ON CERTAIN DEGENERATE AND SINGULAR ELLIPTIC PDES I: NONDIVERGENCE FORM OPERATORS WITH UNBOUNDED DRIFTS AND APPLICATIONS TO SUBELLIPTIC EQUATIONS

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1 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 degenerate and singular elliptic PDEs modeled after certain convex functions and in the presence of unbounded drifts Our main theorem extends the Harnack inequality for the linearized Monge-Ampère equation due to Caffarelli and Gutiérrez and it is related, although under different hypotheses, to a recent work by N Q Le Since our results are shown to apply to the convex functions x p with p and their tensor sums, the degenerate elliptic operators that we can consider include subelliptic Grushin and Grushin-like operators as well as a recent example by A Montanari of a nondivergenceform subelliptic operator arising from the geometric theory of several complex variables In the light of these applications, it follows that the Monge-Ampère quasi-metric structure can be regarded as an alternative to the usual Carnot-Carathéodory metric in the study of certain subelliptic PDEs Introduction and main result Fix an open bounded set Ω R n The purpose of this article is to use the Monge- Ampère real-analysis and PDE techniques developed in [, 3, 8, 9, 0, 6, 4, 5, 6, 7] to prove a Harnack inequality for nonnegative strong) solutions u CΩ) W,n loc Ω) to the degenerate/singular elliptic PDE with L ϕ A u)x) = fx) ae x Ω, ) L ϕ A u)x) := traceax)d ux)) + bx), ϕ ux) + cx)ux), where the first- and second-order terms of L ϕ A are associated to the Hessian of a strictly convex function ϕ C R n ) More precisely, given 0 < λ Λ <, we will assume that, for Lebesgue) ae x Ω, the n n symmetric matrix Ax) in ) satisfies λ D ϕx) Ax) Λ D ϕx) in the sense of non-negative-definite matrices That is, ae x Ω ) λ D ϕx) ξ, ξ Ax)ξ, ξ Λ D ϕx) ξ, ξ, for ae x Ω and every ξ R n, where, denotes the usual dot product in R n We will write A Eλ, Λ, ϕ, Ω) to indicate ) Date: September 4, Mathematics Subject Classification Primary 35J70, 35J96; Secondary 35J75, 3E05 Key words and phrases Degenerate and singular elliptic PDEs, linearized Monge-Ampère operator, Grushin and subelliptic operators Author supported by NSF under grant DMS 36754

2 DIEGO MALDONADO The first-order term of L ϕ A involves the Monge-Ampère gradient ϕ defined as 3) ϕ hx) := D ϕx) / hx) The strictly convex function ϕ C R n ) will also model the geometry shaping the Harnack inequality for L ϕ A by means of its Monge-Ampère sections The Monge-Ampère section of ϕ centered at x R n and with height r > 0 is defined as the open convex set where S ϕ x, r) := {y R n : ϕy) < ϕx) + ϕx), y x + r} 4) δ ϕ x, y) := ϕy) ϕx) ϕx), y x x, y R n The hypotheses on ϕ Our hypotheses on ϕ are the following: H ϕ C R n ) is strictly convex in the sense that its graph contains no line segments H The Monge-Ampère measure associated to ϕ, denoted by µ ϕ and defined as 5) µ ϕ E) := ϕe) for every Borel set E R n where F denotes the Lebesgue measure of F R n ), satisfies the following doubling property on the Monge-Ampère sections: There exist constants C d and α 0, ) such that 6) µ ϕ S ϕ x, r)) C d µ ϕ αs ϕ x, r)) x R n, r > 0 Here αs ϕ x, r) denotes the α-contraction of S ϕ x, r) with respect to its center of mass computed with respect to the Lebesgue measure), see [6, Sections and 3] We will sometimes write µ ϕ DC) ϕ to indicate 6) The hypothesis µ ϕ DC) ϕ turns the triple R n, δ ϕ, µ ϕ ) into a quasi-metric doubling space In particular, there exists a constant K, depending only on C d, α, and n, such that δ ϕ in 4) satisfies the following quasi-triangle inequality 7) δ ϕ x, y) K min{δ ϕ z, x), δ ϕ x, z)} + min{δ ϕ z, y), δ ϕ y, z)}), and the quasi-symmetry inequalities K δ ϕ x, y) δ ϕ y, x) Kδ ϕ x, y), for every x, y, z R n, see [5] and references therein H3 There exists a non-decreasing function ζ : 0, ) 0, ) with lim ζɛ) = 0 such that ɛ 0 + for every x R n and R > 0 we have 8) µ ϕ S ϕ x, + ɛ)r) \ S ϕ x, R)) ζɛ)µ ϕ S ϕ x, R)) ɛ 0, ) H4 D ϕx) > 0 for ae x Ω In particular, D ϕx) and D ϕx) / exist and are positive-definite for ae x Ω Notice that, by A D Alexandrov s theorem, D ϕx) exists for ae x R n due to the convexity of ϕ H5 There exists p > n such that 9) D ϕ L p Ω) Notice that the convexity of ϕ implies that n ϕx) D ϕx) ϕx) for ae x R n, so that 9) is equivalent to ϕ L p Ω) From Remark below we will see that H5 can be weakened to D ϕ L n loc Ω) when b = 0 Through this hypothesis the constant will only depend on p and not on the L p -norm of D ϕ Also, if b = 0, the constants will not depend on the L n loc -norm of D ϕ

3 H6 There exists σ 0, ) such that for every x 0 Ω and r > 0 with S ϕ x 0, Kr) Ω, where K is the quasi-triangle constant in 7), we have 0) D ϕ x 0, x) σδ ϕ x 0, x) for ae x S ϕ x 0, r) \ S ϕ x 0, r) ) where D ϕ x 0, x) := D ϕx) ϕx) ϕx 0 )), ϕx) ϕx 0 )) = D ϕx) ϕx) ϕx0 )) For the sake of convenience we have used the annulus S ϕ x 0, r) \ S ϕ x 0, r) in 0), but it can be replaced by S ϕ x 0, Θr) \ S ϕ x 0, θr) for any constants 0 < θ < Θ Notice that x S ϕ x 0, r) \ S ϕ x 0, r) if and only if r δ ϕ x 0, x) < r, so that a condition equivalent to 0) is the existence of σ 0, ) such that ) D ϕ x 0, x) σr for ae x S ϕ x 0, r) \ S ϕ x 0, r), where σ and σ depend only on one another The hypotheses on the lower-order coefficients Throughout the article we assume that bx) and cx) in ) satisfy the following hypotheses 3) D ϕ) / b L n Ω, dµ ϕ ) and c L n Ω, dµ ϕ ) The hypothesis D ϕ) / b L n Ω) is crucial to the ABP maximum principle, which serves as the cornerstone to the whole approach towards the Harnack inequality for the nondivergence form elliptic operators L ϕ A As discussed in the comments in Section 3, it cannot be improved to D ϕ) / b L n ε Ω) for any ε > 0 Our main result is the following see Section 5 for the definitions of geometric and structural constants) Theorem Suppose that ϕ satisfies H H6 and fix A Eλ, Λ, ϕ, Ω) There exist structural constants ε H 0, ), M H >, and a geometric constant τ 0, ) such that for every section S ϕ z, r) with S Kr := S ϕ z, Kr) Ω and every u CS Kr ) W,n S Kr ) satisfying u 0 in S Kr and 4) L ϕ A u) = traceax)d ux)) + bx), ϕ ux) + cx)ux) = f ae in S Kr, the inequalities ) 5) sup x S ϕz,r) ρ r ρ Sϕ x, ρ) n p D ϕ) / b L n S ϕx,ρ), dµ ϕ) D ϕ L p S ϕx,ρ), dx) ε H and 6) S Kr n c L n S Kr, dµ ϕ) ε H imply the Harnack inequality ) 7) sup u M H inf u + S r n f L S ϕz,τr) S n S Kr, dµ ϕ) ϕz,τr) Remark 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, ϕ u, 3

