# Cordes-Nirenberg type results for nonlocal equations with deforming kernels

Disson dos Prazeres  Universidade Federal de Sergipe, Brazil
<EMAIL_ADDRESS>,  Aelson Sobral  Department of Mathematics, Universidade
Federal da Paraíba, 58059-900 João Pessoa, PB-Brazil
<EMAIL_ADDRESS>and  José Miguel Urbano  King Abdullah
University of Science and Technology (KAUST), Computer, Electrical and
Mathematical Sciences and Engineering Division (CEMSE), Thuwal 23955-6900,
Saudi Arabia and University of Coimbra, CMUC, Department of Mathematics,
3001-501 Coimbra, Portugal<EMAIL_ADDRESS>
###### Abstract.

We derive Cordes-Nirenberg type results for nonlocal elliptic integro-
differential equations with deforming kernels comparable to sections of a
convex solution of a Monge-Ampère equation. Under a natural integrability
assumption on the Monge-Ampère solution, we prove a stability lemma allowing
the ellipticity class to vary. Using a compactness method, we then derive
Hölder regularity estimates for the gradient of the solutions.

###### Key words and phrases:

Integro-differential equations; nonlocal elliptic equations; deforming
kernels; Monge-Ampère equation

###### 2010 Mathematics Subject Classification:

Primary 35B65. Secondary 45H05, 35J96, 35J60, 35D40

##  1\. Introduction

The fully nonlinear elliptic integro-differential Isaacs equation

$\mathcal{I}[u(x),x]:=\inf_{\alpha}\sup_{\beta}L_{\alpha\beta}u(x)=f(x),\quad
x\in\Omega,$ (1)

where

$L_{\alpha\beta}u(x):=\int_{\mathbb{R}^{n}}\left(u(x+y)+u(x-y)-2u(x)\right)\mathcal{K}_{\alpha,\beta}(x,y)dy,$

appears naturally in competitive stochastic games when two players are allowed
to choose from different strategies at every step in order to maximize the
expected value $u(x)$ at the first exit point of the domain
$\Omega\subset\mathbb{R}^{n}$. For arbitrary index sets $\mathcal{A}$ and
$\mathcal{B}$, the kernels
$\\{\mathcal{K}_{\alpha,\beta}(x,y)\\}_{\alpha\in\mathcal{A},\beta\in\mathcal{B}}$
are nonnegative functions measuring the frequency of jumps in the $y$
direction at the point $x$. Further applications and motivations related to
nonlocal equations can be found, for example, in [1, 3, 16].

The large applicability of this type of models reinforces the relevance of
seeking qualitative properties for their solutions. In this paper, we obtain
Cordes-Nirenberg type results for solutions of (1), with $f\in
L^{\infty}(\Omega)$. We argue through an approximation method, associating (1)
to a constant coefficients equation. We are particularly interested in the
case where the level sets of the kernels $\mathcal{K}_{\alpha,\beta}$ are
comparable to sections of a convex solution $\phi$ of a Monge-Ampère equation,
as in [10]. Setting $v_{\phi}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ as

$v_{\phi}(y):=\phi(y)-\phi(0)-D\phi(0)\cdot y,$

we note that, due to convexity, $v_{\phi}\geq 0$, and define the sections
$S_{r}^{\phi}$ of $\phi$ as

$S_{r}^{\phi}:=\left\\{v_{\phi}<r^{2}\right\\}.$

We will study equation (1) for kernels of the particular form

$\mathcal{K}_{\alpha,\beta}(x,y)=(2-\sigma)\frac{b_{\alpha,\beta}(x,y)}{{v_{\phi}}^{\frac{n+\sigma}{2}}(y)},\quad\mbox{with}\quad
b_{\alpha,\beta}:\Omega\times\mathbb{R}^{n}\rightarrow\left[{\Lambda}^{-1},\Lambda\right]$
(2)

for constants $\Lambda\geq 1$ and $\sigma\in(1,2)$. Our main result states
that if

$\sup_{\begin{subarray}{c}\alpha\in\mathcal{A},\beta\in\mathcal{B}\\\
(x,y)\in\Omega\times\mathbb{R}^{n}\end{subarray}}\left|b_{\alpha,\beta}(x,y)-b(y)\right|<\eta,$
(3)

for a small enough $\eta$ and $b\in L^{\infty}(\mathbb{R}^{n})$, then
solutions of (1) are of class $\mathcal{C}^{1,\alpha_{0}}_{loc}(\Omega)$,
where $\alpha_{0}$ is to be specified further.

The regularity theory for fully nonlinear nonlocal elliptic equations was
initiated by Caffarelli and Silvestre in their seminal paper [7]. They treat
the case corresponding to $\phi=|.|^{2}$ and a translation-invariant operator,
and develop a nonlocal version of the ABP estimate, establish a Harnack
inequality and derive Hölder regularity estimates. Additional regularity
assumptions on the kernels lead to Hölder gradient regularity through a
standard cut-off and integration by parts technique. This theory was the
starting point for many other developments in the nonlocal setting. For
example, an anisotropic scenario was considered in [6, 17] and the case of a
general convex $\phi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ satisfying

$\gamma\leq\det(D^{2}\phi)\leq\Gamma\quad\mbox{in $\mathbb{R}^{n}$},$ (4)

for $0<\gamma\leq\Gamma<\infty$, was the object of [10]. In this case, since
the sections of $\phi$ are comparable to ellipsoids (see Fritz John’s lemma,
in [13]), the kernels can be very degenerate and improved estimates that take
into account the deformation of the sections of $\phi$, which is driven by the
Monge-Ampère equation, are required.

As in the local case, a natural extension is to develop an approximation
theory that allows for the inclusion of coefficients in the operator. In
general, dealing with $x-$dependant equations can be rather challenging since
the dependence could break off the effect of the operator. The case
$\phi=|.|^{2}$ was again dealt with by Caffarelli and Silvestre, who were able
to prove in [8] that if two nonlocal operators are close in an appropriate
sense, then the solutions of the corresponding equations have the same
regularity. To prove Hölder gradient regularity for the solutions, however,
the kernels of the constant coefficient operators needed to have a fair amount
of regularity, a restriction that would be removed by Kriventsov in [15] by
means of an approximation argument. Putting it bluntly, our paper is to [10]
what [8] was to [7]. Alternative regularity results are also available in
different scenarios and we steer the interested reader to, e.g., [2, 11, 14,
18, 19].

The heart of the matter in proving gradient Hölder regularity in [8] is to
understand what kind of equation functions of the form

$w_{\lambda}(x)=\frac{1}{\lambda^{1+\alpha}}[u-l](\lambda x)$ (5)

solve, for an affine function $l$, and capture their behaviour as $\lambda$
approaches zero. The first issue is to somehow force a fair scaling behaviour
of the operator, so that (5) still satisfies an elliptic equation in the same
class as that of the equation satisfied by $u$. The second issue is purely
nonlocal: in order to apply compactness arguments, one needs to extend
previous Hölder regularity results for solutions with certain growth at
infinity since this behaviour is always expected. Therefore, finding suitable
weights that control the growth of the solutions is crucial. The weights must
be comparable to the kernels at infinity, so that we can assure the operator
is well-defined at the solutions. In [8], since the order of the equation is
greater than $1$, the weights have the form

$\mathcal{W}(y):=\frac{1}{1+|y|^{n+\sigma}},$

and so the solutions could have growth comparable to
$\zeta_{\alpha_{c}}=|.|^{1+\alpha_{c}}$, for any $\alpha_{c}$ satisfying
$\alpha_{c}<\sigma-1$. This is due to the fact that

$\left\|\zeta_{\alpha_{c}}\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W})}:=\int_{\mathbb{R}^{n}}|y|^{1+\alpha_{c}}\mathcal{W}(y)dy<\infty.$

The main novelty in our problem, with a general convex
$\phi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ satisfying (4), is that the setting
changes in both issues described above. Foremost, scalings of the form (5)
still satisfy an elliptic equation, but in a different class as the original
one. More precisely, we need to scale the kernels (3) as

$\frac{b_{\alpha,\beta}(\lambda-,\lambda-)}{v_{\phi_{\lambda}}^{\frac{n+\sigma}{2}}},\quad\mbox{where}\
\phi_{\lambda}=\lambda^{-2}\phi(\lambda-),$

and it is crucial that, although the ellipticity class changes, the
$\phi_{\lambda}$ still satisfy (4), with the same bounds. Secondly, since
growth at infinity is unavoidable, one needs to beware the choice of the
weights. The reasonable alternative is to choose

$\mathcal{W}_{\phi}(y):=\frac{1}{1+v_{\phi}(y)^{\frac{n+\sigma}{2}}},$

since it controls the behaviour of (2) at infinity, and so, for some
$\alpha_{1}<\sigma-1$, we need

$\left\|\zeta_{\alpha_{1}}\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi})}<\infty.$
(6)

If scalings were not in use, the basic setting would be then settled to
develop the theory. However, because scaling is imperative in our case, we
need to choose scaled weights $\mathcal{W}_{\phi_{\lambda}}$ and find an
exponent $\alpha_{\lambda}<\sigma-1$ such that

$\left\|\zeta_{\alpha_{\lambda}}\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi_{\lambda}})}<\infty.$
(7)

We stress that all these ingredients need to be uniform in $\lambda$, and so a
cautious analysis needs to be performed so that estimates do not degenerate as
$\lambda$ goes to zero.

Unlike in [8], a regularizing effect on the kernels of the equation occurs
through the scaling procedure. More precisely, the following convergence holds
true (see proposition 5), locally uniformly in $\mathbb{R}^{n}$:

$v_{\phi_{\lambda}}(y)\xrightarrow{\ \lambda\rightarrow 0\ }D^{2}\phi(0)y\cdot
y.$

Furthermore, if condition (6) is true, then (7) becomes stable and we are able
to prove a stability result that allows the kernels to vary (see Lemma 3).
Now, the fact that the family $\\{\phi_{\lambda}\\}_{\lambda\in(0,1]}$
satisfies (4) becomes convenient again, since it makes Hölder estimates from
[10] uniform in the parameter $\lambda$. Therefore, we make way to import the
regularity theory valid for a nice equation with kernels comparable, up to a
rotation, to that of the fractional Laplacian, and we may use Kriventsov’s
results from [15]. We prove, as a consequence, $\mathcal{C}^{1,\alpha_{0}}$
regularity estimates for solutions of (1) with kernels satisfying only (2),
(3) and (6), for

$\alpha_{0}<\min\\{\alpha_{1},\alpha_{*}\\},$

where $\alpha_{*}$ is the exponent obtained in [15] and $\alpha_{1}$ is from
(6). In particular, if our equation does not have $x-$dependence, we slightly
improve the gradient regularity from [10] since we demand only (6) for the
kernels (this is automatically true in the setting $\phi=|.|^{2}$).

Moreover, as an additional technical difficult, we do not assume any sort of
symmetry on the kernels. The aftermath is that many computations in the paper,
such as in Proposition 3 and Lemma 3, need to be cautiously done. Condition
(6) is once again instrumental in assuring the estimates are steady with
respect to the parameter $\lambda$. It is also important to mention that all
of our estimates are uniform with respect to the parameter $\sigma$, which
satisfies $\sigma\geq\sigma_{0}>1+\alpha_{1}$, with $\alpha_{1}$ from (6).

The paper is organized as follows: in section 2, after some preliminaries, we
introduce the definitions to be used in the remainder of the paper and our
main assumptions. In section 3, we present the stability result and the
approximation lemma. By applying an iteration procedure, we derive, in section
4, the gradient regularity estimates for solutions of $(\ref{MAINEQ})$.

##  2\. Preliminaries, definitions and main assumptions

We gather in this section some definitions and auxiliary results, in addition
to stating our main assumption. We start with a simple lemma that will be
instrumental in the sequel.

######  Lemma 1.

Let $\Psi_{1},\Psi_{2}\in\mathcal{C}^{2}(\mathbb{R}^{n})$ be two convex
functions such that

$\lambda_{1}<det(D^{2}\Psi_{1}),\quad\lambda_{2}>det(D^{2}\Psi_{2}),$

for $0<\lambda_{1},\lambda_{2}<\infty$ and define
$v_{\Psi_{i}}(z):=\Psi_{i}(z)-\Psi_{i}(0)-D\Psi_{i}(0)\cdot z$ for $i=1,2$.
For every $r>0$, there exists a constant $C=C(r,\Psi_{1},\Psi_{2})$ such that

$v_{\Psi_{1}}(z)\leq Cv_{\Psi_{2}}(z),\quad\forall z\in\overline{B}_{r}.$

###### Proof.

Given $r>0$, consider the $C^{2}$ auxiliary function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined by

$f(z):=v_{\Psi_{1}}(z)-Cv_{\Psi_{2}}(z),$

where

$C=\max_{z\in\overline{B}_{r}}\left\\{\frac{\lambda_{max}(D^{2}\Psi_{1}(z))}{\lambda_{min}(D^{2}\Psi_{2}(z))}\right\\}\geq\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{1/n}.$

Then

$\begin{array}[]{lll}D^{2}f(z)&=&D^{2}v_{\Psi_{1}}(z)-CD^{2}v_{\Psi_{2}}(z)\\\
&&\\\ &=&D^{2}\Psi_{1}(z)-CD^{2}\Psi_{2}(z)\\\ &&\\\
&\leq&(\lambda_{max}(D^{2}\Psi_{1}(z))-C\lambda_{min}(D^{2}\Psi_{2}(z)))I_{d}\leq
0\end{array}$

so $f$ is a concave function in $\overline{B}_{r}$ and, therefore, it stays
below any tangent hyperplane. In particular,

$f(z)\leq f(0)+Df(0)\cdot z=0,\quad\forall z\in\overline{B}_{r}.$

∎

We next define the appropriate notion of solution to our problem.

