The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic divergence-form operators
aa r X i v : . [ m a t h . A P ] J u l THE STRONG MAXIMUM PRINCIPLE ANDTHE HARNACK INEQUALITY FOR A CLASS OFHYPOELLIPTIC DIVERGENCE-FORM OPERATORS
ERIKA BATTAGLIA, STEFANO BIAGI, AND ANDREA BONFIGLIOLI
Abstract.
In this paper we consider a class of hypoelliptic second-order partial differentialoperators L in divergence form on R N , arising from CR geometry and Lie group theory,and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for L .The involved operators are not assumed to belong to the Hörmander hypoellipticity class,nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on the degeneracy ofthe second order part; indeed our results hold true in the infinitely-degenerate case and foroperators which are not necessarily sums of squares. We use a Control Theory result onhypoellipticity in order to recover a meaningful geometric information on connectivity andmaxima propagation, yet in the absence of any Hörmander condition. For operators L with C ω coefficients, this control-theoretic result will also imply a Unique Continuation propertyfor the L -harmonic functions. The (Strong) Harnack Inequality is obtained via the WeakHarnack Inequality by means of a Potential Theory argument, and by a crucial use of theStrong Maximum Principle and the solvability of the Dirichlet problem for L on a basis ofthe Euclidean topology. Introduction and main results
Throughout the paper, we shall be concerned with linear second order partial differentialoperators (PDOs, in the sequel), possibly degenerate-elliptic, of the form(1.1) L := 1 V ( x ) N X i,j =1 ∂∂x i (cid:16) V ( x ) a i,j ( x ) ∂∂x j (cid:17) , x ∈ R N , where V is a C ∞ positive function on R N , the matrix A ( x ) := ( a i,j ( x )) i,j is symmetric and positive semi-definite at every point x ∈ R N , and it has real-valued C ∞ entries. In particular, L is formally self-adjoint on L ( R N , d ν ) with respect to the measure d ν ( x ) = V ( x ) d x , whichclarifies the rôle of V . We tacitly understand these structural assumptions on L throughout.The literature on divergence-form operators like (1.1) in the strictly-elliptic case is so vast thatwe do no attempt to collect the related references. Instead, we mention some papers (relevantfor the topics of the present paper) in the degenerate case.Degenerate-elliptic operators of the form (1.1) were extensively studied by Jerison andSánchez-Calle in the paper [25] (under a suitable subelliptic assumption), where it is also de-scribed how these PDOs naturally intervene in the study of function theory of several complexvariables and CR Geometry (see also [20, 27, 38]). Prototypes for the PDOs (1.1) also arise in thetheory of sub-Laplace operators on real Lie groups (e.g., for Carnot groups, [7]), as well as in Rie-mannian Geometry (e.g., the Laplace-Beltrami operator has the form p | g | − P ∂ i ( p | g | g ij ∂ j ) ).Regularity issues for degenerate-elliptic divergence-form operators comprising the Harnack Ine-quality and the Maximum Principles (to which this paper is devoted) trace back to the 80’s,with the deep investigations by: Fabes, Kenig, Serapioni [16]; Fabes, Jerison, Kenig [14, 15];Gutiérrez [22]. In these papers, operators as in (1.1) are considered (with V ≡ ) with lowregularity assumptions on the coefficients, under the hypothesis that the degeneracy of A ( x ) becontrolled on both sides by some Muckenhoupt weight.Recent investigations on the Harnack inequality for variational operators, comprising (1.1)as a special case, also assume Muckenhoupt weights on the degeneracy; see [12, 41]. Veryrecently, a systematic study of the Potential Theory for the harmonic/subharmonic functions Mathematics Subject Classification.
Primary: 35B50, 35B45, 35H20; Secondary: 35J25, 35J70, 35R03.
Key words and phrases.
