<\beta<\min\left\\{\frac{sp}{p-1},\frac{sp}{p-2},\log_{C_{0}}\left(\frac{2}{2-\eta}\right)\right\\}\quad\mbox{
and
}\quad\int_{1}^{\infty}\frac{(\rho^{\beta}-1)^{p-1}}{\rho^{1+sp}}\,d\rho<\frac{\eta^{p-1}}{2^{p}\mathbf{C}_{1}},$
(7.1)

where $\displaystyle\eta$ is the constant that appears in Proposition 5.1 for
$\displaystyle\alpha=1/2$ and $\displaystyle\mathbf{C}_{1}$ appears in Eq.
7.9. The constant $\displaystyle C_{0}\geq 2$ will be determined in a later
step depending only on the data. The integral can be made small enough by
taking small $\displaystyle\beta$. Also define

$\displaystyle\displaystyle
j_{0}:=\left\lceil\frac{1}{sp-\beta(p-1)}\log_{C_{0}}\left(\frac{2^{p}(\mathbf{C}_{3}+2^{p-1})}{\eta^{p-1}}\right)\right\rceil,$
(7.2)

where $\displaystyle\eta$ is again the constant that appears in Proposition
5.1 for $\displaystyle\alpha=1/2$.

######  Claim 7.1.

For a universal constant $\displaystyle C_{0}\gg 2$ and $\displaystyle b,\eta$
and $\displaystyle\delta$ as obtained in Proposition 5.1 for
$\displaystyle\alpha=1/2$, we claim that there exist non-decreasing sequence
$\displaystyle\\{m_{i}\\}_{i=0}^{\infty}$ and non-increasing sequence
$\displaystyle\\{M_{i}\\}_{i=0}^{\infty}$ such that for any $\displaystyle
i=1,2,\ldots$, we have

$m_{i}\leq u\leq M_{i}\qquad\mbox{ in }\
Q_{R_{i}}(d_{i}):=B_{R_{i}}\times(-d_{i}\,R_{i}^{sp},0)=B_{i}\times I_{i}.$
(7.3)

Here, we have denoted

$\begin{array}[]{rcl}M_{i}-m_{i}&=&C_{0}^{-\beta i}L,\\\
R_{i}&:=&C_{0}^{1-i}R,\\\ L&:=&2\cdot
C_{0}^{\frac{sp}{p-1}\,j_{0}}\|u\|_{L^{\infty}(B_{C_{0}R}\times(-(C_{0}R)^{2s},0))}+\operatorname*{\operatorname{Tail}_{\infty}}(u;C_{0}\,R,0,(-(C_{0}\,R)^{2s},0)),\\\
d_{i}&:=&\frac{b^{p-2}}{(\eta\,C_{0}^{-\beta
i}L)^{p-2}}\delta\frac{2^{p-2}}{(2C_{0})^{sp}}\end{array}$ (7.4)

By a slight abuse of notation, only when $\displaystyle i=0$, we will denote
$\displaystyle Q_{0}=Q_{R_{0}}(d_{0}):=B_{C_{0}R}\times(-(C_{0}R)^{2s},0)$
instead of $\displaystyle B_{C_{0}R}\times(-(C_{0}R)^{sp},0)$

###### Proof of Claim 7.1.

The proof will proceed in two steps, first we show Eq. 7.3 holds for
$\displaystyle i=0,1,2,\ldots,j_{0}$ and then use induction to obtain Eq. 7.3
for all $\displaystyle i>j_{0}$.

Step 1:

Without loss of generality, we can assume $\displaystyle
Q_{R_{1}}(d_{1})\subset B_{C_{0}R}\times(-(C_{0}R)^{2s},0)$, since otherwise,
we would have

$\displaystyle L^{p-2}\leq R^{s(p-2)}\frac{b^{p-2}\delta}{C_{0}^{2s}(\eta
C_{0}^{-\beta})^{p-2}}\frac{2^{p-2}}{(2C_{0})^{sp}},$

which implies oscillation is comparable to the radius.

It is also easy to see that $\displaystyle Q_{R_{i+1}}(d_{i+1})\subseteq
Q_{R_{i}}(d_{i})$ for $\displaystyle i=1,2,\ldots$, since from Eq. 7.1, we
have $\displaystyle\beta(p-2)-sp<0$ which implies $\displaystyle
d_{i+1}R_{i+1}^{sp}\leq d_{i}R_{i}^{sp}$.

Step 2:

For $\displaystyle i=0,1,\ldots,j_{0}$, let us define $\displaystyle
m_{i}:=\tfrac{-C_{0}^{-\beta i}L}{2}$ and $\displaystyle
M_{i}:=\tfrac{C_{0}^{-\beta i}L}{2}$. From (Step 1), we have $\displaystyle
Q_{R_{i}}(d_{i})\subset Q_{0}$ holds and thus we have

$\begin{array}[]{rcl}\|u\|_{L^{\infty}(Q_{R_{i}}(d_{i}))}\leq\|u\|_{L^{\infty}(Q_{0})}=\tfrac{C_{0}^{-\beta
i}}{2}\left(2C_{0}^{\frac{sp}{p-1}j_{0}}\|u\|_{L^{\infty}(Q_{0})}\right)\left(\frac{C_{0}^{\beta
<EMAIL_ADDRESS>
Step 3:

For some $\displaystyle j\geq j_{0}$, we suppose that the sequence
$\displaystyle M_{i}$ and $\displaystyle m_{i}$ have been defined for
$\displaystyle i=1,2,\ldots,j$. Inductively, we will construct $\displaystyle
m_{j+1}$ and $\displaystyle M_{j+1}$ so that Eq. 7.3 holds. Define the
function $\displaystyle v:=u-\tfrac{(M_{j}+m_{j})}{2}$, then using
monotonicity of $\displaystyle M_{j}$ and $\displaystyle m_{j}$, we see that
the following holds

