ON CERTAIN DEGENERATE AND SINGULAR ELLIPTIC PDES I:

NONDIVERGENCE FORM OPERATORS WITH UNBOUNDED DRIFTS

AND APPLICATIONS TO SUBELLIPTIC EQUATIONS

DIEGO MALDONADO

Abstract. We prove a Harnack inequality for nonnegative strong solutions to degenerateand singular elliptic PDEs modeled after certain convex functions and in the presence ofunbounded drifts. Our main theorem extends the Harnack inequality for the linearizedMonge-Ampere equation due to Caffarelli and Gutierrez and it is related, although underdifferent hypotheses, to a recent work by N. Q. Le.

Since our results are shown to apply to the convex functions |x|p with p ≥ 2 and their ten-sor sums, the degenerate elliptic operators that we can consider include subelliptic Grushinand Grushin-like operators as well as a recent example by A. Montanari of a nondivergence-form subelliptic operator arising from the geometric theory of several complex variables.In the light of these applications, it follows that the Monge-Ampere quasi-metric structurecan be regarded as an alternative to the usual Carnot-Caratheodory metric in the study ofcertain subelliptic PDEs.

1. Introduction and main result

Fix an open bounded set Ω ⊂ Rn. The purpose of this article is to use the Monge-Ampere real-analysis and PDE techniques developed in [2, 3, 8, 9, 10, 16, 24, 25, 26, 27] to

prove a Harnack inequality for nonnegative (strong) solutions u ∈ C(Ω) ∩W 2,nloc (Ω) to the

degenerate/singular elliptic PDE

LϕA(u)(x) = f(x) a.e. x ∈ Ω,

with

(1.1) LϕA(u)(x) := trace(A(x)D2u(x)) + 〈b(x),∇ϕu(x)〉+ c(x)u(x),

where the first- and second-order terms of LϕA are associated to the Hessian of a strictlyconvex function ϕ ∈ C1(Rn). More precisely, given 0 < λ ≤ Λ <∞, we will assume that, for(Lebesgue) a.e. x ∈ Ω, the n× n symmetric matrix A(x) in (1.1) satisfies

λD2ϕ(x)−1 ≤ A(x) ≤ ΛD2ϕ(x)−1 a.e. x ∈ Ω

in the sense of non-negative-definite matrices. That is,

(1.2) λ〈D2ϕ(x)−1ξ, ξ〉 ≤ 〈A(x)ξ, ξ〉 ≤ Λ〈D2ϕ(x)−1ξ, ξ〉,for a.e. x ∈ Ω and every ξ ∈ Rn, where 〈·, ·〉 denotes the usual dot product in Rn. We willwrite A ∈ E(λ,Λ, ϕ,Ω) to indicate (1.2).

Date: September 14, 2017.2000 Mathematics Subject Classification. Primary 35J70, 35J96; Secondary 35J75, 31E05.Key words and phrases. Degenerate and singular elliptic PDEs, linearized Monge-Ampere operator, Grushin

and subelliptic operators.Author supported by NSF under grant DMS 1361754.

1

2 DIEGO MALDONADO

The first-order term of LϕA involves the Monge-Ampere gradient ∇ϕ defined as

(1.3) ∇ϕh(x) := D2ϕ(x)−1/2∇h(x).

The strictly convex function ϕ ∈ C1(Rn) will also model the geometry shaping the Harnackinequality for LϕA by means of its Monge-Ampere sections. The Monge-Ampere section of ϕcentered at x ∈ Rn and with height r > 0 is defined as the open convex set

Sϕ(x, r) := y ∈ Rn : ϕ(y) < ϕ(x) + 〈∇ϕ(x), y − x〉+ rwhere

(1.4) δϕ(x, y) := ϕ(y)− ϕ(x)− 〈∇ϕ(x), y − x〉 ∀x, y ∈ Rn.

1.1. The hypotheses on ϕ. Our hypotheses on ϕ are the following:

H1. ϕ ∈ C1(Rn) is strictly convex in the sense that its graph contains no line segments.H2. The Monge-Ampere measure associated to ϕ, denoted by µϕ and defined as

(1.5) µϕ(E) := |∇ϕ(E)|for every Borel set E ⊂ Rn (where |F | denotes the Lebesgue measure of F ⊂ Rn),satisfies the following doubling property on the Monge-Ampere sections: There existconstants Cd ≥ 1 and α ∈ (0, 1) such that

(1.6) µϕ(Sϕ(x, r)) ≤ Cd µϕ(αSϕ(x, r)) ∀x ∈ Rn, r > 0.

Here αSϕ(x, r) denotes the α-contraction of Sϕ(x, r) with respect to its center of mass(computed with respect to the Lebesgue measure), see [16, Sections 1.1 and 3.1]. Wewill sometimes write µϕ ∈ (DC)ϕ to indicate (1.6).

The hypothesis µϕ ∈ (DC)ϕ turns the triple (Rn, δϕ, µϕ) into a quasi-metric doublingspace. In particular, there exists a constant K ≥ 1, depending only on Cd, α, and n,such that δϕ in (1.4) satisfies the following quasi-triangle inequality

(1.7) δϕ(x, y) ≤ K (minδϕ(z, x), δϕ(x, z)+ minδϕ(z, y), δϕ(y, z)) ,and the quasi-symmetry inequalities K−1δϕ(x, y) ≤ δϕ(y, x) ≤ Kδϕ(x, y), for everyx, y, z ∈ Rn, see [25] and references therein.

H3. There exists a non-decreasing function ζ : (0, 1)→ (0,∞) with limε→0+

ζ(ε) = 0 such that

for every x ∈ Rn and R > 0 we have

(1.8) µϕ(Sϕ(x, (1 + ε)R) \ Sϕ(x,R)) ≤ ζ(ε)µϕ(Sϕ(x,R)) ∀ε ∈ (0, 1).

H4. D2ϕ(x) > 0 for a.e. x ∈ Ω. In particular, D2ϕ(x)−1 and D2ϕ(x)−1/2 exist and arepositive-definite for a.e. x ∈ Ω. Notice that, by A. D. Alexandrov’s theorem, D2ϕ(x)exists for a.e. x ∈ Rn due to the convexity of ϕ.

H5. There exists p > n such that

(1.9) ‖D2ϕ‖ ∈ Lp(Ω).

Notice that the convexity of ϕ implies that 1n∆ϕ(x) ≤ ‖D2ϕ(x)‖ ≤ ∆ϕ(x) for a.e.

x ∈ Rn, so that (1.9) is equivalent to ∆ϕ ∈ Lp(Ω). From Remark 21 below we will seethat H5 can be weakened to ‖D2ϕ‖ ∈ Lnloc(Ω) when b = 0. Through this hypothesisthe constant will only depend on p and not on the Lp-norm of ‖D2ϕ‖. Also, if b = 0,the constants will not depend on the Lnloc-norm of ‖D2ϕ‖.

3

H6. There exists σ ∈ (0, 1) such that for every x0 ∈ Ω and r > 0 with Sϕ(x0, 2Kr) ⊂⊂ Ω,where K ≥ 1 is the quasi-triangle constant in (1.7), we have

(1.10) Dϕ(x0, x) ≥ σδϕ(x0, x) for a.e.x ∈ Sϕ(x0, 2r) \ Sϕ(x0, r)

where

Dϕ(x0, x) := 〈D2ϕ(x)−1(∇ϕ(x)−∇ϕ(x0)), (∇ϕ(x)−∇ϕ(x0))〉(1.11)

= |D2ϕ(x)−12 (∇ϕ(x)−∇ϕ(x0))|2.

For the sake of convenience we have used the annulus Sϕ(x0, 2r) \ Sϕ(x0, r) in (1.10),but it can be replaced by Sϕ(x0,Θr) \ Sϕ(x0, θr) for any constants 0 < θ ≤ 1 < Θ.

Notice that x ∈ Sϕ(x0, 2r) \ Sϕ(x0, r) if and only if r ≤ δϕ(x0, x) < 2r, so that acondition equivalent to (1.10) is the existence of σ ∈ (0, 1) such that

(1.12) Dϕ(x0, x) ≥ σr for a.e.x ∈ Sϕ(x0, 2r) \ Sϕ(x0, r),

where σ and σ depend only on one another.

1.2. The hypotheses on the lower-order coefficients. Throughout the article we assumethat b(x) and c(x) in (1.1) satisfy the following hypotheses

(1.13) |(D2ϕ)−1/2b| ∈ Ln(Ω, dµϕ) and c ∈ Ln(Ω, dµϕ).

The hypothesis (D2ϕ)−1/2b ∈ Ln(Ω) is crucial to the ABP maximum principle, which serves asthe cornerstone to the whole approach towards the Harnack inequality for the nondivergenceform elliptic operators LϕA. As discussed in the comments in Section 1.3, it cannot be improved

to (D2ϕ)−1/2b ∈ Ln−ε(Ω) for any ε > 0.Our main result is the following (see Section 5.1 for the definitions of geometric and struc-

tural constants).

Theorem 1. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). There exist struc-tural constants εH ∈ (0, 1), MH > 1, and a geometric constant τ ∈ (0, 1) such that forevery section Sϕ(z, r) with S2Kr := Sϕ(z, 2Kr) ⊂⊂ Ω and every u ∈ C(S2Kr) ∩W 2,n(S2Kr)satisfying u ≥ 0 in S2Kr and

(1.14) LϕA(u) = trace(A(x)D2u(x)) + 〈b(x),∇ϕu(x)〉+ c(x)u(x) = f a.e. in S2Kr,

the inequalities

(1.15) supx∈Sϕ(z,r)

ρ≤r

(ρ−

12 |Sϕ(x, ρ)|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(x,ρ), dµϕ)‖D2ϕ‖12

Lp(Sϕ(x,ρ), dx)

)≤ εH

and

(1.16) |S2Kr|1n ‖c‖Ln(S2Kr, dµϕ) ≤ εH

imply the Harnack inequality

(1.17) supSϕ(z,τr)

u ≤MH

(inf

Sϕ(z,τr)u+ |Sr|

1n ‖f‖Ln(S2Kr, dµϕ)

).

Remark 2. Notice that any first-order term of the form 〈b,∇u〉, involving the usual gradient∇ and a drift b, can be written as

〈b,∇u〉 = 〈b,∇ϕu〉,

4 DIEGO MALDONADO

where b := (D2ϕ)1/2b, so that the condition |(D2ϕ)−1/2b| ∈ Ln(Ω, dµϕ) in (1.13) just means

|b| ∈ Ln(Ω, dµϕ). Along these lines, Theorem 1 can be equivalently stated with the equation(1.14) and the condition (1.15) replaced with

LϕA(u) = trace(A(x)D2u(x)) + 〈b,∇u〉+ c(x)u(x) = f a.e. in S2Kr

and

supx∈Sϕ(z,r)

ρ≤r

(ρ−

12 |Sϕ(x, ρ)|

1n− 1

2p ‖b‖Ln(Sϕ(x,ρ), dµϕ)‖D2ϕ‖12

Lp(Sϕ(x,ρ), dx)

)≤ εH ,

respectively.

Let us briefly put Theorem 1 in the context of some related results.

1.3. A timeline on the interior Harnack inequality for LϕA with unbounded drift.

We first notice that when the function ϕ is taken as the ϕ2(x) := 12 |x|2, for x ∈ Rn, then its

Monge-Ampere measure reduces to Lebesgue measure, its associated Monge-Ampere gradient∇ϕ2 is just the regular gradient ∇, and Dϕ2(x0, x) = δϕ2(x0, x) = |x− x0|2 for every x, x0 ∈Rn. In particular, the sections Sϕ2(x, r) coincide with the Euclidean balls B(x,

√2r) for every

x ∈ Rn and r > 0. Hence, all the hypotheses H1–H6 hold true for ϕ2.In addition, the condition A ∈ E(λ,Λ, ϕ2,Ω) amounts to the uniform ellipticity of A on

Ω. In this case, the Harnack inequality for positive solutions to Lϕ2

A (u) = f in Ω with|b|, c ∈ L∞(Ω) is due to the fundamental work of N. Krylov and M. Safonov [14, 30], see also[15, Sections 9.7-9.8].

More recently, and always in the uniformly elliptic case, Safonov in [31] established aHarnack inequality for nonnegative solutions to trace(AD2u) + 〈b,∇u〉 = 0 in the case of anunbounded drift b under the assumption |b| ∈ Ln(Ω). The condition |b| ∈ Ln(Ω) is criticalin the following sense: if |b| ∈ Lp(Ω) for some p > n a scaling argument lays a path towardsthe Harnack inequality along the lines of the Krylov-Safonov approach in [14, 30], see forinstance [33]. On the other hand, if |b| ∈ Lp(Ω) for some p < n, the Harnack inequality isknown not to hold. Indeed, Safonov’s example (see [31, Remark 1.3] and [29, Section 1])shows that, in any dimension n, the nonnegative function u(x) := 1

2 |x|2 is a strong solution

to ∆u + 〈b,∇u〉 = 0 with b(x) := −n|x|−2x. Notice that b ∈ Ln,∞(B(0, 1)) \ Ln(B(0, 1))and, in particular, that b ∈ Ln−ε(B(0, 1)) for every ε > 0. However, u(x) := 1

2 |x|2 vanishesat x = 0 (and only at x = 0) and therefore it cannot satisfy a Harnack inequality in, say,B(0, 1). Earlier versions of Holder (as opposed to Harnack) estimates under the assumption|b| ∈ Ln(Ω) had been proved by O. Ladyzhenskaya and N. Uraltseva in [19] under a smallnesscondition on ‖b‖Ln(Ω). See [31, Remark 1.4] for further details.

For later reference we mention a recent extension of the elliptic results in [31], under theassumption |b| ∈ Ln(Ω), due to C. Mooney in [29], where a Harnack inequality is proved fornonnegative functions that are solutions only when their gradients are large. Mooney’s resultalso extends the work of C. Imbert and L. Silvestre in [17], who had assumed |b| ∈ L∞(Ω),by implementing the method of “sliding paraboloids”.

The results mentioned above are based on the choice ϕ2(x) := 12 |x|2. Let us now move on

to the non-uniformly elliptic case and the pioneering work of L. Caffarelli and C. Gutierrez. In[3], Caffarelli and Gutierrez considered a function ϕ ∈ C2(Ω) with D2ϕ(x) > 0 for every x ∈ Ωsuch that its Monge-Ampere measure µϕ = detD2ϕ satisfies the so-called µ∞-condition: forevery δ1 ∈ (0, 1) there exists δ2 ∈ (0, 1) such that for every section S := Sϕ(x, r) ⊂⊂ Ω and

5

every measurable set E ⊂ S the implication

|E| < δ2|S| ⇒ µϕ(E) < δ1µϕ(S)

holds true, where |E| stands for the Lebesgue measure of the set E. Then, given a matrixA ∈ E(λ,Λ, ϕ,Ω), they proved a Harnack inequality for classical solutions of LϕA(u) = 0 inthe case |b| = c = 0. The µ∞-condition is a Muckenhupt-type property of µϕ with respect toLebesgue measure which is strictly stronger than the doubling condition (1.6) from hypothesisH2, see [3, Sections 1 and 5] and [10, Section 3].

We point out that in [3] the hypothesis ϕ ∈ C2(Ω) with D2ϕ(x) > 0 for every x ∈ Ω iscrucial in the proof of the “passage to the double section” (that is, [3, Theorem 2], which, inturn, is essential for the weak-Harnack inequality in [3, Theorem 4]). This can be seen on [3,pp.439-445] where a Dirichlet problem having the Monge–Ampere measure as a nonvanishingfactor of the right-hand side is solved for a function wε required to be smooth (at least C2).

In [25], under the assumptions ϕ ∈ C3(Ω) with D2ϕ(x) > 0 for every x ∈ Ω, but assum-ing (the weaker) H2 hypothesis instead of the µ∞-condition, the author proved a Harnackinequality for nonnegative strong solutions of LϕA(u) = 0 ([25, Theorem 1.4]) in the case|b| = c = 0 and λ = Λ. (Notice that, when λ = Λ, then A ∈ E(λ,Λ, ϕ,Ω) just meansA = λD2ϕ−1.) In particular, the doubling property (1.6) from hypothesis H2 makes theHarnack inequality intrinsic to the quasi-metric space (Rn, µϕ, δϕ), with no need of a prioricomparisons between µϕ and Lebesgue measure such as in the µ∞-condition. In [25] theassumption ϕ ∈ C3(Ω) was used to pass from the non-divergence form of trace((D2ϕ)−1D2u)to its divergence form via the identity

(1.18) trace(AϕD2h) = div(Aϕ∇h) ∀h ∈ C2(Ω),

where Aϕ(x) is the matrix of cofactors of the Hessian D2ϕ(x), that is,

(1.19) Aϕ(x) := D2ϕ(x)−1 detD2ϕ(x) ∀x ∈ Ω.

Later on, in [26], under the assumptions ϕ ∈ C2(Ω) with D2ϕ(x) > 0 for every x ∈ Ω andH2, the author proved a Harnack inequality for nonnegative strong solutions to LϕA(u) = f ,also in the case λ = Λ, when the lower-order coefficients and source term satisfy

(1.20) |(D2ϕ)−1/2b| ∈ Ln(Ω), c ≤ 0, and |b|, c, f ∈ L∞(Ω),

see [26, Theorem 1]. The hypothesis ϕ ∈ C2(Ω) with D2ϕ(x) > 0 for every x ∈ Ω allowedfor an alternative proof of the critical density estimate ([26, Theorem 2]) to the original oneby Caffarelli-Gutierrez ([3, Theorem 1]). In order to prove a weak-Harnack inequality, theapproach in [26] was based on the idea from [25] to use the variational side of the PDE. Inthis case a weaker form of (1.18) was used, for which the hypothesis ϕ ∈ C2(Ω) (instead ofϕ ∈ C3(Ω)) sufficed, see [26, Remark 12].

More recently, under the hypothesis ϕ ∈ C2(Ω) with D2ϕ(x) > 0 for every x ∈ Ω (whichplays a crucial role) as well as

(1.21) 0 < λ0 ≤ detD2ϕ(x) ≤ Λ0 ∀x ∈ Ω,

for some constants λ0,Λ0 ∈ (0,∞), N. Q. Le in [20] considered the PDE

LA(u) := trace(AD2u) + 〈b,∇u〉+ cu = f

(notice the gradient ∇, as opposed to ∇ϕ, on the first-order term and keep in mind Remark2) with A ∈ E(λ,Λ, ϕ,Ω) in the general case of 0 < λ ≤ Λ when the lower-order coefficientsand source term satisfy

(1.22) |b| ∈ Lp(Ω) and c, f ∈ Ln(Ω),

6 DIEGO MALDONADO

for some p > n(1 + α∗)/(2α∗), with α∗ ∈ (0, 1] a structural constant, see [20, Theorem1.1]. Le’s approach relies on his implementation of the “sliding Monge-Ampere paraboloids”technique, thus extending the approach by Mooney in [29]. Notice that the inequalities (1.21)make µϕ uniformly comparable to Lebesgue measure, thus implying the µ∞-condition for µϕ.

1.4. Our point of view and the reasons for the hypotheses H1–H6. In all of theresults mentioned above, the hypothesis ϕ ∈ C2(Ω) with D2ϕ(x) > 0 for every x ∈ Ωhas been essential. In particular, those results do not apply, for example, to the functionsϕp(x) := 1

p |x|p for any p > 1 or their tensor sums. Moreover, the current lack of apriori

estimates prevents an approximation argument to go from ϕ ∈ C2(Ω) with D2ϕ > 0 to ϕp.The role of the hypotheses H4–H6 is precisely to compensate for such lack of apriori

estimates and to be able to apply Theorem 1 directly to “rougher” (i.e. ϕ /∈ C2(Ω)) and“flatter” (i.e. D2ϕ vanishing at some points) functions ϕ. We remark that the Lp-integrabilitycondition in H5 is reminiscent of the one adopted by N. Trudinger in [32] in the context ofdegenerate divergence-form elliptic operators, see [32, Section 5].

