Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms
Disson S. dos Prazeres Makson S. Santos






Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms
    Disson S. dos Prazeres and Makson S. Santos
    Received 16 July 2024; accepted 04 September 2024
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§ ABSTRACT  
We prove optimal regularity estimates for viscosity solutions to a class of fully nonlinear nonlocal equations with unbounded source terms. More precisely, depending on the integrability of the source term f ∈ L^p(B_1), we establish that solutions belong to classes ranging from C^σ-d/p to C^σ, at critical thresholds. We use approximation techniques and Liouville-type arguments. These results represent a novel contribution, providing the first such estimates in the context of not necessarily concave nonlocal equations.




Keywords: Nonlocal operators, Unbounded source, Hölder regularity.


AMS Subject Classifications: 35B65,	35D40, 35R11.



Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms
    Disson S. dos Prazeres and Makson S. Santos
    Received 16 July 2024; accepted 04 September 2024
=====================================================================================




















§ INTRODUCTION


In this article, we examine the regularity theory of viscosity solutions to a fully nonlinear nonlocal equation of the form 


    ℐ_σ(u, x) = C(σ)sup_α∈𝒜inf_β∈ℬ∫_ℝ^dδ(u, x, y)y^TA_α,β(x)y|y|^σ+d+2dy = f(x)     B_1,

where σ∈ (0, 2), C(σ) > 0 is a normalizing constant, δ(u, x,y):= u(x+y) + u(x-y) -2u(x), 𝒜 and  ℬ are indexes sets, and the source term f belongs to a suitable Lesbegue space. Moreover, we suppose the following ellipticity-type condition: for each α∈𝒜 and β∈ℬ, A_α, β is a symmetric d× d matrix  (λ, Λ)-elliptic, i.e., 

    λ I ≤ A ≤Λ I      B_1,

for 0 < λ≤Λ. We establish Hölder regularity for solutions (or their gradient), depending on the range of p for which f ∈ L^p(B_1). 

Regularity results for nonlocal operators have been extensively studied over the years. In <cit.>, the author presents an analytical proof of Hölder continuity and introduces more flexible assumptions on the operator than previous studies. But it is only in <cit.>, where such regularity estimates were given uniformly as the degree of the operator σ→ 2, seen as a natural extension of second-order operators. In that paper, the authors also prove properties such as comparison principle, a nonlocal Alexandrov-Bakelman-Pucci estimate (ABP for short), Harnack inequality, and regularity in C^1, α spaces for viscosity solutions to equations of type (<ref>).

Numerous results were uncovered based on the findings in <cit.>. In <cit.>, the authors extended their previous results using perturbative methods, including the C^1, α-regularity for a class of non translation-invariant equations. Around the same time, the authors in <cit.> also worked with non translation-invariant equations, establishing Hölder regularity of solutions, using different assumptions from <cit.>, allowing first-order terms and some degeneracy in the operators. We also mention the work <cit.>, where the author gives C^1, α-estimates for more general kernels than previous results.

Besides the C^1,α estimates, Evans-Krylov-type results have been explored by the community. In this direction, the authors in <cit.> proved that viscosity solutions for concave (or convex) equations of the form

    ℐ_σ(u, x) := inf_α∈𝒜∫_ℝ^dδ(u, x, y)K_α(y)dy = 0     B_1,

are of class C^σ+β. They assume that, for each α∈𝒜, the kernel K_α belongs to the class ℒ_2, i.e., they are of class C^2 away from the origin, symmetric, satisfies the ellipticity condition

    (2-σ)λ|y|^d+σ≤ K_α(y) ≤ (2-σ)Λ|y|^d+σ,

and 

    D^2K_α(y) ≤C|y|^d+2+σ.

In <cit.>, the author extended the result above to equations of type (<ref>) with rough kernels, i.e., where the kernels K_α satisfy (<ref>) but not necessarily (<ref>), for every α∈𝒜. Under the additional (and optimal) assumption of C^α exterior data for the solution, the author provides C^σ+α a priori estimates. Moreover, results involving non translation-invariant kernels are also given, provided the dependency on the variable x is of a C^α-fashion. We also mention the work <cit.>, where the author also provides C^σ+α for viscosity solutions to nonlocal equations of the type

    F(D^σ u)= 0    B_1

provided the quantities u_L^∞(B_1) and u_L^1_σ(^d) are sufficiently small.

A natural question is whether one can prove optimal regularity estimates for fully nonlinear nonlocal equations with an unbounded right-hand side. However, such results are relatively scarce in the literature. One of the primary challenges in this context is the lack of compactness properties for viscosity solutions of (1.1) (or even the simpler equation (1.7) below) when f ∈ L^p(B_1). In addition, stability results are also crucial for implementing the now-classic strategy proposed in <cit.>. In fact, only a few works involving an unbounded norm on the right-hand side are available. Moreover, the existing results consider a class of kernels that differ from the broader ℒ_0 class discussed in <cit.>, which defines the extremal operator

    ℳ^-_ℒ_0(u,x) = (2-σ)inf_λ≤ a(x,y) ≤Λ∫_^dδ(u,x,y)a(x,y)|y|^d+σ+2.

Instead, they work with kernels of the form

    K_α(y) = y^TA_α y|y|^d+σ+2dy,

where A_α is a symmetric d× d matrix satisfying 

    λ I ≤1d+σ(σ A_α +  Tr(A_α)I) ≤Λ I      B_1,

for positive constants 0 < λ≤Λ. These kernels define a class of extremal operators as

    ℳ^-(u,x) = inf_α∫_^dδ(u,x,y)K_α(y)dy.

It is important to note that, because they allow some degeneracy (conform (<ref>)), although this is a smaller class, it is not necessarily contained within the ℳ^-_ℒ_0.

In this direction, the authors in <cit.> established a quantitative ABP estimate for viscosity supersolutions to 

    {ℳ^-(u,x)    ≤ f(x)     B_1 
    
        u(x)    ≥ 0    ^d∖ B_1.
    .

They show that supersolutions to the equation above satisfy

    -inf_B_1u ≤ C(d, λ)(f^+_L^∞(K_u))^(2-σ)/2(f^+_L^d(K_u))^σ/2,

where K_u is the coincidence set between u and a type of fractional convex envelope of u. In <cit.>, the author improved the previous result, by removing the dependency of the L^∞-norm in the estimate above, in case σ is sufficiently close to 2.




In <cit.>, the author removed the restriction on the degree of the operator present in <cit.>, by proving that supersolutions to (<ref>) with f ∈ L^p(B_1), for p ∈ (d-ε_0, ∞), satisfy

    -inf_B_1u ≤ C(d, σ, λ, Λ)f^+_L^p(B_1).
 