4 4 DIEGO MALDONADO where b := D ϕ) / b, so that the condition D ϕ) / b L n Ω, dµ ϕ ) in 3) just means b L n Ω, dµ ϕ ) Along these lines, Theorem can be equivalently stated with the equation 4) and the condition 5) replaced with L ϕ A u) = traceax)d ux)) + b, u + cx)ux) = f ae in S Kr and respectively ) sup x S ϕz,r) ρ r ρ Sϕ x, ρ) n p b L n S ϕx,ρ), dµ ϕ) D ϕ L p S ϕx,ρ), dx) ε H, Let us briefly put Theorem in the context of some related results 3 A timeline on the interior Harnack inequality for L ϕ A with unbounded drift We first notice that when the function ϕ is taken as the ϕ x) := x, for x R n, then its Monge-Ampère measure reduces to Lebesgue measure, its associated Monge-Ampère gradient ϕ is just the regular gradient, and D ϕ x 0, x) = δ ϕ x 0, x) = x x 0 for every x, x 0 R n In particular, the sections S ϕ x, r) coincide with the Euclidean balls Bx, r) for every x R n and r > 0 Hence, all the hypotheses H H6 hold true for ϕ In addition, the condition A Eλ, Λ, ϕ, Ω) amounts to the uniform ellipticity of A on Ω In this case, the Harnack inequality for positive solutions to L ϕ A u) = f in Ω with b, c L Ω) is due to the fundamental work of N Krylov and M Safonov [4, 30], see also [5, Sections 97-98] More recently, and always in the uniformly elliptic case, Safonov in [3] established a Harnack inequality for nonnegative solutions to tracead u) + b, u = 0 in the case of an unbounded drift b under the assumption b L n Ω) The condition b L n Ω) is critical in the following sense: if b L p Ω) for some p > n a scaling argument lays a path towards the Harnack inequality along the lines of the Krylov-Safonov approach in [4, 30], see for instance [33] On the other hand, if b L p Ω) for some p < n, the Harnack inequality is known not to hold Indeed, Safonov s example see [3, Remark 3] and [9, Section ]) shows that, in any dimension n, the nonnegative function ux) := x is a strong solution to u + b, u = 0 with bx) := n x x Notice that b L n, B0, )) \ L n B0, )) and, in particular, that b L n ε B0, )) for every ε > 0 However, ux) := x vanishes at x = 0 and only at x = 0) and therefore it cannot satisfy a Harnack inequality in, say, B0, ) Earlier versions of Hölder as opposed to Harnack) estimates under the assumption b L n Ω) had been proved by O Ladyzhenskaya and N Uraltseva in [9] under a smallness condition on b L n Ω) See [3, Remark 4] for further details For later reference we mention a recent extension of the elliptic results in [3], under the assumption b L n Ω), due to C Mooney in [9], where a Harnack inequality is proved for nonnegative functions that are solutions only when their gradients are large Mooney s result also extends the work of C Imbert and L Silvestre in [7], who had assumed b L Ω), by implementing the method of sliding paraboloids The results mentioned above are based on the choice ϕ x) := x Let us now move on to the non-uniformly elliptic case and the pioneering work of L Caffarelli and C Gutiérrez In [3], Caffarelli and Gutiérrez considered a function ϕ C Ω) with D ϕx) > 0 for every x Ω such that its Monge-Ampère measure µ ϕ = det D ϕ satisfies the so-called µ -condition: for every δ 0, ) there exists δ 0, ) such that for every section S := S ϕ x, r) Ω and

5 every measurable set E S the implication E < δ S µ ϕ E) < δ µ ϕ S) holds true, where E stands for the Lebesgue measure of the set E Then, given a matrix A Eλ, Λ, ϕ, Ω), they proved a Harnack inequality for classical solutions of L ϕ A u) = 0 in the case b = c = 0 The µ -condition is a Muckenhupt-type property of µ ϕ with respect to Lebesgue measure which is strictly stronger than the doubling condition 6) from hypothesis H, see [3, Sections and 5] and [0, Section 3] We point out that in [3] the hypothesis ϕ C Ω) with D ϕx) > 0 for every x Ω is crucial in the proof of the passage to the double section that is, [3, Theorem ], which, in turn, is essential for the weak-harnack inequality in [3, Theorem 4]) This can be seen on [3, pp ] where a Dirichlet problem having the Monge Ampère measure as a nonvanishing factor of the right-hand side is solved for a function w ε required to be smooth at least C ) In [5], under the assumptions ϕ C 3 Ω) with D ϕx) > 0 for every x Ω, but assuming the weaker) H hypothesis instead of the µ -condition, the author proved a Harnack inequality for nonnegative strong solutions of L ϕ A u) = 0 [5, Theorem 4]) in the case b = c = 0 and λ = Λ Notice that, when λ = Λ, then A Eλ, Λ, ϕ, Ω) just means A = λd ϕ ) In particular, the doubling property 6) from hypothesis H makes the Harnack inequality intrinsic to the quasi-metric space R n, µ ϕ, δ ϕ ), with no need of a priori comparisons between µ ϕ and Lebesgue measure such as in the µ -condition In [5] the assumption ϕ C 3 Ω) was used to pass from the non-divergence form of traced ϕ) D u) to its divergence form via the identity 8) tracea ϕ D h) = diva ϕ h) h C Ω), where A ϕ x) is the matrix of cofactors of the Hessian D ϕx), that is, 9) A ϕ x) := D ϕx) det D ϕx) x Ω Later on, in [6], under the assumptions ϕ C Ω) with D ϕx) > 0 for every x Ω and H, the author proved a Harnack inequality for nonnegative strong solutions to L ϕ A u) = f, also in the case λ = Λ, when the lower-order coefficients and source term satisfy 0) D ϕ) / b L n Ω), c 0, and b, c, f L Ω), see [6, Theorem ] The hypothesis ϕ C Ω) with D ϕx) > 0 for every x Ω allowed for an alternative proof of the critical density estimate [6, Theorem ]) to the original one by Caffarelli-Gutiérrez [3, Theorem ]) In order to prove a weak-harnack inequality, the approach in [6] was based on the idea from [5] to use the variational side of the PDE In this case a weaker form of 8) was used, for which the hypothesis ϕ C Ω) instead of ϕ C 3 Ω)) sufficed, see [6, Remark ] More recently, under the hypothesis ϕ C Ω) with D ϕx) > 0 for every x Ω which plays a crucial role) as well as ) 0 < λ 0 det D ϕx) Λ 0 x Ω, for some constants λ 0, Λ 0 0, ), N Q Le in [0] considered the PDE L A u) := tracead u) + b, u + cu = f notice the gradient, as opposed to ϕ, on the first-order term and keep in mind Remark ) with A Eλ, Λ, ϕ, Ω) in the general case of 0 < λ Λ when the lower-order coefficients and source term satisfy ) b L p Ω) and c, f L n Ω), 5