$\displaystyle(M_{j}-m_{j})+2m_{0}\leq M_{j}+m_{j}\leq 2M_{0}-(M_{j}-m_{j}).$

Recalling $\displaystyle m_{0}=-\tfrac{L}{2}$ and $\displaystyle
M_{0}=\tfrac{L}{2}$ along with the choice $\displaystyle
m_{i}:=\tfrac{-C_{0}^{-\beta i}L}{2}$ and $\displaystyle
M_{i}:=\tfrac{C_{0}^{-\beta i}L}{2}$ gives

$-(1-C_{0}^{-\beta\,j})L\leq M_{j}+m_{j}\leq(1-C_{0}^{-\beta\,j})L.$ (7.5)

Thus, we have

$\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}=\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
u\mp\tfrac{M_{j}+m_{j}}{2}\right)_{-}^{p-1}\overset{\lx@cref{creftype~refnum}{Mmest1}}{\leq}\left(|u|+\tfrac{L}{2}\right)^{p-1}\leq
2^{p-1}|u|^{p-1}+L^{p-1}.$ (7.6)

Step 4:

Recalling the notation from Eq. 7.3, we estimate

$\begin{array}[]{rcl}&&\operatorname*{\operatorname{Tail}_{\infty}}\biggr{(}\big{(}\tfrac{C_{0}^{-\beta
j}L}{2}\pm v\big{)}_{-};C_{0}^{1-j}R,0,I_{j}\biggr{)}^{p-1}\\\
&=&C_{0}^{(1-j)sp}R^{sp}\,\operatorname*{ess\,sup}\limits_{I_{j}}\Bigg{[}\int_{B_{0}\setminus
B_{j}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx+\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx\Bigg{]}\\\
&\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{estim1}}}{{\leq}}&C_{0}^{(1-j)sp}R^{sp}\,\Bigg{[}\operatorname*{ess\,sup}\limits_{I_{j}}\int_{B_{0}\setminus
B_{j}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx\\\
&&+2^{p-1}\operatorname*{ess\,sup}\limits_{I_{0}}\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{|u|^{p-1}}{|x|^{N+ps}}\,dx+L^{p-1}\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{1}{|x|^{N+ps}}\,dx\Bigg{]}\\\
&\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{defofL}}}{{\leq}}&C_{0}^{(1-j)sp}R^{sp}\,\Bigg{[}\underbrace{\operatorname*{ess\,sup}\limits_{I_{j}}\int_{B_{0}\setminus
B_{j}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx}_{:=J}+2^{p-1}\frac{L^{p-1}}{(C_{0}\,R)^{sp}}+\mathbf{C}_{3}\frac{L^{p-1}}{(C_{0}\,R)^{sp}}\Bigg{]}.\end{array}$
(7.7)

To estimate $\displaystyle J$, for given $\displaystyle x\in B_{0}\setminus
B_{j}$, there is $\displaystyle l\in\\{0,1,2,\ldots,j-1\\}$ such that
$\displaystyle x\in B_{l-1}\setminus B_{l}$ so that by the monotonicity of the
sequence $\displaystyle m_{i}$ and by the induction hypothesis, for a.e.
$\displaystyle t\in I_{j}$, we have:

$\displaystyle\begin{array}[]{rcl}v(x,t)&\geq&-(M_{l}-m_{l})+\tfrac{M_{j}-m_{j}}{2}\\\
&=&-\left[C_{0}^{-\beta l}-\tfrac{C_{0}^{-\beta j}}{2}\right]L\\\
&\geq&-\left[\left(\tfrac{|x|}{C_{0}R}\right)^{\beta}-\tfrac{C_{0}^{-\beta
j}}{2}\right]L\end{array}$
$\displaystyle\begin{array}[]{rcl}v(x,t)&\leq&(M_{l}-m_{l})-\tfrac{M_{j}-m_{j}}{2}\\\
&=&\left[C_{0}^{-\beta l}-\tfrac{C_{0}^{-\beta j}}{2}\right]L\\\
&\leq&\left[\left(\tfrac{|x|}{C_{0}R}\right)^{\beta}-\tfrac{C_{0}^{-\beta
j}}{2}\right]L\end{array}$ (7.8)

From Eq. 7.8, we get

$\displaystyle\displaystyle\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v(x,t)\right)_{-}^{p-1}\leq\left[\left(\frac{|x|}{C_{0}R}\right)^{\beta}-C_{0}^{-\beta
j}\right]^{p-1}L^{p-1}$

We use this to estimate

$\begin{array}[]{rcl}J&\leq&C_{0}^{-\beta
j(p-1)}L^{p-1}\operatorname*{ess\,sup}_{I_{j}}\int_{\mathbb{R}^{N}\setminus
B_{C_{0}^{1-j}R}}\frac{\left(\left(\frac{|x|}{C_{0}^{1-j}R}\right)^{\beta}-1\right)^{p-1}}{|x|^{N+sp}}\,dx\\\
&=&\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}C_{0}^{(j-1)sp}}{R^{sp}}\int_{\mathbb{R}^{N}\setminus
B_{1}}\frac{\left(|y|^{\beta}-1\right)^{p-1}}{|y|^{N+sp}}\,dy\\\
&\leq&\mathbf{C}_{1}\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}C_{0}^{(j-1)sp}}{R^{sp}}\int_{1}^{\infty}\frac{\left(\rho^{\beta}-1\right)^{p-1}}{\rho^{1+sp}}\,d\rho.\end{array}$
(7.9)