In terms of the functions ϕp, in Sections 11 and 12 we prove that they, as well as their tensorsums, satisfy the hypotheses H1–H6 whenever p ≥ 2. Such choices allow for a number ofapplications to singular and degenerate (including subelliptic) PDEs with unbounded drifts,as shown in Sections 2, 3, and 4.

Of course, other classes of general convex functions are expected to satisfy H1–H6 and tolead, in turn, to corresponding applications.

Finally, we mention that we have stated the “first-order” hypotheses H1–H3 on Rn, whilethe “second-order” hypotheses H4–H6 are stated on Ω. This is only a matter of conveniencesince stating H1–H3 on Rn allows to consider the triple (Rn, δϕ, µϕ) (as opposed to thetriple (Ω, δϕ, µϕ)) as a space of homogeneous type. However, at the expense of a number oftechnicalities that we have preferred to avoid, the hypotheses H1–H3 could also be statedon Ω.

Regarding applications of Theorem 1. A central motivation for the Harnack inequalityfor nonnegative solutions of the linearized Monge-Ampere equation proved by Caffarelli andGutierrez in [3] stemmed from topics of fluid dynamics, see [3, Section 1]. Since then, otherapplications have appeared, for instance, in the contexts of optimal transport [23], [6, Section2] and differential geometry; in particular, to affine geometry [34, 35, 36] and Abreu’s equation[21, 22] and references therein. Hence, Theorem 1 can be applied in all the contexts aboveto the case of “rougher” and “flatter” convex functions ϕ.

However, in Sections 2 and 3, we focus on applications of Theorem 1 and the Monge-Ampere quasi-metric structure to the study of regularity properties for solutions to certainsubelliptic PDEs containing lower-order terms. These subelliptic PDEs will include GrushinPDEs as well as a recent subelliptic PDE studied by Montanari in [28] in the context of thegeometric theory of several complex variables. In particular, we extend Montanari’s Harnackinequality for the subelliptic operator in [28] to a family of degenerate elliptic operatorsinvolving unbounded drifts. As a consequence of these applications, the Monge-Ampere quasi-metric structure can now be regarded as an alternative to the usual Carnot-Caratheodorymetric in the study of such subelliptic PDEs.

In Section 4 we study other degenerate/singular PDEs which extend the Grushin PDEs.To the best of our knowledge, the Harnack inequalities in the presence of unbounded driftsin Theorems 3 and 5 below as well as the ones in Section 4 are all new contributions tosubelliptic PDE literature.

7

Organization of the article. The article is organized as follows: In Sections 2 and 3 weexplore applications of Theorem 1 to certain subelliptic operators. These operators includeGrushin operators with unbounded drifts as well as a recent nondivergence form subellipticoperator introduced by A. Montanari in [28].

In Section 4 we apply Theorem 1 to classes of singular and degenerate elliptic PDEs withunbounded drifts. In particular, those degenerate classes will include the Grushin operatorsfrom Section 2, see Theorems 7 and 8.

In Section 5 we establish some notation and background material, including a Morrey-typeestimate (Lemma 13) that is new to the Monge-Ampere quasi-metric structure.

In Section 6 we show that the Monge-Ampere quasi-metric space (Rn, δϕ) possesses thesegment and segment-prolongation properties (Theorem 16). This then leads to the construc-tion of geodesics on (Rn, δϕ) and to the fact that the Monge-Ampere sections are geodesicallyconvex (Theorem 17). The material in Section 6 and the Morrey-type estimate from Section5 are of independent interest.

In Section 7, under the hypotheses H1, H2, H4, H5, H6 on ϕ, we prove the double-ballproperty for supersolutions (Theorem 20 and Corollary 22).

In Section 8, under the hypotheses H1–H6 on ϕ, we prove a critical-density estimate forsupersolutions (Theorem 25). In Section 9, also under the hypotheses H1–H6 on ϕ, we provemean-value inequalities for subsolutions (Theorem 27 and Corollaries 28 and 29).

In Section 10, the critical density property and the double-ball property combined withVitali’s covering lemma are shown to imply the power-like decay of the distribution function ofnonnegative supersolutions (Theorem 30) and their corresponding weak-Harnack inequalities(Corollary 31). Thus, with all the elements in place, the proof of Theorem 1 is then completedin Section 10.1.

In Section 11 we prove that (all positive multiples of) the functions ϕp(x) := 1p |x|p satisfy

all of the hypotheses H1–H6 when p ≥ 2 (Theorem 33).In Section 12 we prove that the hypotheses H1, H2, H4, H5, H6 are quantitatively

preserved under tensor sums (Theorem 35) and that all the hypotheses H1–H6 are preservedunder tensor sums of the functions ϕp for p ≥ 2.

Finally, Section 13 corresponds to an Appendix where we include the proofs Lemma 13(the Morrey-type estimate), Lemma 15 (the local Vitali covering lemma), and Theorem 27(the mean-value inequality for nonnegative subsolutions).

In a forthcoming article we will address divergence-form operators with second-order termsof the form div(A∇u) with A ∈ E(λ,Λ, ϕ,Ω) and lower-order terms.

2. Applications to subelliptic PDEs I: Grushin operators

Consider n1, n2 ∈ N, x := (x1, x2) ∈ Rn1×Rn2 , and for γ ≥ 0, let Gγ denote the degenerateelliptic Grushin operator

(2.23) Gγ(u)(x) := ∆1u(x) + |x1|γ∆2u(x),

where ∆j denotes the Laplacian on Rnj for j = 1, 2. The literature on Grushin operators isvast, we will only mention the well-known works [11, 12] for related Harnack inequalities fornonnegative solutions to Gγ(u) = 0. However, as mentioned in the introduction, the authoris not aware of any systematic study of Grushin operators with first-order terms previouslyappeared in the literature.

8 DIEGO MALDONADO

By setting n := n1 + n2, the equation Gγ(u) = 0 a.e. in Ω ⊂ Rn can be recast as

(2.24) 0 =1

|x1|γ∆1u(x) + ∆2u(x) = trace(Aγ(x)D2u(x)) a.e. x ∈ Ω,

where Aγ(x) denotes the diagonal n× n matrix

(2.25) Aγ(x) :=

[ 1|x1|γ In1×n1 0

0 In2×n2

].

Hence, by introducing the convex functions ϕ1γ(x1) := 1

γ+2 |x1|γ+2, ϕ2(x2) := 12 |x2|2, and

ϕγ(x) := ϕ1γ(x1) + ϕ2(x2) we get

(2.26) D2ϕγ(x) =

[D2ϕ1

γ(x1) 00 In2×n2

],

where, for every x1 ∈ Rn1 \ 0, D2ϕ1γ(x1) is the n1 × n1 symmetric matrix

(2.27) D2ϕ1γ(x1) = (γ + 1)|x1|γ

(x1

|x1|⊗ x1

|x1|

)+ |x1|γ

(I − x1

|x1|⊗ x1

|x1|

),

which has eigenvalues (γ+1)|x1|γ with multiplicity 1 and |x1|γ with multiplicity n1−1. Thatis,

(2.28) D2ϕ1γ(x1) = P

(γ + 1)|x1|γ 0 · · · 0

0 |x1|γ · · · 0...

.... . . 0

0 0 0 |x1|γ

P t,for some orthogonal n1 × n1 matrix P . It then follows that

(2.29) λγ〈D2ϕγ(x)−1ξ, ξ〉 ≤ 〈Aγ(x)ξ, ξ〉 ≤ Λγ〈D2ϕγ(x)−1ξ, ξ〉,holds for a.e. x ∈ Ω and every ξ ∈ Rn, with λγ := 1 and Λγ := γ + 1. That is, Aγ ∈E(λγ ,Λγ , ϕγ ,Ω).

Now, as proved in Sections 11 and 12, the convex function ϕγ satisfies all of the hypothesesH1–H6 (moreover, ϕγ satisfies H5 for any p > n) and Theorem 1 will apply. More precisely,given a PDE of the form

Gγ(u) + 〈b,∇u〉+ cu = f

we can recast it in the form of (1.14) after dividing by |x1|γ , to get

1

|x1|γ∆1u(x) + ∆2u(x) +

1

|x1|γ〈b(x),∇u(x)〉+

c(x)

|x1|γu(x) =

f(x)

|x1|γwhich can be written as

trace(Aγ(x)D2u(x)) + 〈b(x),∇ϕu(x)〉+ c(x)u(x) = f(x),

where Aγ(x) is given by (2.25), b(x) := |x1|−γD2ϕγ(x)1/2b(x), c(x) := c(x)|x1|−γ , and f(x) :=

f(x)|x1|−γ .On the other hand, notice that (2.26) implies that the Monge-Ampere measure of ϕγ is

given by µϕγ (x) = detD2ϕγ(x) = (γ + 1)|x1|γn1 . Consequently, given a set S ⊂ Ω, we haveˆS|(D2ϕ)−1/2b(x)|ndµϕγ (x) =

ˆS|x1|−nγ |b(x)|n(γ + 1)|x1|γn1 dx

= (γ + 1)

ˆS|b(x)|n|x1|γ(n1−n) dx = (γ + 1)

ˆS|b(x)|n|x1|−γn2 dx.

9

Similarly, c ∈ Ln(S, dµϕγ ) and f ∈ Ln(S, dµϕγ ) translate into c ∈ Ln(S, |x1|−γn2dx) and

f ∈ Ln(S, |x1|−γn2dx), respectively.By bringing all this together, we can now state a regularity result for solutions to Grushin

operators with lower-order coefficients and unbounded drifts. Namely,

Theorem 3. Fix an open bounded set Ω ⊂ Rn, γ ≥ 0, and consider the subelliptic PDE

(2.30) Gγ(u) + 〈b,∇u〉+ cu = f

where Gγ denotes the Grushin operator in (2.23). Then, there exist constants 0 < τ <1 < KH (depending only on γ and n) as well as structural constants 0 < εH < 1 < MH ,such that for every section Sϕγ (z,R) with SKHR := Sϕγ (z,KHR) ⊂⊂ Ω and every u ∈C(SKHR) ∩W 2,n(SKHR) satisfying u ≥ 0 in SKHR and solving the subelliptic PDE (2.30)a.e. in SKHR, we have that the inequalities

(2.31) supy∈Sϕγ (z,R)

ρ≤r

(ρ−

12 |Sϕγ (y, ρ)|

1n− 1

2p ‖b‖Ln(Sϕγ (y,ρ),|x1|−γn2dx)‖D2ϕγ‖12

Lp(Sϕγ (y,ρ), dx)

)≤ εH

and

(2.32) |SKHR|1n ‖c‖Ln(SKHR,|x1|

−γn2dx) ≤ εHimply the Harnack inequality

supSϕγ (z,τR)

u ≤MH

(inf

Sϕγ (z,τR)u+ |SKHR|

1n ‖f‖Ln(SKHR,|x1|

−γn2 dx)

).

Here p, the exponent from hypothesis H5, can be taken as any number bigger than n.

Remark 4. In order to illustrate how Theorem 3 can be used, let us break down the inequality(2.31). A discussion on the sections of convex functions given as tensor sums of other convexfunctions (such as ϕγ above) is included in Sections 11 and 12. In particular, we have that,given y = (y1, y2) ∈ Sϕγ (z,R), the inclusions (11.119) and (12.135) yield

Sϕγ (y, ρ) ⊂ Sϕ1γ(y1, ρ)× Sϕ2(y2, ρ) ⊂ B1(y1, Cγρ

1γ+2 )×B2(y2, (2ρ)1/2)

and

B1(y1, cγρ1

γ+2 )×B2(y2, 2ρ1/2) ⊂ Sϕ1

γ(y1, ρ/2)× Sϕ2(y2, ρ/2) ⊂ Sϕγ (y, ρ),

where B1 and B2 denote Euclidean balls in Rn1 and Rn2 , respectively, and cγ , Cγ > 0depend only on γ and n1. For the sake of conciseness, since the degeneracy of the PDE(2.30) effectively occurs on |z1| = 0, let us assume that z = (z1, z2) with |z1| < 1. Also,we can assume that 0 < R < 1. Hence, given y = (y1, y2) ∈ Sϕγ (z,R), ρ ∈ (0, R), andx = (x1, x2) ∈ Sϕγ (y, ρ) it follows from the inclusions above that |x1| ≤ |x1− y1|+ |y1− z1|+|z1| . ρ

1γ+2 + R

1γ+2 + 1 . 1, where the implied constants depend only on n1 and γ. Now,

|x1| . 1 along with (2.26) and (2.28) yields ‖D2ϕγ‖ ' 1 on Sϕγ (y, ρ). Consequently, for anygiven p > n, we get

|Sϕγ (y, ρ)|−12p ‖D2ϕγ‖

12

Lp(Sϕγ (y,ρ), dx) ' 1.

On the other hand, by taking Lebesgue measure on the inclusions above,

ρ−12 |Sϕγ (y, ρ)| 1n ‖b‖Ln(Sϕγ (y,ρ),|x1|−γn2dx) ' ρ−

12 ρ

1n

(n1γ+2

+n22

)‖b‖Ln(Sϕγ (y,ρ),|x1|−γn2dx),

10 DIEGO MALDONADO

so that the condition (2.31) can be recast as

(2.33) supy∈Sϕγ (z,R)

ρ≤r

(ρ

1n

(n1γ+2

+n22

)− 1

2 ‖b‖Ln(Sϕγ (y,ρ),|x1|−γn2dx)

)≤ εH ,

which would place the drift b in a Morrey space with respect to the weight |x1|−γn2dx.Alternatively, the condition (2.33) can be realized by asking for higher integrability of b withrespect to a weight. More precisely, given β ≥ 1, by Holder’s inequality with exponents βand β′, we have

‖b‖Ln(Sϕγ (y,ρ),|x1|−γn2dx) ≤ ‖b‖Lnβ(Sϕγ (y,ρ),|x1|−βγn2dx)|Sϕγ (y, ρ)|1nβ′

' ‖b‖Lnβ(Sϕγ (y,ρ),|x1|−βγn2dx)ρ1nβ′

(n1γ+2

+n22

),

and then, by choosing β so that 1n

(n1γ+2 + n2

2

)− 1

2 + 1nβ′

(n1γ+2 + n2

2

)= 0, we get

ρ1n

(n1γ+2

+n22

)− 1

2 ‖b‖Ln(Sϕγ (y,ρ),|x1|−γn2dx)

≤ ρ1n

(n1γ+2

+n22

)− 1

2+ 1nβ′

(n1γ+2

+n22

)‖b‖Lnβ(Sϕγ (y,ρ),|x1|−βγn2dx) = ‖b‖Lnβ(Sϕγ (y,ρ),|x1|−βγn2dx),

and (2.33) is then implied by the condition ‖b‖Lnβ(Sϕγ (y,ρ),|x1|−βγn2dx) ≤ ˜εH with

(2.34) β :=

n1γ+2 + n2

2

n22 + n1

(2

γ+2 − 12

) .Notice that β ≥ 1 for every γ ≥ 0 with

(2.35)1

γ + 2≥ 1

4

(1− n2

n1

).

Therefore, the condition (2.31) is equivalent to the weighted Morrey-space estimate (2.33)and it is weaker than the integrability condition ‖b‖Lnβ(Sϕγ (y,ρ),|x1|−βγn2dx) ≤ ˜εH , with β and

γ as in (2.34) and (2.35).

In Section 4 we will extend Theorem 3 to a large class of degenerate PDEs that will includethe Grushin operators in (2.23).

3. Applications to subelliptic PDEs II: Extensions of Montanari’s example

3.1. Montanari’s example. Let x = (x1, x2) ∈ R2 and consider the vector fields X1 := ∂x1and X2 := x1∂x2, in particular, notice that [X1, X2] = ∂x2 . In [28] and motivated by topicson the geometric theory of several complex variables, A. Montanari introduced the subellipticoperator

(3.36) L = a11X21 + 2a12X2X1 + a22X

22 ,

where the coefficient matrix a := (aij)2i,j=1 satisfies, for some constants 0 < λ0 ≤ Λ0, the

uniform ellipticity condition

(3.37) λ0|ξ|2 ≤ 〈a(x)ξ, ξ〉 ≤ Λ0|ξ|2 ∀x, ξ ∈ R2.

Then, by means of a weighted version of the ABP maximum principle, she established aHarnack inequality for nonnegative classical C2-solutions to Lu = 0 with respect to balls ofthe Carnot-Caratheodory metric generated by X1 and X2.

11

In our next application, by means of Theorem 1 and the Monge-Ampere quasi-metricstructure, we extend Montanari’s Harnack inequality to a family of subelliptic operators thatinclude L in (3.36).

3.2. Extensions of Montanari’s example. Fix an open bounded set Ω ⊂ R2 and ν ∈ N0.Define the vector fields X1 := ∂x1 and X2 := xν1∂x2, and introduce the subelliptic operator

(3.38) Lν = a11X21 + 2a12X2X1 + a22X

22 ,

where the coefficient matrix a := (aij)2i,j=1 satisfies (3.37) for a.e. x ∈ Ω.

Notice that in this case we have [X1, X2] = νxν−11 ∂x2 and this commutator will vanish on

x1 = 0 whenever ν > 1. However, the vector fields X1 and X2 will still satisfy Hormander’scondition, but commutators of up to order ν will be required.

Also, notice that, when ν = 1, L1 recovers L from (3.36).Let us write Lν as Lν(u) = x2ν

1 trace(AνD2u) where

Aν(x) = Aν(x1, x2) :=

[a11(x)/x2ν

1 a12(x)/xν1a12(x)/xν1 a22(x)

]and introduce the convex function ϕν : R2 → R as ϕν(x) := 1

(2ν+1)(2ν+2)x2ν+21 + 1

2x22 so that

(3.39) D2ϕν(x) =

[x2ν

1 00 1

]and

D2ϕν(x)1/2Aν(x)D2ϕν(x)1/2 =

[a11 a12

|x1|νxν1

a12|x1|νxν1

a22

].

Then, the ellipticity condition (3.37) implies that

λ0|ξ|2 ≤ 〈D2ϕν(x)1/2Aν(x)D2ϕν(x)1/2ξ, ξ〉 ≤ Λ0|ξ|2

holds for a.e. x ∈ Ω and every ξ ∈ Rn, that is, Aν ∈ E(λ0,Λ0, ϕν ,Ω). Next, let us write anysubelliptic PDE of the form

(3.40) Lν(u) + 〈b,∇u〉+ cu = f

astrace(AνD

2u) + 〈b,∇ϕu〉+ cu = f,

where b(x) := x−2ν1 D2ϕ(x)1/2b(x), c(x) := c(x)x−2ν

1 , and f(x) := f(x)x−2ν1 , and notice that,

since the Monge-Ampere measure of ϕν equals µϕν (x) = detD2ϕν(x) = x2ν1 , given S ⊂ Ω

the condition |(D2ϕν)−1/2b| ∈ L2(S, dµϕν ) from (1.13) translates intoˆS|(D2ϕν)−1/2b(x)|2dµϕν (x) =

ˆSx−4ν

1 |b(x)|2x2ν1 dx =

ˆS|b(x)|2x−2ν

1 dx <∞,

and, similarly, c ∈ L2(S, dµϕν ) and f ∈ L2(S, dµϕν ) translate into c ∈ L2(S, x−2ν1 ) and

f ∈ L2(S, x−2ν1 ), respectively. Thus, by bringing things together, Theorem 1 yields the

following Harnack inequality for nonnegative solutions to the subelliptic PDEs (3.40) withunbounded drifts.