6 6 DIEGO MALDONADO for some p > n + α )/α ), with α 0, ] a structural constant, see [0, Theorem ] Le s approach relies on his implementation of the sliding Monge-Ampère paraboloids technique, thus extending the approach by Mooney in [9] Notice that the inequalities ) make µ ϕ uniformly comparable to Lebesgue measure, thus implying the µ -condition for µ ϕ 4 Our point of view and the reasons for the hypotheses H H6 In all of the results mentioned above, the hypothesis ϕ C Ω) with D ϕx) > 0 for every x Ω has been essential In particular, those results do not apply, for example, to the functions ϕ p x) := p x p for any p > or their tensor sums Moreover, the current lack of apriori estimates prevents an approximation argument to go from ϕ C Ω) with D ϕ > 0 to ϕ p The role of the hypotheses H4 H6 is precisely to compensate for such lack of apriori estimates and to be able to apply Theorem directly to rougher ie ϕ / C Ω)) and flatter ie D ϕ vanishing at some points) functions ϕ We remark that the L p -integrability condition in H5 is reminiscent of the one adopted by N Trudinger in [3] in the context of degenerate divergence-form elliptic operators, see [3, Section 5] In terms of the functions ϕ p, in Sections and we prove that they, as well as their tensor sums, satisfy the hypotheses H H6 whenever p Such choices allow for a number of applications to singular and degenerate including subelliptic) PDEs with unbounded drifts, as shown in Sections, 3, and 4 Of course, other classes of general convex functions are expected to satisfy H H6 and to lead, in turn, to corresponding applications Finally, we mention that we have stated the first-order hypotheses H H3 on R n, while the second-order hypotheses H4 H6 are stated on Ω This is only a matter of convenience since stating H H3 on R n allows to consider the triple R n, δ ϕ, µ ϕ ) as opposed to the triple Ω, δ ϕ, µ ϕ )) as a space of homogeneous type However, at the expense of a number of technicalities that we have preferred to avoid, the hypotheses H H3 could also be stated on Ω Regarding applications of Theorem A central motivation for the Harnack inequality for nonnegative solutions of the linearized Monge-Ampère equation proved by Caffarelli and Gutiérrez in [3] stemmed from topics of fluid dynamics, see [3, Section ] Since then, other applications have appeared, for instance, in the contexts of optimal transport [3], [6, Section ] and differential geometry; in particular, to affine geometry [34, 35, 36] and Abreu s equation [, ] and references therein Hence, Theorem can be applied in all the contexts above to the case of rougher and flatter convex functions ϕ However, in Sections and 3, we focus on applications of Theorem and the Monge- Ampère quasi-metric structure to the study of regularity properties for solutions to certain subelliptic PDEs containing lower-order terms These subelliptic PDEs will include Grushin PDEs as well as a recent subelliptic PDE studied by Montanari in [8] in the context of the geometric theory of several complex variables In particular, we extend Montanari s Harnack inequality for the subelliptic operator in [8] to a family of degenerate elliptic operators involving unbounded drifts As a consequence of these applications, the Monge-Ampère quasimetric structure can now be regarded as an alternative to the usual Carnot-Carathéodory metric 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 drifts in Theorems 3 and 5 below as well as the ones in Section 4 are all new contributions to subelliptic PDE literature

7 Organization of the article The article is organized as follows: In Sections and 3 we explore applications of Theorem to certain subelliptic operators These operators include Grushin operators with unbounded drifts as well as a recent nondivergence form subelliptic operator introduced by A Montanari in [8] In Section 4 we apply Theorem to classes of singular and degenerate elliptic PDEs with unbounded drifts In particular, those degenerate classes will include the Grushin operators from Section, see Theorems 7 and 8 In Section 5 we establish some notation and background material, including a Morrey-type estimate Lemma 3) that is new to the Monge-Ampère quasi-metric structure In Section 6 we show that the Monge-Ampère quasi-metric space R n, δ ϕ ) possesses the segment and segment-prolongation properties Theorem 6) This then leads to the construction of geodesics on R n, δ ϕ ) and to the fact that the Monge-Ampère sections are geodesically convex Theorem 7) The material in Section 6 and the Morrey-type estimate from Section 5 are of independent interest In Section 7, under the hypotheses H, H, H4, H5, H6 on ϕ, we prove the double-ball property for supersolutions Theorem 0 and Corollary ) In Section 8, under the hypotheses H H6 on ϕ, we prove a critical-density estimate for supersolutions Theorem 5) In Section 9, also under the hypotheses H H6 on ϕ, we prove mean-value inequalities for subsolutions Theorem 7 and Corollaries 8 and 9) In Section 0, the critical density property and the double-ball property combined with Vitali s covering lemma are shown to imply the power-like decay of the distribution function of nonnegative supersolutions Theorem 30) and their corresponding weak-harnack inequalities Corollary 3) Thus, with all the elements in place, the proof of Theorem is then completed in Section 0 In Section we prove that all positive multiples of) the functions ϕ p x) := p x p satisfy all of the hypotheses H H6 when p Theorem 33) In Section we prove that the hypotheses H, H, H4, H5, H6 are quantitatively preserved under tensor sums Theorem 35) and that all the hypotheses H H6 are preserved under tensor sums of the functions ϕ p for p Finally, Section 3 corresponds to an Appendix where we include the proofs Lemma 3 the Morrey-type estimate), Lemma 5 the local Vitali covering lemma), and Theorem 7 the mean-value inequality for nonnegative subsolutions) In a forthcoming article we will address divergence-form operators with second-order terms of the form diva u) with A Eλ, Λ, ϕ, Ω) and lower-order terms 7 Applications to subelliptic PDEs I: Grushin operators Consider n, n N, x := x, x ) R n R n, and for γ 0, let G γ denote the degenerate elliptic Grushin operator 3) G γ u)x) := ux) + x γ ux), where j denotes the Laplacian on R n j for j =, The literature on Grushin operators is vast, we will only mention the well-known works [, ] for related Harnack inequalities for nonnegative solutions to G γ u) = 0 However, as mentioned in the introduction, the author is not aware of any systematic study of Grushin operators with first-order terms previously appeared in the literature

8 8 DIEGO MALDONADO By setting n := n + n, the equation G γ u) = 0 ae in Ω R n can be recast as 4) 0 = x γ ux) + ux) = tracea γ x)d ux)) ae x Ω, where A γ x) denotes the diagonal n n matrix [ ] 5) A γ x) := x I γ n n 0 0 I n n Hence, by introducing the convex functions ϕ γx ) := γ+ x γ+, ϕ x ) := x, and ϕ γ x) := ϕ γx ) + ϕ x ) we get [ 6) D D ϕ γ x) = ϕ ] γx ) 0, 0 I n n where, for every x R n \ {0}, D ϕ γx ) is the n n symmetric matrix 7) D ϕ γx ) = γ + ) x γ x x x ) + x γ I x x x x ), x which has eigenvalues γ +) x γ with multiplicity and x γ with multiplicity n That is, γ + ) x γ 0 0 8) D ϕ 0 x γ 0 γx ) = P 0 P t, x γ for some orthogonal n n matrix P It then follows that 9) λ γ D ϕ γ x) ξ, ξ A γ x)ξ, ξ Λ γ D ϕ γ x) ξ, ξ, holds for ae x Ω and every ξ R n, with λ γ := and Λ γ := γ + That is, A γ Eλ γ, Λ γ, ϕ γ, Ω) Now, as proved in Sections and, the convex function ϕ γ satisfies all of the hypotheses H H6 moreover, ϕ γ satisfies H5 for any p > n) and Theorem will apply More precisely, given a PDE of the form G γ u) + b, u + cu = f we can recast it in the form of 4) after dividing by x γ, to get which can be written as x γ ux) + ux) + cx) fx) bx), ux) + ux) = x γ x γ x γ tracea γ x)d ux)) + bx), ϕ ux) + cx)ux) = fx), where A γ x) is given by 5), bx) := x γ D ϕ γ x) / bx), cx) := cx) x γ, and fx) := fx) x γ On the other hand, notice that 6) implies that the Monge-Ampère measure of ϕ γ is given by µ ϕγ x) = det D ϕ γ x) = γ + ) x γn Consequently, given a set S Ω, we have ˆ ˆ D ϕ) / bx) n dµ ϕγ x) = x nγ bx) n γ + ) x γn dx S S ˆ ˆ = γ + ) bx) n x γn n) dx = γ + ) bx) n x γn dx S S