######  Definition 1 (Viscosity solution).

Let $f$ be a bounded and continuous function in $\mathbb{R}^{n}$. We say a
function $u:\mathbb{R}^{n}\rightarrow\mathbb{R}$, upper semicontinuous in
$\overline{\Omega}$, is a viscosity subsolution in $\Omega$ of the equation

$\mathcal{I}[u(x),x]=f,$

and we write $\mathcal{I}[u(x),x]\geq f$, if, whenever $x_{0}\in\Omega$,
$N\subset\Omega$ is a neighbourhood of $x_{0}$, and $\varphi\in
C^{2}(\overline{N})$ satisfies

$\varphi(x_{0})=u(x_{0})\qquad\textrm{and}\qquad\varphi(y)>u(y),\quad\forall
y\in N\backslash\\{x_{0}\\},$

then, if we choose as test function $\tau$ such that

$\tau:=\left\\{\begin{array}[]{lcl}\varphi&\mbox{in}&N\\\
u&\mbox{in}&\mathbb{R}^{n}\backslash N,\end{array}\right.$

we have $\mathcal{I}[\tau(x),x](x_{0})\geq f(x_{0})$.

A supersolution is defined analogously and a solution is a function that is
both a viscosity subsolution and a viscosity supersolution.

The definition of the extremal operators is basically the same as is [10].
These operators are important to define the uniform ellipticity condition with
respect to some class $\mathcal{L}$. Here, $\mathcal{L}$ is a set of linear
operators $L$ of the form

$L[u](x)=\int_{\mathbb{R}^{n}}(u(x+y)+u(x-y)-2u(x))\mathcal{K}(y)dy,$

for some kernel $\mathcal{K}$.

We say that a nonlocal operator $\mathcal{I}$ is elliptic with respect to some
class $\mathcal{L}$ if

$M^{-}_{\mathcal{L}}[w](x)\leq\mathcal{I}[(u+w)(x),x](x)-\mathcal{I}[u(x),x](x)\leq
M^{+}_{\mathcal{L}}[w](x),$ (8)

where

$M^{-}_{\mathcal{L}}[w](x):=\inf_{L\in\mathcal{L}}L[w](x),\quad
M^{+}_{\mathcal{L}}[w](x):=\sup_{L\in\mathcal{L}}L[w](x).$

In [10], in order to obtain the ABP estimate, a Harnack inequality and Hölder
regularity results, the kernels needed to be in the class
$\mathcal{L}^{0}_{\phi}(\sigma)$, which is the class of linear operators with
kernels $\mathcal{K}$ satisfying

$(2-\sigma)\frac{1/\Lambda}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}\leq\mathcal{K}(y)\leq(2-\sigma)\frac{\Lambda}{v_{\phi}(y)^{\frac{n+\sigma}{2}}},$
(9)

for some $\Lambda\geq 1$. It is known that for the class defined by
$(\ref{CLASSOFKJMU})$ the extremal operators have the simple form

$M^{-}_{\mathcal{L}^{0}_{\phi}(\sigma)}[u](x)=(2-\sigma)\int_{\mathbb{R}^{n}}\frac{\frac{1}{\Lambda}\delta^{+}(u,x,y)-\Lambda\delta^{-}(u,x,y)}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy$

and

$M^{+}_{\mathcal{L}^{0}_{\phi}(\sigma)}[u](x)=(2-\sigma)\int_{\mathbb{R}^{n}}\frac{\Lambda\delta^{+}(u,x,y)-\frac{1}{\Lambda}\delta^{-}(u,x,y)}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy,$

where $\delta(u,x,y)=u(x+y)+u(x-y)-2u(x)$, $\delta^{+}(u,x,y)$ denotes its
positive part and $\delta^{-}(u,x,y)$ its negative part. We would like to
point out that if $\gamma=\Gamma=1$ in equation $(\ref{MONGEAMPEREEQ})$ then
Pogorelov’s result (a proof can be found in [12, Theorem 4.18]) states that
$\phi$ is a quadratic polynomial. Then $v_{\phi}(y)=Ay\cdot y$, for a matrix
$A$ satisfying $\det(A)=1$. Equations with kernels in this form were studied
in [4] in a different setting related to a nonlocal version of the Monge-
Ampère equation. The interested reader may also appreciate [9], for a
different approach, and the more recent contribution in [5].

The requirement that $u\in L^{1}(\mathbb{R}^{n},\mathcal{W})$, for some weight
$\mathcal{W}$, allows $u$ to have a certain growth at infinity. For example,
in the classical case of [8], for operators with kernels satisfying the
inequality $(\ref{CLASSOFKJMU})$ with $\phi=|.|^{2}$, the natural choice would
be

$\mathcal{W}(y)=\frac{1}{1+|y|^{n+\sigma_{0}}},$

simply because this weight controls the tails of the kernels at infinity, and
so the integrals are not singular for $\sigma\geq\sigma_{0}$.

Therefore, a natural choice for the weight in our setting is

$\mathcal{W}_{\phi}(y)=\frac{1}{1+v_{\phi}(y)^{\frac{n+\sigma_{0}}{2}}},$ (10)

in order to bound the tails of the kernels $\mathcal{K}$ satisfiying
$(\ref{CLASSOFKJMU})$.

The weight $\mathcal{W}_{\phi}$ satisfies three properties that will play an
important role in our analysis. We next state and prove them.

######  Proposition 1.

Assume $\mathcal{K}$ is a function that satisfies

$\mathcal{K}(y)\leq\Lambda\frac{(2-\sigma)}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}.$

Then for each $r>0$, there exists a constant $C_{r}$ such that

$\mathcal{K}(y)\leq C_{r}\mathcal{W}_{\phi}(y),\quad\mbox{for}\quad y\not\in
S_{r}^{\phi}.$

###### Proof.

Notice that for a kernel $\mathcal{K}$ satisfying the assumed bound, we have

$\begin{array}[]{lll}\mathcal{K}(y)&\leq&\displaystyle\Lambda\frac{(2-\sigma)}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}\\\
&&\\\
&=&\displaystyle\Lambda(2-\sigma)\frac{1+v_{\phi}(y)^{\frac{n+\sigma_{0}}{2}}}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}\mathcal{W}_{\phi}(y)\\\
&&\\\
&=&\displaystyle\Lambda(2-\sigma)\left(\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}+v_{\phi}(y)^{\frac{\sigma_{0}-\sigma}{2}}\right)\mathcal{W}_{\phi}(y)\\\
&&\\\
&\leq&\displaystyle\Lambda(2-\sigma_{0})\left(\frac{1}{r^{n+\sigma}}+r^{\sigma_{0}-\sigma}\right)\mathcal{W}_{\phi}(y)\\\
&&\\\
&\leq&2\Lambda(2-\sigma_{0})\max\left\\{\frac{1}{r^{n+2}},\frac{1}{r^{2-\sigma_{0}}}\right\\}\mathcal{W}_{\phi}(y)\\\
&&\\\ &=&C_{r}\mathcal{W}_{\phi}(y),\end{array}$

for every $y\not\in S_{r}^{\phi}$, with $C_{r}=C(\Lambda,\sigma_{0},r,n)$. ∎

######  Proposition 2.

The weight $\mathcal{W}_{\phi}$ is not singular, i.e., for each $r>0$, there
exists a constant $C_{r}$ such that

$\sup_{z\in B_{r}(y)}\mathcal{W}_{\phi}(z)\leq C_{r}\mathcal{W}_{\phi}(y).$

###### Proof.

$\begin{array}[]{lll}\mathcal{W}_{\phi}(z)&=&\displaystyle\frac{1}{1+v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}=\displaystyle\frac{1+v_{\phi}(y)^{\frac{n+\sigma_{0}}{2}}}{1+v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}\mathcal{W}_{\phi}(y)\\\
&&\\\
&=&\displaystyle\frac{1+(v_{\phi}(y)-v_{\phi}(z)+v_{\phi}(z))^{\frac{n+\sigma_{0}}{2}}}{1+v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}\mathcal{W}_{\phi}(y)\\\
&&\\\
&\leq&\displaystyle\frac{1+(|v_{\phi}(y)-v_{\phi}(z)|+v_{\phi}(z))^{\frac{n+\sigma_{0}}{2}}}{1+v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}\mathcal{W}_{\phi}(y)\\\
&&\\\
&\leq&\displaystyle\frac{1+2^{\frac{n+\sigma_{0}}{2}}(|v_{\phi}(y)-v_{\phi}(z)|^{\frac{n+\sigma_{0}}{2}}+v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}})}{1+v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}\mathcal{W}_{\phi}(y)\\\
&&\\\ &\leq&\displaystyle
2^{\frac{n+\sigma_{0}}{2}}\left(1+\frac{|v_{\phi}(y)-v_{\phi}(z)|^{\frac{n+\sigma_{0}}{2}}}{1+v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}\right)\mathcal{W}_{\phi}(y)\\\
&&\\\
&\leq&C(n,\sigma_{0})\left(1+\rho(r)^{\frac{n+\sigma_{0}}{2}}\right)\mathcal{W}_{\phi}(y)\\\
&&\\\ &=&C_{r}\mathcal{W}_{\phi}(y),\end{array}$

for every $z\in B_{r}(y)$, with $C_{r}=C(\sigma_{0},r,n)$ and where $\rho$ is
the modulus of continuity of $v_{\phi}$.

∎

It is important to notice that the modulus of continuity of $v_{\phi}$ is the
same as that of $\phi$, since $D^{2}\phi=D^{2}v_{\phi}$. Therefore, the
constant $C_{r}$ is the same for any solution $\Psi$ in the Monge-Ampère class

$\gamma<det(D^{2}\Psi)<\Gamma.$

The next statement guarantees that the test functions in Definition 1 are
suitable for our operators. These computations have already been made in [10]
but we need to assure here they are invariant with respect to solutions of the
Monge-Ampère equation.

######  Proposition 3.

Let $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a function such that
$g\in\mathcal{C}^{2}(B_{r}(x))$,

$\left\|g\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi})}\leq M\quad
and\qquad\left|g(y)-l_{Dg(x)}(y)\right|\leq M|y-x|^{2},\quad y\in B_{r}(x),$

where $l_{Dg(x)}(y):=g(x)+Dg(x)\cdot(y-x)$. Then, there exists a constant $C$
such that

$|g(x)|\leq CM$ (11)

and

$\left|L[g](x)\right|\leq CM,\quad\forall L\in\mathcal{L}^{0}_{\phi}(\sigma).$
(12)

###### Proof.

Observe that, for $y\in B_{r}(0)$, we have

$\begin{array}[]{cl}|\delta(g,x,y)|&=|2g(x)-g(x+y)-g(x-y)|\\\ &\\\
&=|2g(x)-g(x+y)-g(x-y)-l_{Dg(x)}(x+y)+l_{Dg(x)}(x+y)|\\\ &\\\
&\leq\left|g(x-y)-l_{Dg(x)}(x-y)\right|+\left|g(x+y)-l_{Dg(x)}(x+y)\right|\\\
&\\\ &\leq 2M|y|^{2}.\end{array}$

Using the reverse triangle inequality and multiplying both sides by the weight
$\mathcal{W}_{\phi}$, we obtain

$2|g(x)|\mathcal{W}_{\phi}(y)\leq
2M|y|^{2}\mathcal{W}_{\phi}(y)+|g(x+y)|\mathcal{W}_{\phi}(y)+|g(x-y)|\mathcal{W}_{\phi}(y).$

Since $\mathcal{W}_{\phi}(y)\leq 1$ and $|y|\leq r$, integrating over
$B_{r}(0)$, we reach

$\begin{array}[]{lll}\displaystyle
2|g(x)|\int_{B_{r}}\mathcal{W}_{\phi}(y)dy&\leq&\displaystyle
2Mr^{2}|B_{r}|+\int_{B_{r}}|g(x+y)|\mathcal{W}_{\phi}(y)dy\\\ &&\\\
&&\displaystyle+\int_{B_{r}}|g(x-y)|\mathcal{W}_{\phi}(y)dy.\end{array}$

We now bound the two integrals on the right-hand side using Proposition 2. To
uniformly bound

$\int_{B_{r}}\mathcal{W}_{\phi}(y)dy$

from below (independently of scalings of the form
$\lambda^{-2}v_{\phi}(\lambda x)$), we use Lemma 1 to get

$1+v_{\phi}(y)^{\frac{n+\sigma_{0}}{2}}=1+\left[\frac{v_{\phi}(y)}{|y|^{2}}\right]^{\frac{n+\sigma_{0}}{2}}|y|^{n+\sigma_{0}}\leq
C(r,\phi,n),$

thus obtaining (11).

To prove (12), we start by choosing $\eta$, depending on the modulus of
continuity of $\phi$, such that $S^{\phi}_{\eta}\subset B_{r}(0)$. Then, if
$L\in\mathcal{L}^{0}_{\phi}(\sigma)$, we have

$\begin{array}[]{lll}\displaystyle\left|\int_{S^{\phi}_{\eta}}\delta(g,x,y)\mathcal{K}(y)dy\right|&\leq&\displaystyle(2-\sigma)\Lambda
M\int_{S^{\phi}_{\eta}}|y|^{2}\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy\\\
&&\\\ &\leq&\displaystyle(2-\sigma)\Lambda
MC\int_{S^{\phi}_{\eta}}|y|^{2}|y|^{-n-\sigma}dy\\\ &&\\\
&\leq&\displaystyle(2-\sigma)\Lambda MC\int_{B_{r}}|y|^{2}|y|^{-n-\sigma}dy\\\
&&\\\ &=&\displaystyle(2-\sigma)\Lambda MC|\partial
B_{1}|\int_{0}^{r}s^{2}s^{-n-\sigma}s^{n-1}ds\\\ &&\\\ &=&\displaystyle
C_{1}(\Lambda,n,C,r)M\\\ \end{array}$

The constant $C=C(r,\phi)$ is from Lemma $(\ref{PREM1})$, which turns out to
be invariant by scalings of the form $\lambda^{-2}\phi(\lambda x)$.

Now we estimate the integral at infinity. Let us separate it in three parts.