Theorem 5. Fix an open bounded set Ω ⊂ R2, ν ∈ N0, and consider the subelliptic PDE

(3.41) Lν(u) + 〈b,∇u〉+ cu = f

where Lν denotes the subelliptic operator in (3.38). Then, there exist constants 0 < τ <1 < KH (depending only on ν) as well as structural constants 0 < εH < 1 < MH , such that

12 DIEGO MALDONADO

for every section Sϕν (z,R) with SKHR := Sϕν (z,KHR) ⊂⊂ Ω and every u ∈ C(SKHR) ∩W 2,2(SKHR) satisfying u ≥ 0 in SKHR and solving the subelliptic PDE (3.41) a.e. in SKHR,we have that the inequalities

(3.42) supy∈Sϕν (z,R)

ρ≤r

(ρ−

12 |Sϕν (y, ρ)|

12− 1

2p ‖b‖L2(Sϕν (y,ρ),|x1|−2νdx)‖D2ϕν‖12

Lp(Sϕν (y,ρ), dx)

)≤ εH

and

(3.43) |SKHR|12 ‖c‖L2(SKHR,|x1|

−2νdx) ≤ εHimply the Harnack inequality

supSϕν (z,τR)

u ≤MH

(inf

Sϕν (z,τR)u+ |SKHR|

12 ‖f‖L2(SKHR,|x1|

−2ν dx)

).

Here p, the exponent from hypothesis H5, can be taken as any number bigger than 2.

Remark 6. A comment on the condition (3.42) follows along the lines of Remark 4. Forinstance, assuming z := (z1, z2) with |z1| < 1 and 0 < R < 1 we get that ‖D2ϕν‖ ' 1 onSϕγ (y, ρ). In this case the condition (3.42) turns out to be equivalent to the Morrey-spaceestimate

(3.44) supy∈Sϕγ (z,R)

ρ≤r

(ρ− ν

4(ν+1) ‖b‖L2(Sϕγ (y,ρ),|x1|−2νdx)

)≤ εH ,

as well as weaker than the integrability ‖b‖L2β(Sϕγ (y,ρ),|x1|−2βνdx) ≤ ˜εH with β := 2+ν2 .

We close this application by mentioning that the weight in the weighted ABP maximumprinciple from [28, Theorem 2.5] is precisely the Monge-Ampere measure x2

1 and that theMonge-Ampere quasi-distance δϕ provides an equivalent gauge to the Carnot-Caratheodoryand Grushin metrics. In the case of ϕν as above, by [9, Theorem 4(iii)] and (12.136) fromSection 12 we have

δϕν ((x1, x2), (y1, y2)) ' (x2ν+11 − y2ν+1

1 )(x1 − y1) + (x2 − y2)2, ∀(x1, x2), (y1, y2) ∈ R2,

where the implied constants depend only on ν. Several equivalent distances and quasi-distances were considered in [28, Section 2].

4. Applications to other singular and degenerate elliptic operators

In this section we extend the class of subelliptic Grushin operators from Section 2, introducea related singular elliptic operator, and establish a Harnack inequality for their correspondingPDEs including unbounded drifts.

Fix m ∈ N, n1, . . . , nm ∈ N, and set n := n1+· · ·nm. Fix Ω ⊂ Rn and for each j = 1, . . . ,m,let Ωj denote the projection of Ω over Rnj and let the functions Γj : Ωj → R satisfy

(4.45) λj |xj |γj ≤ Γj(xj) ≤ Λj |xj |γj a.e. xj ∈ Ωj ,

for some constants 0 < λj ≤ Λj and γ := (γ1, . . . , γm) ∈ [0,∞)m. Notice that only measura-bility is required from the the Γj ’s.

Introduce the convex function ϕγ : Rn → R as the tensor sum

ϕγ(x) :=1

(1 + γ1)(2 + γ1)|x1|2+γ1 + · · ·+ 1

(1 + γm)(2 + γm)|xm|2+γm ,

13

for x = (x1, . . . , xm) ∈ Rn1 × · · ·Rnm = Rn, so that D2ϕγ is a direct sum, for j = 1, . . . ,m,of nj × nj matrices of the form

Pj

(1 + γj)|xj |γj 0 · · · 0

0 |xj |γj · · · 0...

.... . . 0

0 0 0 |xj |γj

P tj ,

for some nj × nj orthogonal matrix Pj . In particular, detD2ϕγ(x) =m∏j=1

(1 + γj)|xj |njγj .

4.1. The degenerate case: Grushin-like operators. Let us consider the following de-generate elliptic operator

(4.46) GΓ(u) := Π1(x)∆1u(x) + · · ·+ Πm(x)∆mu(x),

where ∆j is the Laplacian on Rnj and, for j = 1, . . . ,m,

Πj(x) :=∏k 6=j

Γk(xk).

For instance, if m = 2 and λj = Λj = 1 for j = 1, 2, then we have

GΓ(u) := |x2|γ2∆1u(x) + |x1|γ1∆2u(x),

so that the choice γ2 = 0 recovers the Grushin operator (2.23), and, for general 0 < λj ≤ Λj ,we have

GΓ(u) := Γ2(x2)∆1u(x) + Γ1(x1)∆2u(x),

with λj |xj |γj ≤ Γj ≤ Λj |xj |γj and j = 1, 2. If m = 3 we have

GΓ(u) := Γ2(x2)Γ3(x3)∆1u(x) + Γ1(x1)Γ3(x3)∆2u(x) + Γ1(x1)Γ2(x2)∆3u(x),

etc. Now, given a PDE of the form

GΓ(u) + 〈b,∇u〉+ cu = f,

after dividing it by the product ΠΓ(x) := Πmj=1Γj(xj), it turns into

trace(AΓ(x)D2u(x)) + 〈b(x),∇ϕu(x)〉+ c(x)u(x) = f(x),

where AΓ is the direct sum

(4.47) AΓ(x) :=

1

Γ(x1)In1×n1 0 · · · 0

0 1Γ(x2)In2×n2 · · · 0

......

. . . 00 0 0 1

Γ(xm)Inm×nm

,and b(x) := ΠΓ(x)−1D2ϕγ(x)1/2b(x), c(x) := ΠΓ(x)−1c(x), and f(x) := ΠΓ(x)−1f(x). Inparticular, it follows that AΓ ∈ E(λΓ,ΛΓ, ϕγ ,Ω), for some constants 0 < λΓ ≤ ΛΓ dependingonly on the λj ’s and Λj ’s in (4.45).

14 DIEGO MALDONADO

Notice that, given S ⊂ Ω, the condition |(D2ϕγ)−1/2b| ∈ Ln(S, dµϕγ ) from (1.13) means

ˆS|(D2ϕγ)−1/2b(x)|ndµϕγ (x) =

ˆS

ΠΓ(x)−n|b(x)|nm∏j=1

(1 + γj)|xj |njγj dx

'ˆS|b(x)|n

m∏j=1

|xj |−(n−nj)γj dx <∞,

where the implicit constants depend only on the γj ’s and the λj ’s, Λj ’s from (4.45). Similarlywith c, f ∈ Ln(S, dµϕγ ). Thus, Theorem 1 yields the following Harnack inequality.

Theorem 7. Fix an open bounded set Ω ⊂ Rn, γ = (γ1, . . . , γm) ∈ [0,∞)m, and consider thedegenerate PDE

(4.48) GΓ(u) + 〈b,∇u〉+ cu = f

where GΓ denotes the degenerate operator in (4.46). Then, there exist constants 0 < τ <1 < KH (depending only on γ and n) as well as structural constants 0 < εH < 1 < MH ,such that for every section Sϕγ (z,R) with SKHR := Sϕγ (z,KHR) ⊂⊂ Ω and every u ∈C(SKHR) ∩W 2,n(SKHR) satisfying u ≥ 0 in SKHR and solving the degenerate PDE (4.48)a.e. in SKHR, we have that the inequalities(4.49)

supy∈Sϕγ (z,R)

ρ≤r

ρ− 12 |Sϕγ (y, ρ)|

1n− 1

2p ‖b‖Ln(Sϕγ (y,ρ),

m∏j=1|xj |−(n−nj)γj dx)

‖D2ϕγ‖12

Lp(Sϕγ (y,ρ), dx)

≤ εHand

|SKHR|1n ‖c‖

Ln(SKHR,m∏j=1|xj |−(n−nj)γj dx)

≤ εH

imply the Harnack inequality

supSϕγ (z,τR)

u ≤MH

infSϕγ (z,τR)

u+ |SKHR|1n ‖f‖

Ln(SKHR,m∏j=1|xj |−(n−nj)γj dx)

.

Again, here p is any number bigger than n and, regarding the Lebesgue measure of thesections Sϕγ (z,R), we have

(4.50) |Sϕγ (z,R)| ' Rm∑j=1

nj2+γj ∀z ∈ Rn, R > 0,

where the implicit constants depend only on m and the nj ’s for j = 1, . . . ,m.

4.2. The singular case. In this case let us consider the singular elliptic operator

(4.51) HΓ(u) =1

Γm(x1)∆1u(x) + · · ·+ 1

Γm(xm)∆mu(x),

which can be written as HΓ(u) = trace(AΓD2u) where AΓ is the n × n matrix in (4.47).

Reasoning along the lines of the previous example, we obtain the following Harnack inequalityfor the singular operator HΓ with unbounded drifts.

15

Theorem 8. Fix an open bounded set Ω ⊂ Rn, γ = (γ1, . . . , γm) ∈ [0,∞)m, and consider thesingular PDE

(4.52) HΓ(u) + 〈b,∇u〉+ cu = f

where HΓ denotes the singular operator in (4.51). Then, there exist constants 0 < τ <1 < KH (depending only on γ and n) as well as structural constants 0 < εH < 1 < MH ,such that for every section Sϕγ (z,R) with SKHR := Sϕγ (z,KHR) ⊂⊂ Ω and every u ∈C(SKHR) ∩W 2,n(SKHR) satisfying u ≥ 0 in SKHR and solving the singular PDE (4.52) a.e.in SKHR, we have that the inequalities(4.53)

supy∈Sϕγ (z,R)

ρ≤r

ρ− 12 |Sϕγ (y, ρ)|

1n− 1

2p ‖b‖Ln

(Sϕγ (y,ρ),

m∏j=1|xj |njγj dx

)‖D2ϕγ‖12

Lp(Sϕγ (y,ρ), dx)

≤ εHand

|SKHR|1n ‖c‖

Ln(SKHR,m∏j=1|xj |njγj dx)

≤ εH

imply the Harnack inequality

supSϕγ (z,τR)

u ≤MH

infSϕγ (z,τR)

u+ |SKHR|1n ‖f‖

Ln(SKHR,m∏j=1|xj |njγj dx)

.

Once more, here p is any number bigger than n and the Lebesgue measure of a sectionSϕγ (z,R) behaves as in (4.50).

Remark 9. The conditions (4.49) and (4.53) can be commented upon as done in Remarks 4and 6 and we leave the details to the interested reader.

5. Preliminaries

5.1. Geometric and structural constants. Constants depending only on dimension n,on the constants Cd and α from the doubling condition µϕ ∈ (DC)ϕ in (1.6), and on thefunction ζ from hypothesis H3 will be called geometric constants.

Constants depending only on λ and Λ in (1.2), on the exponent p > n from hypothesis

H5, on the constant σ > 0 from hypothesis H6, on ‖(D2ϕ)−1/2b‖Ln(Ω, dµϕ), as well as ongeometric constants, will be called structural constants.

5.2. Doubling properties. By [3, Lemma 5.2], and this requires only the hypothesis H1,we have the following doubling property for the Lebesgue measure on Monge-Ampere sections

(5.54) |Sϕ(x, 2r)| ≤ 2n|Sϕ(x, r)| ∀x ∈ Rn, r > 0,

which in turn implies

(5.55) |Sϕ(x, s)| ≤ 2n(sr

)n|Sϕ(x, r)| ∀x ∈ Rn, 0 < r < s.

By [27, Lemma 2.1], under hypotheses H1 and H2 there exists a geometric constant KD > 1such that

(5.56) µϕ(Sϕ(x, s)) ≤ KD

(sr

)nµϕ(Sϕ(x, r)) ∀x ∈ Rn, 0 < r < s.

From [8, Theorem 8], there exist geometric constants 0 < κ ≤ 1 ≤ K2 such that

(5.57) κnrn ≤ |Sϕ(x, r)|µϕ(Sϕ(x, r)) ≤ Kn2 r

n ∀x ∈ Rn, r > 0.

16 DIEGO MALDONADO

5.3. On the hypothesis H3. In the literature on real analysis on spaces of homogeneoustype, the inequality (1.8) is sometimes referred to as the annular decay property or ringcondition and it quantifies the relative smallness if thin annuli with respect to µϕ. Typicalchoices for the function ζ include ζ(ε) = Cεα0 or ζ(ε) = C| log(ε)|−α0 for some constantsC0, α0 > 0.

In this section we point out that (1.8) is implied by the following Coifman-Feffermancondition for µϕ (see [3, Section 5]): There exist constants C0, θ0 > 1 such that for everysection S := Sϕ(x, r) and every Borel set E ⊂ S it holds true that

(5.58)µϕ(E)

µϕ(S)≤ C0

( |E||S|

)θ0.

Indeed, by Lemma 5.2(b) from [3] for every x ∈ Rn, t > 0 and δ ∈ (0, 1) we have

|Sϕ(x, t) \ Sϕ(x, δt)| ≤ n(1− δ)|Sϕ(x, t)|,so that, given R > 0 and ε ∈ (0, 1), by taking t := R(1 + ε) and δ := (1 + ε)−1 we get

|Sϕ(x, (1 + ε)R) \ Sϕ(x,R)| = |Sϕ(x, t) \ Sϕ(x, δt)| ≤ n(1− δ)|Sϕ(x, t)|=

nε

1 + ε|Sϕ(x, (1 + ε)R)| ≤ nε|Sϕ(x, (1 + ε)R)|.

Hence, by using (5.58) with S := Sϕ(x, (1 + ε)R) and E := Sϕ(x, (1 + ε)R) \ Sϕ(x,R) we

obtain (1.8) with ζ(ε) := C0nθ0εθ0 for every ε ∈ (0, 1).

The condition (5.58) is equivalent to the µ∞-condition for µϕ mentioned in Section 1.3 ofthe Introduction, see [3, Section 5] for further characterizations. In Sections 11 and 12 wewill see that the condition (5.58) is satisfied by the power functions |x|p, with p > 1, as wellas their tensor sums.

5.4. The Aleksandrov–Bakelman–Pucci maximum principle. We will use the follow-ing version of the ABP maximum principle. See [15, Section 9.1 and Exercise 9.3] and [4,Chapter 6, pp.79-87].

Lemma 10. Suppose that ϕ satisfies H1 and H4. Fix an open, convex, bounded set S ⊂ Rnand consider the operator

L(h)(x) := trace(A(x)D2h(x)) + 〈D2ϕ(x)−12 b(x),∇h(x)〉+ c(x)h(x),

where A(x) is a symmetric, nonnegative-definite n × n matrix and c(x) ≤ 0 for a.e. x ∈ S.Given E ⊂ S define N(b, E) as

(5.59) N(b, E)n := exp

[2n−2

nnωn

(1 +

ˆE

|D2ϕ(x)−12 b(x)|n dx

detA(x)

)]− 1.

Then, there exists a dimensional constant Cabp > 0 such that for every h ∈ C(S) ∩W 2,nloc (S)

satisfying L(h)(x) ≥ g(x) a.e. x ∈ S the following hold true:

(i) If sup∂S

h ≤ 0 then

(5.60) supSh ≤ CabpN(b, S)|S| 1n

∥∥∥∥∥ g

(detA)1n

∥∥∥∥∥Ln(S)

.

17

(ii) If sup∂S

h = 0 then

(5.61) supSh ≤ CabpN(b,Γ+

h (S))|S| 1n∥∥∥∥∥ g

(detA)1n

∥∥∥∥∥Ln(Γ+

h (S))

,

where Γ+h (S) denotes the upper contact set of h in S and defined as

Γ+h (S) := y ∈ S : ∃ p ∈ Rn such that h(x) ≤ h(y) + 〈p, x− y〉 ∀x ∈ S ∩ y ∈ S : h(y) ≥ 0.

Remark 11. The role of the convexity of S in Lemma 10 is to allow for |S| 1n , instead ofthe diameter of S (as in [15, Theorem 9.1] and [4, Theorem 1.9]), on the right-hand sideof (5.61). Indeed, the convexity of S and F. John’s lemma imply the existence of an affine

transformation T : Rn → Rn such that B(0, n−1/2) ⊂ T (S) ⊂ B(0, 1) and the change ofvariable y := Tx yields (5.61).

This technique also proves a version of Lemma 10 on sets of the form S0 := S \ S0 for any

S0 ⊂⊂ S. Namely, if h ∈ C(S0)∩W 2,nloc (S0) with h ≤ 0 on ∂S0 satisfying L(h)(x) ≥ g(x), for

a.e. x ∈ S0, with c(x) ≤ 0 a.e. x ∈ S0, we have

(5.62) supS0

h ≤ CabpN(b, S)|S| 1n∥∥∥∥∥ g

(detA)1n

∥∥∥∥∥Ln(S)

.

Notice the factor |S| 1n (as opposed to |S0| 1n ) on the right-hand side of (5.62). The estimate(5.62) will be used in the proof of Theorem 20 with S = Sϕ(x0, R) and S0 an annulus of theform Sϕ(x0, R) \ Sϕ(x0, r) for some x0 ∈ Rn and 0 < r < R.

Remark 12. Notice the misprint on [15, p.224] (repeated on [4, p.87]) where, instead of theright-hand side of (5.59), it reads

exp

[2n−2

nnωn

ˆΓ+h (S)

(1 +|D2ϕ(x)−

12 b(x)|n

detA(x)

)dx

]− 1,

which would make for a spurious |Γ+h (S)|.

5.5. A Morrey-type estimate. The next lemma is a consequence of the existence of weakPoincare inequalities in quasi-metric spaces, see for instance [1, Proposition 5.48], we includea proof of the Monge-Ampere version in the Appendix in Section 13.

Lemma 13. Suppose that ϕ satisfies H1 and H2. Given q > 2n, there exists a constantCq,K > 0, depending only on q and K, where K ≥ 1 is the quasi-triangle constant in (1.7),such that for every x, y ∈ Ω with Sϕ(x, 2Kδϕ(x, y)) ⊂⊂ Ω and h ∈ C1(Sϕ(x, 2Kδϕ(x, y))) wehave

(5.63) |h(x)− h(y)| ≤ Cq,K δϕ(x, y)12

( Sϕ(x,2Kδϕ(x,y))

|∇ϕh(z)|q dz) 1

q

.

Remark 14. Lemma 13 will be used as follows: For j = 1, . . . , n, notice that the function ϕjsatisfies

|∇ϕϕj |2 = 〈(D2ϕ)−1∇ϕj ,∇ϕj〉 = ϕjj ≤ ∆ϕ,

hence, by applying (5.63) with h = ϕj we obtain

(5.64) |∇ϕ(x)−∇ϕ(y)| ≤ n1/2Cq,Kδϕ(x, y)12

( Sϕ(x,2Kδϕ(x,y))

|∆ϕ(z)|q/2 dz) 1

q

.

18 DIEGO MALDONADO

In particular, given a section Sϕ(x0, 2Kr) ⊂⊂ Ω, the inequality (5.64) with q := 2p, wherep > n is as in the hypothesis H5, implies that

(5.65) |∇ϕ(x0)−∇ϕ(x)| ≤ K(p)r12

( Sϕ(x0,2Kr)

‖D2ϕ(z)‖p dz) 1

2p

,

for every x ∈ Sϕ(x0, 2Kr), where K(p) := n3/2C2p,K . We will repeatedly use the inequality(5.65) to deal with presence of the unbounded drift b in the first-order term of LϕA.