Substituting Eq. 7.9 into Eq. 7.7, we obtain

$\operatorname*{\operatorname{Tail}_{\infty}}\left(\left(\tfrac{C_{0}^{-\beta
j}L}{2}\pm v\right)_{-};C_{0}^{1-j}R,0,I_{j}\right)^{p-1}\\\
\begin{array}[]{rcl}&\leq&C_{0}^{(1-j)sp}R^{sp}\,\left[\mathbf{C}_{1}\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}C_{0}^{(j-1)ps}}{R^{sp}}\int_{1}^{\infty}\frac{\left(\rho^{\beta}-1\right)^{p-1}}{\rho^{1+ps}}\,d\rho+2^{p-1}\frac{L^{p-1}}{(C_{0}\,R)^{sp}}+\mathbf{C}_{3}\frac{L^{p-1}}{(C_{0}\,R)^{sp}}\right]\\\
&\leq&C_{0}^{-\beta
j(p-1)}L^{p-1}\left[\mathbf{C}_{1}\int_{1}^{\infty}\frac{(\rho^{\beta}-1)^{p-1}}{\rho^{1+ps}}\,d\rho+(\mathbf{C}_{3}+2^{p-1})\,C_{0}^{-(sp-\beta(p-1))j}\right]\\\
&\overset{\hypertarget{7.10a}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}}}{\leq}&\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}\eta^{p-1}}{2^{p-1}},\end{array}$ (7.10)

where (a) follows from Eq. 7.1, Eq. 7.2 and the fact that $\displaystyle j\geq
j_{0}$.

Step 5:

Now, one of the following two alternatives must hold:

$\displaystyle\displaystyle\left|\\{v(\cdot,s_{j})\geq 0\\}\cap
B_{C_{0}^{-j}R/2}\right|\geq\frac{1}{2}\left|B_{C_{0}^{-j}R/2}\right|,$ (7.11)
$\displaystyle\displaystyle\left|\\{v(\cdot,s_{j})\geq 0\\}\cap
B_{C_{0}^{-j}R/2}\right|<\frac{1}{2}\left|B_{C_{0}^{-j}R/2}\right|,$ (7.12)

where we have set

$\displaystyle\displaystyle s_{j}:=-\frac{b^{p-2}}{(\eta\,C_{0}^{-\beta
j}L)^{p-2}}\delta R_{j}^{sp}\frac{2^{p-2}}{(2C_{0})^{sp}}.$

Without loss of generality, we assume that Eq. 7.11 holds, noting that if Eq.
7.12 holds, analogous conclusion follows with $\displaystyle v$ replaced by
$\displaystyle-v$.

Step 6:

Now denote $\displaystyle w=\tfrac{C_{0}^{-\beta j}L}{2}+v$, then Eq. 7.11
becomes

$\displaystyle\displaystyle\left|\left\\{w(\cdot,s_{j})\geq\tfrac{C_{0}^{-\beta
j}L}{2}\right\\}\cap
B_{C_{0}^{-j}R/2}\right|\geq\frac{1}{2}\left|B_{C_{0}^{-j}R/2}\right|.$

Moreover from the induction hypothesis, we see that $\displaystyle w\geq 0$ in
$\displaystyle Q_{R_{j}}(d_{j})$ because

$\displaystyle\displaystyle w=v+\tfrac{C_{0}^{-\beta
j}L}{2}=u-\tfrac{M_{j}+m_{j}}{2}+\tfrac{C_{0}^{-\beta
j}L}{2}\overset{\lx@cref{creftype~refnum}{defofL}}{=}u-\tfrac{M_{j}+m_{j}}{2}+\tfrac{M_{j}-m_{j}}{2}=u-m_{j}\geq
0.$

We see that since Eq. 7.10 holds, the Tail alternative in Proposition 5.1 is
satisfied. Therefore, we get

$\displaystyle\displaystyle w\geq\frac{\eta C_{0}^{-\beta j}L}{2}\qquad\mbox{
a.e. }(x,t)\in Q_{R_{j+1}}(d_{j+1}),$ (7.13)

provided $\displaystyle
C_{0}>\left(\tfrac{1}{1-\sigma^{\prime}}\right)^{\frac{1}{\beta(2-p)+sp}}$
where $\displaystyle\sigma^{\prime}\in(0,1)$ is as obtained in Proposition 5.1
for $\displaystyle\alpha=1/2$. Recalling Eq. 7.1 we enforce such a universal
choice of $\displaystyle C_{0}\gg 2$. We calculate

$\begin{array}[]{rcl}u(x,t)&=&\tfrac{M_{j}+m_{j}}{2}+v=\tfrac{M_{j}+m_{j}}{2}+w-\tfrac{C_{0}^{-\beta
j}L}{2}\\\
&\overset{\lx@cref{creftype~refnum}{redofosc}}{\geq}&\tfrac{M_{j}+m_{j}}{2}-\tfrac{C_{0}^{-\beta
j}L}{2}(1-\eta)\\\ &=&M_{j}-\tfrac{M_{j}-m_{j}}{2}-\tfrac{C_{0}^{-\beta
j}L}{2}(1-\eta)\\\
&\overset{\hypertarget{ind}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}}}{=}&M_{j}-\tfrac{C_{0}^{-\beta
j}L}{2}(2-\eta)\\\
&\overset{\lx@cref{creftype~refnum}{choiceofalpha}}{\geq}&M_{j}-C_{0}^{-\beta(j+1)}L,\end{array}$

where (a) follows from the induction hypothesis. Thus if we define
$\displaystyle m_{j+1}:=M_{j}-C_{0}^{-\beta(j+1)}L$ and $\displaystyle
M_{j+1}:=M_{j}$ then the inductive process in (Step 3) is completed.