$\begin{array}[]{lll}\displaystyle\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|g(x+y)|\mathcal{K}(y)dy&\leq&\displaystyle
C_{\eta}\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|g(x+y)|\mathcal{W}_{\phi}(y)dy\\\ &&\\\ &\leq&\displaystyle
C_{\eta}\int_{\mathbb{R}^{n}\backslash S^{\phi}_{\eta}}|g(x+y)|\sup_{z\in
B_{1}(x+y)}\mathcal{W}_{\phi}(z)dy\\\ &&\\\ &\leq&\displaystyle
C_{\eta}C_{1}\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|g(x+y)|\mathcal{W}_{\phi}(x+y)dy\\\ &&\\\
&\leq&C(\eta,n,\sigma_{0},\rho(1))\left\|g\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi})}\\\
&&\\\ &\leq&C(\eta,n,\sigma_{0},\rho(1))M.\par\end{array}$

where $C_{1}$ is the constant from Proposition 2 and $\rho$ stands for the
modulus of continuity of $\phi$.

We may assume that the kernels are symmetric (but see Remark 1 at the end of
this section) and apply a similar reasoning to get

$\displaystyle\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|g(x-y)|\sup_{z\in B_{1}(x-y)}\mathcal{W}_{\phi}(z)dy\leq
C(\eta,n,\sigma_{0},\rho(1))M.$

Finally, using (11), we can estimate

$\begin{array}[]{lll}\displaystyle\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}2|g(x)|\mathcal{K}(y)dy&\leq&\displaystyle
CM\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy.\end{array}$

Notice that we can show that the quantity

$\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy$

is finite employing the elementary layer-cake formula [12, Lemma A.36] and the
fact that $|S_{r}^{\phi}|\approx r^{n}$ (depending only on the bounds of the
Monge-Ampère equation), as shown in [13]. Indeed,

$\begin{array}[]{rl}\displaystyle\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy=&\displaystyle(n+\sigma)\int_{0}^{\infty}t^{n+\sigma-1}\left|(\mathbb{R}^{n}\backslash
S^{\phi}_{\eta})\cap\left\\{\frac{1}{v_{\phi}(y)^{\frac{1}{2}}}>t\right\\}\right|dt\\\
&\\\
=&\displaystyle(n+\sigma)\int_{0}^{\infty}t^{n+\sigma-1}\left|(\mathbb{R}^{n}\backslash
S^{\phi}_{\eta})\cap S_{\frac{1}{t}}\right|dt\\\ &\\\
=&\displaystyle(n+\sigma)\int_{0}^{\frac{1}{\eta}}t^{n+\sigma-1}\left|(\mathbb{R}^{n}\backslash
S^{\phi}_{\eta})\cap S_{\frac{1}{t}}\right|dt\\\ &\\\ \leq&\displaystyle
C(n+\sigma)\int_{0}^{\frac{1}{\eta}}t^{n+\sigma-1}\frac{1}{t^{n}}dt\\\ &\\\
=&C\displaystyle\frac{(n+\sigma)}{\sigma}\frac{1}{\eta^{\sigma}}.\end{array}$

Putting all estimates together, we obtain that

$\left|L[g](x)\right|=\left|\int_{\mathbb{R}^{n}}\delta(g,x,y)\mathcal{K}(y)dy\right|\leq
CM.$

∎

We need to impose an integrability assumption on the solution
$\phi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ of the Monge-Ampère equation.

[A1] The function $\zeta:\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined by
$\zeta(y)=|y|^{1+\alpha_{1}}$ satisfies

$\zeta\in L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi}\right),$ (13)

for some $\alpha_{1}<\sigma_{0}-1$.

A simple example of a function in $\mathbb{R}^{2}$ satisfying the assumption
is

$\phi(y_{1},y_{2})=y_{1}^{2}+y_{2}^{2}+\frac{1}{2}\sin(y_{1}).$

######  Proposition 4.

Let assumption [A1] be in force and let
$\phi_{\lambda}(x)=\lambda^{-2}\phi(\lambda x)$. Then

$\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi_{\lambda}}\right)}\leq
C\left(n,\sigma_{0},\alpha_{1},\phi,\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi}\right)}\right).$

###### Proof.

We have

$\begin{array}[]{lll}\displaystyle\left\|\zeta\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi_{\lambda}})}&=&\displaystyle\int_{\mathbb{R}^{n}}|y|^{1+\alpha_{1}}\frac{1}{1+(\lambda^{-2}v_{\phi}(\lambda
y))^{\frac{n+\sigma_{0}}{2}}}dy\\\ &&\\\
&=&\displaystyle\lambda^{-n-1-\alpha_{1}}\int_{\mathbb{R}^{n}}|z|^{1+\alpha_{1}}\frac{1}{1+(\lambda^{-2}v_{\phi}(z))^{\frac{n+\sigma_{0}}{2}}}dz\\\
&&\\\ &=&\lambda^{-n-1-\alpha_{1}}(A_{1}+A_{2}+A_{3}),\end{array}$

where $A_{1},A_{2},A_{3}$ stand for the integrals of
$|.|^{1+\alpha_{1}}(1+(\lambda^{-2}v_{\phi}(.))^{\frac{n+\sigma_{0}}{2}})^{-1}$
over the sets $B_{\lambda}$, $B_{1}\backslash B_{\lambda}$ and
$\mathbb{R}^{n}\backslash B_{1}$, respectively.

Now,

$A_{1}\leq\int_{B_{\lambda}}|z|^{1+\alpha_{1}}dz\leq
C(n)\lambda^{n+1+\alpha_{1}},$

and so $\lambda^{-n-1-\alpha_{1}}A_{1}\leq C(n)$. For $A_{2}$, we have

$A_{2}\leq\lambda^{n+\sigma_{0}}\int_{B_{1}\backslash
B_{\lambda}}|z|^{1+\alpha_{1}}\frac{1}{v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}dz.$

By Lemma 1, with $r=1$, $\Psi_{1}=|.|^{2}$ and $\Psi_{2}=\phi$, we get a
constant $C(\phi)$ such that

$\frac{|z|^{2}}{v_{\phi}(z)}\leq C(\phi),\quad\mbox{for
$z\in\overline{B}_{1}$}.$

Then

$\displaystyle A_{2}$ $\displaystyle\leq$ $\displaystyle
C(\phi,n)\lambda^{n+\sigma_{0}}\int_{B_{1}\backslash
B_{\lambda}}|z|^{1+\alpha_{1}}\frac{1}{|z|^{n+\sigma_{0}}}dz$ $\displaystyle=$
$\displaystyle
C(\phi,n)\lambda^{n+\sigma_{0}}\int_{\lambda}^{1}r^{\alpha_{1}-\sigma_{0}}dr$
$\displaystyle\leq$ $\displaystyle
C(\phi,n)\lambda^{n+\sigma_{0}}\frac{1}{\sigma_{0}-1-\alpha_{1}}[\lambda^{\alpha_{1}-\sigma_{0}+1}-1].$

Therefore, we get

$\lambda^{-n-1-\alpha_{1}}A_{2}\leq C(\phi,n,\sigma_{0},\alpha_{1}).$

Finally, for $A_{3}$, we have

$\lambda^{-n-1-\alpha_{1}}A_{3}\leq\lambda^{\sigma_{0}-1-\alpha_{1}}\int_{\mathbb{R}^{n}\backslash
B_{1}}|z|^{1+\alpha_{1}}\frac{1}{v_{\phi}(z)^{\frac{n+\sigma_{0}}{2}}}dz.$

Therefore, if assumption [A1] holds, we can get a uniform bound (in the
parameter $\lambda$) for the term $A_{3}$.

∎

######  Remark 1.

If the kernels are not symmetric, we control the quantity

$\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|g(x-y)|\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy,$

using assumption [A1] and a natural growth condition. Indeed, assuming
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a function such that

$|g(y)|\leq|y|^{1+\alpha_{1}}\quad\mbox{for $y\in\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}$},$

for some $\eta>0$, we obtain

$\begin{array}[]{lll}\displaystyle\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|g(x-y)|\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy&\leq&\displaystyle\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|x-y|^{1+\alpha_{1}}\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy\\\
&&\\\ &\leq&\displaystyle C(|x|)\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|y|^{1+\alpha_{1}}\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy\\\
&&\\\ &\leq&\displaystyle C(\eta,|x|)\int_{\mathbb{R}^{n}\backslash
S^{\phi}_{\eta}}|y|^{1+\alpha_{1}}\mathcal{W}_{\phi}(y)dy.\end{array}$

In our main theorem, the growth condition stated above will appear naturally
and is, by no means, restrictive.

##  3\. Approximation results

In this section we are going to deliver the key lemma of this paper,
consisting in showing that if the operators in our class are close in some
suitable sense then so are their solutions.

We start with a standard result that makes use of the ellipticity structure to
restrict the set of test functions in Definition 1, which is important to
simplify the proof of the stability result. The proof is essentially the same
as in [8] but we include it here for the reader’s convenience.

######  Lemma 2.

In Definition 1, it is enough to consider, as test functions $\varphi$,
quadratic polynomials and, as neighbourhoods $N,$ balls centered at the
touching point.

###### Proof.

Let $\varphi\in C^{2}(\overline{N})$ be a test function that touches $u$ from
above at a point $x_{0}$ and, for simplicity, assume $x_{0}=0$. Let
$P_{\epsilon}$ be the following polynomial

$P_{\epsilon}(x)=\frac{1}{2}\left(D^{2}\varphi(0)+\epsilon I_{n}\right)x\cdot
x+\nabla\varphi(0)\cdot x+\varphi(0).$

From Taylor’s expansion,

$\varphi(x)=\varphi(0)+\nabla\varphi(0)\cdot
x+\frac{1}{2}D^{2}\varphi(0)x\cdot
x+r_{2}(x),\quad\mbox{with}\quad\lim\limits_{x\rightarrow
0}\frac{r_{2}(x)}{|x|^{2}}=0,$

so, given $\overline{\epsilon}>0$, there exists $\mu>0$ such that, if
$|x|<\mu$, then

$\left|\varphi(x)-\left(\varphi(0)+\nabla\varphi(0)\cdot
x+\frac{1}{2}D^{2}\varphi(0)x\cdot
x\right)\right|\leq\overline{\epsilon}|x|^{2}$ (14)

which implies

$\varphi(x)\leq P_{\overline{\epsilon}}(x),\quad\forall x\in B_{\mu}.$

Then $P_{\overline{\epsilon}}\geq\varphi\geq u$ in a neighbourhood
$B_{r}\subset N$, with $r<\mu$.

Define

$\tau_{\varphi}(x)=\left\\{\begin{array}[]{ll}\varphi(x)&\mbox{if $x\in N$}\\\
u(x)&\mbox{if $x\not\in N$}.\end{array}\right.$

and

$\tau_{\overline{\epsilon}}(x)=\left\\{\begin{array}[]{ll}P_{\overline{\epsilon}}(x)&\mbox{if
$x\in B_{r}$}\\\ u(x)&\mbox{if $x\not\in B_{r}$}.\end{array}\right.$

With $\mathbbm{1}$ denoting the indicator function, we have

$\tau_{\overline{\epsilon}}(x)\leq\tau_{\varphi}(x)+2\overline{\epsilon}|x|^{2}\mathbbm{1}_{B_{r}},\quad\forall
x\in\mathbb{R}^{n},$

which is obvious if $x\notin B_{r}$; for $x\in B_{r}$, it follows from
$(\ref{INEQLEMMA1})$ that

$-\overline{\epsilon}|x|^{2}\leq\varphi(x)-\left(\varphi(0)+\nabla\varphi(0)\cdot
x+\frac{1}{2}D^{2}\varphi(0)x\cdot x\right)\leq\overline{\epsilon}|x|^{2}.$

So we have,

$\begin{array}[]{lll}\tau_{\varphi}(x)&=&\varphi(x)\\\
&\geq&\varphi(0)+\nabla\varphi(0)\cdot x+\frac{1}{2}D^{2}\varphi(0)x\cdot
x-\overline{\epsilon}|x|^{2}\\\
&=&P_{\overline{\epsilon}}(x)-2\overline{\epsilon}|x|^{2}\\\
&=&\tau_{\overline{\epsilon}}(x)-2\overline{\epsilon}|x|^{2}\mathbbm{1}_{B_{r}}.\end{array}$

Since $P_{\overline{\epsilon}}$ is a quadratic polynomial such that
$P_{\overline{\epsilon}}\geq\varphi\geq u$ and $\varphi$ touches $u$ from
above at $0$, we have $P_{\overline{\epsilon}}(0)=\varphi(0)=u(0)$. Hence,
$P_{\overline{\epsilon}}$ touches $u$ from above at $0$ and so, by the
definition of viscosity subsolution, we have

$\mathcal{I}[\tau_{\overline{\epsilon}}(x),x](0)\geq f(0).$

On the other hand, by uniform ellipticity,

$\begin{array}[]{lll}\mathcal{I}[\tau_{\varphi}(x),x](0)-\mathcal{I}[\tau_{\overline{{\epsilon}}}(x),x](0)&\geq&-M^{+}_{\mathcal{L}^{0}_{\phi}(\sigma)}\left[2\overline{\epsilon}|x|^{2}\mathbbm{1}_{B_{r}}\right](0)\\\
&=&-\sup_{L\in\mathcal{L}^{0}_{\phi}(\sigma)}L\left[2\overline{\epsilon}|x|^{2}\mathbbm{1}_{B_{r}}\right](0)\\\
&\geq&-2\overline{\epsilon}C,\end{array}$

where we have used Proposition 3 in the last inequality. Since

$\mathcal{I}[\tau_{\overline{\epsilon}}(x),x](0)\geq f(0),$

we have

$\mathcal{I}[\tau_{\varphi}(x),x](0)\geq\mathcal{I}[\tau_{\overline{\epsilon}}(x),x](0)-2C\overline{\epsilon},\quad\forall\overline{\epsilon}>0.$

Consequently, $\mathcal{I}[\tau_{\varphi}(x),x](0)\geq f(0)$. ∎

The following is a simple remark concerning the scalings of the solution of
the Monge-Ampère equation.