Notice that if we set ρϕ(x, y) := δϕ(x, y)12 for every x, y ∈ Ω, then the inequality (5.65)

gives that ∇ϕ is locally Lipschitz in Ω with respect to the quasi-distance ρϕ.

5.6. A local Vitali covering lemma under µϕ ∈ (DC)ϕ. Monge-Ampere versions ofVitali’s covering lemma under the hypothesis that µϕ ∼ 1 (in the sense of (1.21)) haveappeared, for instance, in [7, Lemma 1] and [20, Lemma 2.15]. For the sake of completeness,in the Appendix we include a proof for the following local version of Vitali’s covering lemmaunder the assumption µϕ ∈ (DC)ϕ only and with an explicit value of the Vitali constantKV > 1 in terms of the quasi-triangle constant K from (1.7).

Lemma 15. (Local Vitali covering lemma for Monge-Ampere sections.) Suppose that ϕsatisfies H1 and H2. Let Ω ⊂ Rn be an open bounded set. Fix KV > 2K2+K, where K is thequasi-triangle constant from (1.7). Given a section S0 := Sϕ(x0, R0) with Sϕ(x0, 2R0) ⊂⊂ Ω,a subset E ⊂ S0, and a covering Sϕ(x, rx)x∈I of E such that Sϕ(x, 2rx) ⊂⊂ S0 for everyx ∈ I, there exists a finite/countable sub-collection Sϕ(xj , rxj )j∈J of pairwise disjointsections such that ⋃

x∈ISϕ(x, rx) ⊂

⋃j∈J

Sϕ(xj ,KV rxj ).

6. On the construction of Monge-Ampere geodesics and the segmentproperties for δφ

The results proved in this Section 6 will be used in Section 10, but they contribute to thestudy of geometric properties of sections of general convex functions. As such they are ofindependent interest and will require no doubling conditions on the Monge-Ampere measure.

Let Ω0 ⊂ Rn be an open convex set. For a strictly convex function φ ∈ C1(Ω0) set

Sφ(x, r) := x ∈ Ω0 : δφ(x, y) < rwhere

δφ(x, y) := φ(y)− φ(x)− 〈∇φ(x), y − x〉 ∀x, y ∈ Ω0.

Theorem 16. Let Ω0 ⊂ Rn be an open convex set and φ ∈ C1(Ω0) be a strictly convexfunction. Then, the Monge-Ampere quasi-distance δφ possesses the following two properties:

(i) The segment property: Given x, z ∈ Ω0 with Sφ(x, δφ(x, z)) ⊂⊂ Ω0 and 0 < r < δφ(x, z),there exists y ∈ Sφ(x, δφ(x, z)) such that δφ(x, y) = r and

(6.66) δφ(x, y) + δφ(y, z) = δφ(x, z).

(ii) The segment-prolongation property: Given x, z ∈ Ω0 and R > δϕ(x, z) with Sφ(x,R) ⊂⊂Ω0, there exists y ∈ ∂Sφ(x,R) (that is, δφ(x, y) = R) such that

(6.67) δφ(x, y) = δφ(x, z) + δφ(z, y).

19

Proof. In order to prove (6.66), given x, z ∈ Ω0 with Sφ(x, δφ(x, z)) ⊂⊂ Ω0 and 0 < r <δφ(x, z) let P be any hyperplane passing through z and tangent to ∂Sφ(x, r) and let y ∈P ∩ ∂Sφ(x, r), see Figure 1.

xy

z

P ∇φ(y)−∇φ(x)

Sφ(x, r)

Sφ(x, δφ(x, z))

Figure 1. The segment property

Since y ∈ ∂Sφ(x, r) if and only if y belongs to the level set φ(y)−φ(x)−〈∇φ(x), y−x〉 = r,it follows that the vector ∇φ(y) − ∇φ(x) is orthogonal to P and then to z − y. That is,〈∇φ(x)−∇φ(z), z − y〉 = 0, so that

δφ(x, y) + δφ(y, z) = φ(y)− φ(x)− 〈∇φ(x), y − x〉+ φ(z)− φ(y)− 〈∇φ(y), z − y〉= φ(z)− φ(x)− 〈∇φ(x), z − x〉+ 〈∇φ(x), z − x〉 − 〈∇φ(x), y − x〉 − 〈∇φ(y), z − y〉= δφ(x, z) + 〈∇φ(x)−∇φ(z), z − y〉 = δφ(x, z) + 0 = δφ(x, z).

Notice that the choice of y as in (6.66) is by no means unique since each hyperplane P willproduce one such y. Similarly, in order to prove (6.67), given x, z ∈ Ω0 and R > δϕ(x, z)with Sφ(x,R) ⊂⊂ Ω0, let P be the hyperplane tangent to Sφ(x, δφ(x, z)) at z (notice thatz ∈ ∂Sφ(x, δφ(x, z)), that is, z belongs to the level set φ(z) − φ(x) − 〈∇φ(x), z − x〉 = rxz,where rxz := δφ(x, z)), see Figure 2.

x

z

y

P∇φ(z)−∇φ(x)

Sφ(x, δφ(x, z))

Sφ(x,R)

Figure 2. The segment-prolongation property

Now, pick any y ∈ P ∩ ∂Sφ(x,R) to obtain δφ(x, y) = R as well as the fact that the vector∇φ(z)−∇φ(x) is orthogonal to P and then to z − y. That is, 〈∇φ(x)−∇φ(z), y − z〉 = 0,

20 DIEGO MALDONADO

so that

δφ(x, z) + δφ(z, y) = φ(z)− φ(x)− 〈∇φ(x), z − x〉+ φ(y)− φ(z)− 〈∇φ(z), y − z〉= φ(y)− φ(x)− 〈∇φ(x), y − x〉+ 〈∇φ(x), y − x〉 − 〈∇φ(x), z − x〉 − 〈∇φ(z), y − z〉= δφ(x, y) + 〈∇φ(x)−∇φ(z), y − z〉 = δφ(x, y) + 0 = δφ(x, y),

and the proof of Theorem 16 is complete.

Theorem 17. Let Ω0 ⊂ Rn be an open convex set and φ ∈ C1(Ω0) be a strictly convexfunction. Given any section S := Sφ(x0, R) ⊂⊂ Ω0 and y0, y1 ∈ S there exists a continuouscurve γ : [0, 1]→ S such that γ(0) = y0, γ(1) = y1 and

(6.68) δφ(γ(t), γ(t′)) = (t′ − t)δφ(y0, y1) ∀ 0 ≤ t ≤ t′ ≤ 1.

Proof. Let us start by noticing that given any x, z ∈ Sφ(x0, R) it is always possible to finda midpoint y from x to z (in the sense that δφ(x, y) = δφ(x, z)/2) such that y ∈ Sφ(x0, R).Otherwise, the whole section Sφ(x, δφ(x, z)/2) would lie outside Sφ(x0, R) contradicting thefact that x ∈ Sφ(x0, R), see Figure 3.

x0

x

z

y

y′Sφ(x0, R)

Sφ(x, δφ(x, z)/2)

Figure 3. Both y and y′ are midpoints from x to z, but y ∈ Sφ(x0, R).

Now, given y0, y1 ∈ S set r := δφ(y0, y1). Let y1/2 ∈ S be a mid-point between y0 and y1,let y1/4 ∈ S be midpoint between y0 and y1/2, let y3/4 ∈ S a midpoint between y1/2 and y1,and so on. This defines a function γ : Dom(γ) ⊂ [0, 1] → S, where Dom(γ) consists of thenumbers of the form k2−j for j ∈ N0 and k ∈ 0, 1, . . . , 2j, and γ(k2−j) = yk2−j . For a fixedj ∈ N0 and k ≤ k′ we have

δφ(γ(k2−j), γ(k′2−j)) = δφ(yk2−j , yk′2−j ) = (k′ − k)2−jr.

For j, j′ ∈ N0 and k ∈ 0, 1, . . . , 2j and k′ ∈ 0, 1, . . . , 2j′ with k2−j < k′2−j′

we have

k2−j = k2j′−j2−j

′=: k′′2−j

′< k′2−j

′,

so that

δφ(γ(k2−j), γ(k′2−j′)) = δφ(γ(k′′2−j

′), γ(k′2−j

′)) = (k′ − k′′)2−j′r = (k′2−j

′ − k2−j)r.

That is, for every t, t′ ∈ Dom(γ) with t < t′ we have δφ(γ(t), γ(t′)) = (t′ − t)r.The fact that φ is strictly convex makes the topology generated by the sections equivalent

to the Euclidean topology, which makes Sϕ(x0, R) a complete topological subspace of Ω0.Therefore, the mapping γ has a continuous extension to all of [0, 1] (that we also denote asγ) such that the δφ-geodesic condition (6.68) holds.

21

Remark 18. In other words, Theorem 17 says that the Monge-Ampere sections are geodesi-cally convex.

Corollary 19. (Chain condition). For every section S := Sφ(x0, R) ⊂⊂ Ω0 and every

x, y ∈ S and N ∈ N there exists a finite sequence yjNj=0 ⊂ S such that y0 = x, yN = y and

δφ(yj−1, yj) =1

Nδφ(x, y) ∀j = 1, . . . , N.

Proof. Let γ : [0, 1] → S be a geodesic joining x to y as in Theorem 17, then for eachj ∈ 0, . . . , N set yj := γ(j/N) ∈ S.

7. The “passage to the double section” for supersolutions

The hypothesis H6 will play a central role in the proof of the following “doubling property”for inf u.

Theorem 20. Suppose that ϕ satisfies H1, H2, H4, H5, H6 and fix A ∈ E(λ,Λ, ϕ,Ω). LetSr := Sϕ(x0, r) be a section with S2r := Sϕ(x0, 2r) ⊂⊂ Ω and let u ∈ C(S2r) ∩W 2,n(S2r)satisfy u ≥ 0 in S2r and LϕA(u) ≤ f a.e. in S2r. Then, there exist structural constantsε1, γ ∈ (0, 1), such that the inequalities

|S2r|1n ‖f‖Ln(S2r, dµϕ) ≤ ε1,(7.69)

r−12 |S2r|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(S2r, dµϕ)‖D2ϕ‖12

Lp(S2r, dx) ≤ ε1,(7.70)

|S2r|1n ‖c‖Ln(S2r, dµϕ) ≤ ε1,(7.71)

and infSϕ(x0,r)

u ≥ 1 imply infSϕ(x0,3r/2)

u ≥ γ.

Proof. For t > r with Sϕ(x0, t) ⊂⊂ Ω, let Annϕ(x0, t, r) denote the annulus Annϕ(x0, t, r) :=Sϕ(x0, t) \ Sϕ(x0, r). For s > 1 to be determined later, set

v(x) :=

(δϕ(x0, x)

r

)−s− 2−s

so that

0 < v(x) < 1 ∀x ∈ Annϕ(x0, 2r, r),(7.72)

(3/2)−s − 2−s ≤ v(x), ∀x ∈ Annϕ(x0, 3r/2, r),(7.73)

∇v(x) = −sr

(δϕ(x0, x)

r

)−s−1

(∇ϕ(x)−∇ϕ(x0)) ∀x ∈ Annϕ(x0, 2r, r)(7.74)

and, for a.e. x ∈ Annϕ(x0, 2r, r),

D2v(x) =s(s− 1)

r2

(δϕ(x0, x)

r

)−s−2

(∇ϕ(x)−∇ϕ(x0))⊗ (∇ϕ(x)−∇ϕ(x0))

− s

r

(δϕ(x0, x)

r

)−s−1

D2ϕ(x).

Notice that we have v ∈ C(Annϕ(x0, 2r, r)) ∩ W 2,n(Annϕ(x0, 2r, r)) due to the fact that‖D2ϕ‖ ∈ Lp(Ω) with p > n by hypothesis H5.

22 DIEGO MALDONADO

Now, suppose that u ∈ C(S2r) ∩W 2,nloc (S2r) satisfies LϕA(u) ≤ f and u ≥ 0 in S2r with

infSϕ(x0,r)

u ≥ 1. Then if x ∈ ∂S2r we have

v(x)− u(x) ≤ v(x) = 2−s − 2−s = 0

and if x ∈ ∂Sϕ(x0, r), then

v(x)− u(x) ≤ 1− 2−s − u(x) ≤ −2−s ≤ 0.

Therefore, v − u ≤ 0 on ∂(Annϕ(x0, 2r, r)). On the other hand, for a.e. x ∈ Annϕ(x0, 2r, r)

LϕA(v)(x)

=s(s− 1)

r2

(δϕ(x0, x)

r

)−s−2

〈A(x)(∇ϕ(x)−∇ϕ(x0)), (∇ϕ(x)−∇ϕ(x0))〉

− s

r

(δϕ(x0, x)

r

)−s−1

trace(A(x)D2ϕ(x)) + 〈b(x),∇ϕv(x)〉+ c(x)v(x).

From (1.2) we get

〈A(x)(∇ϕ(x)−∇ϕ(x0)), (∇ϕ(x)−∇ϕ(x0))〉≥ λ〈D2ϕ(x)−1(∇ϕ(x)−∇ϕ(x0)), (∇ϕ(x)−∇ϕ(x0))〉 = λDϕ(x0, x)

as well as trace(A(x)D2ϕ(x)) ≤ nΛ for a.e. x ∈ S2r. Hence,

s(s− 1)

r2

(δϕ(x0, x)

r

)−s−2

〈A(x)(∇ϕ(x)−∇ϕ(x0)), (∇ϕ(x)−∇ϕ(x0))〉

− s

r

(δϕ(x0, x)

r

)−s−1

trace(A(x)D2ϕ(x))

≥ s

r2

(δϕ(x0, x)

r

)−s−2

((s− 1)λDϕ(x0, x)− nΛδϕ(x0, x)) .

At this point fix s = s(n, σ,Λ/λ) so that

(7.75) s ≥ nΛ

σλ+ 1,

where σ > 0 is as in (1.10) from hypothesis H6. Thus, (1.10) and (7.75) now yield

Dϕ(x0, x)

δϕ(x0, x)≥ σ ≥ nΛ

λ(s− 1)

and then

(s− 1)λDϕ(x0, x)− nΛδϕ(x0, x) ≥ 0 for a.e.x ∈ Annϕ(x0, 2r, r).

Consequently, by writing c(x) as c(x) = c+(x)− c−(x), for a.e. x ∈ Annϕ(x0, 2r, r) we get

LϕA(v)(x) ≥ 〈b(x),∇ϕv(x)〉+ c(x)v(x)

= −sr

(δϕ(x0, x)

r

)−s−1

〈D2ϕ(x)−1/2b(x), (∇ϕ(x)−∇ϕ(x0))〉+ c(x)v(x)

≥ −sr|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − c−(x),

23

where for the last inequality we used that for x ∈ Annϕ(x0, 2r, r) we have 1 ≤ δϕ(x0,x)r < 2

and 0 < v(x) < 1, see (7.72). Next, since LϕA(u) ≤ f a.e. in S2r, it follows that

LϕA(v − u)(x) = trace(A(x)D2(v − u)(x)) + 〈b(x),∇ϕ(v − u)(x)〉+ c(x)(v − u)(x)

≥ −sr|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − c−(x)− f(x)

a.e. in Annϕ(x0, 2r, r). Therefore, by writing c(x) = c+(x) − c−(x) again and using thatu ≥ 0 in S2r, a.e. in Annϕ(x0, 2r, r) we obtain

trace(A(x)D2(v − u)(x)) + 〈b(x),∇ϕ(v − u)(x)〉 − c−(x)(v − u)(x)

≥ −sr|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − c−(x)− f(x)− c+(x)(v(x)− u(x))

≥ −sr|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − c−(x)− f(x)− c+(x),

which gives

(7.76) trace(A(x)D2(v − u)(x)) + 〈b(x),∇ϕ(v − u)(x)〉 − c−(x)(v − u)(x) ≥ g1(x)

where

g1(x) := −sr|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − |c(x)| − f(x).

Now, by the ABP maximum principle (5.62) applied to (7.76) and v − u on Annϕ(x0, 2r, r)we get

supAnnϕ(x0,2r,r)

(v − u) ≤ CabpN(b, S2r)|S2r|1n

∥∥∥∥∥ g1

(detA)1n

∥∥∥∥∥Ln(S2r)

,

where, from the definition of N(b, S2r) in (5.59) and the first inequality in (1.2) (which yieldsdetA ≥ λn(detD2ϕ)−1),

N(b, S2r)n = exp

[2n−2

nnωn

(1 +

ˆS2r

|D2ϕ(x)−12 b(x)|n dx

detA(x)

)]− 1

≤ exp

[2n−2

nnωn

(1 + λ−n‖D2ϕ−

12 b‖nLn(Ω, dµϕ)

)]− 1 =: C1(n, λ, b)n,(7.77)

with C1(n, λ, b) > 0, depending only on n, λ and ‖(D2ϕ)−1/2b‖Ln(Ω, dµϕ), is a structuralconstant. Again by the first inequality in (1.2),∥∥∥∥∥ g1

(detA)1n

∥∥∥∥∥Ln(S2r)

≤ s

λr‖D2ϕ−1/2b‖Ln(S2r, dµϕ)‖∇ϕ(·)−∇ϕ(x0)‖L∞(S2r)

+1

λ‖c‖Ln(S2r, dµϕ) +

1

λ‖f‖Ln(S2r, dµϕ),

so that, from (7.69) and (7.71),

|S2r|1n

∥∥∥∥∥ g1

(detA)1n

∥∥∥∥∥Ln(S2r)

≤ s|S2r|1n

λr‖D2ϕ−1/2b‖Ln(S2r, dµϕ)‖∇ϕ(·)−∇ϕ(x0)‖L∞(S2r) +

2ε1

λ.

24 DIEGO MALDONADO

On the other hand, the Morrey-type estimate (5.65) and the hypothesis (7.70) give

|S2r|1n

r‖D2ϕ−1/2b‖Ln(S2r, dµϕ)‖∇ϕ(·)−∇ϕ(x0)‖L∞(S2r)

≤ K(p)r12|S2r|

1n

r‖D2ϕ−1/2b‖Ln(S2r, dµϕ)

( S2r

‖D2ϕ(z)‖p dz) 1

2p

= K(p)r−12 |S2r|

1n− 1

2p ‖D2ϕ−1/2b‖Ln(S2r, dµϕ)‖D2ϕ‖12

Lp(S2r, dx) ≤ K(p)ε1.(7.78)

Consequently,

supAnnϕ(x0,2r,r)

(v − u) ≤ CabpC1(n, λ, b)|S2r|1n

∥∥∥∥∥ g1

(detA)1n

∥∥∥∥∥Ln(S2r)

≤ CabpC1(n, λ, b)λ−1(2 + sK(p))ε1.

Thus, using (7.72), for every x ∈ Annϕ(x0, 3r/2, r) ⊂ Annϕ(x0, 2r, r) we have

(7.79) (3/2)−s − 2−s ≤ v(x) ≤ u(x) + CabpC1(n, λ, b)λ−1(2 + sK(p))ε1.

Finally, by defining the structural constant

(7.80) γ := (3/2)−s − 2−s − CabpC1(n, λ, b)λ−1(2 + sK(p))ε1

and choosing ε1 ∈ (0, 1), also structural, such that γ > 0 we obtain infSϕ(x0,3r/2)

u ≥ γ. For

instance, one could choose γ := 2s−13−s−2−(s+1), depending only on s as in (7.75), and thenε1 := γλ[CabpC1(n, λ, b)(2 + sK(p))]−1.