Moreover, the author also gives W^σ, p-estimates for viscosity solutions to concave equations of the form

    ℐ_σ(u, x) = C(σ)inf_α∈𝒜∫_ℝ^dδ(u, x, y)y^TA_α(x)y|y|^σ+d+2dy = f(x)     B_1,

which, in particular, implies that solutions are of class C^σ-d/p. To our knowledge, this is the only result in the literature that gives this type of regularity for fully nonlinear nonlocal equations as in (<ref>), in the presence of an unbounded right-hand side.

The main purpose of this paper is to establish optimal regularity estimates for viscosity solutions to equation (<ref>), where the source term f ∈ L^p(B_1). We emphasize that such estimates have not been previously available for this type of equation. We employ the so-called half-relaxed method, originally introduced in <cit.>, to prove that a sequence (u_k)_k∈ of merely bounded viscosity solutions to (<ref>) converges uniformly to a function u_∞ which solves a suitable equation, see Lemma <ref> below. This method relies on the comparison principle of the operator and has been used previously in the context of Hamilton-Jacobi equations and nonlocal equations with Neumann boundary conditions in a half-space. We refer the reader to <cit.>. Once compactness and stability are available, standard approximation arguments can be applied to show the regularity properties of the viscosity solutions.

As discussed earlier, the regularity of solutions depends on the range of p for which f ∈ L^p(B_1), in the spirit of <cit.>. First, we prove that if p ∈ (d-ε_0, d/σ-1), then solutions are of class C^σ-d/p. The constant ε_0 is known as the Escauriaza's exponent.  Notice that this type of regularity was previously known from <cit.> in the context of a concave equation as in (<ref>). 

The borderline case f ∈ L^d/(σ-1) is particularly significant. In the local case, i.e., σ = 2, this value separates continuity estimates from differentiability properties in the regularity theory. Moreover, this quantity also appears in ABP and Harnack estimates. Meanwhile, in the nonlocal case, this is the first time such a threshold has been explicitly considered, as previous ABP estimates in <cit.> only consider the L^d-norm of the right-hand side. We believe this is the correct threshold for future general ABP estimates for the general class ℒ_0. Nevertheless, in this scenario, we show that viscosity solutions to (<ref>) are Log-Lipschitz, which is better than C^α for every α∈ (0,1). 

For f ∈ L^p for p ∈ (d/σ-1, +∞), we prove that the solutions belong to the class C^1,α, where α is defined in (<ref>). Finally, for the borderline case where f ∈BMO(B_1), we show that viscosity solutions to (<ref>) are locally of class C^σ, which is the best regularity we can hope for without assuming further regularity for f. These results are detailed in Theorems <ref>, <ref>, <ref> and <ref> below.  





The class of kernels in (<ref>) follows the form of (<ref>), as described in <cit.>, but with the additional requirement of uniform ellipticity, as in (<ref>). In particular, the class of kernels that we deal with is a subset of ℒ_0 and enjoys all its properties. Furthermore, because uniform ellipticity as in (<ref>) ensures that the condition (<ref>) is satisfied with the same ellipticity constants, our class of kernels is also included in those described in <cit.>. Developing a similar theory for more general kernels, such as those in ℒ_0, remains an open challenge, primarily due to the need for an appropriate ABP estimate, as discussed earlier.

The remainder of this paper is structured as follows: In section 2 we gather some auxiliary results and present our main results. The proof of the optimal Hölder regularity is the subject of Section 3. In Section 4 we put forward the Log-Lipschitz regularity of solutions. Section 5 is devoted to the proof of Hölder regularity for the gradient of solutions. Finally, in the last section, we investigate the borderline C^σ-regularity.



§ PRELIMINARIES




 §.§ Notations and definitions


This section collects some definitions and notations used throughout the paper. The open ball of radius r and centered at x_0 in ^d is denoted by B_r(x_0). For α∈ (0, 1], the notation u ∈ C^α^-(B_1) means that u ∈ C^β(B_1), for every β < α. We proceed by defining the Log-Lipschitz space.


A function u belongs to C^ Log-Lip(B_r) if there exists a universal constant C > 0 such that

    sup_B_r/2(x_0)|u(x) - u(x_0)| ≤ rlnr^-1.




We say that f ∈ BMO(B_1) if for all B_r(x_0) ⊂ B_1, we have

    f_ BMO(B_1) := sup_0<r≤1_B_r(x_0)|f(x) - ⟨ f⟩_x_0,r |dx,

where ⟨ f⟩_x_0,r := _B_r(x_0)f(x)dx.


Since u is defined in the whole ^d, it can behave very widely as |x| →∞. Hence, we work within a class where we have a certain decay of the solutions as they approach infinity, see also <cit.>. 


We say that a function u:ℝ^d →ℝ belongs to L^1_σ(^d), if

    u_L^1_σ(^d):= ∫_^d |u(x)|11 +|x|^d+σdx < +∞.



In the next, we define viscosity solutions:


We say that an upper (lower) semicontinuous function u ∈ L_σ^1(^d) is a viscosity subsolution (supersolution) to (<ref>), if for any x_0 ∈ B_1 and φ∈ C^2(B_r(x)) such that u-φ has a local maximum (minimum) at x_0, then the function

    v(x):=
    {φ(x)        B_r(x_0) 
    
        u(x)       ^d∖ B_r(x_0),
    .

satisfies

    -ℐ_σ(v,x) ≤ (≥) f(x)    B_1

We say a function u ∈ C(B_1)∩ L_σ^1(^d) is a viscosity solution to (<ref>) if it is simultaneously a viscosity subsolution and supersolution.



Throughout the manuscript, we assume certain smallness conditions on the norms of u and the source term f. We want to stress that such conditions are not restrictive. In fact, if u is a viscosity solution to (<ref>), then for ε >0 the function 

    v(x) := u(x)u_L^∞(B_1) + ε^-1f_L^p(B_1),

satisfy v_L^∞(B_1)≤ 1 and solves

    ℐ_σ(v, x) = f̃(x)     B_1,

where 

    f̃(x) := f(x)u_L^∞(B_1) + ε^-1f_L^p(B_1),
 
is such that f̃_L^∞(B_1)≤ε.




 §.§ Main results



As mentioned earlier, the regularity of solutions depends on the range of p under consideration. The critical cases occur when f ∈ L^d/σ-1(B_1) for C^α-regularity and f ∈BMO for C^σ-regularity. Due to the nonlocal nature of the problem, achieving these critical cases requires f to possess higher regularity compared to the local case. For instance, in the local case, the first critical threshold is at p = d, leading to C^Log-Lip_loc-regularity. Similarly, when f ∈BMO in the local case, it yields C^1,Log-Lip_loc-regularity.

We now present the main results of this article, beginning with a result in Hölder spaces for the case where p is below d/σ-1.


