On the $W^{2,1+\varepsilon }$ W 2 , 1 + ε -estimates for the Monge-Ampère equation and related real analysis

$Page 1: On the $W^{2,1+\varepsilon }$ W 2 , 1 + ε -estimates for the Monge-Ampère equation and related real analysis$
Calc. Var.DOI 10.1007/s00526-013-0629-1 Calculus of Variations

On the W2,1+ε-estimates for the Monge-Ampère equationand related real analysis

Diego Maldonado

Received: 16 October 2012 / Accepted: 25 April 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract We build upon the techniques introduced by De Philippis and Figalli regardingW 2,1+ε bounds for the Monge-Ampère operator, to improve the recent A∞ estimates for‖D2ϕ‖ to A2 ones. Also, we prove a (1, 2)−Poincaré inequality and weak (q, p)−Poincaréinequalities associated to the Monge-Ampère quasi-metric structure. In turn, these Poincaréinequalities are used to prove Harnack’s inequality for non-negative solutions to the linearizedMonge-Ampère under minimal geometric assumptions.

Mathematics Subject Classification (2000) Primary 35J60 · 35D10; Secondary 26B25

1 Introduction and main results

Let ν denote a Borel measure in Rn and, given 0 < �1 ≤ �2 < ∞, let us write ν ∈ B�1,�2

if

�1|E | ≤ ν(E) ≤ �2|E |, ∀E ⊂ Rn Borel set. (1.1)

Here |E | stands for the Lebesgue measure of E .Based on the groundbreaking W 2,1-estimates for the Monge-Ampère equation due to

De Philippis and Figalli in [7], recently De Philippis et al. [8] and, independently, Schmidt[19], improved the W 2,1-estimates to W 2,1+ε0 -estimates for convex solutions to the Monge-Ampère. Namely, they prove that if μϕ , the Monge-Ampère measure of a convex solutionϕ, satisfies μϕ ∈ B�1,�2 , then there exist constants ε0 > 0 and C0 ≥ 1, depending only on�1,�2 and dimension n, such that the reverse-Hölder inequality

Communicated by O. Savin.

D. Maldonado (B)Department of Mathematics , Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506, USAe-mail: [email protected]

123

D. Maldonado

⎛⎝ 1

|S|∫

S

‖D2ϕ(x)‖1+ε0 dx

⎞⎠

11+ε0

≤ C0

|S|∫

S

‖D2ϕ(x)‖ dx, (1.2)

holds true for every section S = Sϕ(x0, t).In Sect. 3 we interpret the techniques by De Philippis and Figalli in [7] in terms of the

theory of Muckenhoupt’s A∞ weights in spaces of homogeneous type. In Sect. 4, by meansof the Legendre transform, we show how to improve from the Aϕ∞ condition [encoded in(1.2)] to an Aϕ2 condition. More precisely, for a non-negative definite matrix M let λmin(M)and λmax (M) denote its smallest and largest eigenvalues, respectively. Also, set wϕ(x) :=λmin(D2ϕ(x)) and Wϕ(x) := λmax (D2ϕ(x)). Notice that D2ϕ(x) exists a.e. x ∈ R

n andthat Wϕ(x) = ‖D2ϕ(x)‖. Our first result reads

Theorem 1.1 If μϕ ∈ B�1,�2 , then wϕ,Wϕ ∈ Aϕ2 . That is, there exists a constant C > 0,depending only on �1,�2 and dimension n, such that

⎛⎝ 1

|S|∫

S

w(x) dx

⎞⎠

⎛⎝ 1

|S|∫

S

w(x)−1 dx

⎞⎠ ≤ C (1.3)

for every section S = Sϕ(x0, t). Here w denotes either Wϕ or wϕ .

As a consequence of Theorem 1.1 the a priori W 2,1+ε0 -estimate (1.1) can be improved byreplacing, on its right-hand side, the arithmetic mean of Wϕ by its harmonic mean. That is,

Corollary 1.2 If μϕ ∈ B�1,�2 , then there exist constants ε0 > and C > 1, depending onlyon �1,�2 and dimension n, such that

⎛⎝ 1

|S|∫

S

w1+ε0(x) dx

⎞⎠

11+ε0

≤ C |S|∫S w(x)

−1 dx, (1.4)


As an application of the underlying real analysis associated to the Monge-Ampère equa-tion, in Sect. 5 we prove a (1, 2)−Poincaré inequality and weak (q, p)−Poincaré inequalitiesassociated to the Monge-Ampère quasi-metric structure. These inequalities complement therecent results on Sobolev-type inequalities associated to the Monge-Ampère structure intro-duced by Tian and Wang in [20] and further carried out in [17]. To the best of our knowledge,Theorem 1.3 below contains the first Poincaré-type inequalities for the Monge-Ampère struc-ture (see Sect. 2 for the appropriate definitions).

Theorem 1.3 Assume thatμϕ ∈ (DC)ϕ . Then there exists a constant C > 0, depending onlyon the (DC)ϕ constants for μϕ and dimension n, such that given any section S := Sϕ(x0, t)and u ∈ C1(S) we have

1

|S|∫

S

|u(x)− uS | dx ≤ Ct12

⎛⎝ 1

|S|∫

S

〈D2ϕ(x)−1∇u(x),∇u(x)〉 dx

⎞⎠

12

, (1.5)

where uS := 1|S|

∫S u(x) dx. In particular, if μϕ ∈ B�1,�2 , we have

1

|S|∫

S

|u(x)− uS | dx ≤ Ct12

�121

⎛⎝ 1

|S|∫

S

〈Aϕ(x)∇u(x),∇u(x)〉 dx

⎞⎠

12

, (1.6)

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On the W 2,1+ε-estimates for the Monge-Ampère

where

Aϕ(x) := D2ϕ(x)−1μϕ(x). (1.7)

Moreover, under the assumption μϕ ∈ B�1,�2 , given 1 < p ≤ q with

0 <1

2p− 1

2q<

1

n, (1.8)

there exists a constant C, depending only on �1, �2, p, q, and dimension n, such that forevery section S := Sϕ(x0, t) ⊂ S∗ := Sϕ(x0, 2K t) [where K is as in (2.5) below] and everyu ∈ C1(S∗) we have

⎛⎝ 1

|S|∫

S

|u(x)− uS |2q dx

⎞⎠

12q

≤ Ct12

⎛⎝ 1

|S|∫

S∗|Aϕ(x) 1

2 ∇u(x)|2p dx

⎞⎠

12p

. (1.9)

In particular, (1.5) says that the Monge-Ampère quasi-metric structure admits a (1, 2)-Poincaré-type inequality. As an application of Theorem 1.3, in Sect. 6 we prove a Harnackinequality for positive solutions to the linearized Monge-Ampère equation. More precisely,let � ⊂ R

n be an open bounded set and consider the possibly-degenerate elliptic operatorLϕ defined as

Lϕ(u)(x) := trace(Aϕ(x)D2u(x)), x ∈ �,

where Aϕ(x) is as in (1.7). Then we have,

Theorem 1.4 Assume that μϕ ∈ (DC)ϕ . Then there exist constants CH ≥ 1 and η ∈(0, 1), depending only on the (DC)ϕ constants and dimension n, such that for every sec-tion Sϕ(x0, t) with Sϕ(x0, 2t) ⊂⊂ � and every solution u > 0 to Lϕ(u) = 0 in �we have

supSϕ(x0,ηt)

u ≤ CH infSϕ(x0,ηt)

u. (1.10)

Harnack’s inequality (1.10) for Lϕ has been proved under the hypothesis μϕ ∈ Aϕ∞ byCaffarelli and Gutiérrez in [5]. Theorem 1.4 extends their theorem to the case of minimalgeometric hypothesis μϕ ∈ (DC)ϕ . By minimal geometric hypothesis we mean that, asmentioned in Sect. 2, the condition μϕ ∈ (DC)ϕ is precisely the one that renders a quasi-metric structure for Monge-Ampère (the bare minimum of structure to carry out real analysis).The gap between (DC)ϕ and Aϕ∞ is comparable to the one between doubling measures andMuckenhoupt A∞ weights (see [11, Section 3]).