This completes the proof of the claim. ∎

A consequence of Claim 7.1 is the following oscillation decay:

$\operatorname*{ess\,osc}_{Q_{R_{i}}(d_{i})}\,u:=\operatorname*{ess\,sup}_{Q_{R_{i}}(d_{i})}\,u-\operatorname*{ess\,inf}_{Q_{R_{i}}(d_{i})}\,u\leq
C_{0}^{-\beta i}L.$ (7.14)

###### Proof of Theorem 2.6.

With notation as in Claim 7.1, let us take $\displaystyle
Q_{0}:=B_{C_{0}R}\times(-(C_{0}R)^{2s},0)$ and define $\displaystyle L$,
$\displaystyle R_{i}$ and $\displaystyle d_{i}$ as Eq. 7.4.

Consider any two points Let $\displaystyle(x_{1},t_{1}),(x_{2},t_{2})\in
B_{R}(0)\times(-d_{1}^{2-p}R^{sp},0)$ such that $\displaystyle x_{1}\neq
x_{2}$ and $\displaystyle t_{1}\neq t_{2}$, then there exist non-negative
integers $\displaystyle n$ and $\displaystyle m$ such that

$R_{n+1}<|x_{1}-x_{2}|\leq R_{n},\qquad\text{and}\qquad
d_{m+1}R_{m+1}^{sp}<|t_{1}-t_{2}|\leq d_{m}R_{m}^{sp}.$ (7.15)

As a result, with $\displaystyle k:=\max\\{n,m\\}$, we obtain

$\displaystyle\displaystyle|u(x_{1},t_{1})-u(x_{2},t_{2})|\leq\operatorname*{ess\,osc}_{Q_{r_{k}}(d_{k})}u\overset{\lx@cref{creftype~refnum}{{oscdecays}}}{\leq}\max\\{C_{0}^{-\beta
n},C_{0}^{-\beta m}\\}L.$ (7.16)

From the first inequality in Eq. 7.15, we deduce

$\frac{|x_{1}-x_{2}|}{R}>C_{0}^{-n}\qquad\Longrightarrow\qquad\left(\frac{|x_{1}-x_{2}|}{R}\right)^{\beta}\,L>C_{0}^{-\beta
n}L.$ (7.17)

On the other hand, from the second inequality in Eq. 7.15, we get

$\mathbf{C}_{4}\,L^{\frac{p-2}{sp}}\frac{|t_{1}-t_{2}|^{\frac{1}{sp}}}{R}>\left(C_{0}^{\beta(m+1)}\right)^{\frac{p-2}{sp}}\,C_{0}^{-m}\geq
C_{0}^{-m}$ (7.18)

where

$\mathbf{C}_{4}=\left(\frac{\eta^{p-2}}{b^{p-2}\delta}\frac{(2C_{0})^{sp}}{2^{p-2}}\right)^{\frac{1}{sp}}$
(7.19)

is a constant depending only on data and the last inequality in Eq. 7.18
follows since $\displaystyle p>2$ which implies the following
$\displaystyle\left(C_{0}^{-\beta(m+1)}\right)^{\frac{2-p}{sp}}\geq 1$ holds
as $\displaystyle C_{0}>1$.

The proof of the theorem follows from substituting Eq. 7.17 and Eq. 7.18 into
Eq. 7.16. ∎

##  8 Hölder regularity for singular parabolic fractional $\displaystyle
p$-Laplace equations

In this section, we present the proof of Theorem 2.7 in the case
$\displaystyle p<2$. The induction argument is similar to the degenerate proof
from Section 7. Since the complete details are not easily available in the
literature, we present all the calculations for the sake of completeness. We
assume without loss of generality that $\displaystyle(x_{0},t_{0})=(0,0)$ and
choose $\displaystyle\beta\in(0,1)$ satisfying

$0<\beta<\min\left\\{sp,\frac{sp}{2-p},\log_{C_{0}}\left(\frac{2}{2-\eta}\right)\right\\}\quad\mbox{
and
}\quad\int_{1}^{\infty}\frac{(\rho^{\frac{\beta}{\Gamma}}-1)^{p-1}}{\rho^{1+sp}}\,d\rho<\frac{\eta^{p-1}}{2^{p}\mathbf{C}_{1}},$
(8.1)

where $\displaystyle\Gamma:=\tfrac{\beta(p-2)}{sp}+1>0$, $\displaystyle\eta$
is the constant that appears in Proposition 6.1 for $\displaystyle\alpha=1/2$
and $\displaystyle\mathbf{C}_{1}$ appears in Eq. 8.10. The constant
$\displaystyle C_{0}\geq 2$ will be determined in a later step depending only
on the data The integral can be made small enough by taking small
$\displaystyle\beta$. We further fix $\displaystyle\varepsilon_{0}>0$ and
define

$\displaystyle\displaystyle
j_{0}:=\left\lceil\frac{1}{(sp-\beta)}\log_{C_{0}}\left(\frac{2^{p}C_{0}^{\varepsilon_{0}sp}(\mathbf{C}_{3}+2^{p-1})}{\eta}\right)\right\rceil,$
(8.2)

where $\displaystyle\eta$ is the constant that appears in Proposition 6.1 for
$\displaystyle\alpha=1/2$.