Remark 21. When b = 0 the hypothesis H5 can be weakened to ‖D2ϕ‖ ∈ Lnloc(Ω) in theproof of Theorem 20. Indeed, the hypothesis that ‖D2ϕ‖ ∈ Lploc(Ω), for some p > n, wasonly used to apply the Morrey-type estimate (5.65). If b = 0, then there is no need to control‖∇ϕ(·)−∇ϕ(x0)‖L∞(S2r).

Notice, however, that even in the case b = 0 the hypothesis ‖D2ϕ‖ ∈ Lnloc(Ω) is required

to guarantee that the test function v − u belongs to W 2,nloc (S2r), as required by the ABP

maximum principle in Lemma 10.

The next corollary quantifies iterations of Theorem 20 to pass from a section Sϕ(y0, ρ) toSϕ(y0, N0ρ) for an arbitrary N0 > 1.

Corollary 22. Suppose that ϕ satisfies H1, H2, H4, H5, H6 and fix A ∈ E(λ,Λ, ϕ,Ω). Forevery N0 > 1 there exists a constant γ(N0) ∈ (0, 1), depending only on the structural constantγ from Theorem 20 as well as on N0, such that for every section S2N0ρ := Sϕ(y0, 2N0ρ) ⊂⊂ Ω

and every u ∈ C(S2N0ρ)∩W 2,n(S2N0ρ) satisfying u ≥ 0 in S2N0ρ and LϕA(u) ≤ f a.e. in S2N0ρ

the assumptions

|S2N0ρ|1n ‖f‖Ln(S2N0ρ

, dµϕ) ≤ ε1γlnN0ln(3/2) ,(7.81)

ρ−12 |S2N0ρ|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(S2N0ρ, dµϕ)‖D2ϕ‖

12

Lp(S2N0ρ, dx) ≤ ε1,(7.82)

|S2N0ρ|1n ‖c‖Ln(S2N0ρ

, dµϕ) ≤ ε1,(7.83)

and infSϕ(y0,ρ)

u ≥ 1 imply infSϕ(y0,N0ρ)

u ≥ γ(N0). Here ε1 ∈ (0, 1) is the structural constant from

Theorem 20.

25

Proof. The proof of Corollary 22 is based on an iteration of Theorem 20. Notice that thehypotheses (7.81), (7.82), and (7.83) imply

(7.84)

|S2t|

1n ‖f‖Ln(S2t, dµϕ) ≤ ε1γ

lnN0ln(3/2) ,

t−12 |S2t|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(S2t, dµϕ)‖D2ϕ‖12

Lp(S2t, dx) ≤ ε1,

|S2t|1n ‖c‖Ln(S2t, dµϕ) ≤ ε1,

for every t ∈ [ρ,N0ρ] (recall that 1n − 1

2p > 0 because p > n from hypothesis H5). Thus,

given N0 > 1, let `0 ∈ N such that (3/2)`0−1 < N0 ≤ (3/2)`0 . Now, Theorem 20 applied to u,f , Sϕ(y0, ρ) (that is, t = ρ in (7.84)) and ε1 gives inf

Sϕ(y0,3ρ/2)u ≥ γ, then Theorem 20 applied

to u/γ, f/γ, ε1γ, and Sϕ(y0, 3ρ/2) (that is, t = 3ρ/2 in (7.84)) gives infSϕ(y0,(3/2)2ρ)

u ≥ γ2,

etc. At the (`0 − 1)-th iteration we apply Theorem 20 to u/γ`0−1, f/γ`0−1, ε1γ`0−1, and

Sϕ(y0, (3/2)`0−1ρ) (that is, t = (3/2)`0−1ρ in (7.84) and notice that 2(3/2)`0−1 < 2N0 so

that Sϕ(y0, 2(3/2)`0−1ρ) ⊂⊂ Ω) to obtain infSϕ(y0,(3/2)`0ρ)

u ≥ γ`0 . Next, since N0 ≤ (3/2)`0 , it

follows that infSϕ(y0,N0ρ)

u ≥ γ`0 ≥ γ1+

lnN0ln(3/2) =: γ(N0). Finally, notice that the definition of `0

implies γlnN0ln(3/2) ≤ γ`0−1.

8. A critical-density estimate for supersolutions

In this section we prove a critical-density estimate for nonnegative supersolutions (Theorem25). The proof differs from the ones for similar previous results (e.g. [3, Theorem 1], [20,Lemma 3.1], and [26, Theorem 2]) in the following ways: the hypothesis ϕ ∈ C2(Ω) withD2ϕ > 0 is not used, the Caffarelli-Gutierrez normalization technique from [3, Section 1]is not used, the first-order term of LϕA is handled by means of the Morrey-type estimate inLemma 13 and the hypothesis H5.

As another difference with previous proofs, by considering infima over the boundariesof sections (see (8.88) below) we first prove Theorem 23, which is a weaker version of thecritical-density estimate better suited to the ABP maximum principle in Lemma 10. Then,by means of Corollary 22 and the hypothesis H3, the condition on the infima over boundariesof sections is removed.

Theorem 23. Suppose that ϕ satisfies H1, H2, H4, H5 and fix A ∈ E(λ,Λ, ϕ,Ω). LetSr := Sϕ(x0, r) be a section with S2r := Sϕ(x0, 2r) ⊂⊂ Ω and let u ∈ C(S2r) ∩W 2,n(S2r)satisfy u ≥ 0 in S2r and LϕA(u) ≤ f a.e. in S2r. Then, there are structural constantsε0, ε2 ∈ (0, 1) such that the inequalities

|S2r|1n ‖f‖Ln(S2r, dµϕ) ≤ ε2,(8.85)

r−12 |S2r|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(S2r, dµϕ)‖D2ϕ‖12

Lp(S2r, dx) ≤ ε2,(8.86)

|S2r|1n ‖c‖Ln(S2r, dµϕ) ≤ ε2,(8.87)

infSru < 1, and inf

∂S2r

u < 1(8.88)

imply µϕ(x ∈ S2r : u(x) < 5) ≥ ε0 µϕ(S2r).

26 DIEGO MALDONADO

Proof. Introduce the test function

(8.89) h(x) := −u(x)− 4

(δϕ(x0, x)

2r− 1

)+ inf∂S2r

u ∀x ∈ S2r.

Then, h ∈ C(S2r) ∩W 2,n(S2r) with D2h(x) = −D2u(x) − 2rD

2ϕ(x) a.e. in S2r and, sinceδϕ(x0, x) = 2r for every x ∈ ∂S2r, we have sup

∂S2r

h = 0. By the hypothesis infSru < 1, there

exists x1 ∈ Sr (i.e., δϕ(x0, x1) < r) such that u(x1) < 1 and then

(8.90) h(x1) = −u(x1)− 4

(δϕ(x0, x1)

2r− 1

)> −1− 4

(1

2− 1

)= 1.

Also,

LϕA(h)(x) = −LϕA(u)(x)− 2

rtrace(A(x)D2ϕ(x))

− 2

r〈b(x), D2ϕ(x)−1/2(∇ϕ(x)−∇ϕ(x0))〉 − 4c(x)

(δϕ(x0, x)

2r− 1

).

Now, by the second inequality in (1.2) we have trace(A(x)D2ϕ(x)) ≤ nΛ a.e. x ∈ S2r and,since δϕ(x0, x) < 2r for every x ∈ S2r and c(x) = c+(x)− c−(x),

0 ≥ −4c(x)

(δϕ(x0, x)

2r− 1

)≥ −4c−(x) a.e. x ∈ S2r.

Therefore, by using that LϕA(u) ≤ f a.e. in S2r we get that, for a.e. x ∈ S2r,

LϕA(h)(x) = trace(A(x)D2h(x)) + 〈b(x),∇ϕh(x)〉+ c(x)h(x)

≥ −f(x)− 2nΛ

r− 2

r|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − 4c−(x),

which implies

trace(A(x)D2h(x)) + 〈b(x),∇ϕh(x)〉 − c−(x)h(x)

≥ −f(x)− 2nΛ

r− 2

r|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − 4c−(x)− c+(x)h(x),

and, since u ≥ 0 in S2r,

c+(x)h(x) = c+(x)

(−u(x)− 4

(δϕ(x0, x)

2r− 1

)+ inf∂S2r

u

)≤ c+(x)(4 + inf

∂S2r

u) < 5c+(x),

where for the last inequality we used (8.88). Hence,

trace(A(x)D2h(x)) + 〈b(x),∇ϕh(x)〉 − c−(x)h(x)

≥ −f(x)− 2nΛ

r− 2

r|D2ϕ(x)−1/2b(x)||∇ϕ(x)−∇ϕ(x0)| − 5|c(x)| =: g,

and by the ABP estimate (5.61) applied to h in S2r it follows that

(8.91) 1 ≤ CabpN(b,Γ+h (S2r))|S2r|

1n

∥∥∥∥∥ g

(detA)1n

∥∥∥∥∥Ln(Γ+

h (S2r))

,

27

where Γ+h (S2r) is the contact set from Lemma 10. In particular, recalling the definitions of h

and Γ+h (S2r), we have the inclusions

Γ+h (S2r) ⊂ x ∈ S2r : h(x) ≥ 0 = x ∈ S2r : u(x) < 4− 2δϕ(x0, x)/r + inf

∂S2r

u

⊂ x ∈ S2r : u(x) < 5,(8.92)

where for the last inclusion we used the hypothesis inf∂S2r

u < 1. Also, notice that the definition

of C1(n, λ, b) in (7.77) gives N(b,Γ+h (S2r)) ≤ N(b, S2r) ≤ C1(n, λ, b).

Using again that the first inequality in (1.2) implies detA ≥ λn(detD2ϕ)−1 we can write

|S2r|1n

∥∥∥∥∥ g

(detA)1n

∥∥∥∥∥Ln(Γ+

h (S2r))

≤ 1

λ|S2r|

1n ‖f‖Ln(S2r, dµϕ) +

2nΛ

λr|S2r|

1nµϕ(Γ+

h (S2r))1n

+2

λr|S2r|

1n ‖D2ϕ−1/2b‖Ln(S2r, dµϕ)‖∇ϕ(·)−∇ϕ(x0)‖L∞(S2r)

+5

λ|S2r|

1n ‖c‖Ln(S2r, dµϕ).

Now, from (5.54) and (5.57) we get

|S2r|1n

r=

(µϕ(S2r)|S2r|)1n

rµϕ(S2r)1n

≤ 2K2

µϕ(S2r)1n

,

which, along with the inclusion (8.92), gives

|S2r|1n

rµϕ(Γ+

h (S2r))1n ≤ 2K2

(µϕ(Γ+

h (S2r))

µϕ(S2r)

) 1n

≤ 2K2

(µϕ(x ∈ S2r : u(x) < 5)

µϕ(S2r)

) 1n

.

On the other hand, from the Morrey-type estimate (5.65) and in the same way we obtained(7.78), but now using the hypothesis (8.86), we get

1

r|S2r|

1n ‖D2ϕ−1/2b‖Ln(S2r, dµϕ)‖∇ϕ(·)−∇ϕ(x0)‖L∞(S2r) ≤ K(p)ε2.

Thus, coming back to (8.91) and by using the hypotheses (8.85) and (8.87), we deduce

1 ≤ CabpC1(n, λ, b)

λ

[(2K(p) + 6)ε2 + 4nK2Λ

(µϕ(x ∈ S2r : u(x) < 5)

µϕ(S2r)

) 1n

],

so that by taking ε2 > 0 in (8.85)–(8.87) as, for instance, the structural constant defined bythe equality CabpC1(n, λ, b)(2K(p) + 6)ε2 := λ/2 we obtain

4nK2CabpC1(n, λ, b)Λ

λ

(µϕ(x ∈ S2r : u(x) < 5)

µϕ(S2r)

) 1n

≥ 1

2

and the theorem follows with ε0 := λn(8nK2CabpC1(n, λ, b)Λ)−n, a structural constant.

Remark 24. As in Remark 21, we point out that when b = 0 the hypothesis H5 can beweakened to ‖D2ϕ‖ ∈ Lnloc(Ω) in the proof of Theorem 23 as well. Again, even in the caseb = 0 the hypothesis ‖D2ϕ‖ ∈ Lnloc(Ω) is required to guarantee that the test function h in

(8.89) belongs to W 2,nloc (S2r), as required by the ABP maximum principle in Lemma 10.

28 DIEGO MALDONADO

The next result strengthens Theorem 23 by removing the hypothesis inf∂S2r

u < 1 from (8.88).

The price to pay is that the critical density increases from (1−ε0) to (1− ε02 ) and that, instead

to having infSru ≥ 1, we obtain inf

Sru ≥ γ0 for some structural constant γ0 ∈ (0, 1). Here the

hypothesis H3 makes its first appearance.

Theorem 25. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). There exist geomet-ric constants κ0 ∈ (0, 1), K3 ≥ 2 and structural constants ε3, γ0 ∈ (0, 1) such that for everysection Sr := Sϕ(x0, r) with SK3r := Sϕ(x0,K3r) ⊂⊂ Ω and every u ∈ C(SK3r)∩W 2,n(SK3r)with u ≥ 0 in SK3r and LϕA(u) ≤ f a.e. in SK3r, the inequalities

|SK3r|1n ‖f‖Ln(SK3r

, dµϕ) ≤ ε3,(8.93)

r−12 |SK3r|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(SK3r, dµϕ)‖D2ϕ‖

12

Lp(SK3r, dx) ≤ ε3,(8.94)

|SK3r|1n ‖c‖Ln(SK3r

, dµϕ) ≤ ε3,(8.95)

and µϕ(x ∈ S2r : u(x) ≥ 5) ≥ (1 − ε02 )µϕ(S2r) imply inf

Sru ≥ γ0. Here ε0 ∈ (0, 1) is the

structural constant from Theorem 23.

Proof. Fix ε0 ∈ (0, 1) such that

ζ(ε0) <ε0

2(1− ε0),

where ζ is the non-decreasing function with ζ(0+) = 0 from the hypothesis H3. Hence, forevery ε ∈ (0, ε0), the annular-decay property (1.8) implies

µϕ(S(1+ε)2r \ S2r)

µϕ(S2r)≤ ζ(ε) ≤ ζ(ε0) <

ε0

2(1− ε0),

which yieldsµϕ(S(1+ε)2r)

µϕ(S2r)<

ε0

2(1− ε0)+ 1 =

1− ε0/2

1− ε0∀ε ∈ (0, ε0).

Then, from the hypothesis µϕ(x ∈ S2r : u(x) ≥ 5) ≥ (1− ε02 )µϕ(S2r), for every ε ∈ (0, ε0)

we get

µϕ(x ∈ S2(1+ε)r : u(x) ≥ 5) ≥ µϕ(x ∈ S2r : u(x) ≥ 5) ≥ (1− ε02 )µϕ(S2r)

> (1− ε0)µϕ(S(1+ε)2r),

so that µϕ(x ∈ S2(1+ε)r : u(x) < 5) < ε0µϕ(S(1+ε)2r) for every ε ∈ (0, ε0). Choose K3 ≥ 2and ε3 ≤ ε2, where ε2 ∈ (0, 1) is the structural constant from Theorem 23, and t = 2r in(8.93), (8.94), and (8.95). Given ε ∈ (0, ε0), by applying Theorem 23 (in its contrapositive)to the section S2(1+ε)r (notice that S2(1+ε)r ⊂ S4r ⊂ SK3r ⊂⊂ Ω) it follows that either

infS(1+ε)r

u ≥ 1 or inf∂S2(1+ε)r

u ≥ 1.

Now, if infS(1+ε)r

u ≥ 1 for some ε ∈ (0, ε0), then infSru ≥ 1 and the theorem gets proved with

γ0 := 1, ε3 := ε2, K3 := 2 and, say, κ0 := 1/2.On the other hand, if inf

S(1+ε)r

u < 1 for every ε ∈ (0, ε0), then we must have inf∂S2(1+ε)r

u ≥ 1

for every ε ∈ (0, ε0). In particular, it follows that u ≥ 1 in the annulus S2(1+ε0)r \ S2r =Sϕ(x0, 2(1 + ε0)r) \ Sϕ(x0, 2r). Next, by Theorem 3.3.10 on [16, p.60] there exists a con-stant κ0 ∈ (0, 1), depending only on ε0 and geometric constants, such that for every y ∈

29

∂Sϕ(x0, 2(1 + ε0/2)r) (the section Sϕ(x0, 2(1 + ε0/2)r) being the intermediate section be-tween Sϕ(x0, 2r) and Sϕ(x0, 2(1 + ε0)r)) we have the inclusion

Sϕ(y, κ0r) ⊂ Sϕ(x0, 2(1 + ε0)r) \ Sϕ(x0, 2r).

Thus, by fixing a y ∈ ∂Sϕ(x0, 2(1 + ε0/2)r), we obtain infSϕ(y,κ0r)

u ≥ 1. Also, since y ∈∂Sϕ(x0, 2(1 + ε0/2)r) ⊂ S3r, the quasi-triangle inequality (1.7) gives the inclusions

(8.96) Sϕ(x0, r) ⊂ Sϕ(y, 4Kr)

and

(8.97) Sϕ(y, 8Kr) ⊂ Sϕ(x0, (8K + 6)Kr).

The idea is to use Corollary 22 with the sections Sϕ(y, κ0r) and Sϕ(y, 4Kr) to pass frominf

Sϕ(y,κ0r)u ≥ 1 to inf

Sϕ(y,4Kr)u ≥ γ0 so that the latter along with the inclusion (8.96) proves

infSru ≥ γ0 and the theorem.

All that is left is the tuning of the constants to apply Corollary 22 to the sections Sϕ(y, κ0r)and Sϕ(y, 4Kr). In the notation from Corollary 22 put y0 := y, ρ := κ0r and N0 := 4K/κ0

so that 4Kr = N0ρ.By choosing K3 := (8K + 6)K and using the inclusion (8.97) we obtain Sϕ(y, 2N0ρ) =

Sϕ(y, 8Kr) ⊂ Sϕ(x0,K3r) ⊂⊂ Ω so that the inequalities (8.93) and (8.95) imply (7.81) and

(7.83), respectively, provided that ε3 ∈ (0, 1) satisfies ε3 ≤ ε1γln(4K/κ0)ln(3/2) and, regarding (7.82),

we get

ρ−12 |Sϕ(y, 2N0ρ)|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(y,2N0ρ), dµϕ)‖D2ϕ‖12

Lp(Sϕ(y,2N0ρ), dx)

≤ (κ0r)− 1

2 |Sϕ(x0,K3r)|1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(x0,K3r), dµϕ)‖D2ϕ‖12

Lp(Sϕ(x0,K3r), dx) ≤ ε1,

where the last inequality holds true provided that ε3 ≤ ε1κ1/20 . That is, the constants K3

and ε3 are given by K3 := (8K + 6)K (geometric) and ε3 := minε2, ε1κ1/20 , ε1γ

ln(4K/κ0)ln(3/2)

(structural). The structural constant γ0 equals γ0(4K/κ0) := γ1+

ln(4K/κ0)ln(3/2) from Corollary 22.

We close this section by combining Theorem 25 and Corollary 22 as follows.

Corollary 26. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). There existsa structural constant ε4 ∈ (0, 1) such that for every N0 ≥ K3 (where K3 ≥ 2 is the geo-metric constant from Theorem 25) there exists M(N0) > 1, depending only on structuralconstants as well as on N0, such that for every section S2N0r := Sϕ(x0, 2N0r) ⊂⊂ Ω and

every w ∈ C(S2N0r) ∩W 2,n(S2N0r) satisfying w ≥ 0 in S2N0r and LϕA(w) ≤ f a.e. in S2N0r

the assumptions

|S2N0r|1n ‖f‖Ln(S2N0r

, dµϕ) ≤ ε4,

r−12 |S2N0r|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(S2N0r, dµϕ)‖D2ϕ‖

12

Lp(S2N0r, dx) ≤ ε4,

|S2N0r|1n ‖c‖Ln(S2N0r

, dµϕ) ≤ ε4,

and infSϕ(x0,N0r)

w < 1 imply µϕ(x ∈ Sϕ(x0, 2r) : w(x) < M(N0)) ≥ ε02 µϕ(Sϕ(x0, 2r)). Here

ε0 ∈ (0, 1) is the structural constant from Theorem 23.