Let u ∈ C(B_1)∩ L^1_σ(^d) be a viscosity solution to (<ref>), with f ∈ L^p(B_1), p ∈(d-ε_0, d/σ-1). Then, u ∈ C^α_loc(B_1) for any

    α∈(0, σ p -d/p].

Moreover, there exists a positive constant C=C(p,d, σ, λ, Λ), such that

    u_C^α(B_1/2)≤ C(u_L^∞(B_1) + u_L^1_σ(^d) + f_L^p(B_1)).



We observe that for σ = 2, we have p ∈ (d - ε_0, d) and α∈ (0, 2p-d/p), recovering the regularity result for the local case  reported in <cit.>. Next, we consider the borderline case p = d/σ - 1. In this scenario, we show that solutions are Log-Lipschitz continuous, achieving the same level of regularity as in <cit.>, but with the requirement of higher regularity for the source term f.


Let u ∈ C(B_1)∩ L^1_σ(^d) be a viscosity solution to (<ref>), with f ∈ L^p(B_1), p = d/σ-1. Then, u ∈ C^Log-Lip_loc(B_1), and there exists a positive constant C=C(d, σ, λ, Λ), such that

    u_C^Log-Lip(B_1/2)≤ C(u_L^∞(B_1) + u_L^1_σ(^d) + f_L^d/σ-1(B_1)).



In what follows, we present our third main theorem. As before, we recover the local regularity in the limit as σ→ 2, demonstrated in <cit.>. Recall that α_0 comes from the C^1, α_0-regularity of ℐ_σ-harmonic functions. 


Let u ∈ C(B_1)∩ L^1_σ(^d) be a viscosity solution to (<ref>), with f ∈ L^p(B_1), p ∈(d/σ-1, +∞). Then, u ∈ C^1, α_loc(B_1) for any

    α∈(0, min(σ - 1 - d/p, α_0^- )].

Moreover, there exists a positive constant C=C(p, d, σ, λ, Λ), such that

    u_C^1, α(B_1/2)≤ C(u_L^∞(B_1) + u_L^1_σ(^d) + f_L^p(B_1)).



Observe that as p →∞, the corresponding Hölder exponent approaches σ - 1, indicating that we achieve C^σ-regularity in the case p = ∞, which is indeed the case as confirmed in <cit.>. However, we also show that this result holds under the weaker assumption that f ∈BMO(B_1), which is a proper subset of L^∞(B_1). This is the content of our last main result.


Let u ∈ C(B_1)∩ L^1_σ(^d) be a viscosity solution to (<ref>), with f ∈ BMO(B_1). Then, u ∈ C^σ_loc(B_1) and there exists a positive constant C=C(d, σ, λ, Λ), such that

    u_C^σ(B_1/2)≤ C(u_L^∞(B_1) + u_L^1_σ(^d) + f_ BMO(B_1)).



The proof of Theorem <ref> differs from the strategies used for the previous theorems. This is primarily because scaling of the form x ↦ρ^-σu(ρ x), for ρ≪ 1, leads to a growth rate of |x|^-σ at infinity, which increases too rapidly to be integrable with respect to the tails of our kernel. Consequently, we employ techniques similar to those in <cit.>, where Liouville-type results are used to establish interior regularity of solutions through blow-up arguments.

















 §.§ Auxiliary results


In this subsection, we prove some results used throughout the paper. Since we did not find any references stating exactly what we needed, we start with a comparison principle for viscosity solutions of (<ref>) (and also (<ref>)), in the case where f ≡ 0. See also <cit.>. 


Let u,v ∈ L^1_σ(^d), u upper semicontinuous and v lower semicontinuous, be respectively a viscosity subsolution and supersolution to the equation

    ℐ_σ(w, x) = 0     B_1,

such that u ≤ v in ℝ^N∖ B_1. Then, u ≤ v in B_1.
 

Suppose by contradiction that 

    θ=sup_B_1(u-v)>0.

For ε>0 we define the auxiliary function

    Φ(x,y)=u(x)-v(y)-|x-y|^2/2ε^2,

and consider (x_ε,y_ε)∈B_1×B_1 such that

    Φ(x_ε,y_ε)=sup_x,y∈ B_1Φ(x,y).

Notice that Φ(x_ε,y_ε)≥sup_x∈ B_1Φ(x,x)=θ, which yields to

    |x_ε-y_ε|/2ε^2≤ u(x_ε)-v(y_ε)-θ.

By compactness, we have that (x_ε,y_ε) → (x̅,y̅)∈B_1×B_1 as ε→ 0, and by using (<ref>) we obtain x̅=y̅. Therefore

    0 ≤lim_ε→ 0|x_ε-y_ε|^2/2ε^2 = u(x̅) - v(x̅) - θ≤ 0,

which implies

    u(x̅) - v(x̅) = θ > 0.

Moreover, since u≤ v on ^d∖ B_1, we have x̅∈ B_1. 
We set φ_1(x):= v(y_ε) + |x-y_ε|/2ε^2 and φ_2(y):= u(x_ε) + |y-x_ε|/2ε^2, and observe that u-φ_1 has a local maximum at x_ε, while v-φ_2 has a local minimum at y_ε. Hence by using that u is a subsolution and v is a supersolution of (<ref>), we have the viscosity inequalities

    sup_α∈𝒜inf_β∈ℬ(∫_B_rδ(φ_1, x_ε, y)y^TA_α,β(x_ε)y|y|^σ+d+2dy  + ∫_^d∖ B_rδ(u, x_ε, y)y^TA_α,β(x_ε)y|y|^σ+d+2dy ) ≥ 0,

and 

    sup_α∈𝒜inf_β∈ℬ(∫_B_rδ(φ_2, y_ε, y)y^TA_α,β(y_ε)y|y|^σ+d+2dy  + ∫_^d∖ B_rδ(v, y_ε, y)y^TA_α,β(y_ε)y|y|^σ+d+2dy ) ≤ 0,

for r small enough. Hence, from the definition of sup and inf, there exist α∈𝒜 and β∈ℬ such that

    ε^-2 o_r(1) + ∫_^d∖ B_rδ(u, x_ε, y)y^TA_α,β(x_ε)y|y|^σ+d+2dy ≥ -γ/2,

and

    ε^-2 o_r(1) + ∫_^d∖ B_rδ(v, y_ε, y)y^TA_α,β(y_ε)y|y|^σ+d+2dy ≤γ/2,







for γ> 0 sufficiently small. Subtracting the inequalities above yields to

    ε^-2 o_r(1) + ∫_^d∖ B_rδ(u, x_ε, y)y^TA_α,β(x_ε)y|y|^σ+d+2dy - ∫_^d∖ B_rδ(v, y_ε, y)y^TA_α,β(y_ε)y|y|^σ+d+2dy ≥ -γ.