######  Claim 8.1.

Let $\displaystyle R\leq 1$ be fixed, then for a universal constant
$\displaystyle C_{0}\gg 2$ and $\displaystyle b,\eta$ as obtained in
Proposition 6.1, we claim that there exist non-decreasing sequence
$\displaystyle\\{m_{i}\\}_{i=0}^{\infty}$ and non-increasing sequence
$\displaystyle\\{M_{i}\\}_{i=0}^{\infty}$ such that for any $\displaystyle
i=1,2,\ldots$, we have

$m_{i}\leq u\leq M_{i}\qquad\mbox{ in }\
Q_{R_{i}}(d_{i}):=B_{d_{i}R_{i}}\times(-R_{i}^{sp},0)=B_{i}\times I_{i}.$
(8.3)

Here, we have denoted

$\begin{array}[]{rcl}M_{i}-m_{i}&=&C_{0}^{-\beta
i}L\overset{\lx@cref{creftype~refnum}{choiceofalphasin}}{\searrow}0,\\\
R_{i}&:=&C_{0}^{1-i}R,\\\ L&:=&2\cdot C_{0}^{-\beta
j_{0}}\|u\|_{L^{\infty}(B_{(C_{0}R)^{1-\varepsilon_{o}}}\times(-(C_{0}R)^{sp},0))}+\operatorname*{\operatorname{Tail}_{\infty}}(u;(C_{0}\,R)^{1-\varepsilon_{o}},0,(-(C_{0}\,R)^{sp},0))+1,\\\
d_{i}&:=&(\eta\,C_{0}^{-\beta i}L)^{\frac{p-2}{sp}}\delta\end{array}$ (8.4)

where $\displaystyle\delta>0$, $\displaystyle b>0$ and
$\displaystyle\eta\in(0,1)$ appears in Proposition 6.1 for $\displaystyle
p<2$.

By a slight abuse of notation, only when $\displaystyle i=0$, we will denote
$\displaystyle
Q_{0}=Q_{R_{0}}(d_{0}):=B_{(C_{0}R)^{1-\varepsilon_{o}}}\times(-(C_{0}R)^{sp},0)=B_{0}\times
I_{0}$.

###### Proof of Claim 8.1.

The proof will proceed in two steps, first we show Eq. 8.3 holds for
$\displaystyle i=0,1,2,\ldots,j_{0}$ and then use induction to obtain Eq. 8.3
for all $\displaystyle i>j_{0}$.

Step 1:

Without loss of generality, we can assume $\displaystyle
Q_{R_{1}}(d_{1})\subset
B_{(C_{0}R)^{1-\varepsilon_{o}}}\times(-(C_{0}R)^{sp},0)$, since otherwise, we
would have

$\displaystyle L^{\frac{2-p}{sp}}\leq R^{\varepsilon_{o}}\frac{(\eta
C_{0}^{-\beta})^{\frac{p-2}{sp}}}{C_{0}^{1-\varepsilon_{o}}},$

which implies oscillation is comparable to the radius.

It is also easy to see that $\displaystyle Q_{R_{i+1}}(d_{i+1})\subseteq
Q_{R_{i}}(d_{i})$ for $\displaystyle i=1,2,\ldots$, since from Eq. 8.1, we
have $\displaystyle\beta(2-p)-sp<0$ which implies $\displaystyle
d_{i+1}R_{i+1}\leq d_{i}R_{i}$.

Step 2:

For $\displaystyle i=0,1,\ldots,j_{0}$, let us define $\displaystyle
m_{i}:=\tfrac{-C_{0}^{-\beta i}L}{2}$ and $\displaystyle
M_{i}:=\tfrac{C_{0}^{-\beta i}L}{2}$. From (Step 1), we have $\displaystyle
Q_{R_{i}}(d_{i})\subset Q_{0}$ holds and thus we have

$\begin{array}[]{rcl}\|u\|_{L^{\infty}(Q_{R_{i}}(d_{i}))}\leq\|u\|_{L^{\infty}(Q_{0})}=\tfrac{C_{0}^{-\beta
i}}{2}\left(2C_{0}^{-\beta
j_{0}}\|u\|_{L^{\infty}(Q_{0})}\right)\left(\frac{C_{0}^{\beta
i}}{C_{0}^{-\beta
<EMAIL_ADDRESS>
Step 3:

For some $\displaystyle j\geq j_{0}$, we suppose that the sequence
$\displaystyle M_{i}$ and $\displaystyle m_{i}$ have been defined for
$\displaystyle i=1,2,\ldots,j$. Inductively, we will construct $\displaystyle
m_{j+1}$ and $\displaystyle M_{j+1}$ so that Eq. 8.3 holds. Define the
function $\displaystyle v:=u-\tfrac{(M_{j}+m_{j})}{2}$, then using
monotonicity of $\displaystyle M_{j}$ and $\displaystyle m_{j}$, we see that
the following holds

$\displaystyle(M_{j}-m_{j})+2m_{0}\leq M_{j}+m_{j}\leq 2M_{0}-(M_{j}-m_{j}).$

Recalling $\displaystyle m_{0}=-\tfrac{L}{2}$ and $\displaystyle
M_{0}=\tfrac{L}{2}$ along with the choice $\displaystyle
m_{i}:=\tfrac{-C_{0}^{-\beta i}L}{2}$ and $\displaystyle
M_{i}:=\tfrac{C_{0}^{-\beta i}L}{2}$ gives

$-(1-C_{0}^{-\beta j})L\leq M_{j}+m_{j}\leq(1-C_{0}^{-\beta j})L.$ (8.5)

Thus, we have

$\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}=\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
u\mp\tfrac{M_{j}+m_{j}}{2}\right)_{-}^{p-1}\overset{\lx@cref{creftype~refnum}{Mmest1sin}}{\leq}\left(|u|+\tfrac{L}{2}\right)^{p-1}\leq
2^{p-1}|u|^{p-1}+L^{p-1}.$ (8.6)

Step 4:

Recalling the notation from Eq. 8.3, we estimate

$\begin{array}[]{rcl}&&\operatorname*{\operatorname{Tail}_{\infty}}\biggr{(}\big{(}\tfrac{C_{0}^{-\beta
j}L}{2}\pm v\big{)}_{-};B_{j},0,I_{j}\biggr{)}^{p-1}\\\
&=&(d_{j}R_{j})^{sp}\,\operatorname*{ess\,sup}\limits_{I_{j}}\Bigg{[}\int_{B_{0}\setminus
B_{j}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx+\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx\Bigg{]}\\\
&\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{estim1sin}}}{{\leq}}&(d_{j}R_{j})^{sp}\,\Bigg{[}\operatorname*{ess\,sup}\limits_{I_{j}}\int_{B_{0}\setminus
B_{j}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx\\\
&&+2^{p-1}\operatorname*{ess\,sup}\limits_{I_{0}}\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{|u|^{p-1}}{|x|^{N+ps}}\,dx+L^{p-1}\int_{\mathbb{R}^{N}\setminus
B_{0}}\frac{1}{|x|^{N+ps}}\,dx\Bigg{]}\\\
&\stackrel{{\scriptstyle\lx@cref{creftype~refnum}{defofLsin}}}{{\leq}}&(d_{j}R_{j})^{sp}\,\Bigg{[}\underbrace{\operatorname*{ess\,sup}\limits_{I_{j}}\int_{B_{0}\setminus
B_{j}}\frac{\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v\right)_{-}^{p-1}}{|x|^{N+ps}}\,dx}_{:=J}+2^{p-1}\frac{L^{p-1}}{(C_{0}\,R)^{sp(1-\varepsilon_{0})}}\\\
&&+\mathbf{C}_{3}\frac{L^{p-1}}{(C_{0}\,R)^{sp(1-\varepsilon_{0})}}\Bigg{]}.\end{array}$
(8.7)

To estimate $\displaystyle J$, for given $\displaystyle x\in B_{0}\setminus
B_{j}$, there is $\displaystyle l\in\\{0,1,2,\ldots,j-1\\}$ such that
$\displaystyle x\in B_{l-1}\setminus B_{l}$ which implies
$\displaystyle|x|\geq d_{l}R_{l}$ which is equivalent to the condition
$\displaystyle\frac{|x|}{C_{0}R}\frac{(\eta L)^{\frac{2-p}{sp}}}{\delta}\geq
C_{0}^{-l\Gamma}$ where $\displaystyle\Gamma=\tfrac{\beta(p-2)}{sp}+1\geq 0$
(by Eq. 8.1) so that by the monotonicity of the sequence $\displaystyle m_{i}$
and by the induction hypothesis, for a.e. $\displaystyle t\in I_{j}$, we have:

$\begin{array}[]{rcl}v(x,t)&\geq&-(M_{l}-m_{l})+\tfrac{M_{j}-m_{j}}{2}\\\
&=&-\left[C_{0}^{-\beta l}-\tfrac{C_{0}^{-\beta j}}{2}\right]L\\\
&\geq&-\left[\left(\frac{|x|}{C_{0}R}\frac{(\eta
L)^{\frac{2-p}{sp}}}{\delta}\right)^{\frac{\beta}{\Gamma}}-\tfrac{C_{0}^{-\beta
j}}{2}\right]L\end{array}$ (8.8)

and

$\begin{array}[]{rcl}v(x,t)&\leq&(M_{l}-m_{l})-\tfrac{M_{j}-m_{j}}{2}\\\
&=&\left[C_{0}^{-\beta l}-\tfrac{C_{0}^{-\beta j}}{2}\right]L\\\
&\leq&\left[\left(\frac{|x|}{C_{0}R}\frac{(\eta
L)^{\frac{2-p}{sp}}}{\delta}\right)^{\frac{\beta}{\Gamma}}-\tfrac{C_{0}^{-\beta
j}}{2}\right]L\end{array}$ (8.9)

From Eq. 8.8 and Eq. 8.9, we get

$\displaystyle\displaystyle\left(\tfrac{C_{0}^{-\beta j}L}{2}\pm
v(x,t)\right)_{-}^{p-1}\leq\left[\left(\frac{|x|}{C_{0}R}\frac{(\eta
L)^{\frac{2-p}{sp}}}{\delta}\right)^{\frac{\beta}{\Gamma}}-C_{0}^{-\beta
j}\right]^{p-1}L^{p-1}$

We use this and the definition of $\displaystyle d_{j}R_{j}$ from Eq. 8.4 to
estimate

$\begin{array}[]{rcl}J&\leq&C_{0}^{-\beta
j(p-1)}L^{p-1}\operatorname*{ess\,sup}_{I_{j}}\int_{\mathbb{R}^{N}\setminus
B_{j}}\frac{\left(\left(\frac{|x|}{d_{l}R_{j}}\right)^{\frac{\beta}{\Gamma}}-1\right)^{p-1}}{|x|^{N+sp}}\,dx\\\
&=&\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}}{(d_{j}R_{j})^{sp}}\int_{\mathbb{R}^{N}\setminus
B_{1}}\frac{\left(|y|^{\frac{\beta}{\Gamma}}-1\right)^{p-1}}{|y|^{N+sp}}\,dy\\\
&=&\mathbf{C}_{1}\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}}{(d_{j}R_{j})^{sp}}\int_{1}^{\infty}\frac{\left(\rho^{\frac{\beta}{\Gamma}}-1\right)^{p-1}}{{\rho}^{1+sp}}\,dy\rho\\\
\end{array}$ (8.10)