2 Preliminaries

Let ϕ : Rn → R be a strictly convex differentiable function. For x, y ∈ R

n define

δϕ(x, y) := ϕ(y)− ϕ(x)− 〈∇ϕ(x), y − x〉. (2.1)

Given x ∈ Rn and t > 0, a section of ϕ centered at x with height t is the open bounded

convex set

Sϕ(x, t) := {y ∈ Rn : δϕ(x, y) < t}.

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D. Maldonado

Let ν be a Borel measure in Rn , following [4,5,14] we write ν ∈ (DP)ϕ if there exists a

constant C > 0 such that

ν(Sϕ(x, 2t)) ≤ Cν(Sϕ(x, t)), x ∈ Rn, t > 0. (DP)

We write ν ∈ (DC)ϕ if there exist constants α ∈ (0, 1) and C > 0 such that

ν(Sϕ(x, t)) ≤ C ν(αSϕ(x, t)), x ∈ Rn, t > 0, (Dc)

where αSϕ(x, t) is the α-contraction of Sϕ(x, t) with respect to its center of mass (the centerof mass being computed with respect to Lebesgue measure).

We write ν ∈ Aϕ∞ if there exist constants K1, K2 > 0 and 0 < θ1 ≤ 1 ≤ θ2 such that forevery Borel set F ⊂ S := Sϕ(x, t) we have

K2

( |F ||S|

)θ2

≤ ν(F)

ν(S)≤ K1

( |F ||S|

)θ1

. (2.2)

Let μϕ denote the Monge-Ampère measure associated to ϕ. Since ϕ ∈ C1(Rn), its Monge-Ampère measure is characterized by

μϕ(E) = |∇ϕ(E)|, ∀E ⊂ Rn Borel set. (2.3)

In their foundational work on the linearized Monge-Ampère operator, Gutiérrez and Caf-farelli [4,5] made extensive use of the hypothesis μϕ ∈ Aϕ∞ [although they used the notationμϕ ∈ (μ∞) instead]. The class Aϕ∞ contains all the densities of the form |p| where p isa polynomial in R

n . Moreover, the constants involved depend only on the degree of thepolynomial and not on its coefficients.

The following inclusions summarize the hierarchy among these properties

B�1,�2 � Aϕ∞ � (DC)ϕ and μϕ ∈ (DC)ϕ ⇒ μϕ ∈ (DP)ϕ. (2.4)

The inclusions in (2.4) are quantitative. A geometric characterization of (DC)ϕ states thatμϕ ∈ (DC)ϕ if and only if the function δϕ in (2.1) yields a quasi-distance in R

n , that is,δ(x, y) = 0 if and only if x = y, and there is a K > 1 such that

1

Kδϕ(y, x) ≤ δϕ(x, y) ≤ K δϕ(y, x), ∀x, y ∈ R

n,

and

δϕ(x, y) ≤ K(δϕ(x, z)+ δϕ(z, y)

), ∀x, y, z ∈ R

n . (2.5)

Moreover, the constant K > 1 and the constants in (DC)ϕ depend one another, and not onthe specific function ϕ. By Lemma 5(a) in [5], Lebesgue measure belongs to (DP)ϕ withrespect to the sections of any convex function ϕ. More precisely,

|Sϕ(x, t)| ≤ 2n |Sϕ(x, t/2)|, ∀x ∈ Rn, t > 0. (2.6)

Consequently, under the hypothesis μϕ ∈ (DC)ϕ the triple (Rn, δϕ, | · |) becomes a spaceof homogeneous type and condition (2.2) represents Muckenhoupt’s A∞ condition adaptedto (Rn, δϕ, | · |). Lastly, we mention that under the hypotheses μϕ ∈ (DC)ϕ and ϕ strictlyconvex, Caffarelli proved in [3] that ϕ ∈ C1,δ(Rn), for some structural δ ∈ (0, 1), (see [10]for an alternative proof). All the properties of ϕ above remain quantitatively invariant undersubtraction of an affine transformation from ϕ, therefore we can (and will) always assume,for instance, that ϕ ≥ 0 in R

n and that ∇ϕ(x0) = 0 at any convenient point x0 ∈ Rn . For all

these background results, see [4,5,9–11,13,14].

123


3 The De Philippis–Figalli lemma revisited

In this section we present a variation of lemmas 3.2 and 3.3 by De Philippis and Figalli in [7],see also [8, Section 3.3], to account for the behavior, as weights, of the largest and smallesteigenvalues of D2ϕ(x) under the hypothesis μϕ ∈ Aϕ∞.

Lemma 3.1 If μϕ ∈ Aϕ∞, then wϕ,Wϕ ∈ Aϕ∞ with constants depending only on the Aϕ∞-constants for μϕ . That is, there exist constants ε0 > 0 and C0 ≥ 1, depending only on theAϕ∞ constants for μϕ and dimension n, such that

⎛⎝ 1

|S|∫

S

w1+ε0(x) dx

⎞⎠

11+ε0

≤ C0

|S|∫

S

w(x) dx, (3.1)


Proof We strictly follow the notation in Section 1 of [5] regarding the normalization tech-nique. In particular, we use φ∗, S∗, and λ as in [5], where

φ∗(y) := 1

λϕ(T −1 y)− l(y)− t

λ,

l is a linear function, S∗ := T (S),

λn := μϕ(S)

| det T | and αn ≤ |S|| det T | ≤ βn,

for some positive dimensional constants αn, βn . Setting μ(y) := det D2φ∗(y), this normal-ization yields μ(S∗) = 1. We will also use the fact that

C3t ≤ λ ≤ C4t, (3.2)

where C3 and C4 depend on the doubling constants in (DC)ϕ and the dimension n (seeTheorem 8 in [9]). From the definition of φ∗ we get

T t D2φ∗(y)T = 1

λD2ϕ(T −1 y). (3.3)

From the first few lines of the proof of Theorem 2 in [5] or Lemma 3.2.1 in [13] or Lemma 3.2in [7], there exists a constant C5 > 0, depending only on the (DC)ϕ constants and dimensionn, such that

∫

S∗�φ∗(x) dx ≤ C5. (3.4)

Now, if y := T x , we control the operator (matrix) norm ‖D2ϕ(x)‖ as follows

‖D2ϕ(x)‖ = λ‖T t D2φ∗(y)T ‖ ≤ λ‖T ‖2‖D2φ∗(y)‖ ≤ λ‖T ‖2�φ∗(y). (3.5)

Use (3.4), (3.2), and (3.5) to obtain

1

|S|∫

S

∥∥D2ϕ(x)∥∥ dx ≤ λ‖T ‖2

|S|∫

S

�φ∗(T x) dx ≤ λ‖T ‖2

|S|| det T |∫

S∗�φ∗(y) dy

≤ C5

αnλ‖T ‖2 =: C2λ‖T ‖2.

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D. Maldonado

Hence,

1

|S|∫

S

Wϕ(x) dx ≤ C2λ‖T ‖2. (3.6)

Also, from the fact that M ≤ trace(M)I for every non-negative-definite matrix M (here I isthe n × n identity matrix), from (3.3) and (3.4) we obtain the matrix inequality

1

|S|∫

S

D2ϕ(x) dx ≤ λ

|S| T t

⎛⎝

∫

S

�φ∗(T x)I dx

⎞⎠ T ≤ C2λT t T . (3.7)

Next, for a.e. x ∈ Sϕ(x0, t), let {vϕj (x)} be an orthonormal basis of Rn made out of eigen-

vectors of D2ϕ(x) associated to its eigenvalues {λϕj (x)}. Let v0 be an eigenvector of T t Tassociated to its minimum eigenvalue, denoted by λmin(T t T ), with ‖v0‖ = 1. Hence, fora.e. x ∈ Sϕ(x0, t) we have

v0 =n∑

j=1

γ j (x)vϕj (x), for some scalars γ j (x) with

n∑j=1

γ 2j (x) = 1

and

〈v0, D2ϕ(x)v0〉 =n∑

i, j=1

γi (x)γ j (x)〈vϕi (x), D2ϕ(x)vϕj (x)〉

=n∑

j=1

γ 2j (x)λ

ϕj (x) ≥ min

j=1,...,n{λϕj (x)} = wϕ(x).