30 DIEGO MALDONADO

Proof. Given N0 ≥ K3, let γ(N0) := γ1+

lnN0ln(3/2) be the structural constant from Corollary 22.

Now, the assumption infSϕ(x0,N0r)

w < 1 means infSϕ(x0,N0r)

γ(N0)w < γ(N0) so that by applying

Corollary 22 to γ(N0)w on the section Sϕ(x0, N0r) (think y0 := x0 and ρ := r in the notationfrom Corollary 22) we obtain inf

Sϕ(x0,r)γ(N0)w < 1, provided that the inequalities (7.81), (7.82),

and (7.83) hold true, and this can be accomplished by choosing ε4 ≤ ε1γlnN0ln(3/2) .

Next, we use that infSϕ(x0,r)

γ(N0)w < 1 to write infSϕ(x0,r)

γ0γ(N0)w < γ0, where γ0 ∈ (0, 1)

is the structural constant from Theorem 25. By applying Theorem 25 to γ0γ(N0)w on thesection Sϕ(x0, r) we obtain

µϕ(x ∈ Sϕ(x0, 2r) : γ0γ(N0)w(x) < 5) ≥ ε02 µϕ(Sϕ(x0, 2r)),

provided that ε4 ≤ ε3 and that

|SK3r|1n ‖γ0γ(N0)f‖Ln(SK3r

, dµϕ) ≤ ε3

which follows from ε4 ≤ ε3 since γ0γ(N0) ∈ (0, 1). Hence, Corollary 26 is proved with

M(N0) := 5γ0γ(N0) and ε4 := min(ε1γ

lnN0ln(3/2) , ε3).

9. Mean-value inequalities for subsolutions

As a consequence of the critical-density estimate in Theorem 25, in this section we provea mean-value property for nonnegative subsolutions. The proof of Theorem 27 below followsfrom a combination of the proofs of Lemmas 5 and 6 from [26] and we sketch it in Section13.3 of the Appendix.

Let ε3 ∈ (0, 1) and K3 > 1 be the structural and geometric constants from Theorem 25.

Theorem 27. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). There existstructural constants 0 < ε5 < 1 < C6 such that for every section Sϕ(z, r) with S2Kr :=

Sϕ(z, 2Kr) ⊂⊂ Ω and every u ∈ C(S2Kr) ∩W 2,n(S2Kr) with u ≥ 0 in S2Kr and LϕA(u) ≥ fa.e. in S2Kr the inequalities

|S2Kr|1n ‖f‖Ln(S2Kr, dµϕ) + |S2Kr|

1n ‖c‖Ln(S2Kr, dµϕ) ≤ ε5,(9.98)

supx∈Sϕ(z,r)

ρ<r

(ρ−

12 |Sϕ(x, ρ)|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(x,ρ), dµϕ)‖D2ϕ‖12

Lp(Sϕ(x,ρ), dx)

)≤ ε5,(9.99)

andfflS2Kr

u(x) dµϕ(x) ≤ 1 imply supSϕ(z,r/2)

u ≤ C6.

Corollary 28. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). Let Sϕ(z, r) be a

section with S2Kr := Sϕ(z, 2Kr) ⊂⊂ Ω and let u ∈ C(S2Kr) ∩W 2,n(S2Kr) satisfy u ≥ 0 in

S2Kr and LϕA(u) ≥ f a.e. in S2Kr. Suppose that |S2Kr|1n ‖c‖Ln(S2Kr, dµϕ) ≤ ε5

2 and that (9.99)hold true. Then,

(9.100) supSϕ(z,r/2)

u ≤ C6

S2Kr

u(x) dµϕ(x) + 2C6ε−15 |S2Kr|

1n ‖f‖Ln(S2Kr, dµϕ).

Proof. Just divide the inequality LϕA(u) ≤ f by Q(u, f), where

Q(u, f) :=

S2Kr

u(x) dµϕ(x) + 2ε−15 |S2Kr|

1n ‖f‖Ln(S2Kr, dµϕ),

31

and apply Theorem 27 to u/Q(u, f).

Corollary 29. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). Then, thereexist geometric constants 0 < κ1 < κ2 ≤ 1 such that for every 0 < q < ∞ there is aconstant C7 > 0, depending only on structural constants as well as on q, so that for everysection Sr := Sϕ(z, r) with S2Kr := Sϕ(z, 2Kr) ⊂⊂ Ω and every u ∈ C(S2Kr) ∩W 2,n(S2Kr)that satisfies u ≥ 0 in S2Kr and LϕA(u) ≥ f a.e. in S2Kr we have that the inequalities

|S2Kr|1n ‖c‖Ln(S2Kr, dµϕ) ≤ ε5

2 and (9.99) imply

supSϕ(z,κ1r)

u ≤ C7

( Sϕ(z,κ2r)

u(x)q dµϕ(x)

) 1q

+ C7|Sr|1n ‖f‖Ln(S2Kr, dµϕ).

Proof. The self-improving properties of inequalities of the type (9.100) are well-known. More-over, the constant κ2 can be chosen close to κ1 at the price of C7 blowing up as a negativepower of κ2 − κ1. In the case of the Monge-Ampere quasi-metric space (Rn, δϕ, µϕ), suchself-improvement has been addressed in Theorem 7 from [26], see also Remark 10 on [26,p.2020].

10. On the power-like decay property and the weak-Harnack inequality

The next theorem establishes a power-like decay property for the distribution function ofnonnegative supersolutions. Its proof is based on Lemma 4 and Theorem 7 from [18] and thesegment properties from Section 6 will play a key role.

Let us choose KV := 2K(K + 1) > 2K2 + K as the Vitali constant in Lemma 15 andintroduce the geometric constants K0, K4, and K5 as

(10.101) K0 := 2K(1 +KKV ), K4 :=1 +KKV

2(K + 1)+

1

2KV, K5 := K(K4KV + 1),

where K3 := (8K + 6)K is the geometric constant from Theorem 25. In particular, it followsthat K3 ≤ K0 ≤ K5 and K4 > K2. Next, put M0 := M(K0) > 1, where M(K0) is as inCorollary 26 for N0 := K0 as in (10.101).

Let ε4 ∈ (0, 1) be the structural constant from Corollary 26.

Theorem 30. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). For every sectionSR := Sϕ(z,R) with S2K5R := Sϕ(z, 2K5R) ⊂⊂ Ω and every u ∈ C(S2K5R) ∩W 2,n(S2K5R)satisfying u ≥ 0 in S2K5R and LϕA(u) ≤ f a.e. in S2K5R the assumptions

|S2K5R|1n ‖f‖Ln(S2K5R

, dµϕ) ≤ ε4,

R−12 |S2K5R|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(S2K5R, dµϕ)‖D2ϕ‖

12

Lp(S2K5R, dx) ≤ ε4,

|S2K5R|1n ‖c‖Ln(S2K5R

, dµϕ) ≤ ε4,

and infSϕ(z,R)

u ≤ 1 imply

(10.102) µϕ(x ∈ Sϕ(z,R) : u(x) > Mk+10 ) ≤ (1− η0)kµϕ(Sϕ(z,R)), ∀k ∈ N0,

for a structural constant η0 ∈ (0, 1). In particular, there exist structural constants β2 > 0and M2 > 0 such that ˆ

Sϕ(z,R)uβ2(x) dµϕ(x) ≤Mβ2

2 .

32 DIEGO MALDONADO

Proof. Set S := Sϕ(z,R) and for k ∈ N0 define

Dk := y ∈ S : u(y) ≤Mk0 .

Notice the hypothesis infSϕ(z,R)

u ≤ 1 makes Dk 6= ∅ for every k ∈ N0. Fix k ∈ N0 and pick

x ∈ S \Dk. It follows that δϕ(x,Dk) < 2KR. Indeed, given x′ ∈ Dk we have

(10.103) δϕ(x,Dk) ≤ δϕ(x, x′) ≤ K(δϕ(z, x) + δϕ(z, x′)) < 2KR.

For x ∈ S \Dk define

(10.104) rx :=KV

2Kδϕ(x,Dk)

so that, in particular, from the fact that K ≥ 1, we have

(10.105) Sϕ(x, rx/KV ) ∩Dk = ∅.Now, either: rx ≤ 2KKV δϕ(x, z) or rx > 2KKV δϕ(x, z). In case of rx ≤ 2KKV δϕ(x, z), bythe segment property in Theorem 16 there exists y ∈ S such that

(10.106)rx

2KKV+ δϕ(y, z) = δϕ(x, y) + δϕ(y, z) = δϕ(x, z) < KR,

where the last inequality is due to fact that x ∈ S = Sϕ(z,R). On the other hand, ifrx > 2KKV δϕ(x, z), by the segment-prolongation property in Theorem 16 there exists y ∈∂Sϕ(x, rx/(2KKV )) such that

(10.107) δϕ(x, z) + δϕ(z, y) = δϕ(x, y) =rx

2KKV.

Next, we claim that in either case we have the inclusions

(10.108) Sϕ

(y,

rx2KKV

)⊂ Sϕ

(x,

rxKV

)⊂ Sϕ(z, 2KR),

so that by intersecting (10.108) with Dk+1 we get

(10.109) Sϕ

(y,

rx2KKV

)∩Dk+1 ⊂ Sϕ

(x,

rxKV

)∩Dk+1 ⊂ S.

In order to prove the first inclusion in (10.108), take x′ ∈ Sϕ(y, rx2KKV

), and by using either

(10.106) or (10.107) (since they both yield δϕ(x, y) = rx/(2KKV )) we get

δϕ(x, x′) ≤ K(δϕ(y, x′) + δϕ(x, y)) < K

(rx

2KKV+

rx2KKV

)=

rxKV

.

To see the second inclusion in (10.108), take x′′ ∈ Sϕ(x, rx/KV ) and do

δϕ(z, x′′) ≤ K(δϕ(z, x) + δϕ(x, x′′))

≤ K(R+ rx/KV ) = K

(R+

δϕ(x,Dk)

2K

)< 2KR,

where we used the definition of rx in (10.104) and (10.103). In addition, along the lines ofthe proof of the inclusions (10.108), by putting t := (1 +KKV )rx, we also have

(10.110) Sϕ (y, t/KV ) ⊂ Sϕ (x,K4rx) ⊂ Sϕ(z,K5R) ⊂⊂ Ω,

33

so that all the sections involved remain strictly inside Ω. The next claim is that, in eithercase (10.106) or (10.107), we have

(10.111) δϕ(y,Dk) ≤t

4KV:=

(1 +KKV )rx4KV

.

In order to prove (10.111), recall that KV = 2K(K + 1) and write

δϕ(y,Dk) ≤ K(δϕ(x, y) + δϕ(x,Dk))

= K

(rx

2KKV+

2KrxKV

)=

(1 + 4K2)rx2KV

≤ (1 +KKV )rx4KV

=:t

4KV.

In particular, (10.111) yields y′ ∈ Sϕ(y, t/(2KV )) such that u(y′) ≤Mk0 (indeed, notice that

for y′′ ∈ ∂Sϕ(y, t/(2KV )) we have δϕ(y, y′′) = t/(2KV ) > t/(4KV ) ≥ δϕ(y,Dk) by (10.111),so that for y′ ∈ Sϕ(y, t/(2KV )) but near ∂Sϕ(y, t/(2KV )), we still have δϕ(y, y′) > δϕ(y,Dk),

which implies u(y′) ≤Mk0 ). Therefore, by setting w := u/Mk

0 we get

infSϕ(y,t/(2KV ))

w ≤ 1.

Hence, since K5 ≥ K0 ≥ K3, we can apply Corollary 26 to w on the section Sϕ(y,K0r)with r := t/(2KVK0) (think x0 := y and N0 := K0 in Corollary 26) and noticing thatK0 ≥ 2K(1 +KKV ) (by its definition in (10.101)) we have

2r =t

KVK0=

(1 +KVK)rxKVK0

≤ rx2KKV

,

so that

µϕ

(Sϕ

(y,

rx2KKV

)∩Dk+1

)≥ µϕ (Sϕ(y, 2r) ∩Dk+1)

= µϕ (Sϕ(y, 2r) ∩ x ∈ S : w(x) ≤M0) = µϕ (x ∈ Sϕ(y, 2r) : w(x) ≤M0)

≥ ε02 µϕ(Sϕ(y, 2r)) = ε0

2 µϕ

(Sϕ

(y, t

K0KV

))≥ ε0

2KDKn0

µϕ(Sϕ(y, t/KV )),

where the second equality is due to the inclusion (10.109) (which implies Sϕ(y, 2r)∩Dk+1 ⊂S), the second inequality is due to Corollary 26 and the last one is due to the doublingproperty (5.56). Thus, by introducing the structural constant

η0 :=ε0

2KDKn0

we obtain

(10.112) µϕ

(Sϕ

(y,

rx2KKV

)∩Dk+1

)≥ η0 µϕ(Sϕ(y, t/KV )).

On the other hand, we have the inclusion

(10.113) Sϕ(x, rx) ⊂ Sϕ(y, t/KV ),

since, given x′ ∈ Sϕ(x, rx),

δϕ(y, x′) ≤ K(δϕ(x, x′) + δϕ(x, y)) ≤ K(rx +

rx2KKV

)=

rxKV

(KKV + 12)

≤ rxKV

(1 +KKV ) =t

KV.

34 DIEGO MALDONADO

Therefore, from the first inclusion in (10.109), the inequality (10.112), and the inclusion(10.113) we get

(10.114) µϕ(Sϕ(x, rx/KV ) ∩Dk+1) ≥ η0 µϕ(Sϕ(x, rx)).

We now look at the collection of sections Sϕ(x, rx/KV )x∈S\Dk and resort to Vitali’s coveringlemma to obtain a pairwise disjoint subcollection Sϕ(xj , rxj/KV )j∈N such that Sϕ(xj , rxj )j∈Ncovers S \Dk. Next, from (10.105), it follows that⋃

j∈NSϕ(xj , rxj/KV ) ∩ Sϕ(z,R) ⊂ S \Dk,

and, after intersecting with Dk+1,⋃j∈N

Sϕ(xj , rxj/KV ) ∩Dk+1 ⊂ Dk+1 \Dk,

and then

µϕ(S \Dk) ≤∑j∈N

µϕ(Sϕ(xj , rxj )) ≤1

η0

∑j∈N

µϕ(Sϕ(xj , rxj/KV ) ∩Dk+1)

=1

η0µϕ

⋃j∈N

Sϕ(xj , rxj/KV ) ∩Dk+1

≤ 1

η0µϕ(Dk+1 \Dk).

That is, for every k ∈ N0, we have obtained η0 µϕ(S \Dk) ≤ µϕ(Dk+1 \Dk), which means

µϕ(S \Dk+1) ≤ (1− η0)µϕ(S \Dk), ∀k ∈ N0,

from whose iterations (10.102) follows.

Corollary 31. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). For every sectionSR := Sϕ(z,R) with S2K5R := Sϕ(z, 2K5R) ⊂⊂ Ω and every u ∈ C(S2K5R) ∩W 2,n(S2K5R)satisfying u ≥ 0 in S2K5R and LϕA(u) ≤ f a.e. in S2K5R the assumptions

R−12 |S2K5R|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(S2K5R, dµϕ)‖D2ϕ‖

12

Lp(S2K5R, dx) ≤ ε4,

and |S2K5R|1n ‖c‖Ln(S2K5R

, dµϕ) ≤ ε4

imply the inequality(ˆSϕ(z,R)

uβ2(x) dµϕ(x)

) 1β2

≤M2 infSϕ(z,R)

u+M2 ε−14 |S2K5R|

1n ‖f‖Ln(S2K5R

, dµϕ),

where β2 ∈ (0, 1) and M2 > 0 are the structural constants from Theorem 30.

Proof. Divide the inequality LϕA(u) ≤ f by Q(f, u) with

Q(f, u) := infSϕ(z,R)

u+ ε−14 |S2K5R|

1n ‖f‖Ln(S2K5R

, dµϕ)

and apply Theorem 30 to u/Q(f, u).

35

10.1. Proof of Theorem 1. Given a section Sr := Sϕ(z, r) with S2Kr = Sϕ(z, 2Kr) ⊂⊂ Ω

such that |S2Kr|1n ‖c‖Ln(S2Kr, dµϕ) ≤ εH and

(10.115)

supx∈Sϕ(z,r)

ρ≤r

(ρ−

12 |Sϕ(x, ρ)|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(x,ρ), dµϕ)‖D2ϕ‖12

Lp(Sϕ(x,ρ), dx)

)≤ εH ,

Corollary 29 applied with q := β2, where β2 ∈ (0, 1) is the structural constant from Corollary31, gives

(10.116) supSϕ(z,κ1r)

u ≤ C7

( Sϕ(z,κ2r)

u(x)β2 dµϕ(x)

) 1β2

+ C7|Sr|1n ‖f‖Ln(S2Kr, dµϕ),

provided that εH ≤ ε5/2. On the other hand, the condition (10.115) used with x = z andρ = r gives

r−12 |Sϕ(z, r)|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(z,r), dµϕ)‖D2ϕ‖12

Lp(Sϕ(z,r), dx) ≤ εHand we can use Corollary 31 with R := r/(2K5) to obtain

(10.117)

(ˆSϕ(z,r/(2K5))

uβ2(x) dµϕ(x)

) 1β2

≤M2 infSϕ(z,r/(2K5))

u+M2 ε−14 |Sr|

1n ‖f‖Ln(Sr, dµϕ),

provided that εH ≤ ε4/(2K5)1/2. Since we can always make κ1 smaller in (10.116), as wellas κ2 close to κ1, we can assume that κ2 ≤ 1/(2K5), so that the inequalities (10.116) and

(10.117) imply the Harnack inequality (1.17) with τ := κ1, εH := minε4/(2K5)1/2, ε5/2,and MH := C7(1 +M2ε

−14 ).

11. Examples of ϕ satisfying H1–H6: The functions |x|p with p ≥ 2

For each p > 1 set

(11.118) ϕp(x) := 1p |x|p ∀x ∈ Rn.

In this section prove that ϕp satisfies H1, H2, H3, and H4 whenever p > 1; H5 wheneverp > 2−n/q for some q > n; and H6 whenever p ≥ 2. Therefore, ϕp satisfies all the hypothesesH1–H6 when p ≥ 2.

When 1 < p < ∞, the strictly convex functions ϕp have been considered in [13] as pro-totypical examples of quasi-uniformly convex functions, see Example 2.4 in [13]. That is,the sections of ϕp are round in the sense that there exist constants 0 < cp ≤ 1 ≤ CP < ∞,depending only on p and n, such that they relate to Euclidean balls through the followinginclusions

(11.119) B(x0, cpR1p ) ⊂ Sϕp(x0, R) ⊂ B(x0, CpR

1p ) ∀x0 ∈ Rn, R > 0.

This means that ∇ϕp : Rn → Rn is a quasi-conformal mapping and, as a consequence (see [13,Corollary 3.8]), the Monge-Ampere measure µϕp satisfies the (µ∞)-condition of Caffarelli–Gutierrez. Namely, there exist constants C0, θ0 > 0, depending only on p and n, such thatfor every section Sϕp ⊂⊂ Ω and every Borel subset E ⊂ Sϕp we have

(11.120)µϕp(E)

µϕp(Sϕp)≤ C0

( |E||Sϕp |

)θ0.