######  Proposition 5.

Let $\phi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ satisfy (4) and define
$\phi_{\lambda}(y)=\lambda^{-2}\phi(\lambda y)$. Then the function
$v_{\phi_{\lambda}}(y)$ converges to $v_{\phi_{0}}:=D^{2}\phi(0)y\cdot y$,
locally uniformly as $\lambda\rightarrow 0$, where
$\phi_{0}(y)=D^{2}\phi(0)y\cdot y$.

###### Proof.

By Taylor’s expansion,

$\begin{array}[]{lll}v_{\phi_{\lambda}}(y)&=&\phi_{\lambda}(y)-\phi_{\lambda}(0)-\nabla\phi_{\lambda}(0)\cdot
y\\\ &&\\\ &=&\lambda^{-2}\left(\lambda^{2}D^{2}\phi(0)y\cdot
y+r\left(|\lambda{y}|^{2}\right)\right)\\\ &&\\\ &=&D^{2}\phi(0)y\cdot
y+\frac{r(|\lambda{y}|^{2})}{\lambda^{2}},\end{array}$

for $y\in B_{\eta}$, with $\eta$ small enough. Therefore, if
$\lambda\rightarrow 0$,

$v_{\phi_{\lambda}}({y})\longrightarrow D^{2}\phi(0)y\cdot y,$

locally uniformly. ∎

We now prove the stability lemma, that will play a key role in the
approximating results. We will consider the simpler linear case just to
highlight the main ideas.

######  Lemma 3.

Let $\Omega$ be open and bounded, and let
$\phi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ be convex and satisfy (4). Consider
the family of scalings $\left\\{v_{\phi_{\lambda}}\right\\}_{\lambda\in(0,1]}$
and assume there exists a sequence

$(\lambda_{k},u_{k},f_{k},b_{k})\subset\mathbb{R}\times\mathcal{C}(\overline{\Omega})\cap
L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi_{\lambda_{k}}}\right)\times
L^{\infty}({\Omega})\times L^{\infty}({\Omega}\times\mathbb{R}^{n})$

such that

  * •

$\mathcal{I}_{k}[u_{k}(x),x]:=\displaystyle\int_{\mathbb{R}^{n}}\delta(u_{k},x,y)\frac{b_{k}(x,y)}{v_{\phi_{\lambda_{k}}}(y)^{\frac{n+\sigma}{2}}}dy=f_{k}(x)$
in the viscosity sense in $\Omega$;

  * •

$u_{k}\rightarrow u$ uniformly in $\overline{\Omega}$ and a.e. in
$\mathbb{R}^{n}$;

  * •

$|u_{k}(x)|\leq(1+\zeta(x))$ in $\mathbb{R}^{n}$, for some $\alpha_{1}$ such
that $\alpha_{1}<\sigma-1$;

  * •

$\lambda_{k}\rightarrow 0$;

  * •

$f_{k}\rightarrow f$, locally uniformly in $\Omega$;

  * •

$b_{k}\rightarrow b$, uniformly in ${\Omega}\times\mathbb{R}^{n}$.

Then,

$\mathcal{I}[u(x),x](x)=\int_{\mathbb{R}^{n}}\delta(u,x,y)\frac{b(x,y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}dy=f(x),\quad
x\in\Omega,$

in the viscosity sense, where $\phi_{0}$ is as in Proposition 5.

###### Proof.

We do the proof only for supersolutions. To prove that

$\mathcal{I}[u(x),x](x)\leq f(x),$

we need to show that, given $\varphi\in C^{2}(x)$, touching $u$ from below at
$x$, we have $\mathcal{I}[\tau(x),x](x)\leq f(x)$, where

$\tau(x)=\left\\{\begin{array}[]{ll}\varphi(x)&\mbox{if $x\in N_{x}$}\\\
u(x)&\mbox{if $x\not\in N_{x}$},\end{array}\right.$

where $N_{x}$ is a neighborhood of $x$. By lemma $(\ref{RA1})$, it is enough
to consider as test functions quadratic polynomials $p$ and neighbourhoods
$N_{x}=B_{r}(x)$. Since $u_{k}$ converges uniformly to $u$ in
$\overline{\Omega}$, for large values of $k$, we can find $x_{k}$ and $d_{k}$
such that $p+d_{k}$ touches $u_{k}$ at $x_{k}$, with $x_{k}\rightarrow x$ and
$d_{k}\rightarrow 0$, when $k\rightarrow\infty$. Since
$\mathcal{I}_{k}[u_{k}(x),x]\leq f_{k}$ in the viscosity sense in $\Omega$, if
we define

$\tau_{k}(x)=\left\\{\begin{array}[]{ll}p+d_{k}&\mbox{in $B_{r}(x)$}\\\
u_{k}&\mbox{in $\mathbb{R}^{n}\backslash B_{r}(x)$},\end{array}\right.$

then $\mathcal{I}_{k}[\tau_{k}(x),x](x_{k})\leq f_{k}(x_{k})$. Clearly,
$\tau_{k}\rightarrow\tau$, uniformly in $B_{r}(x)$.

Now, let $z\in B_{r/4}(x)$. By the triangle inequality,

$\left|\mathcal{I}_{k}[\tau_{k}(x),x](z)-\mathcal{I}[\tau(x),x](z)\right|$
$\leq|\mathcal{I}_{k}[\tau_{k}(x),x](z)-\mathcal{I}_{k}[\tau(x),x](z)|+|\mathcal{I}_{k}[\tau(x),x](z)-\mathcal{I}[\tau(x),x](z)|.$

From the ellipticity condition, and denoting, by simplicity,
$\phi_{\lambda_{k}}$ with $\phi_{k}$, we obtain

$\begin{array}[]{lll}\mathcal{I}_{k}[\tau_{k}(x),x](z)-\mathcal{I}_{k}[\tau(x),x](z)&\leq&\displaystyle
M^{+}_{\mathcal{L}^{0}_{\phi_{k}}(\sigma)}[\tau_{k}-\tau](z)\\\
&\leq&\displaystyle|M^{+}_{\mathcal{L}^{0}_{\phi_{k}}(\sigma)}[\tau_{k}-\tau](z)|\end{array}$

and

$\begin{array}[]{lll}\mathcal{I}_{k}[\tau(x),x](z)-\mathcal{I}_{k}[\tau_{k}(x),x](z)&\leq&\displaystyle
M^{+}_{\mathcal{L}^{0}_{\phi_{k}}(\sigma)}[\tau-\tau_{k}](z)\\\
&\leq&\displaystyle|M^{+}_{\mathcal{L}^{0}_{\phi_{k}}(\sigma)}[\tau-\tau_{k}](z)|.\end{array}$

Hence,

$|\mathcal{I}_{k}[\tau_{k}(x),x](z)-\mathcal{I}_{k}[\tau(x),x](z)|$
$\begin{array}[]{lll}&\leq&\displaystyle\max\left\\{|M^{+}_{\mathcal{L}^{0}_{\phi_{k}}(\sigma)}[\tau_{k}-\tau](z)|,|M^{+}_{\mathcal{L}^{0}_{\phi_{k}}(\sigma)}[\tau-\tau_{k}](z)|\right\\}\\\
&&\\\
&\leq&\displaystyle\sup_{L\in{\mathcal{L}^{0}_{\phi_{k}}(\sigma)}}|L[\tau_{k}-\tau](z)|\\\
&&\\\
&\leq&\displaystyle\int_{\mathbb{R}^{n}}|\delta(\tau_{k}-\tau,z,y)|\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy.\end{array}$

For points $y\in B_{r/2}$, we have $z+y\in B_{r}(x)$. Therefore,
$\tau-\tau_{k}=d_{k}$, which implies $\delta(d_{k},z,y)=0$. Plugging this into
the above inequality, we get

$|\mathcal{I}_{k}[\tau_{k}(x),x](z)-\mathcal{I}_{k}[\tau(x),x](z)|$
$\begin{array}[]{lll}&\leq&\displaystyle\int_{\mathbb{R}^{n}\backslash
B_{r/2}}|\delta(\tau_{k}-\tau,z,y)|\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy\\\
&&\\\ &\leq&\displaystyle\int_{\mathbb{R}^{n}\backslash
S_{\mu}^{\phi_{k}}}|(\tau_{k}-\tau)(z+y)|\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy\\\
&&\\\ &&+\displaystyle\int_{\mathbb{R}^{n}\backslash
S_{\mu}^{\phi_{k}}}|(\tau_{k}-\tau)(z-y)|\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy\\\
&&\\\ &&+\displaystyle 2|(\tau_{k}-\tau)(z)|\int_{\mathbb{R}^{n}\backslash
S_{\mu}^{\phi_{k}}}\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy,\end{array}$

where we choose $\mu=\mu(r)$ such that

$S_{\mu}^{\phi_{k}}\subset B_{r/2},\quad\forall k\in\mathbb{N}.$

Now we apply Propositions 1 and 2 to get

$\int_{\mathbb{R}^{n}\backslash
S^{\phi_{k}}_{\mu}}|(\tau_{k}-\tau)(z+y)|\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy$
$\begin{array}[]{lll}&\leq&\displaystyle C_{\mu}\int_{\mathbb{R}^{n}\backslash
S^{\phi_{k}}_{\mu}}|(\tau_{k}-\tau)(z+y)|\mathcal{W}_{\phi_{k}}(y)dy\\\ &&\\\
&\leq&\displaystyle C_{\mu}C\int_{\mathbb{R}^{n}\backslash
S^{\phi_{k}}_{\mu}}|(\tau_{k}-\tau)(y+z)|\mathcal{W}_{\phi_{k}}(y+z)dy\\\
&&\\\ &\leq&C\left\|\tau_{k}-\tau\right\|_{L^{1}\left(\mathbb{R}^{n}\backslash
S_{\mu}^{\phi_{k}},\mathcal{W}_{\phi_{k}}\right)}.\end{array}$

Furthermore, by computations made in the proof of Proposition (3), we obtain

$\int_{\mathbb{R}^{n}\backslash
S^{\phi_{k}}_{\mu}}\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy\leq\frac{n+2}{\sigma_{0}\mu^{\sigma_{0}}}$

and so

$|\mathcal{I}_{k}[\tau_{k}(x),x](z)-\mathcal{I}_{k}[\tau(x),x](z)|$
$\displaystyle\leq$ $\displaystyle
C_{1}\left\|\tau_{k}-\tau\right\|_{L^{1}\left(\mathbb{R}^{n}\backslash
S^{\phi_{k}}_{\mu},\mathcal{W}_{\phi_{k}}\right)}+C_{2}|\tau_{k}(z)-\tau(z)|$
$\displaystyle+\displaystyle\int_{\mathbb{R}^{n}\backslash
S_{\mu}^{\phi_{k}}}|(\tau_{k}-\tau)(z-y)|\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy.$

Since $u_{k}\rightarrow u$, a.e. in $\mathbb{R}^{n}$, we get that
$\tau_{k}\rightarrow\tau$, a.e. in $\mathbb{R}^{n}$. Therefore, by Proposition
4, assumption [A1] and the growth assumption we obtain, by the dominated
convergence theorem, that

$\left\|\tau_{k}-\tau\right\|_{L^{1}\left(\mathbb{R}^{n}\backslash
S_{\mu}^{\phi_{k}},\mathcal{W}_{\phi_{k}}\right)}\longrightarrow 0.$

Note that, by the same computations made in Remark 1, we get

$\int_{\mathbb{R}^{n}\setminus
S^{\phi_{k}}_{\mu}}\left|(\tau_{k}-\tau)(z-y)\right|\frac{\Lambda(2-\sigma)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}dy$
$\displaystyle\leq$ $\displaystyle
C\left(\mu,|z|,\Lambda\right)\int_{\mathbb{R}^{n}\setminus
S^{\phi_{k}}_{\mu}}|y|^{1+\alpha_{1}}\mathcal{W}_{\phi_{k}}(y)dy,$
$\displaystyle=$ $\displaystyle
C\left(\mu,\Omega,\Lambda\right)\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n}\backslash
S_{\mu}^{\phi_{k}},\mathcal{W}_{\phi_{k}}\right)}$ $\displaystyle\leq$
$\displaystyle\overline{C},$

where we used assumption [A1], along with Proposition 4, to get a uniform
bound on the weighted norm.

Now, notice that, for every $z\in B_{r/4}(x)$, we have
$\tau\in\mathcal{C}^{2}(B_{3r/4}(z))$ and thus

$\mathcal{I}_{k}[\tau(x),x](z)\rightarrow\mathcal{I}[\tau(x),x](z),\quad\mbox{uniformly
in $B_{r/4}(x)$ when $k\rightarrow\infty$}.$

Therefore, we can combine this with Lemma 1, Proposition 4 and assumption [A1]
to get

$|\delta(\tau,z,-)|\left|\frac{b_{k}(z,-)}{v_{\phi_{k}}(-)^{\frac{n+\sigma}{2}}}-\frac{b(z,-)}{v_{\phi_{0}}(-)^{\frac{n+\sigma}{2}}}\right|\in
L^{1}(\mathbb{R}^{n}),$

uniformly in $z$. Since

$\frac{b_{k}(z,y)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}\longrightarrow\frac{b(z,y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}},\quad\mbox{a.e.
$\in\mathbb{R}^{n}$},$

we get, again using the dominated convergence theorem, that

$\left|\mathcal{I}_{k}[\tau(x),x](z)-\mathcal{I}[\tau(x),x](z)\right|$
$\leq\int_{\mathbb{R}^{n}}|\delta(\tau,z,y)|\left|\frac{b_{k}(z,y)}{v_{\phi_{k}}(y)^{\frac{n+\sigma}{2}}}-\frac{b(z,y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}\right|dy\longrightarrow
0.$

We thus obtain

$\mathcal{I}_{k}[\tau_{k}(x),x](z)\rightarrow\mathcal{I}[\tau(x),x](z),$ (15)

uniformly in $B_{r/4}(x)$.