By pre- and post-multiplying (3.7) by v0, we get

1

|S|∫

S

wϕ(x) dx ≤ 1

|S|∫

S

〈v0, D2ϕ(x)v0〉 dx ≤ C2λ

n2 λmin(Tt T ). (3.8)

Now we follow the main idea in Lemma 3.3 from [7]. From Lemma 1.1 in [5], there exists aconstant c1 > 0, depending only on the (DC)ϕ constants and dimension n, such that

0 < c1 ≤ | infS∗ φ

∗|.

Just as in [7], set q(x) := c1(|x |2/n2 − 1)/2 and h(x) := φ∗(x)− q(x), so that

| infS∗ h| = | inf

S∗ (φ∗ − q)| ≥ c1

2.

Let �h denote the convex envelope of h in S∗ and let E := {�h = h} ∩ S∗. Then, by theAlexandrov–Bakelman–Pucci maximum principle,

c1

2≤ | inf

S∗ h| ≤ C(n)∫

{�h=h}det D2�h ≤ C(n)

∫E

det D2φ∗ = C(n)μ(E).

Thus,

μ(E) = μ(E)

μ(S∗)≥ c1

2C(n). (3.9)

123


Recall that μ(y) = det D2φ∗(y) is the Monge-Ampère measure associated to the nor-malized solution φ∗ and, in particular, μ(S∗) = 1. By setting A := T −1(E) ⊂ S and using(3.9) we get

μϕ(A)

μϕ(S)= μϕ(T −1(E))

μϕ(T −1(S∗))= μ(E)

μ(S∗)≥ c1

2C(n)=: c6 > 0. (3.10)

From the definition of h and the fact that D2h ≥ 0 on the contact set E it follows that

D2φ∗(x) ≥ c1

n2 I a.e. x ∈ E . (3.11)

Now, for a.e. x ∈ A (think x = T −1 y), from (3.11) we have the following matrix inequality

D2ϕ(x) = λT t D2φ∗(y)T ≥ c1λ

n2 T t T . (3.12)

By taking norms in (3.12), for a.e. x ∈ A, it follows that

Wϕ(x) = ‖D2ϕ(x)‖ ≥ c1λ

n2 ‖T t T ‖ = c1λ

n2 ‖T ‖2. (3.13)

Next we obtain a lower bound for wϕ(x). For a.e. x ∈ Sϕ(x0, t), let vϕ(x) denote an eigen-vector of D2ϕ(x) associated to wϕ(x) with ‖vϕ(x)‖ = 1. Let {v j } be an orthonormal basisof R

n made out of eigenvectors of T t T associated to its eigenvalues {λ j (T t T )}. Therefore,for a.e. x ∈ Sϕ(x0, t), we have

vϕ(x) =n∑

j=1

β j (x)v j , for some scalars β j (x) withn∑

j=1

β2j (x) = 1,

〈vϕ(x), D2ϕ(x)vϕ(x)〉 = wϕ(x), (3.14)

and

〈vϕ(x), T t T vϕ(x)〉 =n∑

j=1

〈vϕ(x), β j (x)Tt T v j , 〉 =

n∑j=1

〈vϕ(x), β j (x)λ jv j , 〉

=n∑

i, j=1

βi (x)β j (x)λ j 〈vi , v j 〉 =n∑

j=1

β2j (x)λ j ≥ λmin(T

t T ).

Consequently, by pre- and post-multiplying (3.12) by vϕ(x), we get

wϕ(x) ≥ c1λ

n2 λmin(Tt T ), a.e. x ∈ A. (3.15)

Now, from (3.10) and the Aϕ∞ hypothesis (2.2)

K1

( |A||S|

)θ1

≥ μϕ(A)

μϕ(S)≥ c6,

so that

|A||S| ≥ c

1θ16

K1=: α. (3.16)

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D. Maldonado

Letw denote eitherwϕ or Wϕ . By combining (3.6) and (3.13), or (3.8) and (3.15), respectively,we obtain

w(x) ≥(

c1

C2

)1

|S|∫

S

w dx =: β 1

|S|∫

S

w(z) dz, a.e. x ∈ A.

Thus,

|{x ∈ S : w(x) ≥ βwS}| ≥ α|S|, ∀S, (3.17)

where wS := (1/|S|) ∫S w(x) dx .Finally, inequality (3.17) and the theory of Muckenhoupt weights applied to the space

of homogeneous type (Rn, δϕ, | · |) imply the reverse-Hölder inequality (3.1). Indeed, in theEuclidean setting, the fact that (3.17) implies (3.1) was first proved in [6] by means of theCalderón-Zygmund decomposition lemma (see the proof of Theorem IV on pp. 247–8). In ageneral space of homogeneous type, the implication follows along the same lines by using,for instance, the “local” Calderón-Zygmund decomposition in Lemma 4.4 of [16]. ��

4 Proof of Theorem 1.1

Under the hypotheses μϕ ∈ (DC)ϕ (which in the present scenario follows from the assump-tionμϕ ∈ B�1,�2 ) andϕ strictly convex, the Legendre transform or convex conjugate functionof ϕ, denoted byψ , satisfiesψ ∈ (DC)ψ (see [10]). Also,ψ is a strictly convex differentiablefunction whose domain is R

n and

∇ϕ(∇ψ(x)) = ∇ψ(∇ϕ(x)) = x, ∀x ∈ Rn . (4.1)

From Theorem 12 in [10], there exists K0 > 0, depending only on the (DC)ϕ constants forμϕ and dimension n, such that

∇ϕ(Sϕ(x, t/K0)) ⊂ Sψ(∇ϕ(x), t) ⊂ ∇ϕ(Sϕ(x, K0t)), (4.2)

for all x ∈ Rn and t > 0. Since (4.1) implies

D2ϕ(x)D2ψ(∇ϕ(x)) = I, a.e. x ∈ Rn,

for a.e. x ∈ Rn we have

W −1ϕ (x) = 1

‖D2ϕ(x)‖ = λmin((D2ϕ(x))−1) = λmin(D2ψ(∇ϕ(x))) = wψ(∇ϕ(x)).Equivalently,

W −1ϕ (∇ψ(y)) = wψ(y), a.e. y ∈ R

n . (4.3)

Notice that the hypothesis μϕ ∈ B�1,�2 implies μψ ∈ B1/�2,1/�1 . Indeed, given any Borelset E ⊂ R

n , we write E := ∇ϕ(F) with F := ∇ψ(E), so that, by (2.3) applied to ψ and ϕ,

μψ(E) = |∇ψ(E)| = |F | ≤ 1

�1μϕ(F) = 1

�1|∇ϕ(∇ψ(E))| = 1

�1|E |.

Similarly, |E | ≤ �2μψ(E). Hence, μψ ∈ B1/�2,1/�1 as claimed. By (2.4), it follows1 that

μψ ∈ Aψ∞, with constants depending only on �1,�2 and dimension n. Notice that, by

1 A similar argument, but also involving (4.2), proves that μϕ ∈ Aϕ∞ implies μψ ∈ Aψ∞.

123


Lemma 3.1, the facts that μψ ∈ B�1,�2 ⊂ Aϕ∞ and μψ ∈ B1/�2,1/�1 ⊂ Aψ∞ impliy that

Wϕ ∈ Aϕ∞ and wψ ∈ Aψ∞. In particular, there exists ε > 0, depending only on �1,�2 anddimension n, such that

⎛⎜⎝ 1

|Sϕ |∫

Sϕ

Wϕ(x)1+ε(x) dx

⎞⎟⎠

11+ε

� 1

|Sϕ |∫

Sϕ

Wϕ(x) dx, (4.4)

for every section Sϕ of ϕ and

⎛⎜⎝ 1

|Sψ |∫

Sψ

wψ(y)1+ε dy

⎞⎟⎠

11+ε

� 1

|Sψ |∫

Sψ

wψ(y) dy, (4.5)

for every section Sψ of ψ , where the equivalence constants depend only on �1,�2 anddimension n.