Degenerate-elliptic operators; Maximum principles; Harnack inequality; Uniquecontinuation; Divergence form operators. related to operators L as in (1.1) has been carried out in the series of papers [1, 3, 5, 6], underthe assumption that L possesses a (smooth) global positive fundamental solution.We remark that in the present paper we do not require L to be a Hörmander operator,our results holding true in the infinitely-degenerate case as well, nor we make any assumptionof subellipticity or Muckenhoupt-weighted degeneracy (see Example 1.2); furthermore, we donot assume the existence of a global fundamental solution for L . Hence our results are notcontained in any of the aforementioned papers.We now describe the main results of this paper concerning L , namely the Strong MaximumPrinciple and the
Harnack Inequality for L ; gradually as we need to specify them, we introducethe three assumptions under which our theorems are proven. As we shall see in a moment, themain hypothesis is a hypoellipticity assumption.In obtaining our main results we are much indebted to the ideas in the pioneering paper byBony, [8], where Hörmander operators are considered. The main novelty of our framework is thatwe have to renounce to the geometric information encoded in Hörmander’s Rank Condition: thelatter implies a connectivity/propagation property (leading to the Strong Maximum Principle),as well as it implies hypoellipticity, due to the well-known Hörmander’s theorem [23]. In oursetting, the approach is somewhat reversed: hypoellipticity is the main assumption, and we needto derive from it some appropriate connectivity and propagation features, even in the absenceof a maximal rank condition. This will be made possible by exploiting a Control Theory resultby Amano [2] on hypoelliptic PDOs, as we shall describe in detail. Once the Strong MaximumPrinciple is established, the path to the (Strong) Harnack Inequality is traced in [8]: we passthrough the solvability of the Dirichlet problem, the relevant Green kernel and a Weak HarnackInequality. Finally, the gap between the Weak and Strong Harnack Inequalities is filled by anabstract Potential Theory result, due to Mokobodzki and Brelot, [9].In order to describe our results more closely, we first fix some notation and definition: wesay that a linear second order PDO on R N (1.2) L := N X i,j =1 α i,j ( x ) ∂ ∂x i ∂x j + N X i =1 β i ( x ) ∂∂x i + γ ( x ) is non-totally degenerate at a point x ∈ R N if the matrix ( α i,j ( x )) i,j (which will be referredto as the principal matrix of L ) is non-vanishing. We observe that the principal matrix of anoperator L of the form (1.1) is precisely A ( x ) = ( a i,j ( x )) i,j . We also recall that L is said to be( C ∞ -)hypoelliptic in an open set Ω ⊆ R N if, for every u ∈ D ′ (Ω) , every open set U ⊆ Ω andevery f ∈ C ∞ ( U, R ) , the equation Lu = f in U implies that u is (a function-type distributionassociated with) a C ∞ function on U .In the sequel, if Ω ⊆ R N is open, we say that u is L -harmonic (resp., L -subharmonic ) in Ω if u ∈ C (Ω , R ) and Lu = 0 (resp., Lu ≥ ) in Ω . The set of the L -harmonic functions in Ω willbe denoted by H L (Ω) . We observe that, if L is hypoelliptic on every open subset of R N , then H L (Ω) ⊂ C ∞ (Ω , R ) ; under this hypoellipticity assumption, H L (Ω) has important topologicalproperties, which will be crucially used in the sequel (Remark 4.2).In order to introduce our first main result we assume the following hypotheses on L : (NTD): L is non-totally degenerate at every point of R N , or equivalently (recalling that A ( x ) is symmetric and positive semi-definite),(1.3) trace ( A ( x )) > , for every x ∈ R N . (HY): L is C ∞ - hypoelliptic in every open subset of R N .Under these two assumptions we shall prove the Strong Maximum Principle for L .Condition (NTD), if compared with the above mentioned Muckenhoupt-type weights on thedegeneracies of A ( x ) , does not allow a simultaneous vanishing of the eigenvalues of A ( x ) , but ithas the advantage of permitting a very fast vanishing of the smallest eigenvalue (see Example1.2) together with a very fast growing of the largest one (see Example 1.1); both phenomenacan happen at an exponential rate (e.g., like e − /x as x → in the first case, and like e x as x → ∞ in the second case), which is not allowed when Muckenhoupt weights are involved. ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 3
Meaningful examples of operators satisfying hypotheses (NTD) and (HY), providing proto-type PDOs to which our theory applies and a motivation for our investigation, are now describedin the following two examples.