Finally, we have

$\displaystyle|\mathcal{I}_{k}[\tau_{k}(x),x](x_{k})-\mathcal{I}[\tau(x),x](x)|$
$\displaystyle\leq$
$\displaystyle|\mathcal{I}_{k}[\tau_{k}(x),x](x_{k})-\mathcal{I}[\tau(x),x](x_{k})|$
$\displaystyle+|\mathcal{I}[\tau(x),x](x_{k})-\mathcal{I}[\tau(x),x](x)|$

and, using (15)) and the continuity of $\mathcal{I}[\tau(x),x]$, we obtain

$|\mathcal{I}_{k}[\tau_{k}(x),x](x_{k})-\mathcal{I}[\tau(x),x](x)|\rightarrow
0,$

when $k\rightarrow\infty$. Since $x_{k}\rightarrow x$ and $f_{k}\rightarrow f$
locally uniformly,

$\mathcal{I}[\tau(x),x](x)=\lim\limits_{k\rightarrow\infty}\mathcal{I}_{k}[\tau_{k}(x),x](x_{k})\leq\lim\limits_{k\rightarrow\infty}f_{k}(x_{k})=f(x),$

and therefore $\mathcal{I}[\tau(x),x](x)\leq f(x)$. ∎

We are now ready for the approximation lemma. It plays an important role in
the subsequent analysis.

######  Lemma 4.

Assume $\sigma\geq\sigma_{0}>1+\alpha_{1}$ and $\phi\in C^{2}(\mathbb{R}^{n})$
is a convex function satisfying (4) and assumption [A1]. Let $\phi_{0}$ be the
function from Proposition 5. Given $M>0$, a modulus of continuity $\rho$ and
$\epsilon>0$, there exist a small $\eta>0$, $\lambda>0$ and a large $R_{0}>0$
such that, if

  * •

$-\eta<\displaystyle\inf_{\alpha\in\mathcal{A}}\sup_{\beta\in\mathcal{B}}\int_{\mathbb{R}^{n}}\delta(w,-,y)\frac{b_{\alpha\beta}(-,y)}{v_{\phi_{\lambda}}(y)^{\frac{n+\sigma}{2}}}dy<\eta$,
in the viscosity sense in $S_{1}^{\phi_{0}}$;

  * •

for every $\alpha$ and $\beta$, $b(y)-\eta<b_{\alpha\beta}(x,y)<b(y)+\eta$,
$\forall x\in S_{1}^{\phi_{0}}$, $\forall y\in\mathbb{R}^{n}$;

  * •

$|w(y)-w(x)|\leq\rho(|y-x|)$, $\forall x,y\in B_{R_{0}}$;

  * •

$|w(x)|\leq M(1+\zeta(x))$, $\forall x\in\mathbb{R}^{n}$,

then, there exist a function $w_{0}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such
that

  * •

$|w_{0}(x)|\leq M\left(1+\zeta(x)\right)$, $\forall x\in\mathbb{R}^{n}$;

  * •

$\displaystyle\int_{\mathbb{R}^{n}}\delta(w_{0},-,y)\frac{b(y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}dy=0$,
in the viscosity sense in $S_{1}^{\phi_{0}}$;

  * •

$|w-w_{0}|<\epsilon$ in $S_{1}^{\phi_{0}}$.

###### Proof.

Suppose, by contradiction, the lemma is false. Then, there exist sequences
$R_{k},\eta_{k},w_{k},\lambda_{k},b_{k}$ such that

$R_{k}\rightarrow\infty,\quad\eta_{k}\rightarrow 0,\quad\lambda_{k}\rightarrow
0\quad\mbox{and}\quad\quad b_{k}\rightarrow b\quad\mbox{uniformly in
$S_{1}^{\phi_{0}}\times\mathbb{R}^{n}$}$

and

$-\eta_{k}<\displaystyle\int_{\mathbb{R}^{n}}\delta(w_{k},x,y)\frac{b_{k}(x,y)}{v_{\phi_{\lambda_{k}}}(y)^{\frac{n+\sigma}{2}}}dy<\eta_{k}$

in the viscosity sense in $S_{1}^{\phi_{0}}$ but

$\sup|w_{k}-w_{0}|\geq\epsilon\quad\mbox{in $S^{\phi_{0}}_{1}$},$ (16)

for all $w_{0}$ solution of

$\int_{\mathbb{R}^{n}}\delta(w_{0},x,y)\frac{b(y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}dy=0.$

By Proposition 5,

$v_{\phi_{\lambda_{k}}}(x)\longrightarrow D^{2}\phi(0)x\cdot
x=v_{\phi_{0}}(x),$

locally uniformly and thus almost everywhere in $\mathbb{R}^{n}$. On the other
hand, the $w_{k}$ have a uniform modulus of continuity in $B_{R_{k}}$, where
$R_{k}\rightarrow\infty$, so, up to subsequences,
$w_{k}\rightarrow\overline{w}$ uniformly on compacts and thus almost
everywhere in $\mathbb{R}^{n}$. By Lemma 3, we pass to the limit to get

$\int_{\mathbb{R}^{n}}\delta(\overline{w},x,y)\frac{b(y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}dy=0\quad\mbox{in
$S_{1}^{\phi_{0}}$}.$

Since $w_{k}\rightarrow\overline{w}$ uniformly in $S_{1}^{\phi_{0}}$, for $k$
sufficiently large we have a contradiction with (16). ∎

##  4\.  Regularity for variable coefficients

This section is devoted to the proof of the main theorem of the paper.

######  Theorem 1.

Let $\sigma\geq\sigma_{0}>1$, let $\phi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ be
a convex function satisfying $(\ref{MONGEAMPEREEQ})$ and assumption [A1], and
let $u$ be a viscosity solution to $(\ref{MAINEQ})$ in $S^{\phi}_{2}$. There
exists $0<\eta\ll 1$ such that if

$\sup_{\begin{subarray}{c}\alpha\in\mathcal{A},\beta\in\mathcal{B}\\\ (x,y)\in
S^{\phi}_{2}\times\mathbb{R}^{n}\end{subarray}}\left|b_{\alpha,\beta}(x,y)-b(y)\right|<\eta,$

for $b\in L^{\infty}(\mathbb{R}^{n})$, then
$u\in\mathcal{C}^{1,\alpha_{0}}(S^{\phi}_{1})$ and the following estimate
holds

$\left\|u\right\|_{\mathcal{C}^{1,\alpha_{0}}\left(S^{\phi}_{1}\right)}\leq
C\left(\left\|u\right\|_{L^{\infty}\left(\mathbb{R}^{n}\right)}+\left\|f\right\|_{L^{\infty}\left(S^{\phi}_{2}\right)}\right),$

for $0<\alpha_{0}<\min\\{\alpha_{*},\alpha_{1}\\}$, where $\alpha_{1}$ is from
assumption [A1] and $\alpha_{*}$ is from Theorem 3. The constant $C$ depends
only on $n,\Lambda,\sigma_{0},\alpha_{\ast},\gamma$ and $\Gamma$.

The idea is to iterate the approximation lemma (Lemma 4) to quotients of the
form

$\frac{[u-l](\lambda-)}{\lambda^{1+\alpha_{0}}},$

where $l$ is an affine function and $\lambda>0$ and $\alpha_{0}$ are to be
chosen in the sequel. To assure that Lemma 4 can be applied for every small
enough $\lambda$, the first step is to grant a modulus of continuity to
solutions that grow at infinity. This is done using a standard technique.

######  Theorem 2.

Assume $\sigma>\sigma_{0}>1$ and
$w\in\mathcal{C}\left(\overline{S_{2}^{\phi}}\right)$ satisfies, for $M>0$ and
$\zeta$ from [A1],

$|w(x)|\leq M(1+\zeta(x)),\quad x\in\mathbb{R}^{n},$

and solves

$M^{+}_{\mathcal{L}^{0}_{\phi}(\sigma)}w\geq-C_{0},\quad
M^{-}_{\mathcal{L}^{0}_{\phi}(\sigma)}w\leq C_{0}$

in the viscosity sense in $S_{2}^{\phi}$. Then, there exists $\alpha>0$ and a
radius $\overline{\rho}$ such that
$w\in\mathcal{C}^{\alpha}\left(S_{\overline{\rho}}^{\phi}\right)$ and

$\left\|w\right\|_{\mathcal{C}^{\alpha}\left(S_{\overline{\rho}}^{\phi}\right)}\leq
C^{*}\left(\sup_{S_{2}^{\phi}}|w|+C_{0}+C\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi}\right)}\right)$

for some universal constant $C^{*}$.

###### Proof.

Define $\overline{w}(x)=\mathbbm{1}_{S^{\phi}_{2}}(x)w(x)$ and notice that for
$x\in S^{\phi}_{1}$,

$\begin{array}[]{lll}L\overline{w}(x)&=&\displaystyle\int_{\mathbb{R}^{n}}\delta(\overline{w},x,y)\mathcal{K}(y)dy\\\
&&\\\ &=&\displaystyle
Lw(x)+\int_{\mathbb{R}^{n}}(w(x+y)\mathbbm{1}_{S^{\phi}_{2}}(x+y)-w(x+y))\mathcal{K}(y)dy\\\
&&\\\
&&\displaystyle+\int_{\mathbb{R}^{n}}(w(x-y)\mathbbm{1}_{S^{\phi}_{2}}(x-y)-w(x-y))\mathcal{K}(y)dy\\\
&&\\\ &=&\displaystyle
Lw(x)-\int_{\mathbb{R}^{n}\backslash\Omega^{+}}w(x+y)\mathcal{K}(y)dy-\int_{\mathbb{R}^{n}\backslash\Omega^{-}}w(x-y)\mathcal{K}(y)dy,\end{array}$

where

$\Omega^{+}=\\{y\in\mathbb{R}^{n}:x+y\in
S^{\phi}_{2}\\}\quad\mbox{and}\quad\Omega^{-}=\\{y\in\mathbb{R}^{n}:x-y\in
S^{\phi}_{2}\\}.$

Now, if $d=dist(\partial S^{\phi}_{1},\partial S^{\phi}_{2})$, we have

$B_{d}(0)\subset\Omega^{+},\Omega^{-}.$

Therefore, from the growth assumption for $w$, we can estimate

$\begin{array}[]{lll}\displaystyle\left|\int_{\mathbb{R}^{n}\backslash\Omega^{+}}w(x+y)\mathcal{K}(y)dy\right|&\leq&\displaystyle
C\int_{\mathbb{R}^{n}\backslash
B_{d}}|y|^{1+\alpha_{1}}\frac{1}{v_{\phi}(y)^{\frac{n+\sigma}{2}}}dy\\\ &&\\\
&\leq&C(d)\left\|\zeta\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi})}.\end{array}$

The same computations are valid for the other term and we get

$M^{+}_{\mathcal{L}^{0}_{\phi}(\sigma)}\overline{w}(x)\geq
M^{+}_{\mathcal{L}^{0}_{\phi}(\sigma)}w(x)-C\left\|\zeta\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi})}\geq-(C_{0}+C\left\|\zeta\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi})}),$

for $x\in S^{\phi}_{1}$. The same calculations apply to
$M^{-}_{\mathcal{L}^{0}_{\phi}(\sigma)}$ leading to

$M^{-}_{\mathcal{L}^{0}_{\phi}(\sigma)}\overline{w}(x)\leq
C_{0}+C\left\|\zeta\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi})},$

for $x\in S^{\phi}_{1}$. We now apply a scaled version of [10, Theorem 6.2],
to get the existence of $\alpha\in(0,1)$ and a small radius $\overline{\rho}$,
depending on $\lambda,\Lambda,\sigma_{0}$ and dimension, such that

$\left\|\overline{w}\right\|_{\mathcal{C}^{\alpha}\left(S^{\phi}_{\overline{\rho}}\right)}\leq
C^{*}\left(\sup_{\mathbb{R}^{n}}|\overline{w}|+C_{0}+C\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi}\right)}\right).$

Since $\overline{w}=w$ in $S^{\phi}_{2}$, we also have

$\left\|w\right\|_{\mathcal{C}^{\alpha}\left(S^{\phi}_{\overline{\rho}}\right)}\leq
C^{*}\left(\sup_{S^{\phi}_{2}}|w|+C_{0}+C\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi}\right)}\right).$

This constant $C^{*}$ depends on the constant $\overline{\tau}$ from [10,
Proposition 3.1], the norm of the normalization mapping of the section
$S^{\phi}_{\overline{\gamma}\rho/2},\Lambda,\sigma_{0}$ and dimension, where
$\overline{\gamma}$ stands for the standard engulfing constant and $\rho$ is
the constant from [10, Theorem 5.1]. ∎

######  Remark 2.

We will apply the theorem above for the scaled family $\\{\phi_{\lambda}\\}$,
with $\lambda\in(0,1]$. Since we need uniform estimates, we must assure that
the estimate above does not degenerate as $\lambda$ varies in the interval
$(0,1]$. From Proposition 4 we can uniformly bound the quantity
$\left\|\zeta\right\|_{L^{1}(\mathbb{R}^{n},\mathcal{W}_{\phi_{\lambda}})}$.
On the other hand, the quantity $d_{\lambda}=dist(\partial
S_{1}^{\phi_{\lambda}},\partial S_{2}^{\phi_{\lambda}})$ is uniformly bounded
above and below away from zero. We can also bound uniformly the norm of the
normalization mapping of $S^{\phi_{\lambda}}_{\overline{\gamma}\rho/2}$ by
making use of [12, Corollary 4.7].