Next we compare averages of some powers of W −1ϕ over sections of ϕ with averages of

the same powers of wψ over sections of ψ . Let p ∈ [1, 1 + ε]. Given any section Sϕ(x0, t),set y0 := ∇ϕ(x0), by (4.2) and (4.3),

∫

Sϕ(x0,t)

W −pϕ (x)dx ≤

∫

∇ψ(Sψ (y0,K0t))

W −pϕ (x) dx

=∞∫

0

|{x ∈ ∇ψ(Sψ(y0, K0t)) : W −pϕ (x) > r}| dr

=∞∫

0

|∇ψ({y ∈ Sψ(y0, K0t) : W −pϕ (∇ψ(y)) > r})| dr

≤ 1

�1

∞∫

0

|{y ∈ Sψ(y0, K0t) : W −pϕ (∇ψ(y)) > r}| dr

= 1

�1

∫

Sψ (y0,K0t)

W −pϕ (∇ψ(y)) dy

= 1

�1

∫

Sψ (y0,K0t)

wpψ(y) dy ≤ C7

�1

∫

Sψ (y0,t)

wpψ(y) dy,

where C7 > 1 depends only on K and the doubling constant forw pψ (which, in turn, depends

only on �1,�2, and dimension n). Similarly, by (4.2) and starting from the inequality∫

Sϕ(x0,t)

W −pϕ (x) dx ≥

∫

∇ψ(Sψ (y0,t/K0))

W −pϕ (x) dx,

it follows that ∫

Sϕ(x0,t)

W −pϕ (x) dx ≥ c8

�2

∫

Sψ (y0,t)

wpψ(y) dy,

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D. Maldonado

where c8 > 0 depends only on K0 and the doubling constant for wψ . That is,∫

Sϕ(x0,t)

W −pϕ (x) dx �

∫

Sψ (y0,t)

wpψ(y) dy, p ∈ [1, 1 + ε], (4.6)

where the equivalence constants ultimately depend only on �1,�2, dimension n, and p.Also, by (4.2),

|Sϕ(x0, t)| � |Sψ(y0, t)|,where the equivalence constants depend only on�1,�2, and dimension n. Next, always forp ∈ [1, 1 + ε], we put everything together as follows

⎛⎜⎝ 1

|Sϕ(x0, t)|∫

Sϕ(x0,t)

W −pϕ (x) dx

⎞⎟⎠

1p

�⎛⎜⎝ 1

|Sψ(y0, t)|∫

Sψ (y0,t)

wpψ(y) dy

⎞⎟⎠

1p

� 1

|Sψ(y0, t)|∫

Sψ (y0,t)

wψ(y) dy

� 1

|Sϕ(x0, t)|∫

Sϕ(x0,t)

W −1ϕ (x) dx .

Hence, W −1ϕ ∈ Aϕ∞, which along with the fact that Wϕ ∈ Aϕ∞, yields Wϕ ∈ Aϕ2 . A similar

argument, interchanging the roles of ϕ and ψ , proves that wϕ ∈ Aϕ2 . ��We close this section by recalling that the Muckenhoupt theory as well as the techniques

in this note are local in nature. In particular, the hypotheses in Theorem 1.1 and Corollary1.2 can be localized by assuming �1|E | ≤ μϕ(E) ≤ �2|E | for every Borel set E ⊂ �,where � is a open bounded convex set with ϕ|∂� = 0, and then inequalities (1.3) and (1.4)will hold for every section S := Sϕ(x0, t) such that S(x0,Ct) ⊂⊂ �, where C is a constantdepending only on �1,�2 and dimension n.


The proof of the Poincaré inequality (1.5) will be an immediate consequence of the tech-niques in Sect. 3, the Caffarelli-Gutiérrez normalization procedure, and the usual EuclideanPoincaré inequality. Indeed, given a section S := Sϕ(x0, t), let T be an affine transformationnormalizing S. By the usual (1, 1)-Poincaré inequality, for any u ∈ C1(T (S)) we have

∫

T (S)

|u(y)− uT (S)| dy ≤ Cabs

∫

T (S)

|∇u(y)| dy, (5.1)

where Cabs is an absolute numeric constant [recall that T (S) ⊂ B(0, 1) and T (S) is a convexset]. After changing variables y = T x in (5.1), we obtain

∫

S

|u(x)− uS | dx ≤ Cabs

∫

S

|(T −1)t∇u(x)| dx, (5.2)

123


where u(x) := u(T x), x ∈ S. Next, notice that

‖(T −1)t D2ϕ(x)12 ‖2 = ‖(T −1)t D2ϕ(x)T −1‖.

Now, from (3.3) we get

‖(T −1)t D2ϕ(x)T −1‖ ≤ λ�φ∗(T x),

and from (3.4) [just as in the proof of (3.6), under only the assumption μϕ ∈ (DC)ϕ] and(3.2) we obtain

1

|S|∫

S

‖(T −1)t D2ϕ(x)T −1‖ dx ≤ C2λ ≤ C2C4t. (5.3)

Consequently,∫

S

|(T −1)t∇u(x)| dx =∫

S

|(T −1)t D2ϕ(x)12 D2ϕ(x)−

12 ∇u(x)| dx

≤∫

S

‖(T −1)t D2ϕ(x)12 ‖|D2ϕ(x)−

12 ∇u(x)| dx

≤⎛⎝

∫

S

‖(T −1)t D2ϕ(x)12 ‖2 dx

⎞⎠

12⎛⎝

∫

S

|D2ϕ(x)−12 ∇u(x)|2 dx

⎞⎠

12

=⎛⎝

∫

S

‖(T−1)t D2ϕ(x)T t‖dx

⎞⎠

12⎛⎝

∫

S

〈D2ϕ(x)−1∇u(x),∇u(x)〉 dx

⎞⎠

12

≤ (C2C4)12 t

12 |S| 1

2

⎛⎝

∫

S

〈D2ϕ(x)−1∇u(x),∇u(x)〉 dx

⎞⎠

12

, (5.4)

and (1.5) follows from (5.4) and (5.2). ��The proof of the Poincaré inequalities (1.9) will be more involved. We will first show that

inequality (5.2) implies a representation formula for the oscillation of a function with respectto its mean value over sections. More precisely, given any section S := Sϕ(x0, t) and anyx ∈ S, we have

|u(x)− uS |2 ≤ Ct1−ε Iϕ(|D2ϕ− 1

2 ∇u|2χSϕ(x0,2K t)

)(x), (5.5)

where K denotes the quasi-triangle constant in (2.5), Iϕ is an appropriate potential operator,ε ∈ (0, 1) is as in (5.10) below, and the constant C > 0 depends only on�1,�2, dimensionn, and ε. Our proof for (5.5) is inspired by the argument in the main theorem of [12] (seeTheorem 1), where Franchi, Lu, and Wheeden show certain Poincaré-type inequalities tobe equivalent to representation formulas involving potential operators. At several stages,however, our proof of (5.5) must necessarily diverge from the techniques in [12]. The mainreason being the absence of a priori L∞-estimates for ‖(T −1)t D2ϕ(x)T −1‖ in the generalMonge-Ampère context.2 All we have at our disposal are the L1-estimates developed in the

2 Of course, such L∞-estimates are obvious in the Euclidean setting. In that case we have ϕ(x) = |x |2/2,so that sections reduce to the usual balls and any normalizing T is just a composition of a dilation and atranslation, taking some ball B(x0, r) into B(0, 1).