Notice that, When ε→ 0, by the contradiction hypotheses we have

    ∫_B_1∖ B_rδ(u, x̅, y)y^TA_α,β(x̅)y|y|^σ+d+2dy-∫_B_1∖ B_rδ(v, x̅, y)y^TA_α,β(x̅)y|y|^σ+d+2dy≤ 0.

Moreover,

    ∫_^d∖ B_1δ(u, x̅, y)y^TA_α,β(x̅)y|y|^σ+d+2dy-∫_^d∖ B_1δ(v, x̅, y)y^TA_α,β(x̅)y|y|^σ+d+2dy≤ - λθ∫_ℝ^d∖ B_1  |z|^-(d + σ)dz.

Therefore

    -λθ∫_ℝ^d∖ B_1  |z|^-(d + σ)dz ≥ -γ,

which is a contradiction for γ small enough. This finishes the proof.












    



































































































We now focus on one of the main contributions of this paper: the stability of solutions to (<ref>) when f ∈ L^p(B_1). To the best of our knowledge, this is the first time such a result has been established in the fully nonlinear nonlocal context. For comparison, see <cit.>.


Let u_k be a normalized viscosity solution to

    ℐ_σ(u_k, x) = f_k     B_1.
 
Suppose that there exists a positive constant M such that 

    |u_k(x)| ≤ M(1+|x|)^1+α    x ∈^d.

Suppose further that

    f_k_L^p(B_1)→ 0,

for p∈ (d-ε_0,+∞). Then there exists u_∞∈ C(B_1)∩ L_σ^1(^d) such that 

    u_k-u_∞_L^∞(B_4/5)→ 0.

Moreover, u_∞ solves

    ℐ_σ(u_∞, x) = 0     B_1.




First, observe that given R > 0, we obtain from (<ref>) 

    |u_k(x)| ≤ C(R)     x ∈ B_R.

Hence, Given any compact set Ω⊂^d, we have that the a.e. limits

    u̅(x) := lim sup_k→∞, y_k→ x u_k(y_k),    x  Ω,

and

    u(x) := lim inf_k→∞, y_k→ x u_k(y_k),    x  Ω,

are well-defined. Since the a.e. convergence holds for every compact set of ^d, we also have the a.e. convergence in the whole ^d. Using this fact and once again (<ref>), the Dominated Convergence Theorem ensures that

    u̅-u_k_L^1_σ(^d)→ 0,

and 

    u-u_k_L^1_σ(^d)→ 0,

through the respective subsequences.











We are going to show that u̅ is a viscosity subsolution to (<ref>), and 
u is a viscosity supersolution to (<ref>). We will prove the subsolution case since the supersolution case is analogous. Let x_0 ∈ B_1 and φ∈ C^2(B_r(x_0)) be such that u̅ - φ has a maximum at x_0. Without loss of generality, we can assume that φ is defined by

    φ = {
    P,         B_r(x_0),
    u̅,        B_r(x_0)^c,
    .

for some paraboloid P. We need to show that 

    ℐ_σ(φ, x) ≥ 0     B_1.

Suppose by contradiction that 

    ℐ_σ(φ, x) < -η,

for some η > 0. Now, let ψ_k be a viscosity solution to

    {ℳ^+_λ^*, 1(ψ_k, x)      =     -|f_k(x)|        B_r(x_0) 
    ψ_k     =     0       ∂ B_r(x_0),    
    .

for some λ^* < 1 to be chosen later. Here, the maximal operator ℳ^+ is defined with respect to the class ℒ_0, as in <cit.> (and defined below). We have 

    ℐ_σ(φ + ψ_k,x) - ℐ_σ(φ, x)    ≤ℳ^+_λ, Λ(ψ_k, x) 
        = Λ∫_ℝ^nδ^+(ψ_k, x, y)|y|^n+σdy - λ∫_ℝ^nδ^-(ψ_k, x, y)|y|^n+σdy 
        = Λ∫_ℝ^nδ^+(ψ_k, x, y)|y|^n+σdy - λλ^*(∫_ℝ^nδ^+(ψ_k, x, y)|y|^n+σdy + |f_k(x)|) 
        = (Λ - λλ^*)∫_ℝ^nδ^+(ψ_k, x, y)|y|^n+σdy - λλ^*|f_k(x)|,

where we have used (<ref>) to conclude

    ∫_ℝ^nδ^+(ψ_k, x, y)|y|^n+σdy - λ^*∫_ℝ^nδ^-(ψ_k, x, y)|y|^n+σdy = -|f_k(x)|.

Now, by choosing λ^* such that Λ - λλ^*≤ 0 and -λλ^*≤ -1, we obtain

    ℐ_σ(φ + ψ_k,x) ≤ℐ_σ(φ, x) + f_k(x).

Let P_k be defined by

    φ_k={
    			P,         B_r(x_0),
    
    			u_k,        B_r(x_0)^c,
    		.

and 

    P={
    			c|x-x_0|^2,         B_r(x_0),
    
    			0,        B_r(x_0)^c.
    		.

By using the ABP estimates in <cit.> we have ψ_k_∞→ 0, as k→∞. Then, there exists a x_k∈ B_r(x_0) such that φ_k+ψ_k+P touch u_k by above in B_r(x_0). Therefore, we have the viscosity inequality

    ℐ_σ(φ_k+ψ_k+P)≥ f(x_k).

By ellipticity and using (<ref>) and (<ref>) we obtain that

    ℐ_σ(φ_k+ψ_k+P) ≤ℐ_σ(φ_k+ψ_k)+ℳ^+_σ(P)
     ≤ℐ_σ(φ_k+ψ_k)-ℐ_σ(φ+ψ_k)-η+ f(x_k)+ℳ^+_σ(P).







We observe that

    |∫_^Nδ(φ_k+ψ_k,x,y)dy  -∫_^Nδ(φ+ψ_k,x,y)dy|    ≤∫_^N∖ B_r(x_0)|δ(u_k,x,y)-δ(u̅,x,y)|dy 
       ≤ C(r)u_k-u̅_L^1_σ(^d),

and by using (<ref>), we can conclude that for k sufficiently large

    |ℐ_σ(φ_k+ψ_k)-ℐ_σ(φ+ψ_k)| ≤η4.

Hence, from (<ref>) and (<ref>), we obtain

    ℐ_σ(φ_k+ψ_k+P)≤η4  -η +f(x_k)+ℳ^+_σ(P).

Now, choose c(Λ,N,σ) sufficiently small so that

    ℳ^+_σ(P)≤η/4.