36 DIEGO MALDONADO

In turn, as shown in [3, p.426], the condition (11.120) implies µϕp ∈ (DC)ϕp . Therefore, for

every p > 1 we have µϕp ∈ (DC)ϕp , with doubling constants depending only on p and n, and

the hypothesis H2 is met. Also, from (11.120) and the discussion in Section 5.3 it followsthat ϕp satisfies H3 as well.

Next, regarding the local Lq-integrability of D2ϕp, since ‖D2ϕp(x)‖ ∼ ∆ϕp(x) ∼ |x|p−2,we have that D2ϕp ∈ Lq(B(0, 1)) if and only if q(p − 2) > −n, that is, p > 2 − n/q. Inparticular, if p ≥ 2, then D2ϕp ∈ Lq(B(0, 1)) for every q > 0. On the other hand, sincethe required Lq-integrability asks for some q > n, such q can be found (close enough to n)whenever p > 1.

In order to prove condition (1.10) when p ≥ 2 we will use the next observation.

Lemma 32. Given v, w ∈ Rn \ 0 let Nv,w : (0,∞)→ [0,∞) be the function

(11.121) Nv,w(q) :=

∣∣∣∣ v|v| −( |w||v|

)q w

|w|

∣∣∣∣ .If |v| < |w| then N ′v,w(q) > 0 for every q ∈ (0,∞). In particular, since N ′v,w ≡ 0 when|v| = |w|, it follows that N ′v,w(q) ≥ 0 whenever |v| ≤ |w|.Proof. Notice that, given q ∈ (0,∞), we have Nv,w(q) = 0 if and only if v = w, so that thehypothesis |v| < |w| implies Nv,w(q) > 0. Now, by considering Nv,w(q)2 we get

d(Nv,w(q)2)

dq=

n∑j=1

d

dq

[vj|v| −

( |w||v|

)q wj|w|

]2

= 2

n∑j=1

[vj|v| −

( |w||v|

)q wj|w|

] [−wj|w|

( |w||v|

)qlog

( |w||v|

)]

= 2

( |w||v|

)qlog

( |w||v|

)[−〈v, w〉|v||w| +

( |w||v|

)q].

Next, from the hypothesis |v| < |w| and the Cauchy-Schwarz inequality we obtain( |w||v|

)q> 1 ≥ 〈v, w〉|v||w| ,

log(|w|/|v|) > 0 and Nv,w(q) > 0 for every q ∈ (0,∞), which make N ′v,w > 0 as claimed.

Theorem 33. For every p ≥ 2 there exists σ(p, n) > 0, depending only on p and dimensionn, such that

(11.122) Dϕp(x0, x) ≥ σ(p, n)δϕp(x0, x) ∀x0 ∈ Rn, x ∈ Rn \ 0.Proof. Since µϕp ∈ (DC)ϕp with constants depending only on p and n, by Theorems 1 and 4from [9] there exists a constant κp,n ∈ (0, 1), also depending only on p and n, such that

(11.123) κp,n δϕp(x, y) ≤ 〈∇ϕp(x)−∇ϕp(y), x− y〉 ∀x, y ∈ Rn.

On the other hand,

Dϕp(x0, x) := 〈D2ϕp(x)−1(∇ϕp(x0)−∇ϕp(x)),∇ϕp(x0)−∇ϕp(x)〉.Now, we have ∇ϕp(0) = 0 and, given x ∈ Rn \ 0, ∇ϕp(x) = |x|p−1 x

|x| and

D2ϕp(x) = (p− 1)|x|p−2

(x

|x| ⊗x

|x|

)+ |x|p−2

(I − x

|x| ⊗x

|x|

).

37

Thus, for x ∈ Rn \ 0, the eigenvalues of D2ϕp(x) are (p− 1)|x|p−2 with multiplicity 1 and

eigenvector x and |x|p−2 with multiplicity n− 1 and eigenspace x⊥.Let us first consider the case x0 = 0. In this case we have

Dϕp(0, x) = 〈D2ϕp(x)−1∇ϕp(x),∇ϕp(x)〉 =1

(p− 1)|x|p−2|∇ϕp(x)|2 =

|x|pp− 1

and

〈∇ϕp(x)−∇ϕp(0), x− 0〉 = 〈∇ϕp(x), x〉 = |x|p,so that by (11.123) we get

(11.124) Dϕp(0, x) ≥ κp,np− 1

δϕp(0, x) ∀x ∈ Rn \ 0.

Next, consider the case x, x0 ∈ Rn \ 0 with |x| < |x0|. We have

〈∇ϕp(x)−∇ϕp(x0), x− x0〉 ≤∣∣∣∣|x|p−1 x

|x| − |x0|p−1 x0

|x0|

∣∣∣∣ |x− x0|

= |x|p∣∣∣∣∣ x|x| −

( |x0||x|

)p−1 x0

|x0|

∣∣∣∣∣∣∣∣∣ x|x| − x0

|x|

∣∣∣∣ = |x|p∣∣∣∣ x|x| − x0

|x|

∣∣∣∣Nx,x0(p− 1),(11.125)

where Nx,x0 is as in (11.121). By using that the smallest eigenvalue of D2ϕp(x)−1 is 1(p−1)|x|p−2

we can write

Dϕp(x0, x) = 〈D2ϕp(x)−1(∇ϕp(x0)−∇ϕp(x)),∇ϕp(x0)−∇ϕp(x)〉

≥ 1

(p− 1)|x|p−2|∇ϕp(x0)−∇ϕp(x)|2 =

∣∣∣|x|p−1 x|x| − |x0|p−1 x0

|x0|

∣∣∣2(p− 1)|x|p−2

,(11.126)

and by expressing p ≥ 2 as p− 1 ≥ 1 and using Lemma 32 to get Nx,x0(p− 1) > Nx,x0(1), weobtain∣∣∣|x|p−1 x

|x| − |x0|p−1 x0|x0|

∣∣∣2(p− 1)|x|p−2

=

|x|2(p−1)

∣∣∣∣ x|x| − ( |x0||x| )p−1x0|x0|

∣∣∣∣2(p− 1)|x|p−2

=|x|p

(p− 1)Nx,x0(p− 1)2

>|x|p

(p− 1)Nx,x0(1)Nx,x0(p− 1) =

|x|p(p− 1)

∣∣∣∣ x|x| − x0

|x|

∣∣∣∣Nx,x0(p− 1),

which combined with (11.125), (11.126), and (11.123), gives

(11.127) Dϕp(x0, x) >κp,np− 1

δϕp(x0, x) ∀x, x0 ∈ Rn \ 0,with |x| < |x0|.

Next, we split the case 0 < |x0| ≤ |x| into two subcases: namely, Nx,x0(p − 1) ≥ 1/2 andNx,x0(p− 1) < 1/2. If Nx,x0(p− 1) ≥ 1/2, as in (11.126) we write

Dϕ(x0, x) ≥

∣∣∣|x|p−1 x|x| − |x0|p−1 x0

|x0|

∣∣∣2(p− 1)|x|p−2

=|x|p

(p− 1)Nx,x0(p− 1)2 ≥ |x|p

4(p− 1),

and, as in (11.125) and using that |x0| ≤ |x|,

〈∇ϕp(x)−∇ϕp(x0), x− x0〉 ≤ |x|p∣∣∣∣∣ x|x| −

( |x0||x|

)p−1 x0

|x0|

∣∣∣∣∣∣∣∣∣ x|x| − x0

|x|

∣∣∣∣ ≤ 4|x|p.

38 DIEGO MALDONADO

Thus, in the subcase 0 < |x0| ≤ |x| and Nx,x0(p− 1) ≥ 1/2, we obtain

(11.128) Dϕ(x0, x) ≥ 1

16(p− 1)〈∇ϕp(x)−∇ϕp(x0), x− x0〉.

Now, for the subcase 0 < |x0| ≤ |x| and Nx,x0(p−1) < 1/2 we first notice that Nx,x0(p−1) <1/2 implies that

(11.129)

( |x0||x|

)p−1

>1

2

because

0 ≤ 1−( |x0||x|

)p−1

≤∣∣∣∣∣ x|x| −

( |x0||x|

)p−1 x0

|x0|

∣∣∣∣∣ = Nx,x0(p− 1) <1

2.

Again as in (11.126) we have

Dϕ(x0, x) ≥

∣∣∣|x|p−1 x|x| − |x0|p−1 x0

|x0|

∣∣∣2(p− 1)|x|p−2

,

but now we use Nx0,x (as opposed to Nx,x0) as well as (11.129) and Lemma 32 to write∣∣∣|x|p−1 x|x| − |x0|p−1 x0

|x0|

∣∣∣2(p− 1)|x|p−2

=|x0|2(p−1)

(p− 1)|x|p−2

∣∣∣∣∣( |x||x0|

)p−1 x

|x| −x0

|x0|

∣∣∣∣∣2

=|x0|p

(p− 1)

( |x0||x|

)p−2

Nx0,x(p− 1)2 >|x0|p

2p−2p−1 (p− 1)

Nx0,x(p− 1)Nx0,x(1).

On the other hand,

〈∇ϕp(x)−∇ϕp(x0), x− x0〉 ≤∣∣∣∣|x|p−1 x

|x| − |x0|p−1 x0

|x0|

∣∣∣∣ |x− x0|

= |x0|p∣∣∣∣∣( |x||x0|

)p−1 x

|x| −x0

|x0|

∣∣∣∣∣∣∣∣∣ x|x0|

− x0

|x0|

∣∣∣∣ = |x0|pNx0,x(p− 1)Nx0,x(1).

Thus, in the subcase 0 < |x0| ≤ |x| and Nx,x0(p− 1) < 1/2, we obtain

(11.130) Dϕ(x0, x) ≥ 1

2p−2p−1 (p− 1)

〈∇ϕp(x)−∇ϕp(x0), x− x0〉.

Hence, (11.122) follows from (11.124), (11.127), (11.128), and (11.130) with

(11.131) σ(p, n) :=κp,n

(p− 1)min2−4, 2

− p−2p−1 .

Remark 34. The condition (1.10) fails for ϕp when 1 < p < 2. Indeed, given x0 ∈ Rn \ 0,put r := δϕp(x0, 0) > 0 so that 0 ∈ Sϕ(x0, 2r) \ Sϕ(x0, r). Next, notice that, when 1 < p < 2,we have that Dϕp(x0, x) tends to 0 as x→ 0. Therefore, (1.10) cannot happen for any valueof σ > 0.

39

12. The preservation of H1, H2, H4, H5, and H6 under tensor sums

In this section we prove that the hypotheses H1, H2, H4, H5, and H6 are preserved,quantitatively, under tensor sums. Also, we point out that H3 is preserved under tensorsums of (positive multiples of) the functions ϕp in (11.118).

Fix m ∈ N. For each j = 1, . . . ,m, consider nj ∈ N and a convex function ϕj : Rnj → R.

Let ϕ be the tensor sum of the ϕj ’s, that is, ϕ : Rn → R, with n :=m∑j=1

nj , and

(12.132) ϕ(x) :=m∑j=1

ϕj(xj), x = (x1, . . . , xn) ∈ Rn, xj ∈ Rnj , j = 1, . . . ,m.

It is immediate to see that if the functions ϕj , j = 1, . . . ,m, satisfy H1 and H4, then so doesϕ. In order to consider H2, H5, and H6, given an open, bounded set Ω ⊂ Rn, let Ωj denotethe projection of Ω on Rnj . Then we have the following

Theorem 35. If the functions ϕj, j = 1, . . . ,m, satisfy H2 then so does ϕ, quantitatively.Moreover, δϕ satisfies a quasi-triangle inequality with constant K := max

j=1,...,mKj, where

Kj ≥ 1 is the quasi-triangle constant for δϕj .

If the functions ϕj, j = 1, . . . ,m, satisfy H5 in Ωj with some p > n, then ϕ satisfies H5in Ω with the same p > n.

If the functions ϕj, j = 1, . . . ,m, satisfy H6 in the following sense: For each j = 1, . . . ,m,there exists σj > 0 such that for every x0,j ∈ Rnj and r > 0 with Sϕj (x0,j , 2Kjr) ⊂⊂ Ωj, wehave

(12.133) Dϕj (x0,j , xj) ≥ σjδϕj (x0,j , xj) a.e. xj ∈ Sϕj (x0,j , 2mr) \ Sϕj (x0,j , r).

Then, ϕ satisfies H6 with σ := 12m min

j=1,...,mσj in the sense that for every x0 ∈ Rn and r > 0

with Sϕ(x0, 2Kmr) ⊂⊂ Ω we have

(12.134) Dϕ(x0, x) ≥ σδϕ(x0, x) a.e. x ∈ Sϕ(x0, 2r) \ Sϕ(x0, r).

Proof. If the functions ϕjmj=1 satisfy the hypothesis H2, each with some pairs of constants

(Cd,j , αj) in (1.6), then so does ϕ with some constants (Cd, α) depending only on the constantsCd,j , αj , njmj=1. This is so because of the characterization of the doubling property (1.6) in

terms of the so-called engulfing property of the Monge-Ampere sections (see [8, Theorem 8]and [9, Theorem 4]) and the fact that the sections of ϕ and the ones for ϕjmj=1 are relatedby the inclusions

(12.135) Sϕ(x, t) ⊂ Sϕ1(x1, t)× · · · × Sϕm(xm, t) ⊂ Sϕ(x,mt),

for every x = (x1, . . . , xn) ∈ Rn, xj ∈ Rnj , j = 1, . . . ,m, and t > 0, see [10, Lemma 6]. Also,the fact that ∇ϕ = ⊕mj=1∇ϕj and the definition of δϕ in (1.4) imply that

(12.136) δϕ(x0, x) =m∑j=1

δϕj (x0,j , xj);

in particular, δϕ satisfies a quasi-triangle inequality with constant K := maxj=1,...,m

Kj.Next, assume that each function ϕj , j = 1, . . . ,m, satisfies H5 in Ωj with an exponent

p > n. Since ∆ϕ(x) =m∑j=1

∆jϕj(xj), where ∆j denotes the Laplacian in Rnj , by Fubini’s

40 DIEGO MALDONADO

theorem we get(12.137)ˆ

Ω∆ϕ(x)p dx 'm,p

m∑j=1

ˆΩ

∆jϕj(xj)

p dx =m∑j=1

(ˆΩj

∆jϕj(xj)

p dxj

)Ln−nj (Ωj) <∞,

where Ωj denotes the projection of Ω over Rn1×· · ·×Rnj−1×Rnj+1×· · ·×Rnm and Ln−nj (Ωj)denotes its Lebesgue measure. Thus, ϕ satisfies H5 in Ω with the same p > n.

Finally, regarding H6 for ϕ, suppose that each ϕj satisfies (12.133). Given x0 ∈ Rn andr > 0 with Sϕ(x0, 2Kmr) ⊂⊂ Ω, we first notice that the inclusions (12.135) imply thatSϕj (x0,j , 2Kjr) ⊂⊂ Ωj for every j = 1, . . . ,m. On the other hand, since D2ϕ = ⊕mj=1D

2ϕj ,it follows that for every x, x0 ∈ Ω

Dϕ(x0, x) := 〈D2ϕ(x)−1(∇ϕ(x)−∇ϕ(x0)),∇ϕ(x)−∇ϕ(x0)〉

=

m∑j=1

〈D2ϕj(xj)−1(∇ϕ(xj)−∇ϕ(x0,j)),∇ϕ(xj)−∇ϕ(x0,j)〉 =

m∑j=1

Dϕj (x0,j , xj).

Fix x ∈ Sϕ(x0, 2r)\Sϕ(x0, r) ⊂⊂ Ω. From the inclusion (12.135) we get that xj ∈ Sϕj (x0,j , 2r)for every j = 1, . . . ,m, and that there exists j∗ such that xj∗ /∈ Sϕj∗ (x0,j∗ , r/m). That is,

δϕj∗ (x0,j∗ , xj∗) > r/m. Then, by recalling that σ := 12m min

j=1,...,mσj,

Dϕ(x0, x) =m∑j=1

Dϕj (x0,j , xj) ≥ Dϕj∗ (x0,j∗ , xj∗) ≥ σj∗δϕj∗ (x0,j∗ , xj∗)

> σj∗r

m>σj∗2m

δϕ(x0, x) ≥ σδϕ(x0, x),

which proves (12.134).

Remark 36. (On the preservation of the µ∞-condition under tensor sums). It is not clear tothe author whether the hypothesis H3 is preserved under tensor sums. However, H3 willbe preserved, quantitatively, by tensor sums of the functions ϕp in (11.118). This is due tothe fact that given strictly convex functions ϕj ∈ C1(Rnj ), j = 1, . . . ,m, each satisfying theµ∞-condition, then its tensor sum (12.132) will also satisfy the µ∞-condition, quantitatively(see Lemma 6(ii) from [10]).

Now, given p1, . . . , pm ∈ (1,∞), each function ϕpj satisfies (11.120), which is equivalentto the µ∞-condition (see [3, Section 5]). Then, their tensor sum will also satisfy (11.120),quantitatively and, from the discussion in Section 5.3, it will satisfy H3 as well.

13. Appendix

13.1. Proof of Lemma 13 (Morrey-type estimate). By [25, Theorem 1.3], there existsa geometric constant KP > 0 such that for every section S := Sϕ(z0, t) such that for everyh ∈ C1(S) the following Poincare inequality holds true

(13.138)1

|S|

ˆS|h(z)− hS | dz ≤ KP t

12

(1

|S|

ˆS|∇ϕh(z)|2 dz

) 12

,

where ∇ϕh(z) := D2ϕ(z)−1/2∇h(z) and

hS :=1

|S|

ˆSh(z) dz.

41

Given x, y ∈ Ω and j ∈ N0, set Sxj := Sϕ(x, 2−jδϕ(x, y)) and Syj := Sϕ(y, 2−jδϕ(y, x)). Also,

define SxK := Sϕ(x, 2Kδϕ(x, y)) so that the K-quasi-triangle inequality gives the inclusions

Sϕ(x, δϕ(x, y)/(2K)) ⊂ Sy0 = Sϕ(y, δϕ(y, x))(13.139)

⊂ Sϕ(x, 2Kδϕ(x, y)) = SxK .

In particular, (13.139) and the doubling property (5.55) imply the inequalities

|SxK | ≤ 2n(2K)2n|Sϕ(x, δϕ(x, y)/(2K))| ≤ 2n(2K)2n|Sy0 |,(13.140)

|SxK | ≤ 2n(2K)n|Sϕ(x, δϕ(x, y))|.By the Lebesgue differentiation theorem applied to h at x, the doubling property (5.54) andthe Poincare inequality (13.138) we have∣∣h(x)− hSx0

∣∣ ≤ ∑j∈N0

|hSxj − hSxj+1| ≤

∑j∈N0

Sxj+1

∣∣∣h(z)− hSxj∣∣∣ dz

≤ 2n∑j∈N0

Sxj

∣∣∣h(z)− hSxj∣∣∣ dz ≤ 2nKP δϕ(x, y)

12

∑j∈N0

2−j/2

(1

|Sxj |

ˆSxj

|∇ϕh(z)|2 dz) 1

2

.

Next, given q > 2n ≥ 2, by the doubling property (5.55)

∑j∈N0

2−j2

(1

|Sxj |

ˆSxj

|∇ϕh(z)|2 dz) 1

2

≤∑j∈N0

2−j/2

(1

|Sxj |

ˆSxj

|∇ϕh(z)|q dz) 1

q

≤ 2n/q∑j∈N0

2−j/22jn/q

(1

|Sx0 |

ˆSxj

|∇ϕh(z)|q dz) 1

q

≤ 2n/q

∑j∈N0

2−j(1/2−n/q)

( 1

|Sx0 |

ˆSx0

|∇ϕh(z)|q dz) 1

q

.