9 Similarly, c L n S, dµ ϕγ ) and f L n S, dµ ϕγ ) translate into c L n S, x γn dx) and f L n S, x γn dx), 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 R n, γ 0, and consider the subelliptic PDE 30) G γ u) + b, u + cu = f where G γ denotes the Grushin operator in 3) Then, there exist constants 0 < τ < < K H depending only on γ and n) as well as structural constants 0 < ε H < < M H, such that for every section S ϕγ z, R) with S KH R := S ϕγ z, K H R) Ω and every u CS KH R) W,n S KH R) satisfying u 0 in S KH R and solving the subelliptic PDE 30) ae in S KH R, we have that the inequalities ) 3) sup y S ϕγ z,r) ρ r ρ Sϕγ y, ρ) n p b L n S ϕγ y,ρ), x γn dx) D ϕ γ L p S ϕγ y,ρ), dx) ε H and 3) S KH R n c L n S KH R, x γn dx) ε H imply the Harnack inequality u M H sup S ϕγ z,τr) inf S ϕγ z,τr) u + S K H R n f L n S KH R, x γn 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 3) A discussion on the sections of convex functions given as tensor sums of other convex functions such as ϕ γ above) is included in Sections and In particular, we have that, given y = y, y ) S ϕγ z, R), the inclusions 9) and 35) yield and S ϕγ y, ρ) S ϕ γ y, ρ) S ϕ y, ρ) B y, C γ ρ γ+ ) B y, ρ) / ) B y, c γ ρ γ+ ) B y, ρ / ) S ϕ γ y, ρ/) S ϕ y, ρ/) S ϕγ y, ρ), where B and B denote Euclidean balls in R n and R n, respectively, and c γ, C γ > 0 depend only on γ and n For the sake of conciseness, since the degeneracy of the PDE 30) effectively occurs on z = 0, let us assume that z = z, z ) with z < Also, we can assume that 0 < R < Hence, given y = y, y ) S ϕγ z, R), ρ 0, R), and x = x, x ) S ϕγ y, ρ) it follows from the inclusions above that x x y + y z + z ρ γ+ + R γ+ +, where the implied constants depend only on n and γ Now, x along with 6) and 8) yields D ϕ γ on S ϕγ y, ρ) Consequently, for any given p > n, we get S ϕγ y, ρ) p D ϕ γ L p S ϕγ y,ρ), dx) On the other hand, by taking Lebesgue measure on the inclusions above, ρ Sϕγ y, ρ) n b L n S ϕγ y,ρ), x γn dx) ρ ρ n ) n γ+ + n ) b L n S ϕγ y,ρ), x γn dx), 9

10 0 DIEGO MALDONADO so that the condition 3) can be recast as ρ n 33) sup y S ϕγ z,r) ρ r ) n γ+ + n b L n S ϕγ y,ρ), x γn dx) ) ε H, which would place the drift b in a Morrey space with respect to the weight x γn dx Alternatively, the condition 33) can be realized by asking for higher integrability of b with respect to a weight More precisely, given β, by Hölder s inequality with exponents β and β, we have b L n S ϕγ y,ρ), x γn dx) b L nβ S ϕγ y,ρ), x βγn dx) S ϕ γ y, ρ) nβ and then, by choosing β so that n n ρ n b L nβ S ϕγ y,ρ), x βγn dx) ρ nβ n γ+ + n ), ) γ+ + n ) n γ+ + n b L n S ϕγ y,ρ), x γn dx) ρ n + nβ n γ+ + n ) = 0, we get ) n γ+ + n + n nβ γ+ + n ) b L nβ S ϕγ y,ρ), x βγn dx) = b L nβ S ϕγ y,ρ), x βγn dx), and 33) is then implied by the condition b L nβ S ϕγ y,ρ), x βγn dx) ε H with 34) β := Notice that β for every γ 0 with 35) n γ+ + n n + n γ+ γ + 4 ) n ) n Therefore, the condition 3) is equivalent to the weighted Morrey-space estimate 33) and it is weaker than the integrability condition b L nβ S ϕγ y,ρ), x βγn dx) ε H, with β and γ as in 34) and 35) In Section 4 we will extend Theorem 3 to a large class of degenerate PDEs that will include the Grushin operators in 3) 3 Applications to subelliptic PDEs II: Extensions of Montanari s example 3 Montanari s example Let x = x, x ) R and consider the vector fields X := x and X := x x, in particular, notice that [X, X ] = x In [8] and motivated by topics on the geometric theory of several complex variables, A Montanari introduced the subelliptic operator 336) L = a X + a X X + a X, where the coefficient matrix a := a ij ) i,j= satisfies, for some constants 0 < λ 0 Λ 0, the uniform ellipticity condition 337) λ 0 ξ ax)ξ, ξ Λ 0 ξ x, ξ R Then, by means of a weighted version of the ABP maximum principle, she established a Harnack inequality for nonnegative classical C -solutions to Lu = 0 with respect to balls of the Carnot-Carathéodory metric generated by X and X

11 In our next application, by means of Theorem and the Monge-Ampère quasi-metric structure, we extend Montanari s Harnack inequality to a family of subelliptic operators that include L in 336) 3 Extensions of Montanari s example Fix an open bounded set Ω R and ν N 0 Define the vector fields X := x and X := x ν x, and introduce the subelliptic operator 338) L ν = a X + a X X + a X, where the coefficient matrix a := a ij ) i,j= satisfies 337) for ae x Ω Notice that in this case we have [X, X ] = νx ν x and this commutator will vanish on x = 0 whenever ν > However, the vector fields X and X will still satisfy Hörmander s condition, but commutators of up to order ν will be required Also, notice that, when ν =, L recovers L from 336) Let us write L ν as L ν u) = x ν tracea νd u) where [ a x)/x A ν x) = A ν x, x ) := ν a x)/x ν ] a x)/x ν a x) and introduce the convex function ϕ ν : R R as ϕ ν x) := [ ] 339) D x ν ϕ ν x) = 0 0 and D ϕ ν x) / A ν x)d ϕ ν x) / = Then, the ellipticity condition 337) implies that ν+)ν+) xν+ + x [ x a a ] ν a x ν x ν a λ 0 ξ D ϕ ν x) / A ν x)d ϕ ν x) / ξ, ξ Λ 0 ξ x ν so that holds for ae x Ω and every ξ R n, that is, A ν Eλ 0, Λ 0, ϕ ν, Ω) Next, let us write any subelliptic PDE of the form 340) L ν u) + b, u + cu = f as tracea ν D u) + b, ϕ u + cu = f, where bx) := x ν D ϕx) / bx), cx) := cx)x ν, and fx) := fx)x ν, and notice that, since the Monge-Ampère measure of ϕ ν equals µ ϕν x) = det D ϕ ν x) = x ν, given S Ω the condition D ϕ ν ) / b L S, dµ ϕν ) from 3) translates into ˆ ˆ ˆ D ϕ ν ) / bx) dµ ϕν x) = x 4ν bx) x ν dx = bx) x ν dx <, S S and, similarly, c L S, dµ ϕν ) and f L S, dµ ϕν ) translate into c L S, x ν ) and f L S, x ν ), respectively Thus, by bringing things together, Theorem yields the following Harnack inequality for nonnegative solutions to the subelliptic PDEs 340) with unbounded drifts Theorem 5 Fix an open bounded set Ω R, ν N 0, and consider the subelliptic PDE 34) L ν u) + b, u + cu = f where L ν denotes the subelliptic operator in 338) Then, there exist constants 0 < τ < < K H depending only on ν) as well as structural constants 0 < ε H < < M H, such that S