Example . The following PDOs satisfy the assumptions (NTD) and (HY).(a.) If R N is equipped with a Lie group structure G = ( R N , ∗ ) , and if we fix a set X := { X , . . . , X m } of Lie-generators for the Lie algebra g of G (this means that the smallestLie algebra containing X is equal to g ), then a direct computation shows that(1.4) L X := − m X j =1 X ∗ j X j is of the form (1.1), where V ( x ) is the density of the Haar measure ν on G , and ( a i,j ) i,j is equalto S S T , where S is the N × m matrix whose columns are given by the coefficients of the vectorfields X , . . . , X m ; here X ∗ j denotes the (formal) adjoint of X j in the Hilbert space L ( R N , d ν ) .Most importantly, L X in (1.4) satisfies the assumptions (NTD) and (HY) above. Indeed: • The non-total-degeneracy is a consequence of X being a set of Lie-generators of g . • L X is a Hörmander operator, of the form P mj =1 X j + X , where X is a linear combi-nation (with smooth coefficients) of X , . . . , X m . Therefore L X is hypoelliptic due toHörmander’s Hypoellipticity Theorem, [23], jointly with the cited fact that X is a setof Lie-generators of g .The density V need not be identically as for example for the Lie group ( R , ∗ ) , where ( x , x ) ∗ ( y , y ) = ( x + y e x , x + y ) , since in this case V ( x ) = e − x . The left-invariant PDO associated with the set of generators X = { e x ∂∂x , ∂∂x } has fast-growing coefficients: L X = e x ∂ ∂x + ∂ ∂x − ∂∂x . Note that the eigenvalues of the principal matrix of L X are e x and , so that the largesteigenvalue cannot be controlled (for x > ) by any integrable weight.(b.) More generally (arguing as above), if X = { X , . . . , X m } is a family of smooth vectorfields in R N satisfying Hörmander’s Rank Condition, if d ν ( x ) = V ( x ) d x is the Radon measureassociated with any positive smooth density V on R N , then the operator − P mj =1 X ∗ j X j is ofthe form (1.1) and it satisfies (NTD) and (HY). Here X ∗ j denotes the formal adjoint of X j in L ( R N , d ν ) . As already observed, PDOs of this form naturally arise in CR Geometry and inthe function theory of several complex variables (see [25]).The above examples show that geometrically meaningful PDOs belonging to the class of ourconcern actually fall in the hypoellipticity class of the Hörmander operators. Nonetheless, hy-potheses (NTD) and (HY) are general enough to comprise non-Hörmander and non-subelliptic PDOs, as it is shown in the next example. Applications to this kind of infinitely-degenerate
PDOs also furnish one of the main motivation for our study.
Example . Let us consider the class of operators in R defined by(1.5a) L a = ∂ ∂x + (cid:16) a ( x ) ∂∂x (cid:17) , with a ∈ C ∞ ( R , R ) , a even, nonnegative, nondecreasing on [0 , ∞ ) and vanishing only at .Then L a satisfies (NTD) (obviously) and (HY), thanks to a result by Fedi ˘ ı, [17]. Note that L a does not satisfy Hörmander’s Rank Condition at x = 0 if all the derivatives of a vanish at , as for a ( x ) = exp( − /x ) . Other examples of operators satisfying our assumptions (NTD)and (HY) but failing to be Hörmander operators can be found, e.g., in the following papers:Bell and Mohammed [4]; Christ [10, Section 1]; Kohn [28]; Kusuoka and Stroock [30, Theorem ERIKA BATTAGLIA, STEFANO BIAGI, AND ANDREA BONFIGLIOLI ∂ ∂x + (cid:16) exp( − / | x | ) ∂∂x (cid:17) + (cid:16) exp( − / | x | ) ∂∂x (cid:17) in R ,(1.5b) ∂ ∂x + (cid:16) exp( − / p | x | ) ∂∂x (cid:17) + ∂ ∂x in R ,(1.5c) ∂ ∂x + (cid:16) x ∂∂x (cid:17) + ∂ ∂x + (cid:16) exp( − / p | x | ) ∂∂x (cid:17) in R . (1.5d)For the hypoellipticity of (1.5b) see [10]; for (1.5c) see [30]; for (1.5d) see [35]. Later on, inproving the Harnack Inequality, we shall add another hypothesis to (NTD) and (HY) and, aswe shall show, the operators from (1.5a) to (1.5d) (and those in Example 1.1) will fulfil thisassumption as well. Hence the main results of this paper (except for the Unique Continuationresult in Section 3, proved for operators with C ω coefficients) fully apply to these PDOs.Moreover, since the PDOs (1.5a)-to-(1.5d) are not subelliptic (see Remark 1.6), they do notfall in the class considered by Jerison and Sánchez-Calle in [25]. Finally, note that the smallesteigenvalue in all the above examples vanishes very quickly (like exp( − / | x | α ) for x → , withpositive α ) and it cannot be bounded from below by any weight w ( x ) with locally integrablereciprocal function.Our first main result under conditions (NTD) and (HY) is the following one. Theorem 1.3 ( Strong Maximum Principle for L ) . Suppose that L is an operator of theform (1.1) , with C ∞ coefficients V > and ( a i,j ) i,j ≥ , and that it satisfies (NTD) and (HY) .Let Ω ⊆ R N be a connected open set. Then, the following facts hold. (1) Any function u ∈ C (Ω , R ) satisfying L u ≥ on Ω and attaining a maximum in Ω isconstant throughout Ω . (2) If c ∈ C ∞ ( R N , R ) is nonnegative on R N , and if we set (1.6) L c := L − c, then any function u ∈ C (Ω , R ) satisfying L c u ≥ on Ω and attaining a nonnegativemaximum in Ω is constant throughout Ω . The rôle of the nonnegativity of the zero-order term c in the above statement (2) in obtainingStrong Maximum Principles is well-known (see e.g., Pucci and Serrin [37]). Remark . (a.) Obviously, the Strong Maximum Principle (SMP, shortly) in Theorem 1.3will immediately provide the Weak