123

D. Maldonado

previous sections. Namely, that given any section S := Sϕ(x0, t) and an affine transformationT normalizing S, it follows that

1

|S|∫

S

‖(T −1)t D2ϕ(x)T −1‖ dx � t, (5.6)

where the equivalence constants depend only on the Aϕ∞ constants for μϕ and dimensionn [but bear in mind that, due to (5.3), the inequality � in (5.6) depends only on the (DC)ϕconstants and dimension n]. In addition, the condition (1.b) in [12] has to be replaced by auniform estimate on the size of the sections provided by inequalities in (5.7) below. It mustbe noticed that the quantity ‖(T −1)t D2ϕ(x)T −1‖ cannot be regarded as a weight definedpointwise in R

n because T depends on each section being considered. The final step in theproof of (1.9) will be based on L p − Lq bounds for the potential operator Iϕ in (5.5). Inour proof of (5.5), the operator Iϕ has been designed to fit within the framework of potentialoperators in spaces of homogeneous type as developed by Sawyer and Wheeden in [18].

Let us start the proof of (5.5) by mentioning that the hypothesis μϕ ∈ B�1,�2 implies theexistence of constants 0 < c′ < 1 < C ′, depending only on �1, �2, and dimension n, suchthat

c′tn2 ≤ |S(x, t)| ≤ C ′t

n2 , ∀x ∈ R

n,∀t > 0, (5.7)

(see, for instance, Theorem 8 in [9] or Corollary 3.2.4 in [13]). As mentioned, in this proofthe relations (5.7) will play the role of the inequality (1.b) in [12]. Now, given any sectionS := Sϕ(x0, t) set S∗ := Sϕ(x0, 2K t). Fix x ∈ S and define S0 := Sϕ(x, t). Then,

|u(x)− uS | ≤ |uS − uS0 | + |u(x)− uS0 |. (5.8)

Since K is the quasi-triangle constant for δϕ , we have S0 ⊂ S∗ (in addition to S ⊂ S∗) andwe estimate the first summand in (5.8) as follows

|uS − uS0 | ≤ |uS0 − uS∗ | + |uS∗ − uS |≤ 1

|Sϕ(x, t)|∫

Sϕ(x,t)

|u(y)− uSϕ(x0,2K t)| dy

+ 1

|Sϕ(x0, t)|∫

Sϕ(x0,t)

|u(y)− uSϕ(x0,2K t)| dy

≤ C

|Sϕ(x0, 2K t)|∫

Sϕ(x0,2K t)

|u(y)− uSϕ(x0,2K t)| dy,

where the constant C is structural, depending only on the doubling constant for Lebesguemeasure | · | on sections (which, in fact, equals 2n as shown in Lemma 5.2(b) of [5]) and thequasi-triangle constant K . We now use (5.2) to do

∫

Sϕ(x0,2K t)

|u(y)− uSϕ(x0,2K t)| dy ≤ Cabs

∫

Sϕ(x0,2K t)

|(T −1)t∇u(y)| dy,

and notice that for y ∈ Sϕ(x0, 2K t) we have

δϕ(x, y) ≤ K [δϕ(x, x0)+ δϕ(x0, y)] ≤ K (1 + 2K )t. (5.9)

123


Given 1 ≤ p < q as in (1.8), define

ε := n

2− n

2q− n

2p′ = n

2

(1

p− 1

q

). (5.10)

The facts that ε belongs to the interval (0, 1) and that ε−n/2 < 0 will be important throughoutthe proof. By using (5.7) with Sϕ(x0, 2K t) and (5.9), we obtain

1

|Sϕ(x0, 2K t)|∫

Sϕ(x0,2K t)

|(T −1)t∇u(y)| dy

≤ Ct−n2

∫

Sϕ(x0,2K t)

|(T −1)t∇u(y)| dy

= Ct−n2

∫

Sϕ(x0,2K t)

|(T −1)t∇u(y)|δϕ(x, y)12 (

n2 −ε)δϕ(x, y)

12 (ε− n

2 )dy

≤ C(n, K , ε)t−n2 + 1

2 (n2 −ε)

∫

Sϕ(x0,2K t)

|(T −1)t∇u(y)|δϕ(x, y)12 (ε− n

2 )dy,

where the constant C(n, K , ε) > 0 is of the form

C(n, K , ε) = C(�1,�2, n)[K (1 + 2K )] 12 (

n2 −ε).

Next, define the kernel

Kϕ(x, y) := δϕ(x, y)ε−n2 , ∀x, y ∈ R

n, (5.11)

and the associated potential operator Iϕ as

Iϕ( f )(x) :=∫

Rn

Kϕ(x, y) f (y) dy.

By Cauchy-Schwarz and (5.6) we get∫

Sϕ(x0,2K t)

|(T −1)t∇u(y)|δϕ(x, y)12 (ε− n

2 ) dy

=∫

Sϕ(x0,2K t)

|(T −1)t D2ϕ(y)12 D2ϕ(y)−

12 ∇u(y)|δϕ(x, y)

12 (ε− n

2 ) dy

≤∫

Sϕ(x0,2K t)

‖(T −1)t D2ϕ(y)12 ‖|D2ϕ(y)−

12 ∇u(y)|δϕ(x, y)

12 (ε− n

2 ) dy

≤⎛⎜⎝

∫

Sϕ(x0,2K t)

‖(T −1)t D2ϕ(y)12 ‖2 dy

⎞⎟⎠

12

×⎛⎜⎝

∫

Sϕ(x0,2K t)

|D2ϕ(y)−12 ∇u(y)|2 Kϕ(x, y) dy

⎞⎟⎠

12

≤ Ct12 |Sϕ(x0, 2K t)| 1

2 Iϕ(|D2ϕ− 12 ∇u|2χSϕ(x0,2K t))

12 (x).

123

D. Maldonado

Thus, putting everything together and using (5.7) again, for every x ∈ S we get

|uS − uS∗ | � t−n2 + 1

2 (n2 −ε)

∫

Sϕ(x0,2K t)

|(T −1)t∇u(y)|δϕ(x, y)12 (ε− n

2 )dy

� t−n2 + 1

2 (n2 −ε)t

12 |Sϕ(x0, 2K t)| 1

2 Iϕ(|D2ϕ− 1

2 ∇u|2χSϕ(x0,2K t)

) 12(x)

� t−n2 + 1

2 (n2 −ε)t

12 + n

4 Iϕ(|D2ϕ− 1

2 ∇u|2χSϕ(x0,2K t)

) 12(x)

= t1−ε

2 Iϕ(|D2ϕ− 1

2 ∇u|2χSϕ(x0,2K t)

) 12(x),

where the implicit constants depend only on �1,�2, dimension n, and ε. In particular, forevery x ∈ S we have

|uS − uS∗ |2 ≤ Ct1−ε Iϕ(|D2ϕ− 1

2 ∇u|2χS∗)(x), (5.12)

where C > 0 depends only on �1,�2, dimension n, and ε.Next, we estimate |u(x) − uS0 | in (5.8) by resorting to Lebesgue’s differentiation theo-

rem in the space of homogeneous type (Rn, δϕ, | · |). For k ∈ N0, set Sk := Sϕ(x, 2−k t),then

|u(x)− uS0 | ≤∞∑

k=0

|uSk+1 − uSk | ≤∞∑

k=0

1

|Sk+1|∫

Sk+1

|u(y)− uSk | dy

≤ C∞∑

k=0

1

|Sk |∫

Sk

|u(y)− uSk | dy ≤ CCabs

∞∑k=0

1

|Sk |∫

Sk

|(T −1k )t∇u(y)| dy,

where Tk is an affine transformation normalizing Sk so that, due to (5.6) and (5.7),

∫

Sk

‖(T −1k )t D2ϕ(x)T −1

k ‖ dx � 2−k t |Sk | � (2−k t)1+ n2 , ∀k ∈ N0, (5.13)

where the equivalence constants depend only on �1,�2 and dimension n. By (5.7), we get

∞∑k=0

1

|Sk |∫

Sk

|(T −1k )t∇u(y)| dy

=∞∑

k=0

1

|Sk |∫

S∗|(T −1

k )t∇u(y)|χ{δϕ(x,y)<2−k t}(y) dy

=∞∑

k=0

1

|Sk |∫

S∗|(T −1

k )t∇u(y)|χ{δϕ(x,y)<2−k t}(y)δϕ(x, y)12 (

n2 −ε)δϕ(x, y)

12 (ε− n

2 ) dy

≤ C∞∑

k=0

(2−k t)−n2 + 1

2 (n2 −ε)

∫

S∗|(T −1


n2 −ε) dy.