Substituting Eq. 8.10 into Eq. 8.7, we obtain

$\operatorname*{\operatorname{Tail}_{\infty}}\left(\left(\tfrac{C_{0}^{-\beta
j}L}{2}\pm v\right)_{-};B_{j},0,I_{j}\right)^{p-1}\\\
\begin{array}[]{rcl}&\leq&(d_{j}R_{j})^{sp}\,\left[\mathbf{C}_{1}\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}}{(d_{j}R_{j})^{sp}}\int_{1}^{\infty}\frac{\left(\rho^{\frac{\beta}{\Gamma}}-1\right)^{p-1}}{{\rho}^{1+sp}}\,d\rho+2^{p-1}\frac{L^{p-1}}{(C_{0}\,R)^{sp(1-\varepsilon_{0})}}+\mathbf{C}_{3}\frac{L^{p-1}}{(C_{0}\,R)^{sp(1-\varepsilon_{0})}}\right]\\\
&\leq&\mathbf{C}_{1}C_{0}^{-\beta
j(p-1)}L^{p-1}\int_{1}^{\infty}\frac{\left(\rho^{\frac{\beta}{\Gamma}}-1\right)^{p-1}}{{\rho}^{1+sp}}\,d\rho+(2^{p-1}+\mathbf{C}_{3})(d_{j}R_{j})^{sp}\frac{L^{p-1}}{(C_{0}\,R)^{sp(1-\varepsilon_{0})}}\\\
&\overset{\hypertarget{7.10asin}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}}}{\leq}&\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}\eta^{p-1}}{2^{p-1}},\end{array}$ (8.11)

where (a) follows from Eq. 8.1,Eq. 8.2,Eq. 8.4 and the fact that
$\displaystyle j\geq j_{0}$, $\displaystyle L\geq 1$ and $\displaystyle
R,\delta\leq 1$. Indeed, from Eq. 8.1 we have

$\displaystyle\mathbf{C}_{1}C_{0}^{-\beta
j(p-1)}L^{p-1}\int_{1}^{\infty}\frac{\left(\rho^{\frac{\beta}{\Gamma}}-1\right)^{p-1}}{{\rho}^{1+sp}}\,d\rho\leq\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}\eta^{p-1}}{2^{p}},$

and

$\displaystyle(2^{p-1}+\mathbf{C}_{3})(d_{j}R_{j})^{sp}\frac{L^{p-1}}{(C_{0}\,R)^{sp(1-\varepsilon_{0})}}\leq(2^{p-1}+\mathbf{C}_{3})\eta^{p-2}C_{0}^{\varepsilon_{0}sp}C_{0}^{j(\beta-
sp)}L^{p-1}C_{0}^{-\beta j(p-1)}$

where we used the definition of $\displaystyle d_{j}R_{j}$ from Eq. 8.4,
$\displaystyle\delta^{sp}\leq 1$, $\displaystyle R^{\varepsilon_{0}}\leq 1$
and $\displaystyle L^{p-2}\leq 1$ as $\displaystyle L\geq 1$ and
$\displaystyle p<2$. Using Eq. 8.2 and that $\displaystyle j\geq j_{0}$ we get

$\displaystyle C_{0}^{j(\beta-
sp)}\leq\left(\frac{2^{p}C_{0}^{\varepsilon_{0}sp}(\mathbf{C}_{3}+2^{p-1})}{\eta}\right)^{-1}$

so that

$\displaystyle(2^{p-1}+\mathbf{C}_{3})(d_{j}R_{j})^{sp}\frac{L^{p-1}}{(C_{0}\,R)^{sp(1-\varepsilon_{0})}}\leq\frac{C_{0}^{-\beta
j(p-1)}L^{p-1}\eta^{p-1}}{2^{p}},$

Step 5:

Now, one of the following two alternatives must hold:

$\displaystyle\displaystyle\left|\\{v(\cdot,s_{j})\geq
0\\}\cap\tfrac{1}{2}B_{j+1}\right|\geq\frac{1}{2}\left|\tfrac{1}{2}B_{j+1}\right|,$
(8.12) $\displaystyle\displaystyle\left|\\{v(\cdot,s_{j})\geq
0\\}\cap\tfrac{1}{2}B_{j+1}\right|<\frac{1}{2}\left|\tfrac{1}{2}B_{j+1}\right|,$
(8.13)

where we have set

$\displaystyle\displaystyle s_{j}:=-\left(\frac{C_{0}^{-\beta
j}L}{2}\right)^{2-p}\left(\frac{d_{j+1}R_{j+1}}{2}\right)^{sp}\delta.$

Without loss of generality, we assume that Eq. 8.12 holds, noting that if Eq.
8.13 holds, analogous conclusion follows with $\displaystyle v$ replaced by
$\displaystyle-v$. Before proceeding, let us ensure that the time level
$\displaystyle s_{j}$ is in the interval $\displaystyle(-R_{j}^{sp},0)$ so
that we can apply Proposition 6.1 in the next step. Indeed, we want

$\displaystyle-R_{j}^{sp}\leq-s_{j}$

which is equivalent to

$\displaystyle R_{j}^{sp}\geq\left(\frac{C_{0}^{-\beta
j}L}{2}\right)^{2-p}\left(\frac{d_{j+1}R_{j+1}}{2}\right)^{sp}\delta$

Plugging in the definition of $\displaystyle d_{j}$ from Eq. 8.4 we get

$\displaystyle
C_{0}^{sp+\beta(p-2)}\geq\frac{\delta^{1+sp}}{\eta^{2-p}2^{2-p}2^{sp}}.$

Since $\displaystyle sp+\beta(p-2)>0$ we can enforce such a universal choice
of $\displaystyle C_{0}>1$.