Maximum Principle (WMP, shortly) for operators L and L − c , for any nonnegative zero-order term c (and any bounded open set Ω ), see Corollary 2.3for the precise statement.(b.) We will show that, in order to obtain the SMP and WMP for L − c , it is also sufficientto replace the hypothesis on the hypoellipticity of L with the (more natural hypothesis of the)hypoellipticity of L − c , still under assumption (NTD) and the divergence-form structure of L ;see Remark 2.4 for the precise result.Our proof of the SMP in Theorem 1.3 follows a rather classical scheme, in that it rests on aHopf Lemma for L (see Lemma 2.1). However, the passage from the Hopf Lemma to the SMPis, in general, non-trivial and the same is true in our framework. For example, in the paper [8]by Bony, where Hörmander operators are considered, this passage is accomplished by means ofa maximum propagation principle, crucially based on Hörmander’s Rank Condition, the latterensuring a connectivity property (the so-called Chow’s Connectivity Theorem for Hörmandervector fields). The novelty in our setting is that, since hypotheses (NTD) and (HY) do not necessarily imply that L is a Hörmander operator (see for instance Example 1.2), we have tosupply for a lack of geometric information. Due to this main novelty, we describe more closelyour argument in deriving the SMP.As anticipated, we are able to supply the lack of Hörmander’s Rank Condition by using anotable control-theoretic property (seemingly long-forgotten in the PDE literature), encoded inthe hypoellipticity assumption (HY), proved by Amano in [2]: indeed, thanks to the hypothesis ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 5 (NTD), we are entitled to use [2, Theorem 2] which states that (HY) ensures the controllability of the ODE system ˙ γ = ξ X ( γ ) + N X i =1 ξ i X i ( γ ) , ( ξ , ξ , . . . , ξ N ) ∈ R N , on every open and connected subset of R N . Here X , . . . , X N denote the vector fields associatedwith the rows of the principal matrix of L , whereas X is the drift vector field obtained bywriting L (this being always possible) in the form L u = N X i =1 ∂∂x i ( X i u ) + X u. By definition of a controllable system, Amano’s controllability result provides another geometric connectivity property (a substitute for Chow’s Theorem): any couple of points can be joinedby a continuous path which is piece-wise an integral curve of some vector field Y belongingto span R { X , X , . . . , X N } . The SMP will then follow if we show that there is a propaga-tion of the maximum of any L -subharmonic function u along all integral curves γ Y of every Y ∈ span R { X , X , . . . , X N } . In other words, we need to show that if the set F ( u ) of themaximum points of u intersects any such γ Y , then γ Y is wholly contained in F ( u ) : briefly, ifthis happens we say that F ( u ) is Y -invariant. In its turn, this Y -invariance property can becharacterized (see Bony, [8, §2]) in terms of a tangentiality property of Y with respect to F ( u ) (the reader is referred to Section 2 below for this notion of tangentiality).Now, the self-adjoint structure of our PDO L in (1.1) ensures that X is a linear combinationwith smooth coefficients of X , . . . , X N . Hence, by the very definition of tangentiality (see e.g.,(2.10)), the tangentiality of X w.r.t. F ( u ) will be inherited from the tangentiality of X , . . . , X N w.r.t. F ( u ) . By means of the above argument of controllability/propagation, this allows us toreduce the proof of the SMP to showing that any of the vector fields X , . . . , X N is tangentto F ( u ) . Luckily, this tangentiality is a consequence of the choice of X , . . . , X N as derivingfrom the rows of the principal matrix of L , together with the Hopf-type Lemma 2.1 for L . Thisargument is provided, in all detail, in Section 2.The use of the above ideas, plus the classical Holmgren’s Theorem, will allow us to provethat, when L has real-analytic coefficients, a Unique Continuation result holds true for L :any L -harmonic function defined on a connected open set U which vanishes on some non-voidopen subset is necessarily null on