123


By Cauchy-Schwarz and (5.13), we estimate the integral on each summand above asfollows

∫

S∗|(T −1


n2 −ε) dy

≤∫

S∗‖(T −1

k )t D2ϕ(y)12 ‖|D2ϕ(y)−

12 ∇u(y)|χ{δϕ(x,y)<2−k t}(y)δϕ(x, y)

12 (

n2 −ε) dy

≤⎛⎝

∫

S∗|D2ϕ(y)−

12 ∇u(y)|2 Kϕ(x, y) dy

⎞⎠

12⎛⎜⎝

∫

Sk

‖(T −1k )t D2ϕ(y)

12 ‖2

⎞⎟⎠

12

≤ C(2−k t)12 + n

4

⎛⎝

∫

S∗|D2ϕ(y)−

12 ∇u(y)|2 Kϕ(x, y) dy

⎞⎠

12

.

Consequently, always for x ∈ S,

|u(x)− uS∗ | �∞∑

k=0

1

|Sk |∫

Sk

|(T −1k )t∇u(y)| dy

�∞∑

k=0

(2−k t)−n2 + 1

2 (n2 −ε)

∫

S∗|(T −1


n2 −ε) dy

�

⎛⎝

∫

S∗|D2ϕ(y)−

12 ∇u(y)|2 Kϕ(x, y) dy

⎞⎠

12 ∞∑

k=0

(2−k t)−n2 + 1

2 (n2 −ε)+ 1

2 + n4

� t1−ε

2

⎛⎝

∫

S∗|D2ϕ(y)−

12 ∇u(y)|2 Kϕ(x, y) dy

⎞⎠

12

,

where the implicit constants depend only on �1,�2, dimension n, and ε. Hence, for allx ∈ S,

|u(x)− uS∗ |2 ≤ Ct1−ε Iϕ(|D2ϕ− 1

2 ∇u|2χS∗)(x), (5.14)

which, together with (5.12), yields (5.5). At this point, we focus on the operator Iϕ . Indeed,by adapting the construction in [18] to the space of homogeneous type (Rn, δϕ, | · |), weaim at putting ourselves in position to apply Theorem 3, part (A), in [18] (in the unweightedcase). Given a section S := Sϕ(x0, t), define

Fϕ(S) := sup{

Kϕ(x, y) : x, y ∈ S and δϕ(x, y) ≥ t},

so that our definition of Kϕ in (5.11) yields

Fϕ(S) = tε−n2 , (5.15)

123

D. Maldonado

[recall that (5.10) implies ε − n/2 < 0]. Now, given another section S′ := S(x ′, t ′), from(5.15) and (5.7), we have

|S′||S| ≤ C

Fϕ(S)

Fϕ(S′)

(t ′

t

)ε, (5.16)

where C > 0 depends only on �1,�2, and dimension n. Notice that inequality (5.16) isprecisely condition (1.20) in [18] and that, from (5.10) and (5.7), for every S := Sϕ(x0, t)we have

Fϕ(S)|S| 1q + 1

p′ ≤ Ctε− n

2 + n2q + n

2p′ = C, (5.17)

where the constant C > 0 depends only on �1,�2, and dimension n. Thus, inequality(5.17) represents condition (1.19) in [18] in the context of the space of homogeneous type(Rn, δϕ, | · |). Therefore, by Theorem 3, part (A), in [18], we have

Iϕ : L p(Rn, | · |) → Lq(Rn, | · |), (5.18)

and the Poincaré inequalities (1.9) follow from the representation formula (5.5), the fact that(5.10) and (5.7) yield

t1−ε = t1+ n2q − n

2p � t |S| 1q − 1

p ,

and the mapping properties for Iϕ in (5.18). ��


In this section assume that ϕ ∈ C2(Rn) as is customary when dealing with the linearizedMonge-Ampère operator. As in the Introduction, let � ⊂ R

n be an open bounded set andconsider the possibly-degenerate elliptic operator Lϕ defined as

Lϕu(x) := trace(Aϕ(x)D2u(x)), x ∈ �,

Recall that the columns of Aϕ are divergence free, so that

div(Aϕ∇g) = trace(AϕD2g), ∀g ∈ C2. (6.1)

Throughout this section we only assume μϕ ∈ (DC)ϕ and constants depending only on the(DC)ϕ constants and dimension n will be called structural constants. We stress that all theestimates that follow are a priori estimates and the main (and only) point is to show that theconstants involved are structural constants.

Harnack’s inequality (1.10) will follow from the Theorems 6.1 and 6.2 below. Theorem6.1 is essentially a re-phrasing of [11, Theorem 3] and it accounts for a weak mean-valueinequality for positive sub-solutions. Theorem 6.2 is an estimate on the energy of the logarithmof super-solutions with respect to the quadratic form associated to Aϕ . The key element inthe proof, however, will rely on exploiting the divergence-form structure of the linearizedMonge-Ampère equation and the identity (6.3) below expressing the duality between thespaces of homogeneous type (Rn, μϕ, δϕ) and (Rn, μψ, δψ) [recall the definitions of thequasi-metrics and the Monge-Ampère measures from (2.1) and (2.3)]. This duality argumentgoes as follows. Let E ⊂ R

n be a Borel set and given differentiable functions u(x) and h(x)define v(y) and H(y) by

u(x) =: v(∇ϕ(x)) and h(x) =: H(∇ϕ(x)), (6.2)

123


so that

∇u(x) = D2ϕ(x)∇v(∇ϕ(x)) and ∇h(x) = D2ϕ(x)∇ H(∇ϕ(x)).Now, by a change of variables where it will be useful to think in terms of y := ∇ϕ(x)

[and, from (4.1), x = ∇ψ(y)] and to bear in mind that, also from (4.1),

μϕ (∇ψ(y)) μψ(y) = det D2ϕ (∇ψ(y)) det D2ψ(y) = det(I ) = 1, ∀y ∈ Rn,

we have∫

E

〈Aϕ(x)∇u(x),∇h(x)〉 dx =∫

∇ψ(∇ϕ(E))〈(D2ϕ(x))−1∇u(x),∇h(x)〉μϕ(x) dx

=∫

∇ψ(∇ϕ(E))〈(D2ϕ(x))−1 D2ϕ(x)∇v(∇ϕ(x)), D2ϕ(x)∇ H(∇ϕ(x))〉μϕ(x) dx

=∫

∇ψ(∇ϕ(E))〈∇v(∇ϕ(x)), D2ϕ(x)∇ H(∇ϕ(x))〉μϕ(x) dx

=∫

∇ϕ(E)〈∇v(y), D2ϕ(∇ψ(y))∇ H(y)〉

=1︷︸︸︷μϕ(∇ψ(y))μψ(y) dy

=∫

∇ϕ(E)〈∇v(y), (D2ψ(y))−1∇ H(y)〉 dy

=∫

∇ϕ(E)〈(D2ψ(y))−1∇v(y),∇ H(y)〉 dy.