Hence,

(13.141)∣∣h(x)− hSx0

∣∣ ≤ C1δϕ(x, y)12

(1

|Sx0 |

ˆSx0

|∇ϕh(z)|q dz) 1

q

,

with C1 := 2n+n/qKP∑j∈N0

2−j(1/2−n/q) < ∞, since q > 2n. Switching the roles of x and y

gives

(13.142)∣∣∣h(y)− hSy0

∣∣∣ ≤ C1δϕ(y, x)12

(1

|Sy0 |

ˆSy0

|∇ϕh(z)|q dz) 1

q

.

Hence,

|hSx0 − hSy0 | ≤ |hSx0 − hSxK |+ |hSxK − hSy0 | ≤ 2

SxK

|h(z)− hSxK | dz

≤ 2KP K1/2δϕ(x, y)

12

(1

|SxK |

ˆSxK

|∇ϕh(z)|q dz) 1

q

.(13.143)

42 DIEGO MALDONADO

Finally, (5.63) follows from (13.141), (13.142), (13.143), and (13.140) with

Cq,K := (22nKn)1/qC1[1 + (2K)n/q] + 2KPK1/2.

13.2. Proof of Lemma 15 (Local Vitali covering lemma). The following proof of Vi-tali’s covering lemma is modeled after Theorem 1.2 on [5, p.69]. Notice that the inclusionSϕ(x, 2rx) ⊂⊂ S0 := Sϕ(x0, R0) for every x ∈ I implies that sup

x∈Irx < ∞. More precisely,

given x ∈ I and y ∈ ∂Sϕ(x, rx), from the quasi-triangle inequality (1.7) we get

rx = δϕ(x, y) ≤ K[δϕ(x0, x) + δϕ(x0, y)] < 2KR0.

Now, given KV > 2K2 +K fix ε > 0 such that

(13.144) Kε :=

(2− ε1− ε

)K +

1

1− ε <KV

K

and put κ := κ(ε) :=(

2−ε1−ε

)K > 1. Next, define E1 := E and, inductively, for k ∈ N choose

xk ∈ Ek := E \⋃k−1j=1 Sϕ(xj , κrxj ) with

(13.145) rxk > (1− ε) supx∈Ek

rx.

Notice that for j ≤ k we have Ek ⊂ Ej so that

(13.146) rxk ≤ supx∈Ek

rx ≤ supx∈Ej

rx <1

1− εrxj .

It now follows that for j < k we have Sϕ(xj , rxj ) ∩ Sϕ(xk, rxk) = ∅. Otherwise, there wouldexist z ∈ Sϕ(xj , rxj ) ∩ Sϕ(xk, rxk) and the quasi-triangle inequality (1.7), the inequality(13.145), and the definition of κ would yield

δϕ(xj , xk) ≤ K[δϕ(xk, z) + δϕ(xj , z)] < K(rxk + rxj ) ≤ K(

1 +1

1− ε

)rxj = κrxj ,

contradicting the fact that xk /∈ Sϕ(xj , κrxj ). If the process of choosing xk’s stops afterfinitely many steps, then the results follows. Let us then assume that there is an xk foreach k ∈ N. In such case we must have lim

k→∞rxk = 0. Otherwise, there would exist ρ > 0

such that rxj > ρ for infinitely many values of j. That is, there would be infinitely manypairwise-disjoint sections Sϕ(xj , rxj )j∈N with

(13.147) Sϕ(xj , ρ) ⊂ Sϕ(xj , rxj ) ⊂ Sϕ(xj , κrxj ) ⊂ S0 ∀j ∈ N.

Let us now see that (13.147) leads to a contradiction. Indeed, given any two of these disjointsections Sϕ(xj , ρ) and Sϕ(xk, ρ), we claim that

(13.148) Sϕ(xj , ρ) ⊂ Sϕ(xk,K(ρ+ 2KR0))

so that the doubling property (5.56) yields

µϕ(Sϕ(xj , ρ)) ≤ µϕ(Sϕ(xk,K(ρ+ 2KR0))) ≤ KD

(K(ρ+ 2KR0))

ρ

)nµϕ(Sϕ(xk, ρ)),

which makes the Monge-Ampere measures of the disjoint sections Sϕ(xj , ρ) and Sϕ(xk, ρ)comparable. Consequently, since (13.147) implies the inclusion

⋃j∈N Sϕ(xj , ρ) ⊂ S0, it follows

that µϕ(S0) =∞, a contradiction.

43

In order to prove the inclusion (13.148), given y ∈ Sϕ(xj , ρ) the quasi-triangle inequality(1.7) gives

δϕ(y, xk) ≤ K[δϕ(xj , y) + δϕ(xj , xk)] ≤ K[ρ+ δϕ(xj , xk)]

and, on the other hand, from(1.7) and (13.147)

δϕ(xj , xk) ≤ K[δϕ(x0, xj) + δϕ(x0, xk)] < 2KR0,

and (13.148) follows. Finally, let us see that⋃x∈I Sϕ(x, rx) ⊂ ⋃j∈N Sϕ(xj ,KV rxj ). Given

z ∈ ⋃x∈I Sϕ(x, rx), there exists a pair (xz, rz) with z ∈ Sz := Sϕ(xz, rz). The next claim isthat Sz ∩ Sϕ(xj , κrxj ) 6= ∅ for some j ∈ N. Indeed, by construction, if Sz ∩ Sϕ(xj , κrxj ) = ∅for every j = 1, . . . , k − 1, then

(13.149) rz <1

1− εrxk .

Now, the fact that limk→∞

rxk = 0 prevents (13.149) from happening for infinitely many k’s in

N. Therefore, there is a first ` ∈ N such that Sz ∩ Sϕ(xj , κrxj ) = ∅ for every j = 1, . . . , `− 1

and Sz ∩ Sϕ(x`, κrx`) 6= ∅. In particular, rz < (1 − ε)−1rx` . The final step is to prove thatz ∈ Sϕ(x`,KV r`) which follows from taking y ∈ Sz ∩ Sϕ(x`, κrx`) 6= ∅ and writing

δϕ(x`, z) ≤ K[δϕ(x`, y) + δϕ(z, y)] < K(κrx` + rz)

< K

(κ+

1

1− ε

)rx` = KKεrx` < KV rx` .

13.3. Sketch of the proof of Theorem 27. The first step is to prove the following coun-terpart to Lemma 5 from [26] based on Theorem 25. The idea is to replace the inequality(4.43) and Theorem 2 from [26] with (5.56) and Theorem 25 in this article) and we will justsketch it. Also, the role of “(1 − ε0)” in the inequality (3.24) of [26] will be played by the“ε0/2” from Theorem 25.

Let γ0, ε3 ∈ (0, 1) and K3 > 1 be the structural constants from Theorem 25 and putM1 := 5/γ0 and ν := 2M1/(2M1 − 1) > 1.

Lemma 37. Suppose that ϕ satisfies H1–H6 and fix A ∈ E(λ,Λ, ϕ,Ω). Let Sϕ(z, r) be a

section with S2Kr := Sϕ(z, 2Kr) ⊂⊂ Ω and let u ∈ C(S2Kr) ∩W 2,n(S2Kr) satisfy u ≥ 0 inS2Kr and LϕA(u) ≥ f a.e. in S2Kr. Assume that the inequalities (9.98) and (9.99) and that

(13.150)

Sϕ(z,2Kr)

u dµϕ < M1.

Suppose that there exists x0 ∈ Sϕ(z, r), j ∈ N, and ρ ∈ (0, r) such that u(x0) ≥ νj−1M1 andthat

(13.151) µϕ(Sϕ(x0, ρ)) ≥(

2n+2K2DK

nKn3

νjε0

)µϕ(Sϕ(z, r)).

Then supSϕ(x0,ρ)

u > νjM1.

Proof. By contradiction, assume that supSϕ(x0,ρ)

u ≤ νjM1 and define

(13.152) wj(x) :=γ0(νjM1 − u(x))

νj−1(ν − 1)M1

44 DIEGO MALDONADO

so that wj ≥ 0 in Sϕ(x0, ρ). Since x0 ∈ Sϕ(z, r) the triangle-inequality (1.7) gives Sϕ(x0, ρ) ⊂Sϕ(z, 2Kr). Then, using that LϕA(u) ≥ f a.e. in S2Kr, for a.e. in Sϕ(x0, ρ) we get

LϕA(wj)(x) ≤ −γ0

νj−1(ν − 1)M1f(x) + 10c(x) =: fj(x),

where we have used that, by definition, ν/(ν − 1) = 2M1 and γ0M1 = 5. In addition, theassumption u(x0) ≥ νj−1M1 gives wj(x0) ≤ γ0, which implies inf

Sϕ(x0,ρ/(2K3)wj ≤ γ0. We want

to apply Theorem 25 to wj and fj on the section Sϕ(x0, ρ/(2K3)) (think r := ρ2K3

in the

notation of Theorem 25). Since Sϕ(x0, ρ) ⊂ Sϕ(z, 2Kr) ⊂⊂ Ω, from the hypothesis (9.98),for r := ρ

2K3we have

|Sϕ(x0,K3r)|1n ‖fj‖Ln(Sϕ(x0,K3r), dµϕ) = |Sϕ(x0, ρ/2)| 1n ‖fj‖Ln(Sϕ(x0,ρ/2), dµϕ)

≤ |Sϕ(z, 2Kr)| 1n(

γ0

νj−1(ν − 1)M1‖f‖Ln(Sϕ(z,2Kr), dµϕ) + 10‖c‖Ln(Sϕ(z,2Kr), dµϕ)

)≤ |Sϕ(z, 2Kr)| 1n

(γ0

(ν − 1)M1‖f‖Ln(Sϕ(z,2Kr), dµϕ) + 10‖c‖Ln(Sϕ(z,2Kr), dµϕ)

)≤ 11ε5,

similarly,

|Sϕ(x0,K3r)|1n ‖c‖Ln(Sϕ(x0,K3r), dµϕ) = |Sϕ(x0, ρ/2)| 1n ‖c‖Ln(Sϕ(x0,ρ/2), dµϕ)

≤ |Sϕ(z, 2Kr)| 1n ‖c‖Ln(Sϕ(z,2Kr), dµϕ) ≤ ε5

and, since x0 ∈ Sϕ(z, r) and ρ < r, from (9.99)

r−12 |Sϕ(x0,K3r)|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(x0,K3r), dµϕ)‖D2ϕ‖12

Lp(Sϕ(x0,K3r), dx)

= (2K3)12 ρ−

12 |Sϕ(x0, ρ/2)|

1n− 1

2p ‖(D2ϕ)−1/2b‖Ln(Sϕ(x0,ρ/2), dµϕ)‖D2ϕ‖12

Lp(Sϕ(x0,ρ/2), dx)

≤ (2K3)12 ε5.

Thus, by taking ε5 := (2K3)−12 ε3, Theorem 25 applied to wj and fj on the section Sϕ(x0, ρ/(2K3))

(and here is where the hypothesis (9.99) comes into play) gives

(13.153) µϕ (x ∈ Sϕ(x0, ρ/K3) : wj(x) ≥ 5) ≤ (1− ε02 )µϕ(Sϕ(x0, ρ/K3)).

Next, set

A1 :=x ∈ Sϕ(z, 2Kr) : u(x) ≥ νjM1/2A2 :=x ∈ Sϕ(x0, ρ/K3) : wj(x) ≥ 5 ⊂ Sϕ(z, 2Kr).

Then, the inclusion Sϕ(x0, ρ/K3) ⊂ A1 ∪ A2 holds true since given x1 ∈ Sϕ(x0, ρ/K3) \ A1

it follows that wj(x1) > γ0ν2ν−1 = γ0M1 = 5, so that x1 ∈ A2. Consequently, Chebyshev’s

inequality, the inequalities (13.150) and (13.153) give

µϕ(Sϕ(x0, ρ/K3)) ≤ µϕ(A1) + µϕ(A2) ≤ 2

νjM1

ˆSϕ(z,2Kr)

u dµϕ + (1− ε02 )µϕ(Sϕ(x0, ρ/K3))

<2µϕ(Sϕ(z, 2Kr))

νj+ (1− ε0

2 )µϕ(Sϕ(x0, ρ/K3)),

45

and then µϕ(Sϕ(x0, ρ/K3)) < 4νjε0M1

µϕ(Sϕ(z, 2Kr)), which, along with the doubling property

(5.56), implies

µϕ(Sϕ(x0, ρ))

KDKn3

≤ µϕ(Sϕ(x0, ρ/K3)) <4

νjε0µϕ(Sϕ(z, 2Kr)) ≤ 4KDK

n

νjε0µϕ(Sϕ(z, r)),

thus contradicting (13.151).

The proof of Theorem 27 now follows (along the same lines as the one for [26, Lemma6]) from Lemma 37 and the argument on [26, pp.2002-2004]. The corresponding structuralconstant C4 (in the notation of [26, p.2002]) now can be taken as

C4 := (2K2)

(2n+2K2

DKnKn

3

ε0

)1/n

,

and the rest of the proof follows the (purely geometric) argument on [26, pp.2002-2004].

Acknowledgements

The author is grateful to Nam Q. Le for comments and corrections to an early version ofthis article. The author is supported by the National Science Foundation under grant DMS1361754.

References

[1] A. Bjorn and J. Bjorn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics 17,European Mathematical Society, Zurich, 2011.

[2] L. Caffarelli and C. Gutierrez, Real analysis related to the Monge-Ampere equation, Trans. Amer. Math.Soc. 348 (1996), 1075–1092.

[3] L. Caffarelli and C. Gutierrez, Properties of the solutions of the linearized Monge-Ampere equation, Amer.J. Math. 119(2) (1997), 423–465.

[4] Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, Translations ofMathematical Monographs, vol. 174, AMS, 1998.

[5] R. R. Coifman and G. Weiss. Analyse harmonique non-commutative sur certains espaces homogenes.Lecture Notes in Mathematics, Vol. 242. Springer-Verlag, Berlin, 1971.

[6] G. De Philippis and A. Figalli, The Monge-Ampere equation and its link to optimal transportation, Bull.Amer. Math. Soc. 51 (2014), 527–580.

[7] G. De Philippis, A. Figalli, and O. Savin, A note on interior W 2,1+ε estimates for the Monge-Ampereequation, Math. Ann., 357(1), (2013), 11–22.

[8] L. Forzani and D. Maldonado, On geometric characterizations for Monge-Ampere doubling measures, J.Math. Anal. Appl. 275(2) (2002), 721–732.

[9] L. Forzani and D. Maldonado, Properties of the solutions to the Monge-Ampere equation, Nonlinear Anal.57(5-6) (2004), 815–829.

[10] L. Forzani and D. Maldonado, A mean-value inequality for non-negative solutions to the linearized Monge-Ampere equation, Potential Anal. 30 (3), (2009), 251-270.

[11] B. Franchi, C.E. Gutierrez and R.L. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin typeoperators, Comm. Partial Differential Equations, 19, (1994), 523–604.

[12] B. Franchi and E. Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fieldsand Harnack inequality, Comm. Partial Differential Equations, 9(13), (1984),1237–1264.

[13] L. V. Kovalev and D. Maldonado, Mappings with convex potentials and the quasiconformal Jacobianproblem, Illinois J. Math. 49 (2005), no. 4, 1039–1060.

[14] N.V. Krylov and M.V. Safonov, A certain property of solutions of parabolic equations with measurablecoefficients, Izvestia Akad. Nauk SSSR, ser. Matem. 44(1980), 161–175 (in Russian). English transl. inMath. USSR Izvestija, 16 (1981), 151–164.

[15] D. Gilbarg and M. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer Verlag,2001.

46 DIEGO MALDONADO

[16] C. Gutierrez, The Monge-Ampere Equation. Progress in Nonlinear Differential Equations and TheirApplications, volume 44. Birkauser, 2001.

[17] C. Imbert and L. Silvestre, Estimates on elliptic equations that hold only where the gradient is large, JEur. Math. Soc., 18 (6), (2016), 1321–1338.

[18] S. Indratno, D. Maldonado, and S. Silwal, On the axiomatic approach to Harnack’s inequality in doublingquasi-metric spaces, J. Differ. Equations, 254 (8) (2013), 3369–3394.

[19] O. A. Ladyzhenskaya and N. N. Uraltseva, Estimates of the Holder constant for functions satisfying auniformly elliptic or a uniformly parabolic quasilinear inequality with unbounded coefficients, Zap. Nauchn.Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 147 (1985), 72–94 (in Russian). English transl. in J.Soviet Math. 37 (1987), 837–851.

[20] N. Q. Le, On the Harnack inequality for degenerate and singular elliptic equations with unbounded lowerorder terms via sliding paraboloids, Comm. Contemp. Math, doi: 10.1142/S0219199717500122.

[21] N. Q. Le, W 4,p solution to the second boundary value problem of the prescribed affine mean curvature andAbreu’s equations, J. Differential Equations. 260 (2016), no. 5, 4285–4300.

[22] N. Q. Le, Global second derivative estimates for the second boundary value problem of the prescribed affinemean curvature and Abreu’s equations, Int. Math. Res. Notices, 11 (2013) 2421–2438.

[23] G. Loeper, On the regularity of the polar factorization for time dependent maps, Calc. Var. Partial Dif-ferential Equations, 22 (2005), 343–374.

[24] D. Maldonado, The Monge-Ampere quasi-metric structure admits a Sobolev inequality, Math. Res. Lett.,20(3), (2013), 527–536.

[25] D. Maldonado, On the W 2,1+ε-estimates for the Monge-Ampere equation and related real analysis, Calc.Var. Partial Differential Equations, 50, (2014), 94–114.

[26] D. Maldonado, Harnack’s inequality for solutions to the linearized Monge-Ampere operator with lower-order terms, J. Differential Equations, 256 (6), (2014), 1987–2022.

[27] D. Maldonado, W 1,pϕ -estimates for Green’s functions of the linearized Monge-Ampere operator, to appear

in Manuscripta Math., 152 (3), (2017), 539–554.[28] A. Montanari, Harnack inequality for a subelliptic PDE in nondivergence form, Nonlinear Anal. 109,

(2014), 285–300.[29] C. Mooney, Harnack inequality for degenerate and singular elliptic equations with unbounded drift, J.

Differential Equations, 258 (5), (2015), 1577–1591.[30] M.V. Safonov, Harnack inequality for elliptic equations and the Holder property of their solutions, Zap.

Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 96 (1980), 272–287 (in Russian). Englishtransl. in J. Soviet Math. 21, no. 5 (1983), 851–863.

[31] M.V. Safonov, Non-divergence elliptic equations of second order with unbounded drift, Amer. Math. Soc.Transl. Ser. 2, vol. 229, Amer. Math. Soc., Providence, RI, 2010, pp. 211–232.

[32] N. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (3), 27,(1973), 265–308.

[33] N. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasi-linear equations, Invent. Math. 61 (1), (1980), 67–79.

[34] N. Trudinger and X.-J. Wang, The Bernstein problem for affine maximal hypersurfaces. Invent. Math.140 (2000), no. 2, 399–422.

[35] N. Trudinger and X.-J Wang, Affine complete locally convex hypersurfaces, Invent. Math. 150 (2002), no.1, 45–60.

[36] N. Trudinger and X.-J. Wang, The Monge-Ampere equation and its geometric applications. Handbook ofgeometric analysis. No. 1, 467–524, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, 2008.

Diego Maldonado, Kansas State University, Department of Mathematics. 138 CardwellHall, Manhattan, KS 66506, USA.

E-mail address: dmaldona@math.ksu.edu