######  Corollary 1.

Assume $\sigma>\sigma_{0}>1$ and
$w\in\mathcal{C}\left(\overline{S^{\phi}_{2r}}\right)$ satisfies

$M^{+}_{\mathcal{L}^{0}_{\phi}(\sigma)}w\geq-C_{0},\quad
M^{-}_{\mathcal{L}^{0}_{\phi}(\sigma)}w\leq C_{0}\quad\mbox{in
$S^{\phi}_{2r}$},$

with $|w|\leq M(1+\zeta)$ in $\mathbb{R}^{n}$. Then, there exists $\alpha>0$
such that $w\in\mathcal{C}^{\alpha}\left(S^{\phi}_{r/2}\right)$ and

$\left\|w\right\|_{\mathcal{C}^{\alpha}\left(S^{\phi}_{r\overline{\rho}}\right)}\leq
C^{*}\left(\sup_{S^{\phi}_{2r}}|w|+C_{0}+C\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi}\right)}\right),$
(17)

for some constant $C^{*}$.

###### Proof.

Let $w^{\ast}$ be the following auxiliary function

$w^{\ast}(x):=\frac{1}{\max\\{1,r^{1+\alpha_{1}},r^{\sigma}\\}}w(rx).$

Then

$M^{+}_{\mathcal{L}^{0}_{\phi_{r}}(\sigma)}w^{\ast}\geq-C_{0},\quad
M^{-}_{\mathcal{L}^{0}_{\phi_{r}}(\sigma)}w^{\ast}\leq C_{0}\quad\mbox{in
$S^{\phi_{r}}_{2}$},$

for $\phi_{r}(x)=r^{-2}\phi(rx)$ and $|w^{\ast}|\leq M(1+\zeta)$ in
$\mathbb{R}^{n}$. By the previous theorem,

$\left\|w^{\ast}\right\|_{\mathcal{C}^{\alpha}\left(S^{\phi_{r}}_{\overline{\rho}}\right)}\leq
C^{*}\left(\sup_{S^{\phi_{r}}_{2}}|w^{\ast}|+C_{0}+C\left\|\zeta\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi}\right)}\right).$

Rescaling back to $w$, we get (17), where $C^{*}$ has the same dependence as
in the previous theorem, replacing for the norm of the normalization mapping
of the section $S^{\phi_{r}}_{\overline{\gamma}\rho/2}$.

∎

We also need gradient Hölder regularity estimates for solutions of the limit
equation

$\int_{\mathbb{R}^{n}}\delta(u,x,y)\frac{b(y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}dy=0,\quad
x\in S^{\phi_{0}}_{1}.$ (18)

It is interesting to note that we will not make use of the
$\mathcal{C}^{1,\alpha}$ regularity results from [10], since our kernels do
not necessarily satisfy the assumptions therein. Instead, due to the
regularizing effect of the scalings, we will make use of the results from
[15], after a suitable change of variables.

######  Theorem 3.

Let $\sigma\geq\sigma_{0}>1$. Assume
$v\in\mathcal{C}\left(\overline{S}^{\phi_{0}}_{1}\right)\cap
L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi_{0}}\right)$ is a viscosity
solution of

$\int_{\mathbb{R}^{n}}\delta(v,x,y)\frac{b(y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}dy=0\quad\mbox{in
$S^{\phi_{0}}_{1}$},$

where
$\mathcal{K}(y)=b(y)/v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}\in\mathcal{L}^{0}_{\phi_{0}}(\sigma)$.
Then, there exists $\alpha_{*}$ such that
$v\in\mathcal{C}^{1,\alpha_{*}}\left(S^{\phi_{0}}_{1/2}\right)$ and

$\left\|v\right\|_{\mathcal{C}^{1,\alpha_{*}}(S^{\phi_{0}}_{1/2})}\leq
C\left(\sup_{S^{\phi_{0}}_{1}}|v|+\left\|v\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi_{0}}\right)}\right),$

where $C>0$ is a universal constant.

###### Proof.

We remark that $v_{\phi_{0}}(y)=D^{2}\phi(0)y\cdot y$. Since $\phi$ solves
$(\ref{MONGEAMPEREEQ})$, we get that, in particular, the matrix $D^{2}\phi(0)$
has only positive eigenvalues, and so is invertible. Therefore, it may be
decomposed in the following form $D^{2}\phi(0)=SDS^{t}$, where $S$ is an
orthogonal matrix and $D$ is diagonal with the eigenvalues of $D^{2}\phi(0)$.
Since the eigenvalues are nonnegative, the matrix $D$ has a square root, which
we will denote by $B:=\sqrt{D}$. Then

$D^{2}\phi(0)y\cdot y=Qy\cdot Qy,$

where $Q:=BS^{t}$. Notice that

$x\in S_{1}^{\phi_{0}}\iff D^{2}\phi(0)x\cdot x<1\iff|Qx|<1\iff x\in
Q^{-1}(B_{1}),$

that is, $Q(S_{1}^{\phi_{0}})=B_{1}$. Now, define $w=v\circ Q^{-1}$. Since
$v\in\mathcal{C}(\overline{S}^{\phi_{0}}_{1})$, we obtain
$w\in\mathcal{C}(\overline{B}_{1})$. For the growth condition, we have

$\begin{array}[]{lll}\displaystyle\int_{\mathbb{R}^{n}}|w(z)|\frac{1}{1+|z|^{n+\sigma_{0}}}dz&=&\displaystyle\int_{\mathbb{R}^{n}}|v(Q^{-1}z)|\frac{1}{1+|z|^{n+\sigma_{0}}}dz\\\
&&\\\
&=&\displaystyle|\det(Q)|\int_{\mathbb{R}^{n}}|v(y)|\frac{1}{1+|Qy|^{n+\sigma_{0}}}dy\\\
&&\\\
&=&\displaystyle|\det(Q)|\int_{\mathbb{R}^{n}}|v(y)|\mathcal{W}_{\phi_{0}}(y)dy,\end{array}$

that is

$\left\|w\right\|_{L^{1}\left(\mathbb{R}^{n},\frac{1}{1+|.|^{n+\sigma_{0}}}\right)}=\det(Q)\left\|v\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi_{0}}\right)}.$

We then obtain that $w\in\mathcal{C}(\overline{B}_{1})\cap
L^{1}\left(\mathbb{R}^{n},\frac{1}{1+|.|^{n+\sigma_{0}}}\right)$. It is
straightforward to check that the function $w$ solves in the viscosity sense

$\int_{\mathbb{R}^{n}}\delta(w,-,z)\frac{\overline{b}(y)}{|y|^{\frac{n+\sigma}{2}}}dy=0\quad\mbox{in
$B_{1}$},$

where $\overline{b}(y)=b(Q^{-1}y)$. By [15, Theorem 4.1], there exists
$\alpha_{*}$ such that

$\left\|w\right\|_{\mathcal{C}^{1,\alpha_{*}}(B_{1/2})}\leq
C\left(\left\|w\right\|_{L^{1}\left(\mathbb{R}^{n},\frac{1}{1+|.|^{n+\sigma_{0}}}\right)}+\left\|w\right\|_{L^{\infty}(B_{1})}\right).$

Rescalling back to $v$ we obtain

$\left\|v\right\|_{\mathcal{C}^{1,\alpha_{*}}\left(S^{\phi_{0}}_{1/2}\right)}\leq\overline{C}\left(\left\|v\right\|_{L^{1}\left(\mathbb{R}^{n},\mathcal{W}_{\phi_{0}}\right)}+\left\|v\right\|_{L^{\infty}\left(S^{\phi_{0}}_{1}\right)}\right),$

where $\overline{C}(C,|Q|,|Q^{-1}|)$. ∎

We are now ready to deliver the proof of the main theorem in its discrete
version.

######  Theorem 4.

Let $\sigma\geq\sigma_{0}>1$, let $\phi\in\mathcal{C}^{2}(\mathbb{R}^{n})$
satisfy $(\ref{MONGEAMPEREEQ})$ and assumption [A1], and let $u$ be a
viscosity solution to $(\ref{MAINEQ})$ in $S^{\phi}_{2R}$. There exists a
small $\eta>0$ and a large $R>0$ such that if

$\sup_{y\in\mathbb{R}^{n}}\left|b_{\alpha\beta}(x,y)-b(y)\right|<\eta,\quad\forall\alpha\in\mathcal{A},\forall\beta\in\mathcal{B},\quad\forall
x\in S^{\phi}_{2R},$ $\|u\|_{L^{\infty}(\mathbb{R}^{n})}\leq 1$

and

$\|f\|_{L^{\infty}(S^{\phi}_{2R})}\leq\eta,$

then we can find universal constants $\theta,\lambda,C_{2}>0$ and a sequence
of linear functions $l_{k}=a_{k}+b_{k}\cdot x$ such that

$\displaystyle\sup_{B_{\theta\lambda^{k}}}|u-l_{k}|$ $\displaystyle\leq$
$\displaystyle\lambda^{k(1+\alpha_{0})}$ $\displaystyle|a_{k+1}-a_{k}|$
$\displaystyle\leq$ $\displaystyle\lambda^{k(1+\alpha_{0})}$
$\displaystyle\left|b_{k+1}-b_{k}\right|$ $\displaystyle\leq$ $\displaystyle
C_{2}\lambda^{k\alpha_{0}},$

for every $k\in\mathbb{N}$. The exponent $\alpha_{0}$ satisfies
$\alpha_{0}<\min\\{\alpha_{*},\alpha_{1}\\}$, where $\alpha_{1}$ is from
assumption [A1] and $\alpha_{*}$ is from Theorem 3.

###### Proof.

We proceed by induction. For step $k=0$, take $l_{0}=l_{1}=0$. Assume the
result holds up to order $k$ and let us show it also holds for $k+1$. Define

$w_{k}(x)=\frac{1}{\lambda^{k(1+\alpha_{0})}}[u-l_{k}](\lambda^{k}x),\quad
x\in\mathbb{R}^{n}$

and consider $\theta$ so small that

$B_{\theta}\subset S_{1}^{\phi_{0}}.$

Observe first that the scaled function

$\phi_{\lambda^{k}}(y):=\lambda^{-2k}\phi(\lambda^{k}y),\quad
y\in\mathbb{R}^{n}$

solves the same Monge-Ampère equation

$\det\left(D^{2}\phi_{\lambda^{k}}(y)\right)=F(\lambda^{k}y)=\overline{F}(y),$

with the right-hand side satisfying the same bounds as $F$,
$\gamma<\overline{F}<\Gamma$, and that the level sets of $\phi_{\lambda^{k}}$
are related with the level sets of $\phi$ by

$v_{\phi_{\lambda^{k}}}(y)=\lambda^{-2k}v_{\phi}(\lambda^{k}y).$

Now $w_{k}$ solves an equation with the same ellipticity constants (but
rescaled kernels), namely

$\inf_{\alpha\in\mathcal{A}}\sup_{\beta\in\mathcal{B}}\int_{\mathbb{R}^{n}}\delta(w_{k},x,y)\frac{b_{\alpha\beta}(\lambda^{k}x,\lambda^{k}y)}{v_{\phi_{\lambda^{k}}}(y)^{\frac{n+\sigma}{2}}}dy=\lambda^{k(\sigma-1-\alpha_{0})}f(\lambda^{k}x)$

for $x\in\lambda^{-k}S^{\phi}_{2R}=S^{\phi_{\lambda^{k}}}_{2R\lambda^{-k}}$.
Since, due to (13),

$\sigma-1-\alpha_{0}\geq\sigma_{0}-1-\alpha_{0}>\alpha_{1}-\alpha_{0}>0$

and $\lambda<1$, we have

$\left\|\lambda^{k(\sigma-1-\alpha_{0})}f(\lambda^{k}x)\right\|_{L^{\infty}(\lambda^{-k}S^{\phi}_{2R})}\leq\|f\|_{L^{\infty}\left(S^{\phi}_{2R}\right)}\leq\eta.$

Therefore, $w_{k}$ solves

$-\eta<\inf_{\alpha\in\mathcal{A}}\sup_{\beta\in\mathcal{B}}\int_{\mathbb{R}^{n}}\delta(w_{k},x,y)\frac{b_{\alpha\beta}(\lambda^{k}x,\lambda^{k}y)}{v_{\phi_{\lambda^{k}}}(y)^{\frac{n+\sigma}{2}}}dy<\eta$
(19)

for $x\in\lambda^{-k}S^{\phi}_{2R}=S^{\phi_{\lambda^{k}}}_{2R\lambda^{-k}}$.

Let $\phi_{0}$ be the function from Proposition 5 and take $R$ large such that
$S_{1}^{\phi_{0}}\subset S^{\phi}_{2R}$. Then, since $\phi_{0}$ is
2-homogeneous,

$S_{2R\lambda^{-k}}^{\phi_{\lambda^{k}}}=\lambda^{-k}S^{\phi}_{2R}\supset\lambda^{-k}S_{1}^{\phi_{0}}=S^{\phi_{0}}_{\lambda^{-k}}\supset
S_{1}^{\phi_{0}},$

so that $(\ref{EQUATIONFORWK})$ holds for $S_{1}^{\phi_{0}}$ for every
$k\in\mathbb{N}$. From Corollary 1, $w_{k}$ is Hölder continuous in
$S^{\phi_{\lambda^{k}}}_{R\overline{\rho}}$. Notice that, by Lemma 1 with
$\Psi_{1}=\phi_{\lambda^{k}},\Psi_{2}=|.|^{2}$ and radius $R_{0}$, we obtain
for large $R$

$B_{R_{0}}\subset S^{\phi_{\lambda^{k}}}_{R\overline{\rho}},$

and so $w_{k}$ is Hölder continuous in $B_{R_{0}}$, where $R_{0}$ is from
Lemma 4. We can now apply Lemma 4 to the function $w_{k}$, finding
$h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ satisfying
$h\in\mathcal{C}(S_{1}^{\phi_{0}})$, $|h|\leq(1+\zeta)$ in $\mathbb{R}^{n}$,

$\int_{\mathbb{R}^{n}}\delta(h,x,y)\frac{b(y)}{v_{\phi_{0}}(y)^{\frac{n+\sigma}{2}}}dy=0,$

in the viscosity sense in $S_{1}^{\phi_{0}}$, and $|w_{k}-h|<\epsilon$ in
$S_{1}^{\phi_{0}}$, for some small $\epsilon$ to be chosen later. By Theorem
3, we obtain

$\left\|h\right\|_{\mathcal{C}^{1,\alpha_{*}}(S^{\phi_{0}}_{1/2})}\leq C_{2}.$

Letting $\bar{l}(x)=h(0)+\nabla h(0)\cdot x$, we have, by the mean value
theorem,

$\begin{array}[]{lll}|h(x)-\bar{l}(x)|&\leq&\displaystyle[\nabla
h]_{C^{\alpha_{\ast}}(S_{1/2}^{\phi_{0}})}|x|^{1+\alpha_{\ast}}\\\ &&\\\
&\leq&\displaystyle C_{2}|x|^{1+\alpha_{\ast}}\end{array}$

for $x\in B_{\theta/2}$.