12 DIEGO MALDONADO for every section S ϕν z, R) with S KH R := S ϕν z, K H R) Ω and every u CS KH R) W, S KH R) satisfying u 0 in S KH R and solving the subelliptic PDE 34) ae in S KH R, we have that the inequalities 34) sup y S ϕν z,r) ρ r and ρ Sϕν y, ρ) p b L S ϕν y,ρ), x ν dx) D ϕ ν L p S ϕν y,ρ), dx) 343) S KH R c L S KH R, x ν dx) ε H imply the Harnack inequality u M H inf u + S K H R f L S ϕν z,τr) S KH R, x ν dx) sup S ϕν z,τr) ) ) ε H Here p, the exponent from hypothesis H5, can be taken as any number bigger than Remark 6 A comment on the condition 34) follows along the lines of Remark 4 For instance, assuming z := z, z ) with z < and 0 < R < we get that D ϕ ν on S ϕγ y, ρ) In this case the condition 34) turns out to be equivalent to the Morrey-space estimate ) 344) sup ρ ν 4ν+) b L S ϕγ y,ρ), x ν dx) ε H, y S ϕγ z,r) ρ r as well as weaker than the integrability b L β S ϕγ y,ρ), x βν dx) ε H with β := +ν We close this application by mentioning that the weight in the weighted ABP maximum principle from [8, Theorem 5] is precisely the Monge-Ampère measure x and that the Monge-Ampère quasi-distance δ ϕ provides an equivalent gauge to the Carnot-Carathéodory and Grushin metrics In the case of ϕ ν as above, by [9, Theorem 4iii)] and 36) from Section we have δ ϕν x, x ), y, y )) x ν+ y ν+ )x y ) + x y ), x, x ), y, y ) R, Several equivalent distances and quasi- where the implied constants depend only on ν distances were considered in [8, Section ] 4 Applications to other singular and degenerate elliptic operators In this section we extend the class of subelliptic Grushin operators from Section, introduce a related singular elliptic operator, and establish a Harnack inequality for their corresponding PDEs including unbounded drifts Fix m N, n,, n m N, and set n := n + n m Fix Ω R n and for each j =,, m, let Ω j denote the projection of Ω over R n j and let the functions Γ j : Ω j R satisfy 445) λ j x j γ j Γ j x j ) Λ j x j γ j ae x j Ω j, for some constants 0 < λ j Λ j and γ := γ,, γ m ) [0, ) m Notice that only measurability is required from the the Γ j s Introduce the convex function ϕ γ : R n R as the tensor sum ϕ γ x) := + γ ) + γ ) x +γ γ m ) + γ m ) x m +γm,

13 for x = x,, x m ) R n R nm = R n, so that D ϕ γ is a direct sum, for j =,, m, of n j n j matrices of the form + γ j ) x j γ j x j γ j 0 P j 0 P j t, x j γ j for some n j n j orthogonal matrix P j In particular, det D ϕ γ x) = m + γ j ) x j n jγ j 4 The degenerate case: Grushin-like operators Let us consider the following degenerate elliptic operator 446) G Γ u) := Π x) ux) + + Π m x) m ux), where j is the Laplacian on R n j and, for j =,, m, Π j x) := k j Γ k x k ) j= 3 For instance, if m = and λ j = Λ j = for j =,, then we have G Γ u) := x γ ux) + x γ ux), so that the choice γ = 0 recovers the Grushin operator 3), and, for general 0 < λ j Λ j, we have with λ j x j γ j Γ j Λ j x j γ j G Γ u) := Γ x ) ux) + Γ x ) ux), and j =, If m = 3 we have G Γ u) := Γ x )Γ 3 x 3 ) ux) + Γ x )Γ 3 x 3 ) ux) + Γ x )Γ x ) 3 ux), etc Now, given a PDE of the form G Γ u) + b, u + cu = f, after dividing it by the product Π Γ x) := Π m j= Γ jx j ), it turns into tracea Γ x)d ux)) + bx), ϕ ux) + cx)ux) = fx), where A Γ is the direct sum 447) A Γ x) := Γx ) I n n Γx ) I n n Γx m) I n m n m, and bx) := Π Γ x) D ϕ γ x) / bx), cx) := Π Γ x) cx), and fx) := Π Γ x) fx) In particular, it follows that A Γ Eλ Γ, Λ Γ, ϕ γ, Ω), for some constants 0 < λ Γ Λ Γ depending only on the λ j s and Λ j s in 445)

14 4 DIEGO MALDONADO Notice that, given S Ω, the condition D ϕ γ ) / b L n S, dµ ϕγ ) from 3) means ˆ S ˆ D ϕ γ ) / bx) n dµ ϕγ x) = ˆ S S m Π Γ x) n bx) n + γ j ) x j n jγ j dx j= m bx) n x j n n j)γ j dx <, where the implicit constants depend only on the γ j s and the λ j s, Λ j s from 445) Similarly with c, f L n S, dµ ϕγ ) Thus, Theorem yields the following Harnack inequality Theorem 7 Fix an open bounded set Ω R n, γ = γ,, γ m ) [0, ) m, and consider the degenerate PDE j= 448) G Γ u) + b, u + cu = f where G Γ denotes the degenerate operator in 446) Then, there exist constants 0 < τ < < K H depending only on γ and n) as well as structural constants 0 < ε H < < M H, such that for every section S ϕγ z, R) with S KH R := S ϕγ z, K H R) Ω and every u CS KH R) W,n S KH R) satisfying u 0 in S KH R and solving the degenerate PDE 448) ae in S KH R, we have that the inequalities 449) sup y S ϕγ z,r) ρ r and ρ Sϕγ y, ρ) n p b L n S ϕγ y,ρ), m imply the Harnack inequality sup u M H S ϕγ z,τr) j= S KH R n c L n S KH R, m ε x j n n j )γj H dx) j= inf u + S K H R n f L S ϕγ z,τr) n S KH R, m x j n n j )γj dx) D ϕ γ L p S ϕγ y,ρ), dx) ε H x j n n j )γj dx) j= Again, here p is any number bigger than n and, regarding the Lebesgue measure of the sections S ϕγ z, R), we have 450) S ϕγ z, R) R m j= n j +γ j z R n, R > 0, where the implicit constants depend only on m and the n j s for j =,, m 4 The singular case In this case let us consider the singular elliptic operator 45) H Γ u) = Γ m x ) ux) + + Γ m x m ) mux), which can be written as H Γ u) = tracea Γ D u) where A Γ is the n n matrix in 447) Reasoning along the lines of the previous example, we obtain the following Harnack inequality for the singular operator H Γ with unbounded drifts