That is,∫

E

〈Aϕ(x)∇u(x),∇h(x)〉 dx =∫

∇ϕ(E)〈(D2ψ(y))−1∇v(y),∇ H(y)〉 dy. (6.3)

Theorem 6.1 Assume μϕ ∈ (DC)ϕ and suppose that u ∈ C2(�), u > 0, satisfies

Lϕ(u) = trace(AϕD2u) ≥ 0 in �. (6.4)

For y ∈ Rn set v(y) := u(∇ψ(y)). Then, for every p > 0 there exists a constant C > 0,

depending on p and the structure, such that for every section Sϕ(x0, t)with Sϕ(x0, 2K t) ⊂⊂�, we have

supSψ (x0,t)

v ≤ C

⎛⎜⎝ 1

|Sψ(x0, 2K t)|∫

Sψ (x0,2K t)

v(y)p dy

⎞⎟⎠

1p

, (6.5)

where we have set Sψ(x, t) := ∇ϕ(Sϕ(x, t)).

Proof Inequality (6.5) will be a revisit of [11, Theorem 3]. The motivation comes from thefact that Caffarelli and Gutiérrez proved in [5, Theorem 1] that if ϕ ∈ (DC)ϕ and a non-negative function w ∈ C2 satisfies trace(Aϕ(x)D2w(x)) ≤ 0 in a section S with 2S ⊂⊂ �,

123

D. Maldonado

then w enjoys a uniform critical density property (with structural constants). In particular, ifu satisfies

Lϕ(u) = trace(AϕD2u) ≥ 0

in a section S and if τ is a number with τ ≥ u in S, then (thinking of w := τ − u ≥ 0 in S),

trace(AϕD2(τ − u)) = −trace(AϕD2u) = −Lϕ(u) ≤ 0 in S.

Therefore, the function τ − u (as well as any positive multiple of it) has the uniform criticaldensity property. By [11, Theorem 3] this implies the following weak mean-value property

supS1/2

u ≤ C ′

μϕ(S)

∫

S

u(x)μϕ(x) dx, ∀S ⊂ 2S ⊂⊂ �, (6.6)

where S1/2 denotes the section with the same center as S and half its height, and C ′ ≥ 1is a structural constant.3 The tight connection between the critical density property andthe mean-value inequality (6.6) has been studied in the context of spaces of homogeneoustype in [15]. An interpolation argument shows that (6.6) implies that for every p > 0we have

supx∈Sϕ(x0,t)

u(x) ≤ C ′′′

⎛⎜⎝ 1

μϕ(Sϕ(x0, 2K t))

∫

Sϕ(x0,2K t)

u(x)pμϕ(x) dx

⎞⎟⎠

1p

, (6.7)

where C ′′′ ≥ 1 depends on the structure and p. Along the same lines of the proof of (6.3),we next express (6.7) in terms of v(y) := u(x), with y = ∇ϕ(x), to obtain

supy∈∇ϕ(Sϕ(x0,t))

v(y) ≤ C ′′′

⎛⎜⎝ 1

μϕ(Sϕ(x0, 2K t)))

∫

∇ϕ(Sϕ(x0,2K t))

v(y)p dy

⎞⎟⎠

1p

. (6.8)

But, since by definition Sψ(x0, t) = ∇ϕ(Sϕ(x0, t)) and, consequently, μϕ(Sϕ(x0, t)) =|∇ϕ(Sϕ(x0, t))| = |Sψ(x0, t)|, the desired inequality (6.5) follows from (6.8). ��Theorem 6.2 Assume μϕ ∈ (DC)ϕ . Then there exists a structural constant C9 > 0 suchthat for every function u > 0 that satisfies

Lϕ(u) = div(Aϕ∇u) ≤ 0 in �, (6.9)

and every section S := Sϕ(x0, t0) with Sϕ(x0, 2t0) ⊂⊂ �, we have∫

S

〈Aϕ(x)∇u(x),∇u(x)〉 1

u(x)2dx ≤ C9

μϕ(S)

t0.

Proof Given a section S := Sϕ(x0, t0), after subtracting an affine transformation, we canassume ϕ ≥ 0 in R

n and ∇ϕ(x0) = 0. Set 2S := Sϕ(x0, 2t0). For any given h ∈ C10 (2S),

3 Theorem 3 in [11] is stated for solutions, but the proof actually works for sub-solutions. Indeed, by Cheby-shev’s inequality one can take q = 1 in line 9 on page 265 of [11] and avoid he use of Proposition 13 in[11].

123


after multiplying (6.9) by h2/u, integrating over 2S, using (6.1) and the divergence theoremwe have

−∫

2S

〈Aϕ∇u,∇h〉2h

udx +

∫

2S

〈Aϕ∇u,∇u〉h2

u2 dx ≤ 0,

and, by Cauchy-Schwartz,

∫

2S

〈Aϕ(x)∇u(x),∇u(x)〉h2(x)

u2(x)dx ≤ 2

∫

2S

〈Aϕ(x)∇u(x),∇h(x)〉h(x)

u(x)dx

≤ 2

⎛⎝

∫

2S

〈Aϕ(x)∇u(x),∇u(x)〉h2(x)

u2(x)dx

⎞⎠

12⎛⎝

∫

2S

〈Aϕ(x)∇h(x),∇h(x)〉 dx

⎞⎠

12

,

yielding

∫

2S

〈Aϕ(x)∇u(x),∇u(x)〉h2(x)

u2(x)dx ≤ 4

∫

2S

〈Aϕ(x)∇h(x),∇h(x)〉 dx . (6.10)

Next, let γ : R → [0, 1] be a smooth function supported in [0, 2] with γ ≡ 1 on [0, 1] andset h(x) := γ (ϕ(x)/t0) so that h ∈ C1

0 (2S) with h ≡ 1 on S and

∇h(x) = 1

t0γ ′(ϕ(x)/t)∇ϕ(x).

Consequently, integrating by parts again and by the null-Lagrangian property (6.1),

∫

2S

〈Aϕ(x)∇h(x),∇h(x)〉 dx = 1

t20

∫

2S

γ ′(ϕ(x)/t0)2〈Aϕ(x)∇ϕ(x),∇ϕ(x)〉 dx

≤ ‖γ ′‖2∞t20

∫

2S

〈Aϕ(x)∇ϕ(x),∇ϕ(x)〉 dx

= ‖γ ′‖2∞t20

∫

2S

〈Aϕ(x)∇(2t0 − ϕ(x)),∇(2t0 − ϕ(x))〉 dx

= −‖γ ′‖2∞t20

∫

2S

div[Aϕ(x)∇(2t0 − ϕ(x))](2t0 − ϕ(x)) dx

= ‖γ ′‖2∞t20

∫

2S

div[Aϕ(x)∇ϕ(x)](2t0 − ϕ(x)) dx

= ‖γ ′‖2∞t20

∫

2S

trace(Aϕ(x)D2ϕ(x))(2t0 − ϕ(x)) dx

= n‖γ ′‖2∞t20

∫

2S

(2t0 − ϕ(x))μϕ(x) dx ≤ 2n‖γ ′‖2∞t0

μϕ(2S).

123

D. Maldonado

Now, using that h ≡ 1 on S, (6.10), the fact thatμϕ ∈ (D P)ϕ [and let us denote the structural

(D P)ϕ-doubling constant by Cdpϕ ], and the inequalities above, we get

∫

S

〈Aϕ(x)∇u(x),∇u(x)〉 1

u2(x)dx ≤

∫

2S

〈Aϕ(x)∇u(x),∇u(x)〉h2(x)

u2(x)dx

≤ 4∫

2S

〈Aϕ(x)∇h(x),∇h(x)〉 dx

≤ 2n‖γ ′‖∞t0

μϕ(2S)

≤ 2nCdpϕ ‖γ ′‖∞

t0μϕ(S) =: C9

μϕ(S)

t0.