Since $|w_{k}|\leq 1$ in $B_{\theta}$, we have $|h|\leq 1+\epsilon$ in
$B_{\theta}$ and then $|h(0)|\leq 1+\epsilon$. Also, by $C^{1,\alpha_{\ast}}$
estimates, we have $|\nabla h(0)|\leq C_{2}$. Therefore, we have the following
estimates

$\begin{array}[]{lll}|w_{k}-\bar{l}|&\leq&|w_{k}-h|+|h-\bar{l}|\\\ &&\\\
&\leq&\epsilon(\eta,R)+C_{2}|x|^{1+\alpha_{\ast}}\quad\mbox{in}\quad
B_{\frac{\theta}{2}}\\\ \\\ |w_{k}-\bar{l}|&\leq&|w_{k}-h|+|h|+|\bar{l}|\\\
&&\\\ &\leq&3\epsilon(\eta,R)+2+C_{2}|x|\quad\mbox{in}\quad
B_{\theta}\backslash B_{\frac{\theta}{2}}\\\ \\\
|w_{k}-\bar{l}|&\leq&|w_{k}|+|\bar{l}|\\\ &&\\\
&\leq&|x|^{1+\alpha_{1}}+|h(0)|+|\nabla h(0)|\;|x|\\\ &&\\\
&\leq&|x|^{1+\alpha_{1}}+1+\epsilon(\eta,R)+C_{2}|x|\quad\mbox{in}\quad\mathbb{R}^{n}\backslash
B_{\theta}.\end{array}$

Let us now fix $\lambda>0$, to be chosen later, such that
$\epsilon\leq\lambda^{1+\alpha_{\ast}}$. Define

$l_{k+1}(x):=l_{k}(x)+\lambda^{k(1+\alpha_{0})}\bar{l}(\lambda^{-k}x),$

and

$\begin{array}[]{lll}w_{k+1}(x)&=&\displaystyle\frac{1}{\lambda^{(k+1)(1+\alpha_{0})}}[u-l_{k+1}](\lambda^{k+1}x)\\\
&&\\\ &=&\displaystyle\frac{1}{\lambda^{1+\alpha_{0}}}[w_{k}-\bar{l}](\lambda
x).\end{array}$

We have the following estimates for the scaling above:

$\begin{array}[]{lll}|w_{k+1}(x)|&\leq&\lambda^{-1-\alpha_{0}}(\epsilon+C_{2}\lambda^{1+\alpha_{\ast}}|x|^{1+\alpha_{\ast}})\quad\mbox{in}\quad\lambda^{-1}B_{\frac{\theta}{2}}\\\
\\\
|w_{k+1}(x)|&\leq&\lambda^{-1-\alpha_{0}}(3\epsilon+2+C_{2}\lambda|x|)\quad\mbox{in}\quad\lambda^{-1}B_{\theta}\backslash\lambda^{-1}B_{\frac{\theta}{2}}\\\
\\\
|w_{k+1}(x)|&\leq&\lambda^{-1-\alpha_{0}}(\lambda^{1+\alpha_{1}}|x|^{1+\alpha_{1}}+1+\epsilon+C_{2}\lambda|x|)\quad\mbox{in}\quad\mathbb{R}^{n}\backslash\lambda^{-1}B_{\theta}.\end{array}\vskip
12.0pt plus 4.0pt minus 4.0pt$

Using the fact that $\epsilon\leq\lambda^{1+\alpha_{\ast}}\leq 1$, we obtain

$\begin{array}[]{lll}|w_{k+1}(x)|&\leq&\lambda^{\alpha_{\ast}-\alpha_{0}}(1+C_{2}|x|^{1+\alpha_{\ast}})\quad\mbox{in}\quad\lambda^{-1}B_{\frac{\theta}{2}}\\\
\\\
|w_{k+1}(x)|&\leq&5\lambda^{-1-\alpha_{0}}+C_{2}\lambda^{-\alpha_{0}}|x|\quad\mbox{in}\quad\lambda^{-1}B_{\theta}\backslash\lambda^{-1}B_{\frac{\theta}{2}}\\\
\\\
|w_{k+1}(x)|&\leq&\lambda^{\alpha_{1}-\alpha_{0}}|x|^{1+\alpha_{1}}+2\lambda^{-1-\alpha_{0}}+C_{2}\lambda^{-\alpha_{0}}|x|\quad\mbox{in}\quad\mathbb{R}^{n}\backslash\lambda^{-1}B_{\theta}.\end{array}$

Notice now that for $\lambda\leq 1/2$, we obtain

$B_{\lambda^{-1}\theta/2}\supset B_{\theta}.$

Recalling that $\alpha_{0}<\min\\{\alpha_{1},\alpha_{*}\\}$, we have, by the
first estimate above,

$\begin{array}[]{lll}|w_{k+1}(x)|&\leq&\lambda^{\alpha_{*}-\alpha_{0}}(1+C_{2}|x|^{1+\alpha_{*}})\\\
&\leq&\lambda^{\alpha_{*}-\alpha_{0}}(1+C_{2})\\\ &\leq&1\end{array}$

in $B_{\theta}$, as long as $\lambda$ is chosen such that

$\lambda\leq\left(\frac{1}{1+C_{2}}\right)^{\frac{1}{\alpha_{*}-\alpha_{0}}}.$

We will choose $\lambda$ so small that $|w_{k+1}(x)|\leq|x|^{1+\alpha_{1}}$
outside $B_{\theta}$ as well. A point $x\in\mathbb{R}^{n}\backslash
B_{\theta}$ must be in one of the sets

$B_{\lambda^{-1}\theta/2},\quad B_{\lambda^{-1}\theta}\backslash
B_{\lambda^{-1}\theta/2}\quad\mbox{or}\quad\mathbb{R}^{n}\backslash
B_{\lambda^{-1}\theta}.$

If, for instance, $x\in B_{\lambda^{-1}\theta/2}$, we have
$\theta\leq|x|<\lambda^{-1}\theta/2$, and so

$\begin{array}[]{lll}|w_{k+1}(x)|&\leq&\lambda^{\alpha_{*}-\alpha_{0}}(1+C_{2}|x|^{1+\alpha_{*}})\\\
&\leq&\lambda^{\alpha_{*}-\alpha_{0}}(\theta^{-1-\alpha_{1}}+C_{2}|x|^{\alpha_{*}-\alpha_{1}})|x|^{1+\alpha_{1}}\\\
&\leq&3\theta^{-1-\alpha_{1}}\lambda^{\alpha_{1}-\alpha_{0}}(1+C_{2})|x|^{1+\alpha_{1}}\\\
&\leq&\lambda^{\alpha_{*}-\alpha_{0}}\theta^{-1-\alpha_{1}}|x|^{1+\alpha_{1}}+C_{2}\lambda^{\alpha_{*}-\alpha_{0}}|x|^{\alpha_{*}-\alpha_{1}}|x|^{1+\alpha_{1}}\\\
&\leq&\left(\lambda^{\alpha_{*}-\alpha_{0}}\theta^{-1-\alpha_{1}}+C_{2}\theta^{\alpha_{*}-\alpha_{1}}.\right.\\\
&&\left..\max\left\\{\left(\frac{1}{2}\right)^{\alpha_{*}-\alpha_{1}}\lambda^{\alpha_{1}-\alpha_{0}},\lambda^{\alpha_{*}-\alpha_{0}}\right\\}\right)|x|^{1+\alpha_{1}}\\\
&\leq&|x|^{1+\alpha_{1}},\end{array}$

as long as $\lambda$ is chosen small such that

$\left(\lambda^{\alpha_{*}-\alpha_{0}}\theta^{-1-\alpha_{1}}+C_{2}\theta^{\alpha_{*}-\alpha_{1}}\max\left\\{\left(\frac{1}{2}\right)^{\alpha_{*}-\alpha_{1}}\lambda^{\alpha_{1}-\alpha_{0}},\lambda^{\alpha_{*}-\alpha_{0}}\right\\}\right)\leq
1,$

recall that both $\alpha_{*}-\alpha_{0}$ and $\alpha_{1}-\alpha_{0}$ are
positive and that $\alpha_{*}-\alpha_{1}$ may be positive or negative.

Now, if $x\in B_{\lambda^{-1}\theta}\backslash B_{\lambda^{-1}\theta/2}$, we
have $\lambda^{-1}\theta/2\leq|x|<\lambda^{-1}\theta$. By the second estimate,
we obtain

$\begin{array}[]{lll}|w_{k+1}(x)|&\leq&5\lambda^{-1-\alpha_{0}}+C_{2}\lambda^{-\alpha_{0}}|x|\\\
&=&5\lambda^{\alpha_{1}-\alpha_{0}}\lambda^{-1-\alpha_{1}}+C_{2}\lambda^{\alpha_{1}-\alpha_{0}}\lambda^{-\alpha_{1}}|x|\\\
&\leq&2\lambda^{\alpha_{1}-\alpha_{0}}\max\\{5,C_{2}\\}\left(\frac{\theta}{2}\right)^{-1-\alpha_{1}}|x|^{1+\alpha_{1}}\\\
&\leq&|x|^{1+\alpha_{1}},\end{array}$

as long as we choose $\lambda$ so small that

$2\lambda^{\alpha_{1}-\alpha_{0}}\max\\{5,C_{2}\\}\left(\frac{\theta}{2}\right)^{-1-\alpha_{1}}\leq
1.$

Finally, if $x\in\mathbb{R}^{n}\backslash B_{\lambda^{-1}\theta}$, we have
$|x|\geq\lambda^{-1}\theta$. By the third estimate, we get

$\begin{array}[]{lll}|w_{k+1}(x)|&\leq&\lambda^{\alpha_{1}-\alpha_{0}}|x|^{1+\alpha_{1}}+2\lambda^{-1-\alpha_{0}}+C_{2}\lambda^{-\alpha_{0}}|x|\\\
&\leq&\lambda^{\alpha_{1}-\alpha_{0}}\left(1+2\theta^{-1-\alpha_{1}}+C_{2}\theta^{-\alpha_{1}}\right)|x|^{1+\alpha_{1}}\\\
&\leq&|x|^{1+\alpha_{1}},\end{array}$

as long as we take $\lambda$ small such that

$\lambda^{\alpha_{1}-\alpha_{0}}\left(1+2\theta^{-1-\alpha_{1}}+C_{2}\theta^{-\alpha_{1}}\right)\leq
1.$

We choose $\lambda$ such that all of the above conditions hold and we get

$|w_{k+1}(x)|\leq|x|^{1+\alpha_{1}},\qquad x\in\mathbb{R}^{n}\backslash
B_{\theta},$

and so $|w_{k+1}|\leq(1+\zeta)$ in $\mathbb{R}^{n}$ as desired.

Notice as well that, for $x\in B_{\lambda^{k+1}\theta}$ and $|w_{k+1}|\leq 1$
in $B_{\theta}$, we have

$\begin{array}[]{lll}|u(x)-l_{k+1}(x)|&=&\lambda^{(k+1)(1+\alpha_{0})}|w_{k+1}(\lambda^{-(k+1)}x)|\\\
&\leq&\lambda^{(k+1)(1+\alpha_{0})}.\end{array}$

By the definition of $l_{k+1}$, we have
$a_{k+1}=a_{k}+\lambda^{k(1+\alpha_{0})}h(0)$ and
$b_{k+1}=b_{k}+\lambda^{k\alpha_{0}}\nabla h(0)$ and so the theorem is proven.
∎

Now, the full regularity estimate of Theorem 1 follows as in [8], using a
covering argument.

Acknowledgments. D.P. partially supported by Capes-Fapitec and CNPq. A.S.
supported by a Capes PhD scholarship. J.M.U. partially supported by the King
Abdullah University of Science and Technology (KAUST), by Fundação para a
Ciência e a Tecnologia, through project PTDC/MAT-PUR/28686/2017 and by the
Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded
by the Portuguese Government through FCT/MCTES).

## References

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# Predicting dominant hand from spatiotemporal context varying physiological
data

Jorge Neira-García, and Sudip Vhaduri  Polytechnic Institute
Purdue University
West Lafayette, Indiana, USA
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Health metrics from wrist-worn devices demand an automatic dominant hand
prediction to keep an accurate operation. The prediction would improve
reliability, enhance the consumer experience, and encourage further
development of healthcare applications. This paper aims to evaluate the use of
physiological and spatiotemporal context information from a two-hand
experiment to predict the wrist placement of a commercial smartwatch. The main
contribution is a methodology to obtain an effective model and features from
low sample rate physiological sensors and a self-reported context survey.
Results show an effective dominant hand prediction using data from a single
subject under real-life conditions.

###### Index Terms:

Commercial smartwatch, contextual information, dominant hand prediction, heart
rate, step-count, wrist-worn device placement.

##  I Related Work / Literature Review

Smartwatches are getting higher technology for increasingly affordable prices
and their popular health-tracking applications demand accuracy [1].
Alternatives to overcome wrist placement inaccuracies are required to support
well-informed health promotion for smartwatch consumers [2]. Automatic
dominant hand prediction would reduce negative impacts from an incorrect setup
process or wrist placement changes. The spatiotemporal user context and its
connection with physiological variations provide a potential solution for
dominant hand prediction and accuracy improvements.