15 Theorem 8 Fix an open bounded set Ω R n, γ = γ,, γ m ) [0, ) m, and consider the singular PDE 45) H Γ u) + b, u + cu = f where H Γ denotes the singular operator in 45) Then, there exist constants 0 < τ < < K H depending only on γ and n) as well as structural constants 0 < ε H < < M H, such that for every section S ϕγ z, R) with S KH R := S ϕγ z, K H R) Ω and every u CS KH R) W,n S KH R) satisfying u 0 in S KH R and solving the singular PDE 45) ae in S KH R, we have that the inequalities 453) and sup y S ϕγ z,r) ρ r ρ Sϕγ y, ρ) n p b L n imply the Harnack inequality sup u M H S ϕγ z,τr) S ϕγ y,ρ), m x j n j γj dx j= S KH R n c L n S KH R, m ε x j n j γj H dx) j= inf u + S K H R n f L S ϕγ z,τr) n S KH R, m ) D ϕ γ L p S ϕγ y,ρ), dx) ε H x j n j γj dx) j= Once more, here p is any number bigger than n and the Lebesgue measure of a section S ϕγ z, R) behaves as in 450) Remark 9 The conditions 449) and 453) can be commented upon as done in Remarks 4 and 6 and we leave the details to the interested reader 5 Preliminaries 5 Geometric and structural constants Constants depending only on dimension n, on the constants C d and α from the doubling condition µ ϕ DC) ϕ in 6), and on the function ζ from hypothesis H3 will be called geometric constants Constants depending only on λ and Λ in ), on the exponent p > n from hypothesis H5, on the constant σ > 0 from hypothesis H6, on D ϕ) / b L n Ω, dµ ϕ), as well as on geometric constants, will be called structural constants 5 Doubling properties By [3, Lemma 5], and this requires only the hypothesis H, we have the following doubling property for the Lebesgue measure on Monge-Ampère sections 554) S ϕ x, r) n S ϕ x, r) x R n, r > 0, which in turn implies 555) S ϕ x, s) n s ) n Sϕ x, r) x R n, 0 < r < s r By [7, Lemma ], under hypotheses H and H there exists a geometric constant K D > such that 556) µ ϕ S ϕ x, s)) K D s r ) n µϕ S ϕ x, r)) x R n, 0 < r < s From [8, Theorem 8], there exist geometric constants 0 < κ K such that 557) κ n r n S ϕ x, r) µ ϕ S ϕ x, r)) K n r n x R n, r > 0 5

16 6 DIEGO MALDONADO 53 On the hypothesis H3 In the literature on real analysis on spaces of homogeneous type, the inequality 8) is sometimes referred to as the annular decay property or ring condition and it quantifies the relative smallness if thin annuli with respect to µ ϕ Typical choices for the function ζ include ζɛ) = Cɛ α 0 or ζɛ) = C logɛ) α 0 for some constants C 0, α 0 > 0 In this section we point out that 8) is implied by the following Coifman-Fefferman condition for µ ϕ see [3, Section 5]): There exist constants C 0, θ 0 > such that for every section S := S ϕ x, r) and every Borel set E S it holds true that 558) µ ϕ E) µ ϕ S) C 0 E S ) θ0 Indeed, by Lemma 5b) from [3] for every x R n, t > 0 and δ 0, ) we have S ϕ x, t) \ S ϕ x, δt) n δ) S ϕ x, t), so that, given R > 0 and ɛ 0, ), by taking t := R + ɛ) and δ := + ɛ) we get S ϕ x, + ɛ)r) \ S ϕ x, R) = S ϕ x, t) \ S ϕ x, δt) n δ) S ϕ x, t) = nɛ + ɛ S ϕx, + ɛ)r) nɛ S ϕ x, + ɛ)r) Hence, by using 558) with S := S ϕ x, + ɛ)r) and E := S ϕ x, + ɛ)r) \ S ϕ x, R) we obtain 8) with ζɛ) := C 0 n θ 0 ɛ θ 0 for every ɛ 0, ) The condition 558) is equivalent to the µ -condition for µ ϕ mentioned in Section 3 of the Introduction, see [3, Section 5] for further characterizations In Sections and we will see that the condition 558) is satisfied by the power functions x p, with p >, as well as their tensor sums 54 The Aleksandrov Bakelman Pucci maximum principle We will use the following version of the ABP maximum principle See [5, Section 9 and Exercise 93] and [4, Chapter 6, pp79-87] Lemma 0 Suppose that ϕ satisfies H and H4 Fix an open, convex, bounded set S R n and consider the operator Lh)x) := traceax)d hx)) + D ϕx) bx), hx) + cx)hx), where Ax) is a symmetric, nonnegative-definite n n matrix and cx) 0 for ae x S Given E S define Nb, E) as [ 559) Nb, E) n n ˆ )] D ϕx) bx) n dx := exp n n + ω n det Ax) Then, there exists a dimensional constant C abp > 0 such that for every h CS) W,n loc S) satisfying Lh)x) gx) ae x S the following hold true: i) If sup h 0 then S 560) sup h C abp Nb, S) S n S E g det A) n L n S)

17 7 ii) If sup h = 0 then S 56) sup h C abp Nb, Γ + h S)) S n S g, det A) n L n 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 hx) hy) + p, x y x S} {y S : hy) 0} Remark The role of the convexity of S in Lemma 0 is to allow for S n, instead of the diameter of S as in [5, Theorem 9] and [4, Theorem 9]), on the right-hand side of 56) Indeed, the convexity of S and F John s lemma imply the existence of an affine transformation T : R n R n such that B0, n / ) T S) B0, ) and the change of variable y := T x yields 56) This technique also proves a version of Lemma 0 on sets of the form S 0 := S \ S 0 for any S 0 S Namely, if h CS 0 ) W,n loc S0 ) with h 0 on S 0 satisfying Lh)x) gx), for ae x S 0, with cx) 0 ae x S 0, we have 56) sup h C abp Nb, S) S n S 0 g det A) n L n S) Notice the factor S n as opposed to S 0 n ) on the right-hand side of 56) The estimate 56) will be used in the proof of Theorem 0 with S = S ϕ x 0, R) and S 0 an annulus of the form S ϕ x 0, R) \ S ϕ x 0, r) for some x 0 R n and 0 < r < R Remark Notice the misprint on [5, p4] repeated on [4, p87]) where, instead of the right-hand side of 559), it reads [ n ˆ ) ] exp n n + D ϕx) bx) n dx, ω n det Ax) Γ + h S) which would make for a spurious Γ + h S) 55 A Morrey-type estimate The next lemma is a consequence of the existence of weak Poincaré inequalities in quasi-metric spaces, see for instance [, Proposition 548], we include a proof of the Monge-Ampère version in the Appendix in Section 3 Lemma 3 Suppose that ϕ satisfies H and H Given q > n, there exists a constant C q,k > 0, depending only on q and K, where K is the quasi-triangle constant in 7), such that for every x, y Ω with S ϕ x, Kδ ϕ x, y)) Ω and h C S ϕ x, Kδ ϕ x, y))) we have 563) hx) hy) C q,k δ ϕ x, y) ϕ hz) q dz S ϕx,kδ ϕx,y)) Remark 4 Lemma 3 will be used as follows: For j =,, n, notice that the function ϕ j satisfies ϕ ϕ j = D ϕ) ϕ j, ϕ j = ϕ jj ϕ, hence, by applying 563) with h = ϕ j we obtain 564) ϕx) ϕy) n / C q,k δ ϕ x, y) ) q ϕz) q/ dz S ϕx,kδ ϕx,y)) ) q