Finally, for k sufficiently large we get

    ℐ_σ(φ_k+ψ_k+P,x_k)≤ -η/2 +f(x_k),

which is a contradiction with (<ref>). This finishes the proof of (<ref>), i.e., u̅ solves in the viscosity sense

    ℐ_σ(u̅, x) ≥ 0     B_1,

We similarly show that

    ℐ_σ(u, x) ≤ 0     B_1.

Now, from the definition of u̅, u and the viscosity inequalities above we can infer from Proposition <ref>, that in fact 

    u̅ = u = u_∞,

and hence up to a subsequence, u_k → u_∞ locally uniformly in B_1 (see for instance <cit.>). Moreover, from the viscosity inequalities satisfied by u̅ and u, we conclude that u_∞ solves

    ℐ_σ(u_∞, x) = 0     B_1,

in the viscosity sense.


Using the stability result above, we can prove the following Approximation Lemma, which relates the solutions to our problem with ℐ_σ-harmonic functions. 


Let u ∈ C(B_1)∩ L^1_σ(^d) be a normalized viscosity solution to (<ref>), with p ∈ (d-ε_0, +∞). Suppose that 

    |u(x)|≤ M(1+|x|)^1+α     x∈^d.

Given δ >0 there exist ε >0, such that if

    f_L^p(B_1)≤ε,

we can find a function h∈ C^1,α_0(B_4/5) satisfying 

    sup_B_3/4|u-h|≤δ.




Suppose not, then there exist δ_0 > 0 and sequences (u_k)_k∈ℕ, (f_k)_k∈ℕ such that 

    ℐ_σ(u_k, x) = f_k     B_1,
  

    f_k_L^p(B_1)≤1k,

and 

    |u_k(x)|≤ M(1+|x|)^1+α     x∈^d,

but,

    |u_k-h|>δ_0,

for all h∈ C^1,α_0(B_4/5). From the contradiction hypotheses (<ref>), (<ref>), (<ref>) and Proposition <ref>, we can guarantee the existence of a function u_∞∈ C(B_1)∩ L^1_σ(^d) such that u_k → u_∞ locally uniformly in B_1 satisfying




    ℐ_σ(u_∞, x) = 0     B_1.
  
Now, the regularity available for (<ref>), see <cit.>, implies that u_∞∈ C^1,α_0(B_4/5). By taking h≡ u_∞, we reach a contradiction with (<ref>) for k sufficiently large.




§ HÖLDER REGULARITY


In this section, we detail the proof of Theorem <ref>, namely, the optimal C^α_loc-regularity, for

    α∈(0, σ p -d/p],

where p ∈(d-ε_0, d/σ-1). We start by applying Lemma <ref> and showing the existence of a constant close to u in sufficiently small balls. 


Let u ∈ C(B_1)∩ L^1_σ(^d) be a normalized a viscosity solution to (<ref>), with p ∈(d-ε_0, d/σ-1).
Assume that 

    |u(x)|≤ M(1+|x|)^1+α     x∈^d.

If

    f_L^p(B_1)≤ε,

then there exist constants 0 < ρ≪ 1/2 and A satisfying, |A| ≤ C and  

    sup_B_ρ|u-A| ≤ρ^α,

where C> 0 is a universal constant.



Fix δ > 0 (to be chosen later) and let h be the function from Lemma <ref>. Since h ∈ C^1, α_0(B_4/5), for ρ sufficiently small, we have

    |h - h(0)| ≤ Cρ.

Now, from Lemma <ref> and the Triangular inequality we obtain

    sup_B_ρ|u - h(0)|    ≤sup_B_ρ|u - h| + sup_B_ρ|h - h(0)|  
       ≤δ + Cρ.

Now, we make the universal choices 

    ρ = min[(1/2C)^1/1-α, (1/(1+C)100)^1/α_0-α]      δ = ρ^α/2,

and by setting A = h(0), we conclude that

    sup_B_ρ|u - A| ≤ρ^α.

Notice that the choice of δ determines the value of ε via Lemma <ref>.


In what follows, we iterate the previous proposition to find a sequence of constants that approaches u at the origin. 


Let u ∈ C(B_1)∩ L^1_σ(^d) be a normalized a viscosity solution to (<ref>), with p ∈(d-ε_0, d/σ-1). Assume that

    |u(x)|≤ M(1+|x|)^1+α     x∈^d.

If

    f_L^p(B_1)≤ε,

then we can find a sequence (A_k)_k ∈ℕ satisfying

    sup_B_ρ^k|u(x)-A_k| ≤ρ^kα,

with

    |A_k+1 - A_k| ≤ Cρ^kα.
    



We argue by an induction argument. By setting A_0=A_1 = 0, the case k=0 follows immediately. Suppose we have verified the statement for k = 1, …, n, and let us prove the case k=n+1. We introduce the auxiliary function v_k : ^d ⟶ 

    v_k(x) := u(ρ^kx) - A_k/ρ^kα.

Notice that by (<ref>) we have |v_k(x)| ≤ 1 in B_1. In addition, v_k solves

    ℐ_σ(v,x) = f̃(x)    B_1,

where f̃(x) = ρ^k(σ-α)f(ρ^kx). Moreover, our choice of α in (<ref>) assures f̃_L^p(B_1)≤ε. Next, we are going to show that v_k satisfies

    |v(x)| ≤ M(1+|x|^1+α_0)      x ∈^d,

for some universal constant M. In fact, we resort again to an induction argument. For k=0, we have v_0=u and (<ref>) is verified. Now, assume that the case k=1, …, n is already verified. We shall prove the case k=n+1. Observe that

    v_n+1 = v_n(ρ x) - Ã_nρ^α,

where Ã_n comes from Lemma (<ref>) applied to v_n. Now, for 2|x|ρ > 1 we estimate

    |v_n+1(x)|    ≤ρ^-α(|v_n(ρ x)| +|Ã_n|)
       ≤ρ^-(1+α)[(1+ ρ^1+α_0|x|^1+α_0) + C(1 + ρ|x|)]
       ≤ρ^(α_0 - α)(5 +9C)|x|^1 + α_0
       ≤  |x|^1+ α_0,

where in the last inequality we used (<ref>). On the other hand, if 2|x|ρ≤ 1, we obtain

    |v_n+1(x)|    ≤ρ^-α(|v_n(ρ x) - h̃(ρ x)| + |h̃(ρ x) -Ã_n| 
       ≤ρ^-α(ρ^α2+ Cρ|x|) 
       ≤1/2 + C/2ρ^α
       ≤ M(1+|x|^1+α_0),

where M:= 1/2 + C/(2ρ^α), and hence (<ref>) is proved. Finally, we now can apply Proposition <ref> to v_k and we obtain

    sup_B_ρ|v_k - Ã_k| ≤ρ^α,

and rescaling back to u we conclude

    sup_B_ρ^k+1|u - A_k+1| ≤ρ^(k+1)α,

where A_k+1 = A_k + ρ^kαÃ_k, which satisfies (<ref>). This finishes the proof.