Addressing concerns in terms of health impact, three popular commercial wrist-
worn devices were recently reviewed in [3]. Results showed accurate heart rate
measurements, but poor energy expenditure estimations. It is consistent with
results for previous models presented in [4], six years ago. Professionals and
users are warned to proceed with caution using those devices for training or
nutritional purposes. Manufacturers are asked to improve algorithms addressing
potential error sources.

Device placement is one of the factors influencing the accuracy of
accelerometer data during human activity recognition [5]. In particular,
wrist-worn devices are known to be susceptible to dominant or non-dominant
wrist placement with up to 8.5% variations [2]. Some manufacturers seem aware
of this problem and claim to include measures that counteract its effects, see
[6]. However, the corrective measures depend on user input during initial
setup or setting adjustments [7]. An automatic dominant hand prediction would
improve the user experience, with added reliability towards accurate health
information.

Developing an effective automatic solution faces various challenges to detect
a dominant hand. Only low sample rate sensor information is available from
consumer-grade devices. That limits elaborated physiological measures such as
Heart Rate Variability (HRV) requiring up to 250 Hz [8]. The prediction should
also work under real-life unconstrained conditions. However, the dominant hand
is only distinguished in activities including writing, eating, cooking,
dishwashing, or generally using tools demanding strength and precision [9].

Machine learning algorithms are already used for different activity
recognition tasks [10, 11]. So, merging human context and sensor data from
different sources is a strategy with proven potential to deal with the
established challenges. A remarkable example effectively estimates stress
levels using limited physiological data and activity information to improve
its performance [12].

##  II Approach / Method

The objective is to use machine learning concepts to predict the dominant hand
of a user wearing a commercial smartwatch. A two-hand experiment was proposed
to gather physiological data simultaneously from both hands for over 2-weeks.
A survey was designed for the user to keep a self-report of contextual
information including location, activity, and subjective physiological
arousal. The raw data was downloaded using the official export tools from the
manufacturer. Algorithms using python and Matlab were proposed to clean,
analyze and calculate features from the raw data. Finally, using feature
selection algorithms a relevant subset was used to train and select an
effective classification model.

###  II-A Two-hand experiment conditions

Smartwatches are getting higher technology sensors and processing units for
increasingly affordable prices. Sensors commonly include GPS, gyroscopes,
pedometers, temperature, blood oxygen (SpO2), and advanced heart monitoring.
The available data enables gesture recognition, physical activity
identification, and estimation of relevant health metrics. These wearable
devices are now a popular alternative to easily monitor wellness information,
such as energy expenditure, sleep quality, heart rhythm assessment, and stress
levels.

The proposed two-hand experiment takes advantage of those characteristics to
develop the dominant hand prediction algorithm. The subject wore a Fitbit
Sense device on each wrist following its recommended tightness and location
for all-day wear with one finger width above the wrist bone, see Figure 1.
Given the long 2-week time frame, the smartwatches were removed to keep
consistent measurement conditions. This was done for recharging the devices, a
daily cleaning procedure, and also to prevent interaction with water.

After the initial setup, the settings were kept with default values. All the
available features were turned on, and the devices had an active Fitbit
Premium subscription during the experiment. Each wristband was connected to a
different smartphone due to a one-device restriction for the official
application. Both device pairs were kept reasonably close to ensure a
consistent Bluetooth link during the experiment.

Lastly, it is important to highlight that the subject was instructed to keep
routines unchanged. Other than wearing the pair of smartwatches, keep them
charged, and clean. The data should record typical unconstrained real-life
conditions for a graduate student without mobility restrictions.

Figure 1: Smartwatch placement for two-hand experiment

###  II-B Self-report survey for contextual information

A simple and short survey was designed to easily gather detailed information
on the subject’s context. The user was prompted to make only between 1 to 4
selections among different options of interest. Those options included 10
common locations, 16 activities, 5 levels of physiological arousal, and a flag
to indicate when the device was removed. The survey was implemented using
Google Forms as shown in Figure 2.

Figure 2: Self-report survey implementation

The suggested procedure was filling out the survey before any change of
activity or location. A shortcut access was added on the home screen of the
subject’s main smartphone to remember the self-report task. However, the
subject remarked serious difficulties to keep on the self-report during the
experiment. It was difficult to establish the routine, the task was easily
forgotten even with an expressed high level of engagement with the experiment.
Unreliable internet connection and slow smartphone responses further
contributed to the problems.

The survey demanded unrealistic commitment under a highly dynamic real-life
routine. So, the subject completed some information using the survey, but also
manually included additional entries to complete the spreadsheet at the end of
each day. The manual updates considered the inspection of additional sources
of information to increase accuracy on the context labels. That includes using
GPS information from Google Timeline records, heart rate, and step count from
the Fitbit dataset. During the experiment 256 context updates were completed
using the survey, and 241 additional updates were included manually.

It is worth clarifying that the purpose of the survey is just to facilitate
the research process. However, a fully practical implementation would require
the extraction of contextual information from the available sources. For
instance, location can be automatically extracted from GPS information, and
activity can be estimated from the numerous sensors on both smartphone and
smartwatch devices.

###  II-C Data processing

The process starts using the official data export tool from the Fitbit website
to download the most detailed raw data available. Files for heart rate, step
count, calories, and altitude are stored in separate folders for each hand.
The manufacturer encodes the information in multiple JSON-formatted files.

A python code with inspiration on [13] is developed to convert the JSON files
into a single CSV file for each hand. The CSV files are further processed
using Matlab to remove unnecessary information, adjust nonuniform date-stamps,
and re-sampling to a consistent 1-second interval dataset. Missing information
was filled using linear interpolation for heart rate, and zero value for the
remaining variables. Clean and consistent datasets for each hand are stored as
timetables on MAT files.

For the self-reported contextual information. The spreadsheet containing the
data is directly processed using Matlab. A timetable structure is also created
and string data types are converted into categorical. A MAT file is saved with
a re-sampling of a 1-second interval consistent with the physiological data.
Another one is saved with a greater re-sampling under the window interval used
for feature calculation. A new time categorical column is created to divide
time into four more meaningful ranges (Noon, Morning, Afternoon, and Evening).

Time synchronization is then performed using the Matlab timetables to merge
both physiological and contextual data with the uniform 1-second intervals.
The physiological data is further cleaned eliminating time frames where the
Fitbit devices are removed. Finally, the corresponding dominant and non-
dominant hand labels are assigned to the clean, uniform, and organized
datasets ready for feature extraction.

###  II-D Feature Extraction

The objective on dominant hand prediction was clearly defined only after the
experiment design and its data collection. So, some of the collected data is
dropped as irrelevant given the context of this problem. For instance,
altitude from the Fitbit data, and subjective physiological arousal form the
self-report survey. The confidence for each heart rate sample was also omitted
given the similar distribution of its values for both dominant and non-
dominant hand, see Figure 3.

Figure 3: Histogram for heart rate confidence values

The features for the remaining data were calculated using different window
sizes of 1, 5, 10, 20, and 40 minutes. For heart rate 3 frequency-based, and
17 time-based statistical features are calculated. The cumulative sum was the
only feature considered for step count and calories. The 3 categorical
features were not further processed and completed a set with 25 features in
total.

The experiment was extended two additional weeks to test two different
settings available on Fitbit devices. The first one with both devices on
default non-dominant hand setting. The second one with the devices configured
accordingly to the dominant or non-dominant hand setting. The features are
calculated for both conditions and stored in separated files.

###  II-E Context Information and Feature Selection

The Minimum Redundancy Maximum Relevance (MRMR) Algorithm, available on
Matlab, is initially used for feature selection. Following an heuristic
approach, it provides significant scores for classification tasks with both
categorical and continuous features. The results show low relevance for most
of the heart rate and categorical features.

Figure 4: 10-minute window features ranked by MRMR score

For the example shown in Figure 4, features with top-five MRMR scores include
calories, step count, and heart rate values for range, minimum, maximum and
number of peaks. The results vary depending on the window size, but also on
the Fitbit setting being used, something evidenced on the example shown.
However, it is important to remark that most categorical features got low or
zero scores on almost every tested condition.

Notwithstanding the MRMR scores, categorical features contain some valuable
contextual information such as activity, location, and time. From literature,
see [9], it is clear that certain activities are strongly connected with
dominant hand preference. Therefore, a further feature filter accounting to
context and problem knowledge is proposed.

The context-based feature filter eliminates windows potentially providing
indistinguishable data for dominant prediction. That includes activities
involving reduced body movement, such as sleeping, movies, or meetings.
Physiological data suggesting similar conditions, like zero step or zero
calories counts. Also, including means to filter windows with specific
activities or conditions facilitating dominant hand prediction.

###  II-F Classification algorithms

The Matlab Classification Learner toolbox was used to evaluate a broad variety
of models. It includes up to 32 different model configurations such as
decision trees, discriminant analysis, logistic regression, naive Bayes,
support vector machines (SVM), k-nearest neighbor (KNN), kernel
approximations, ensemble alternatives, and neural networks. The models
supporting both categorical and continuous features are strongly limited, but
most of the tests included continuous features only. The main configurations
across all the different evaluations were set to a 5-fold cross-validation
method and 10% proportion for the test dataset. Access to the data and codes
is available on the following link:
https://drive.google.com/drive/folders/1kdaSetBQWjWsV2pbKZ-Y41dZirmyWskS?usp=sharing

##  III Dataset Description

Two sets with a commercial smartphone paired with a Fitbit smartwatch were
used for the project. All the information is gathered for a single individual
without constraints on its daily life behavior. The devices record timestamped
physiological data with a low second sample rate and the following
specifications for each hand:

  * •

Heart Rate (HR) ($\sim$355000 data points with 7 s average sample time)

  * •

Steps ($\sim$19000 data points with 2 min average sample time, only 5000 non-
zero entries)

  * •

Calories ($\sim$43000 data points with 1 min average sample time)

  * •

Altitude ($\sim$140 data points with 4 h average sample time)

The self-report survey records include 497 data points for:

  * •

Wearing/Removing smartwatch flags

  * •

Location (10 common points of interest)

  * •

Activity (16 common options)

  * •

Subjective physiological arousal (5 levels)

##  IV Analysis/Results/Evaluations

The baseline evaluation considered default non-dominant Fitbit setting for
both hands, a window size of 1 min, 4 out of 26 top-scoring features using
MRMR, all the recorded data without context-based feature filter. From the 32
classification models, neural networks, SVM, and Naive Bayes required the
highest training time reaching over the 500 s mark. The top-5 validation
accuracy models are decision trees and ensemble versions reaching up to 87.8%
accuracy.

The results suggest that some feature is providing biased information. From
the unconstrained real life context, it is not expected to easily identify the
dominant hand under most of the activities. From the MRMR scores the calories
count have a value around 100 times higher compared to the next feature in the
list, the peak counts for heart rate. A test omitting the calories count for
the highest ranked classifier further confirms the impact of the feature on
model performance dropping best accuracy to 50.4%. The following evaluations
omit the calories feature, and change other conditions to reveal a more
realistic result.

Figure 5: Confusion Matrix for Quadratic SVM 1 min window, exercising only

Another study evaluates the value of using the proposed context-based filter
to clean the dataset. It is performed only using data from all activities with
potential to identify dominant hand, and with step-counts different from zero.
The results show a consistent accuracy between validation and test data for
the Coarse Decision Tree and the Ensemble Boosted Decision Tree reaching up to
59.5%. The improvement on classification performance, motivated additional
individual evaluations for different relevant activities.

The study for individual evaluations start with the exercising activity. It is
worth mentioning that MRMR feature selection is used and 3 out of 21 top-
scoring features are chosen. The top features usually include heart rate peak
count, step-count, and an statistical heart rate measure like range. For these
conditions, a quadratic SVM results with consistent accuracy around 70% for
both validation (see Figure 5) and test data. For other individual activities
such as doing chores, and walking lower accuracy of 58.6% and under 50% values
are respectively obtained for the best performing models.

The remaining evaluations are performed as modifications from the best
performing conditions. Using the context-based filter to work with only
exercising activity, without zero step-counts, ignoring calories count, and
selecting top-scoring features with MRMR. Evaluating the effect of using a
wider window size. The window was changed from 1 min to 5 min. As a result,
MRMR selection showed a dominant score for the heart rate peak counts.
Training models with just two features resulted in accuracy values near 80%
for quadratic SVM, but scatter plots evidence almost exclusive contribution
from the peaks feature, see Figure 6.

Figure 6: Scatter plot for features used with 5 min window evaluation

##  V Discussion / Limitations

From the results it is clear that a dominant hand prediction can be obtained
with up to 70% accuracy. It is critical to carefully select the available
physiological and contextual information. In particular, the results were
better obtained when the individual is exercising. The MRMR algorithm
recognize 3 relevant features including step-count, heart rate peak count, and
range.

It is important to highlight that only data for one person was used across the
study. The data was collected under no activity restrictions, and detailed
information for the most relevant activities on dominant-hand prediction was
strongly limited. The labels for the evaluated activities were manually
recorded and not detailed. The physiological information has low sample rates
and it is limited to manufacturer logs.

##  VI Conclusion & Future Work

The impact and importance of contextual information is evidenced through the
different analysis. Dominant hand detection seems viable using limited
commercial smartwatch data and it is constrained to reliable activity
detection. There is a long list of further tests and aspects to consider, some
including:

  * •

Further model selection, speed concerns

  * •

Improved data processing and feature calculation

  * •

Sensor fusion Smartphone Step-count

  * •

Significant activity selection

  * •

Automatic activity recognition

  * •

Record and usage of accelerometer data

  * •

Experiments collecting data for more persons

  * •

Additional impact and mitigation of inaccuracies when dominant hand is
detected

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