18 8 DIEGO MALDONADO In particular, given a section S ϕ x 0, Kr) Ω, the inequality 564) with q := p, where p > n is as in the hypothesis H5, implies that 565) ϕx 0 ) ϕx) Kp)r D ϕz) p dz S ϕx 0,Kr) for every x S ϕ x 0, Kr), where Kp) := n 3/ C p,k We will repeatedly use the inequality 565) to deal with presence of the unbounded drift b in the first-order term of L ϕ A Notice that if we set ρ ϕ x, y) := δ ϕ x, y) for every x, y Ω, then the inequality 565) gives that ϕ is locally Lipschitz in Ω with respect to the quasi-distance ρ ϕ 56 A local Vitali covering lemma under µ ϕ DC) ϕ Monge-Ampère versions of Vitali s covering lemma under the hypothesis that µ ϕ in the sense of )) have appeared, for instance, in [7, Lemma ] and [0, Lemma 5] For the sake of completeness, in the Appendix we include a proof for the following local version of Vitali s covering lemma under the assumption µ ϕ DC) ϕ only and with an explicit value of the Vitali constant K V > in terms of the quasi-triangle constant K from 7) Lemma 5 Local Vitali covering lemma for Monge-Ampère sections) Suppose that ϕ satisfies H and H Let Ω R n be an open bounded set Fix K V > K +K, where K is the quasi-triangle constant from 7) Given a section S 0 := S ϕ x 0, R 0 ) with S ϕ x 0, R 0 ) Ω, a subset E S 0, and a covering {S ϕ x, r x )} x I of E such that S ϕ x, r x ) S 0 for every x I, there exists a finite/countable sub-collection {S ϕ x j, r xj )} j J of pairwise disjoint sections such that S ϕ x j, K V r xj ) x I S ϕ x, r x ) j J 6 On the construction of Monge-Ampère geodesics and the segment properties for δ φ The results proved in this Section 6 will be used in Section 0, but they contribute to the study of geometric properties of sections of general convex functions As such they are of independent interest and will require no doubling conditions on the Monge-Ampère measure Let Ω 0 R n be an open convex set For a strictly convex function φ C Ω 0 ) set where S φ x, r) := {x Ω 0 : δ φ x, y) < r} δ φ x, y) := φy) φx) φx), y x x, y Ω 0 Theorem 6 Let Ω 0 R n be an open convex set and φ C Ω 0 ) be a strictly convex function Then, the Monge-Ampère quasi-distance δ φ possesses the following two properties: i) The segment property: Given x, z Ω 0 with S φ x, δ φ x, z)) Ω 0 and 0 < r < δ φ x, z), there exists y S φ x, δ φ x, z)) such that δ φ x, y) = r and 666) δ φ x, y) + δ φ y, z) = δ φ x, z) ii) The segment-prolongation property: Given x, z Ω 0 and R > δ ϕ x, z) with S φ x, R) Ω 0, there exists y S φ x, R) that is, δ φ x, y) = R) such that 667) δ φ x, y) = δ φ x, z) + δ φ z, y) ) p,

19 Proof In order to prove 666), given x, z Ω 0 with S φ x, δ φ x, z)) Ω 0 and 0 < r < δ φ x, z) let P be any hyperplane passing through z and tangent to S φ x, r) and let y P S φ x, r), see Figure 9 S φ x, δ φ x, z)) S φ x, r) x y z P φy) φx) Figure The segment property Since y S φ x, r) if and only if y belongs to the level set φy) φx) φx), y x = r, it follows that the vector φy) φx) is orthogonal to P and then to z y That is, φx) φz), z y = 0, so that δ φ x, y) + δ φ y, z) = φy) φx) φx), y x + φz) φy) φy), z y = φz) φx) φx), z x + φx), z x φx), y x φy), z y = δ φ x, z) + φx) φz), z y = δ φ x, z) + 0 = δ φ x, z) Notice that the choice of y as in 666) is by no means unique since each hyperplane P will produce one such y Similarly, in order to prove 667), given x, z Ω 0 and R > δ ϕ x, z) with S φ x, R) Ω 0, let P be the hyperplane tangent to S φ x, δ φ x, z)) at z notice that z S φ x, δ φ x, z)), that is, z belongs to the level set φz) φx) φx), z x = r xz, where r xz := δ φ x, z)), see Figure φz) φx) P z x y S φ x, δ φ x, z)) S φ x, R) Figure The segment-prolongation property Now, pick any y P S φ x, R) to obtain δ φ x, y) = R as well as the fact that the vector φz) φx) is orthogonal to P and then to z y That is, φx) φz), y z = 0,

20 0 DIEGO MALDONADO so that δ φ x, z) + δ φ z, y) = φz) φx) φx), z x + φy) φz) φz), y z = φy) φx) φx), y x + φx), y x φx), z x φz), y z = δ φ x, y) + φx) φz), y z = δ φ x, y) + 0 = δ φ x, y), and the proof of Theorem 6 is complete Theorem 7 Let Ω 0 R n be an open convex set and φ C Ω 0 ) be a strictly convex function Given any section S := S φ x 0, R) Ω 0 and y 0, y S there exists a continuous curve γ : [0, ] S such that γ0) = y 0, γ) = y and 668) δ φ γt), γt )) = t t)δ φ y 0, y ) 0 t t Proof Let us start by noticing that given any x, z S φ x 0, R) it is always possible to find a midpoint y from x to z in the sense that δ φ x, y) = δ φ x, z)/) such that y S φ x 0, R) Otherwise, the whole section S φ x, δ φ x, z)/) would lie outside S φ x 0, R) contradicting the fact that x S φ x 0, R), see Figure 3 z x 0 y S φ x 0, R) x y S φ x, δ φ x, z)/) Figure 3 Both y and y are midpoints from x to z, but y S φ x 0, R) Now, given y 0, y S set r := δ φ y 0, y ) Let y / S be a mid-point between y 0 and y, let y /4 S be midpoint between y 0 and y /, let y 3/4 S a midpoint between y / and y, and so on This defines a function γ : Domγ) [0, ] S, where Domγ) consists of the numbers of the form k j for j N 0 and k {0,,, j }, and γk j ) = y k j For a fixed j N 0 and k k we have δ φ γk j ), γk j )) = δ φ y k j, y k j) = k k) j r For j, j N 0 and k {0,,, j } and k {0,,, j } with k j < k j we have so that k j = k j j j =: k j < k j, δ φ γk j ), γk j )) = δ φ γk j ), γk j )) = k k ) j r = k j k j )r That is, for every t, t Domγ) with t < t we have δ φ γt), γt )) = t t)r The fact that φ is strictly convex makes the topology generated by the sections equivalent to the Euclidean topology, which makes S ϕ x 0, R) a complete topological subspace of Ω 0 Therefore, the mapping γ has a continuous extension to all of [0, ] that we also denote as γ) such that the δ φ -geodesic condition 668) holds

21 Remark 8 In other words, Theorem 7 says that the Monge-Ampère sections are geodesically convex Corollary 9 Chain condition) For every section S := S φ x 0, R) Ω 0 and every x, y S and N N there exists a finite sequence {y j } N j=0 S such that y 0 = x, y N = y and δ φ y j, y j ) = N δ φx, y) j =,, N Proof Let γ : [0, ] S be a geodesic joining x to y as in Theorem 7, then for each j {0,, N} set y j := γj/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 0 Suppose that ϕ satisfies H, H, H4, H5, H6 and fix A Eλ, Λ, ϕ, Ω) Let S r := S ϕ x 0, r) be a section with S r := S ϕ x 0, r) Ω and let u CS r ) W,n S r ) satisfy u 0 in S r and L ϕ A u) f ae in S r Then, there exist structural constants ε, γ 0, ), such that the inequalities 769) 770) S r n f L n S r, dµ ϕ) ε, r Sr n p D ϕ) / b L n S r, dµ ϕ) D ϕ L p S r, dx) ε, 77) and inf u imply inf u γ S ϕx 0,r) S ϕx 0,3r/) S r n c L n S r, dµ ϕ) ε, Proof For t > r with S ϕ x 0, t) Ω, let Ann ϕ x 0, t, r) denote the annulus Ann ϕ x 0, t, r) := S ϕ x 0, t) \ S ϕ x 0, r) For s > to be determined later, set ) δϕ x 0, x) s vx) := s r so that 77) 773) 774) 0 < vx) < x Ann ϕ x 0, r, r), 3/) s s vx), x Ann ϕ x 0, 3r/, r), ) δϕ x 0, x) s ϕx) ϕx 0 )) x Ann ϕ x 0, r, r) vx) = s r and, for ae x Ann ϕ x 0, r, r), r D vx) = ) ss ) δϕ x 0, x) s r ϕx) ϕx 0 )) ϕx) ϕx 0 )) r ) δϕ x 0, x) s D ϕx) s r r Notice that we have v CAnn ϕ x 0, r, r)) W,n Ann ϕ x 0, r, r)) due to the fact that D ϕ L p Ω) with p > n by hypothesis H5