We are now ready to prove Theorem <ref>.


Notice that from (<ref>) we have that (A_k)_k∈ is a Cauchy sequence, and hence there exists A_∞ such that A_k → A_∞, as k→∞. Moreover, we also have from (<ref>)

    |A_k-A_∞| ≤ Cρ^kα.

Now, fix 0< r ≪ 1 and let k ∈ be such that ρ^k+1≤ r ≤ρ^k. We estimate,

    sup_B_r|u(x)-A_∞|    ≤sup_B_ρ^k|u(x)-A_k| + sup_B_ρ^k|A_k - A_∞| 
       ≤ρ^kα + Cρ^kα
       ≤(C+1)/ρ^αρ^(k+1)α
       ≤ Cr^α.

By taking the limit as k →∞ in (<ref>), we obtain A_∞ = u(0). This finishes the proof.




§ LOG-LIPSCHITZ CONTINUITY


This section addresses the first critical case p = d/σ - 1, which yields the desired Log-Lipschitz regularity. In particular, solutions are of class C_loc^α for every α∈ (0, 1). As before, we begin by demonstrating the existence of a linear approximation of u within sufficiently small balls.


Let u ∈ C(B_1)∩ L^1_σ(^d) be a normalized viscosity solution to (<ref>), with p =d/σ-1. Suppose further that

    |u(x)| ≤ M(1+|x|)^1+α_0     x∈^d.

If

    f_L^p(B_1)≤ε,

then, there exist a constant 0 < ρ≪ 1/2 and an affine function ℓ of the form

    ℓ(x) = A + B· x,

satisfying |A|, |B| ≤ C and 

    sup_B_ρ|u(x)-ℓ(x)| ≤ρ.
    



The proof is similar to Proposition <ref>. We fix δ >0 to be determined later. For ρ≪ 1/2, we have that

    |h(x)-h(0)-Dh(0)· x| ≤ Cρ^1+α_0,

where h ∈ C^1,α_0(B_3/4) comes from Lemma <ref>. By setting ℓ(x) = h(0) + Dh(0)· x, we obtain from the Triangular inequality that

    sup_B_ρ|u(x)-ℓ(x)|    ≤sup_B_ρ|u(x)-h(x)| + sup_B_ρ|h(x)-ℓ(x)| 
       ≤δ + Cρ^1+α_0.

As before, we make universal choices

    ρ = (1/2C)^1/α_0    δ = ρ/2,

which determines the value of ε through Lemma <ref>. Therefore

    sup_B_ρ|u(x)-ℓ(x)| ≤ρ.




Let u ∈ C(B_1)∩ L^1_σ(^d) be a normalized viscosity solution to (<ref>), with p=d/σ-1. Suppose further that

    |u(x)| ≤ M(1+|x|)^1+α_0     x∈^d.

If

    f_L^p(B_1)≤ε,

then, there exists a sequence of affine function (ℓ_k)_k ∈ of the form

    ℓ_k(x) = A_k + B_k· x,

satisfying

    |A_k+1-A_k|/ρ^k + |B_k+1-B_k| ≤ C,

and

    sup_B_ρ|u(x)-ℓ(x)| ≤ρ^k.
    


As before, we resort to an induction argument. By considering ℓ_0=ℓ_1=0, the case k=0 follows trivially. Now, suppose that the cases k=1, …, n have been verified, and let us prove the case k=n+1. We define the auxiliary function v_k: ^d ⟶ by

    v_k(x) := u(ρ^kx)-ℓ(ρ^kx)/ρ^k.

We have that v_k(x) ≤ 1 in B_1, and solves

    ℐ_σ(x,v) = f̃(x)      B_1,

where f̃(x) = ρ^k(σ-1)f(ρ^kx). Notice that as σ-1>0, we have f̃_L^p(B_1)≤ε. Arguing similarly as in Proposition <ref>, we can also show that 

    |v_k(x)| ≤ 1 + |x|^1+α_0.

Hence, we can apply Proposition <ref> to v_k to conclude that there exists ℓ̃_k=Ã+B̃x such that

    sup_B_ρ|v_k(x)-ℓ̃_k(x)| ≤ρ.

Rescaling back to u, we obtain

    sup_B_ρ|u(x)-ℓ_k+1(x)| ≤ρ^k,

where ℓ_k+1 = ℓ_k(x) + ρ^kℓ̃_k(ρ^-1x). Observe that 


    |A_k+1-A_k|=|Ãρ^k|≤ Cρ^k,

and

    |B_k+1-B_k|=|B̃|≤ C

which prove the condition (<ref>).


We now present the proof of Theorem <ref>


Notice that, by (<ref>), (A_k)_k∈ is a Cauchy sequence, and hence, there exists A_∞ such that A_k → A_∞, as k→∞. Moreover, we have

    |A_k-A_∞| ≤ Cρ^kα.

Now, fix 0< r ≪ 1 and let k ∈ be such that ρ^k+1≤ r ≤ρ^k. We have,

    sup_B_r|u(x)-A_∞|    ≤sup_B_ρ^k|u(x)-A_k-B_kx| + sup_B_ρ^k|B_kx| 
       ≤ρ^k + Ckρ^k 
       ≤1/ρ(ρ^k+1+Ckρ^k+1) 
       ≤ Cr+lnrlnρCr
       ≤ -Crlnr.

Finally, by taking the limit as k →∞ in (<ref>), we obtain A_∞ = u(0). This finishes the proof.




§ HÖDER CONTINUITY OF THE GRADIENT


In this section, we give the proof of Theorem <ref>, in which we prove C_loc^1, α-regularity for

    α∈(0, min(σ-1-d/p , α_0^-)].

The proof follows the general lines of the proof of Theorem <ref>, but now at the gradient level.


Let u ∈ C(B_1)∩ L^1_σ(^d) be a normalized viscosity solution to (<ref>), with p ∈(d/σ-1, +∞). Suppose further that

    |u(x)| ≤ M(1+|x|)^1+α_0     x∈^d.

If

    f_L^p(B_1)≤ε,

then, there exist a constant 0 < ρ≪ 1/2 and an affine function ℓ of the form

    ℓ(x) = A + B· x,

satisfying |A|, |B| ≤ C and 

    sup_B_ρ|u(x)-ℓ(x)| ≤ρ^1+α.
    