Step 6:

Now denote $\displaystyle w=\tfrac{C_{0}^{-\beta j}L}{2}+v$, then Eq. 8.12
becomes

$\displaystyle\displaystyle\left|\left\\{w(\cdot,s_{j})\geq\tfrac{C_{0}^{-\beta
j}L}{2}\right\\}\cap\tfrac{1}{2}B_{j+1}\right|\geq\frac{1}{2}\left|\tfrac{1}{2}B_{j+1}\right|.$

Moreover from the induction hypothesis, we see that $\displaystyle w\geq 0$ in
$\displaystyle Q_{R_{j}}(d_{j})$ because

$\displaystyle\displaystyle w=v+\tfrac{C_{0}^{-\beta
j}L}{2}=u-\tfrac{M_{j}+m_{j}}{2}+\tfrac{C_{0}^{-\beta
j}L}{2}\overset{\lx@cref{creftype~refnum}{defofLsin}}{=}u-\tfrac{M_{j}+m_{j}}{2}+\tfrac{M_{j}-m_{j}}{2}=u-m_{j}\geq
0.$

We see that since Eq. 8.11 holds, the Tail alternative in Proposition 6.1 is
satisfied. Therefore, we get

$\displaystyle\displaystyle w\geq\frac{\eta C_{0}^{-\beta j}L}{2}\qquad\mbox{
a.e. }(x,t)\in Q_{R_{j+1}}(d_{j+1}),$ (8.14)

which, by computations similar to the previous step holds provided:

$\displaystyle C_{0}^{-\beta(p-2)}\geq
2^{(2-p)+sp}\frac{\eta^{2-p}}{\delta^{1+sp}\varepsilon},$

where $\displaystyle\varepsilon\in(0,1)$ is the as obtained in Proposition 6.1
for $\displaystyle\alpha=1/2$. Since $\displaystyle p<2$ we can further
enforce such a universal choice of $\displaystyle C_{0}>1$.

We calculate

$\begin{array}[]{rcl}u(x,t)&=&\tfrac{M_{j}+m_{j}}{2}+v=\tfrac{M_{j}+m_{j}}{2}+w-\tfrac{C_{0}^{-\beta
j}L}{2}\\\
&\overset{\lx@cref{creftype~refnum}{redofoscsin}}{\geq}&\tfrac{M_{j}+m_{j}}{2}-\tfrac{C_{0}^{-\beta
j}L}{2}(1-\eta)\\\ &=&M_{j}-\tfrac{M_{j}-m_{j}}{2}-\tfrac{C_{0}^{-\beta
j}L}{2}(1-\eta)\\\
&\overset{\hypertarget{indsin}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0})}}}{=}&M_{j}-\tfrac{C_{0}^{-\beta
j}L}{2}(2-\eta)\\\
&\overset{\lx@cref{creftype~refnum}{choiceofalphasin}}{\geq}&M_{j}-C_{0}^{-\beta(j+1)}L,\end{array}$

where (a) follows from the induction hypothesis. Thus if we define
$\displaystyle m_{j+1}:=M_{j}-C_{0}^{-\beta(j+1)}L$ and $\displaystyle
M_{j+1}:=M_{j}$ then the inductive process in (Step 3) is completed.

This completes the proof of the claim. ∎

A consequence of Claim 8.1 is the following oscillation decay:

$\operatorname*{ess\,osc}_{Q_{R_{i}}(d_{i})}\,u:=\operatorname*{ess\,sup}_{Q_{R_{i}}(d_{i})}\,u-\operatorname*{ess\,inf}_{Q_{R_{i}}(d_{i})}\,u\leq
C_{0}^{-\beta i}L.$ (8.15)

###### Proof of Theorem 2.7.

With notation as in Claim 8.1, let us take $\displaystyle
Q_{0}:=B_{C_{0}R}\times(-(C_{0}R)^{2s},0)$ and define $\displaystyle L$,
$\displaystyle R_{i}$ and $\displaystyle d_{i}$ as Eq. 8.4.

Consider any two points Let $\displaystyle(x_{1},t_{1}),(x_{2},t_{2})\in
Q_{1}(d_{1})$ such that $\displaystyle x_{1}\neq x_{2}$ and $\displaystyle
t_{1}\neq t_{2}$, then there exist non-negative integers $\displaystyle n$ and
$\displaystyle m$ such that

$d_{n+1}R_{n+1}<|x_{1}-x_{2}|\leq d_{n}R_{n},\qquad\text{and}\qquad
R_{m+1}^{sp}<|t_{1}-t_{2}|\leq R_{m}^{sp}.$ (8.16)

As a result, with $\displaystyle k:=\max\\{n,m\\}$, we obtain

$\displaystyle\displaystyle|u(x_{1},t_{1})-u(x_{2},t_{2})|\leq\operatorname*{ess\,osc}_{Q_{r_{k}}(d_{k})}u\overset{\lx@cref{creftype~refnum}{{oscdecayssin}}}{\leq}\max\\{C_{0}^{-\beta
n},C_{0}^{-\beta m}\\}L.$ (8.17)

From the first inequality in Eq. 8.16, we deduce

$\mathbf{C}_{4}L^{\frac{2-p}{sp}}\frac{|x_{1}-x_{2}|}{R}>C_{0}^{-n}$ (8.18)

where

$\mathbf{C}_{4}=\frac{\eta^{\frac{2-p}{sp}}}{\delta}$ (8.19)

On the other hand, from the second inequality in Eq. 8.16, we get

$\frac{|t_{1}-t_{2}|^{\frac{1}{sp}}}{R}>C_{0}^{-m}$ (8.20)

The proof of the theorem follows from substituting Eq. 8.18 and Eq. 8.20 into
Eq. 8.17. ∎

######  Remark 8.2.

We note that the proof provides the following relationship between the Holder
exponents for space and time:

| x | t
---|---|---
Degenerate | $\displaystyle\beta$ | $\displaystyle\tfrac{\beta}{sp}$
Singular | $\displaystyle\beta$ | $\displaystyle\tfrac{\beta}{sp}$

In the degenerate case, the condition $\displaystyle\beta(p-1)<sp$ forces
$\displaystyle\beta<sp$.

######  Remark 8.3.

We further note that the space Holder exponent is $\displaystyle\beta$ while
the time exponent is $\displaystyle\beta/sp$. In particular, if we start with
$\displaystyle sp<1$ time regularity beats space regularity - such a thing
does not occur in the local case and is an instance of a purely nonlocal
phenomena.

## Acknowledgements

The authors thank Agnid Banerjee and Prashanta Garain for helpful discussions.

## References

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