��

In the proof of Theorem (6.2) we strongly used the null-Lagrangian property (6.1) of Aϕ toobtain a bound for an expression involving the quadratic form associated to Aϕ . On the otherhand, the Poincaré inequality (1.5) involves the quadratic form associated to D2ϕ(x)−1, notAϕ(x). If we were assuming μϕ ∈ B�1,�2 , these quadratic forms would be equivalent. Butwe are assuming onlyμϕ ∈ (DC)ϕ . The way around this involves equality (6.3) which, as wewill see, allows us to connect the Poincaré inequality (1.5) applied to ψ and the Aϕ-energybound in Theorem 6.2.

Next we prove that given a positive super-solution u and defining v(y) = u(x) withy = ∇ϕ(x) we obtain that v belongs to B M O(∇ϕ(�)) with norm bounded by a structuralconstant. Indeed, by using the fact that Lebesgue measure is doubling on the sections ofany strictly convex function [recall (2.6)], and the Poincaré inequality (1.5) associated to(Rn, δψ , μψ), there is a structural constant C > 0 such that for any section Sψ(y0, t) with∇ψ(Sψ(y0, 2K 3

0 t)) ⊂⊂ �, we have

1

|Sψ(y0, t)|∫

Sψ (y0,t)

| log v(y)− (log v)Sψ (y0,t)| dy

≤ Ct12

⎛⎜⎝ 1

|Sψ(y0, t)|∫

Sψ (y0,t)

〈D2ψ(y)−1∇ log v(y),∇ log v(y)〉 dy

⎞⎟⎠

12

= Ct12

⎛⎜⎝ 1

|Sψ(y0, t)|∫

∇ψ(Sψ (y0,t))

〈Aϕ(x)∇ log u(x),∇ log u(x)〉 dx

⎞⎟⎠

12

,

where in the last line we have used (6.3) with E := ∇ψ(Sψ(y0, t)) and h = u. Settingy0 =: ∇ϕ(x0), by (4.2) we obtain

∇ψ(Sψ(y0, t)) ⊂ Sϕ(x0, K0t),

123


which, combined with Theorem 6.2 applied with t0 := K0t , yields∫

∇ψ(Sψ (y0,t))


≤∫

Sϕ(x0,K0t)


=∫

Sϕ(x0,K0t)

〈Aϕ(x)∇u(x),∇u(x)〉 1

u(x)2dx

≤ C9μϕ(Sϕ(x0, K0t))

K0t.

But, from the definition of the Monge-Ampère measure, the inclusions in (4.2), and (2.6),there is a structural constant C11 > 0 such that

μϕ(Sϕ(x0, K0t)

) = |∇ϕ (Sϕ(x0, K0t)

) | ≤ |Sψ(y0, K 20 t)| ≤ C11|Sψ(y0, t)|.

Therefore,

1|Sψ (y0,t)|

∫

Sψ (y0,t)

| log v(y)− (log v)Sψ (y0,t)| dy ≤ C(C9C11)12

K0=: C12. (6.11)

Finally, given u > 0 a solution to div(Aϕ∇u) = 0 we have that both u and 1/u are sub-solutions and inequality (6.5) can be applied to both v and 1/v. Then, by a localized versionof the John-Nirenberg inequality in metric spaces, see for instance [1, Theorem 3.15, p.70], itfollows from (6.11) that there exists a structural number p0 > 0 such that the arithmetic andharmonic means of v p0 are uniformly comparable with structural constants [in other words,v p0 ∈ Aψ2 (∇ϕ(�))]. Bear in mind that, from (4.2),

Sψ (∇ϕ(x), t/K0) ⊂ Sψ(x, t) = ∇ϕ (Sϕ(x, t)

) ⊂ Sψ (∇ϕ(x), K0t) , ∀x ∈ Rn, t > 0.

It is then routine to obtain Harnack’s inequality for v, namely,

sup∇ϕ(Sϕ(x,t))

v ≤ CH inf∇ϕ(Sϕ(x,t))v, (6.12)

and, after the change y = ∇ϕ(x), Theorem 1.4 follows.A more delicate argument allows to prove a Harnack inequality for positive solutions to

equations with lower-order terms, but this will appear elsewhere. ��Acknowledgements Part of this work was carried out while the author was visiting the Instituto de Matemáti-cas de la Universidad de Sevilla (IMUS), Spain, in June 2012. I would like to thank Professor Carlos Pérezfor his kind invitation and warm hospitality. I am also thankful to Professor Andrei Lerner for insightfulconversations on the Muckenhoupt classes and to Professor Alessio Figalli for his careful reading of an earlierversion of the manuscript. I would also like to thank the anonymous referee for the detailed report and forsuggesting corrections that improved the presentation of the results.

References

1. Björn, A., Björn, J.: Nonlinear potential theory on metric spaces. In: EMS Tracts in Mathematics, vol.17. European Mathematical Society, Zurich (2011)

123

D. Maldonado

2. Caffarelli, L.: Interior a priori estimates for solutions to fully nonlinear elliptic equations. Ann. Math.130, 189–213 (1989)

3. Caffarelli, L.: Some regularity properties of solutions of Monge-Ampère equation. Comm. Pure Appl.Math. 44, 965–969 (1991)

4. Caffarelli, L., Gutiérrez, C.: Real analysis related to the Monge-Ampère equation. Trans. Am. Math. Soc.348, 1075–1092 (1996)

5. Caffarelli, L., Gutiérrez, C.: Properties of the solutions of the linearized Monge-Ampère equation. Am.J. Math. 119(2), 423–465 (1997)

6. Coifman, R.R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals.Studia Math. 51, 241–250 (1974)

7. De Philippis, G., Figalli, A.: W 2,1 regularity for solutions of the Monge-Ampère equation. To appear inInvent. Math. Preprint available at arXiv:1111.7207v1 [math.AP] (2013)

8. De Philippis, G., Figalli, A., Savin, O.: A note on interior W 2,1+ estimates for the Monge-Ampèreequation. Preprint available at arXiv:1202.5566v2 [math.AP]

9. Forzani, L., Maldonado, D.: On geometric characterizations for Monge-Ampère doubling measures. J.Math. Anal. Appl. 275(2), 721–732 (2002)

10. Forzani, L., Maldonado, D.: Properties of the solutions to the Monge-Ampère equation. Nonlinear Anal.57(5–6), 815–829 (2004)

11. Forzani, L., Maldonado, D.: A mean-value inequality for non-negative solutions to the linearized Monge-Ampère equation. Potential Anal. 30(3), 251–270 (2009)

12. Franchi, B., Lu, G., Wheeden,R.: A relationship between Poincaré-type inequalities and representationformulas in spaces of homogeneous type. Int. Math. Res. Not., 1–14 (1996)

13. Gutiérrez, C.: The Monge-Ampère Equation. In: Progress in Nonlinear Differential Equations and TheirApplications, vol. 44. Birkhäuser, Berlin (2001)

14. Gutiérrez, C., Huang, Q.: Geometric properties of the sections of solutions to the Monge-Ampère equation.Trans. Am. Math. Soc. 352(9), 4381–4396 (2000)

15. Indratno, S., Maldonado, D., Silwal, S.: On the axiomatic approach to Harnack’s inequality in doublingquasi-metric spaces. J. Differ. Equ. 254(8), 3369–3394 (2013)

16. MacManus, P., Pérez, C.: Generalized Poincaré inequalities: sharp self-improving properties, IMRN. 2,101–116 (1998)

17. Maldonado, D.: The Monge-Ampère quasi-metric structure admits a Sobolev inequality. (Preprint 2012)18. Sawyer, E., Wheeden, R.: Weighted inequalities for fractional integrals on Euclidean and homogeneous

spaces. Am. J. Math. 114(4), 813–874 (1992)19. Schmidt, T.: W 2,1+ estimates for the Monge-Ampère equation. (Preprint 2012)20. Tian, G., Wang, X.-J.: A class of Sobolev type inequalities. Methods Appl. Anal. 15, 257–270 (2008)

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On the $W^{2,1+\varepsilon }$ W 2 , 1 + ε -estimates for the Monge-Ampère equation and related real analysis