We fix δ >0 to be determined later. For ρ≪ 1/2, we have that

    |h(x)-h(0)-Dh(0)· x| ≤ Cρ^1+α_0,

where h ∈ C^1,α_0(B_3/4) comes from Lemma <ref>. By setting ℓ(x) = h(0) + Dh(0)· x, we obtain from the Triangular inequality that

    sup_B_ρ|u(x)-ℓ(x)|    ≤sup_B_ρ|u(x)-h(x)| + sup_B_ρ|h(x)-ℓ(x)| 
       ≤δ + Cρ^1+α_0.

As before, we make universal choices

    ρ = min[(1/2C)^1/α_0-α, (1/(1+C)100)^1/α_0-α]     δ = ρ^1+α/2,

which determines the value of ε through Lemma <ref>. Therefore

    sup_B_ρ|u(x)-ℓ(x)| ≤ρ^1+α.




Let u ∈ C(B_1)∩ L^1_σ(^d) be a normalized viscosity solution to (<ref>), with p ∈(d/σ-1, +∞). Suppose further that

    |u(x)| ≤ M(1+|x|)^1+α_0     x∈^d.

If

    f_L^p(B_1)≤ε,

then, there exists a sequence of affine function (ℓ_k)_k ∈ of the form

    ℓ_k(x) = A_k + B_k· x,

satisfying

    |A_k+1-A_k| + ρ^k|B_k+1-B_k| ≤ Cρ^k(1+α),

and

    sup_B_ρ^k|u(x)-ℓ_k(x)| ≤ρ^k(1+α).
    



As before, we resort to an induction argument. By considering ℓ_0=ℓ_1=0, the case k=0 follows trivially. Now, suppose that the cases k=1, …, n have been verified, and let us prove the case k=n+1. We define the auxiliary function v_k: ^d ⟶ by

    v_k(x) := u(ρ^kx)-ℓ(ρ^kx)/ρ^k(1+α).

We have that v_k(x) ≤ 1 in B_1, and solves

    ℐ_σ(x,v) = f̃(x)      B_1,

where f̃(x) = ρ^k(σ-α-1)f(ρ^kx). Notice that for our choice of α in (<ref>), we have f̃_L^p(B_1)≤ε. Arguing similarly as in Proposition <ref>, we can also show that 

    |v_k(x)| ≤ 1 + |x|^1+α_0.

Hence, we can apply Proposition <ref> to v_k to conclude that there exists ℓ̃_k such that

    sup_B_ρ|v_k(x)-ℓ̃_k(x)| ≤ρ^1+α.

Rescaling back to u, we obtain

    sup_B_ρ|u(x)-ℓ_k+1(x)| ≤ρ^1+α,

where ℓ_k+1 = ℓ_k(x) + ρ^k(1+α)ℓ̃_k(ρ^-1x). From the definition of ℓ_k+1, it is immediate that condition (<ref>) is also satisfied. This finishes the proof.



The proof follows the same lines as in Theorem 1.1 and Theorem 1.2. Notice that from (<ref>) we have that (A_k)_k∈ and (B_k)_k∈ are a Cauchy sequence, and hence, we can find A_∞ and B_∞ satisfying A_k → A_∞ and B_k → B_∞, as k→∞. Moreover, we have

    |A_k-A_∞| ≤ Cρ^k(1+α)       |B_k-B_∞| ≤ Cρ^kα.

Now, fix 0< r ≪ 1 and let k ∈ be such that ρ^k+1≤ r ≤ρ^k. We have,

    sup_B_r|u(x)-A_∞-B_∞· x|    ≤sup_B_ρ^k|u(x)-A_k-B_k· x| + sup_B_ρ^k|A_k - A_∞| + ρ^ksup_B_ρ^k|B_k - B_∞| 
       ≤ρ^k(1+α) + Cρ^k(1+α) + Cρ^k(1+α)
       ≤C/ρ^1+αρ^(k+1)(1+α)
       ≤ Cr^1+α.

Finally, by taking the limit as k →∞ in (<ref>), we obtain A_∞ = u(0). We can also show that B_∞ = Du(0), see for instance <cit.>. This finishes the proof.




§ THE BORDERLINE CASE


This section deals with the C^σ-regularity for viscosity solutions of (<ref>). As previously discussed, since 1+α = σ, we can no longer follow the strategy employed above. In this case, we follow the ideas put forward in <cit.> (see also<cit.>). We begin with a technical lemma, that can be found in <cit.>. 


Let 0<α<α<1 and u∈ C^1,α(B_1). If u(0)=|Du(0)|=0 and

    sup_0<r<1/2r^α-α[u]_C^1,α(B_r)≤ A,

for a constant A>0, then

    [u]_C^1+α(B_1/2)≤ 2A.




Let u ∈ C^1+α(B_1)∩ L^1_σ(^d), with α< σ-1, be a normalized viscosity solution to (<ref>) with f∈ BMO(B_1). Suppose further that

    u_C^1+α(B_1)≤ M     f_ BMO(B_1)≤ε,

then u∈ C^σ(B_1/2) and

    u_C^σ(B_1/2)≤ C.




We argue by contradiction. Suppose that the result is false, then we can find sequences (u_k)_k∈ℕ, (f_k)_k∈ℕ, such that for every k∈ℕ, 

    [u_k]_C^1+α(B_1)≤ M 
    f_k_ BMO(B_1) < ε_k,

but

    u_k_C^σ(B_1/2) > k,

where ε_k → 0, as k →∞. For k ∈, we define the quantity 

    θ_k(r')=sup_r'<r<1/2sup_z∈ B_1/2r^1+α - σ[u]_C^1+α(B_r(z))

and observe that

    lim_r'→ 0θ_k(r')=sup_r'>0θ_k(r').

Moreover, if r_1 ≤ r_2, then θ_k(r_2) ≤θ_k(r_1). By the Lemma <ref> we have that

    sup_r'>0θ_k(r')≥ k/2,

therefore there exists a r_k>1/k and z_k∈ B_1/2 such that

    r_k^1+α - σ[u_k]_C^1+α(B_r_k(z_k))>θ_k(1/k)>θ_k(r_k)→∞.

Since from (<ref>) [u_k]_C^1+α(B_1)<M, we have r_k→ 0 as k →∞. Now, for R ∈[1, 1/2r_k], we define the blow-up v_k:B_R→ℝ

    v_k(x)=1/θ_k(r_k)1/r_k^σu_k(r_kx+z_k).

Notice that

    [v_k]_ C^1+α(B_R)    =r_k^1+α-σ/θ_k(r_k)[u_k]_C^1+α(B_r_kR(z_k))
         = (Rr_k)^1+α-σ[u_k]_B_r_kR(z_k)/θ_k(r_k)R^σ-1-α
       ≤ R^σ-1-α.

Consider the auxiliary function w_k: ^d → defined as

    w_k(x) := (v_k-l_k)(x),

where  

    l_k=v_k(0)+Dv_k(0)x.