22 DIEGO MALDONADO Now, suppose that u CS r ) W,n loc S r) satisfies L ϕ A u) f and u 0 in S r with inf S ϕx 0,r) u Then if x S r we have and if x S ϕ x 0, r), then vx) ux) vx) = s s = 0 vx) ux) s ux) s 0 Therefore, v u 0 on Ann ϕ x 0, r, r)) On the other hand, for ae x Ann ϕ x 0, r, r) L ϕ A v)x) ) ss ) δϕ x 0, x) s = r Ax) ϕx) ϕx 0 )), ϕx) ϕx 0 )) r s ) δϕ x 0, x) s traceax)d ϕx)) + bx), ϕ vx) + cx)vx) r r From ) we get Ax) ϕx) ϕx 0 )), ϕx) ϕx 0 )) λ D ϕx) ϕx) ϕx 0 )), ϕx) ϕx 0 )) = λd ϕ x 0, x) as well as traceax)d ϕx)) nλ for ae x S r Hence, ) ss ) δϕ x 0, x) s r Ax) ϕx) ϕx 0 )), ϕx) ϕx 0 )) r s ) δϕ x 0, x) s traceax)d ϕx)) r r s ) δϕ x 0, x) s r s )λd ϕ x 0, x) nλδ ϕ x 0, x)) r At this point fix s = sn, σ, Λ/λ) so that 775) s nλ σλ +, where σ > 0 is as in 0) from hypothesis H6 Thus, 0) and 775) now yield and then D ϕ x 0, x) δ ϕ x 0, x) σ nλ λs ) s )λd ϕ x 0, x) nλδ ϕ x 0, x) 0 for ae x Ann ϕ x 0, r, r) Consequently, by writing cx) as cx) = c + x) c x), for ae x Ann ϕ x 0, r, r) we get L ϕ A v)x) bx), ϕ vx) + cx)vx) = s ) δϕ x 0, x) s D ϕx) / bx), ϕx) ϕx 0 )) + cx)vx) r r s r D ϕx) / bx) ϕx) ϕx 0 ) c x),

23 where for the last inequality we used that for x Ann ϕ x 0, r, r) we have δϕx 0,x) r < and 0 < vx) <, see 77) Next, since L ϕ A u) f ae in S r, it follows that L ϕ A v u)x) = traceax)d v u)x)) + bx), ϕ v u)x) + cx)v u)x) s r D ϕx) / bx) ϕx) ϕx 0 ) c x) fx) ae in Ann ϕ x 0, r, r) Therefore, by writing cx) = c + x) c x) again and using that u 0 in S r, ae in Ann ϕ x 0, r, r) we obtain which gives traceax)d v u)x)) + bx), ϕ v u)x) c x)v u)x) s r D ϕx) / bx) ϕx) ϕx 0 ) c x) fx) c + x)vx) ux)) s r D ϕx) / bx) ϕx) ϕx 0 ) c x) fx) c + x), 776) traceax)d v u)x)) + bx), ϕ v u)x) c x)v u)x) g x) where g x) := s r D ϕx) / bx) ϕx) ϕx 0 ) cx) fx) Now, by the ABP maximum principle 56) applied to 776) and v u on Ann ϕ x 0, r, r) we get sup v u) C abp Nb, S r ) S r g n, Ann ϕx 0,r,r) det A) n L n S r ) where, from the definition of Nb, S r ) in 559) and the first inequality in ) which yields det A λ n det D ϕ) ), [ )] Nb, S r ) n n D = exp n n + ω n ˆSr ϕx) bx) n dx det Ax) [ n ) exp ] n n + λ n D ϕ b n 777) ω L n Ω, dµ ϕ) =: C n, λ, b) n, n with C n, λ, b) > 0, depending only on n, λ and D ϕ) / b L n Ω, dµ ϕ), is a structural constant Again by the first inequality in ), g s det A) n λr D ϕ / b L n S r, dµ ϕ) ϕ ) ϕx 0 ) L S r ) L n S r ) so that, from 769) and 77), S r g n s S r n det A) n λr L n S r ) + λ c L n S r, dµ ϕ) + λ f L n S r, dµ ϕ), D ϕ / b L n S r, dµ ϕ) ϕ ) ϕx 0 ) L S r ) + ε λ 3

24 4 DIEGO MALDONADO On the other hand, the Morrey-type estimate 565) and the hypothesis 770) give 778) Consequently, S r n r D ϕ / b L n S r, dµ ϕ) ϕ ) ϕx 0 ) L S r ) Kp)r S r n D ϕ / b r L n S r, dµ ϕ) ) D ϕz) p p dz S r = Kp)r Sr n p D ϕ / b L n S r, dµ ϕ) D ϕ L p S r, dx) Kp)ε sup v u) C abp C n, λ, b) S r n Ann ϕx 0,r,r) g det A) n C abp C n, λ, b)λ + skp))ε L n S r ) Thus, using 77), for every x Ann ϕ x 0, 3r/, r) Ann ϕ x 0, r, r) we have 779) 3/) s s vx) ux) + C abp C n, λ, b)λ + skp))ε Finally, by defining the structural constant 780) γ := 3/) s s C abp C n, λ, b)λ + skp))ε and choosing ε 0, ), also structural, such that γ > 0 we obtain inf u γ For S ϕx 0,3r/) instance, one could choose γ := s 3 s s+), depending only on s as in 775), and then ε := γλ[c abp C n, λ, b) + skp))] Remark When b = 0 the hypothesis H5 can be weakened to D ϕ L n loc Ω) in the proof of Theorem 0 Indeed, the hypothesis that D ϕ L p loc Ω), for some p > n, was only used to apply the Morrey-type estimate 565) If b = 0, then there is no need to control ϕ ) ϕx 0 ) L S r ) Notice, however, that even in the case b = 0 the hypothesis D ϕ L n loc Ω) is required to guarantee that the test function v u belongs to W,n loc S r), as required by the ABP maximum principle in Lemma 0 The next corollary quantifies iterations of Theorem 0 to pass from a section S ϕ y 0, ρ) to S ϕ y 0, N 0 ρ) for an arbitrary N 0 > Corollary Suppose that ϕ satisfies H, H, H4, H5, H6 and fix A Eλ, Λ, ϕ, Ω) For every N 0 > there exists a constant γn 0 ) 0, ), depending only on the structural constant γ from Theorem 0 as well as on N 0, such that for every section S N0 ρ := S ϕ y 0, N 0 ρ) Ω and every u CS N0 ρ) W,n S N0 ρ) satisfying u 0 in S N0 ρ and L ϕ A u) f ae in S N 0 ρ the assumptions 78) 78) 783) S N0 ρ n f L n S N0 ρ, dµ ϕ) ε γ ln N 0 ln3/), ρ SN0 ρ n p D ϕ) / b L n S N0 ρ, dµ ϕ) D ϕ L p S N0 ρ, dx) ε, S N0 ρ n c L n S N0 ρ, dµ ϕ) ε, and inf u imply inf u γn 0) Here ε 0, ) is the structural constant from S ϕy 0,ρ) S ϕy 0,N 0 ρ) Theorem 0