Since that [l_k]_ C^1+α(B_R)=0, it follows from (<ref>) that

    [w_k]_ C^1+α(B_R)≤ R^σ-1-α.

In particular, for R=1 we have

    [Dw_k]_ C^α(B_1)≤ 1,

therefore

    |Dw_k(x)|=|Dw_k(x)-Dw_k(0)|≤ |x|^α,      x∈ B_1,

which implies

    |w_k(x)|=|w_k(x)-w_k(0)|≤Dw_k_L^∞(B_|x|)|x|≤ |x|^1+α,

for all x ∈ B_1. Let η be a smooth function such that η=1 in B_1/2 and η=0 outside B_1. For e∈𝕊^n we have

    ∫_B_1η· D_e w_k dx=∫_B_1D_e η· w_k dx≤ C(n).

Hence, there exists z∈ B_1 such that |Dw_k(z)|≤ C(n) and by (<ref>) we have

    |D_ew_k(x)-D_ew_k(z)|≤ R^σ-1-α|x-z|^α.

Therefore, for x∈ B_R and 1≤ R ≤1/2r_k,

    |D_ew_k(x)|≤ C(n)+ R^σ-1-α|x-z|^α≤ CR^σ-1.

Our goal now is to show that

    [w_k]_C^β(B_R)≤ R^σ-β,

for all β∈ [0,1+α] and R ∈[1, 1/2r_k].

Case β =0: Notice that for 1≤ |x|≤ R, we have from <ref>

    |w_k(x)|     = |w_k(x)-w_k(0)| 
       ≤D_ew_k(x)_L^∞(B_|x|)|x| 
       ≤ CR^σ.

The estimate for x ∈ B_1 follows from (<ref>).

Case β∈ (0, 1): In this case, we estimate for x,x̅∈ B_R

    |w_k(x)-w_k(x̅)|    ≤Dw_k_L^∞(B_R)|x-x̅| 
       ≤ R^σ-1|x-x̅|^1-β|x-x̅|^β
       ≤ CR^σ-β|x-x̅|^β,

which gives

    [w_k]_ C^β(B_R)≤ CR^σ-β.


Case β∈ [1, 1+α]: Finally, we have

    |Dw_k(x)-Dw_k(x̅)|    ≤ [Dw_k]_C^α(R)|x-x̅|^α
       ≤ CR^σ-1-αR^α-β+1|x-x̅|^β-1,

which implies

    [Dw_k]_ C^β-1(B_R)≤ CR^σ-β,

or equivalently

    [w_k]_ C^β(B_R)≤ CR^σ-β.

Therefore, (<ref>) follows from (<ref>), (<ref>) and (<ref>). Thus, there exists a w∈ C^1+α(ℝ^d) such that w_k→ w locally uniformly in the C^1+α-norm. Now, from (<ref>), we have that

    [w_k]_C^1+α(B_1)≥1/2

and therefore

    [w]_C^1+α(B_1)≥1/2.

Moreover, from (<ref>),

    [w]_C^β(^d)≤ R^σ-β,

for all β∈ [0,1+α] and R > 1. Notice that w_k solves 

    ℐ_σ(w_k, x) = f̃_k(x),

where f̃_k(x) = 1θ_k(r_k)f_k(r_kx+z_k),
for p≥ d, we have

    f̃_k(x)_L^p(B_1   =     ( ∫_B_1 |f̃_k(x)|^p dx  )^1/p
       =     |B_1|^1/p ( B_r_k(z_k) |f_k(x)|^p dx  )^1/p
       ≤    |B_1|^1/pf_k_ BMO(B_1)≤ |B_1|^1/pε_k.

Hence,

    ℳ^+_ℒ_0(w_k(· + h) - w_k)    ≥ℐ_σ(w_k, x+h) - ℐ_σ(w_k, x) 
        = 1θ_k(r_k)[ f_k(r_k(x+h) + z_k) - f_k(r_kx + z_k) ] .

Moreover,

    |w_k(x + h) - w_k|≤ [w_k]_ C^α(B_R)|h|^α≤ C|x|^σ-α

 and as [f_k]_ BMO(B_1)→ 0 (and therefore f_k_L^p(B_1)→ 0), we have from Proposition <ref> that

    ℳ^+_ℒ_0(w(· + h) - w) ≥ 0      ^d.

Similarly, we can prove that

    ℳ^-_ℒ_0(w(· + h) - w) ≤ 0     ^d.

Therefore, 

    ℳ^-_ℒ_0(w(· + h) - w) ≤ 0 ≤ℳ^+_ℒ_0(w(· + h) - w)      ^d.

Finally, for

    w̃_k= w_k(· + h)dμ(h) - w_k,

we have that

    |w̃_k|≤ |w_k(· + h) - w_k|dμ(h)≤ C|x|^σ-1-α.

The concavity of ℐ_σ yields 

    ℳ^+_ℒ_0( w̃_k(· + h)dμ(h) - w_k , x)    ≥ℐ_σ(w̃_k(· + h)dμ(h), x) - ℐ_σ(w_k, x) 
       ≥ℐ_σ(w_k, x+h)dμ(h) - f̃_k(x) 
        = f̃_k(x+h) - f̃_k(x) dμ(h).

Hence, by passing the limit as k →∞ we get

    ℳ^+_ℒ_0(  w(· + h)dμ(h) - w , x ) ≥ 0.

Therefore, from (<ref>), (<ref>) and (<ref>), we can apply Theorem <cit.>, to conclude that w is a polynomial of degree 1, which is a contradiction with (<ref>).



 Recall that by Remark <ref> we can assume f_BMO(B_1)≤ε, where  ε>0 is the constant from the previous lemma. Since f_L^p(B_1)≤ Cf_ BMO(B_1), we can use Theorem <ref> to conclude 

    u_C^1, α(B_1/2)≤ C(u_L^∞(B_1) + u_L^1_σ(^d) + f_ BMO(B_1)).

Therefore from Proposition <ref> we have

    u_C^σ(B_1/2)≤ C(u_L^∞(B_1) + u_L^1_σ(^d) + f_BMO(B_1)),

proving the result.


Acknowledgement: D. dos Prazeres was partially supported by CNPq and CAPES/Fapitec. M. Santos was partially supported by the Portuguese government through FCT-Fundação para a Ciência e a Tecnologia, I.P., under the projects UID/MAT/04459/2020, and PTDC/MAT-PUR/1788/2020. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001.


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Makson S. Santos

Departamento de Matemática do Instituto Superior Técnico

Universidade de Lisboa

1049-001 Lisboa, Portugal






Disson dos Prazeres

Department of Mathematics

Universidade Federal de Sergipe - UFS, 

49100-000, Jardim Rosa Elze, São Cristóvão - SE, Brazil