the whole of U (see Theorem 3.1). We observe that the C ω assumption is satisfied, for example, if L is a left invariant operator on a Lie group (e.g., asub-Laplacian on a Carnot group, as in [7]), since, as it is well-know, any Lie group can beendowed with a compatible C ω structure. Remark . We explicitly remark that, as it is proved by Amano in [2, Theorem 1], the abovecontrollability property ensures the validity of the Hörmander Rank Condition only on an open dense subset of R N which may fail to coincide with the whole of R N . This actual possiblelack of the Hörmander Rank Condition is clearly exhibited in Example 1.2 (of non-Hörmanderoperators which nonetheless satisfy our assumptions (NTD) and (HY), and hence the SMP).To the best of our knowledge, Amano’s controllability result for hypoelliptic non-totally-degenerate operators has been long forgotten in the literature; only recently, it has been usedby the third-named author and B. Abbondanza [1] in studying the Dirichlet problem for L , andin obtaining Potential Theoretic results for the harmonic sheaf related to L .In order to give the second main result of the paper (namely, the Harnack Inequality for L ), we shall need a further assumption, very similar to (HY) (and, indeed, equivalent to it inmany important cases), together with some technical results on the solvability of the Dirichletproblem related to L . Our next assumption is the following one: (HY) ε : There exists ε > such that L − ε is C ∞ -hypoelliptic in every open subset of R N .For operators L satisfying hypotheses (NTD), (HY) and (HY) ε we are able to prove the HarnackInequality (see Theorem 1.10).We postpone the description of the relationship between assumptions (HY) and (HY) ε (andtheir actual equivalence for large classes of operators: for subelliptic PDOs, for instance) in ERIKA BATTAGLIA, STEFANO BIAGI, AND ANDREA BONFIGLIOLI
Remark 1.6 below. Instead, we anticipate the rôle of the perturbation L − ε of the operator L :this is motivated by a crucial comparison argument (which we generalize to our setting), dueto Bony [8, Proposition 7.1, p.298], giving the lower bound(1.7) u ( x ) ≥ ε Z Ω u ( y ) k ε ( x , y ) V ( y ) d y ∀ x ∈ Ω , for every nonnegative L -harmonic function u on the open set Ω which possesses a Green kernel k ε ( x, y ) relative to the perturbed operator L − ε (see Theorem 1.9 for the notion of a Greenkernel, and see Lemma 5.1 for the proof of (1.7)). This lower bound, plus some topologicalfacts on hypoellipticity, is the key ingredient for a Weak
Harnack Inequality related to L , as weshall explain shortly.Some remarks on assumption (HY) ε are now in order. Remark . Hypothesis (HY) ε is implicit in hypothesis (HY) for notable classes of operators,whence our assumptions for the validity of the Harnack Inequality for L reduce to (NTD) and(HY) solely: namely, (HY) implies (HY) ε in the following cases: • for Hörmander operators, and, more generally, for second order subelliptic operators (inthe usual sense of fulfilling a subelliptic estimate, see e.g., [25, 28]); indeed, any operator L in these classes of PDOs is hypoelliptic (see Hörmander [23], Kohn and Nirenberg[29]), and L still belongs to these classes after the addition of a smooth zero-order term; • for operators with real-analytic coefficients. Indeed, in the C ω case, one can applyknown results by Ole˘ınik and Radkevič ensuring that, for a general C ω operator L asin (1.2), hypoellipticity is equivalent to the verification of Hörmander’s Rank Conditionfor the vector fields X , X , . . . , X N obtained by rewriting L as P Ni =1 ∂ i ( X i ) + X + γ ;this condition is clearly invariant under any change of the zero-order term γ of L sothat (HY) and (HY) ε are indeed equivalent.The problem of establishing, in general, whether (HY) implies (HY) ε seems non-trivial andit is postponed to future investigations. In this regard we recall that, for example, in thecomplex coefficient case the presence of a zero-order term (even a small ε ) may drastically alterhypoellipticity (see for instance the example given by Stein in [39]).We explicitly remark that the operators (1.5a)-to-(1.5d) are not subelliptic (nor C ω ), yetthey satisfy hypotheses (NTD), (HY) and (HY) ε . The lack of subellipticity is a consequenceof the characterization of the subelliptic PDOs due to Fefferman and Phong [18, 19] (see also[28, Prop.1.3] or [25, Th.2.1 and Prop.2.1], jointly with the presence of a coefficient with a zeroof infinite order in (1.5a)-to-(1.5d)). The second assertion concerning the verification of (HY) ε (the other hypotheses being already discussed) derives from the following result by Kohn, [28]:any operator of the form L + λ ( x ) L in R nx × R my is hypoelliptic, where λ ∈ C ∞ ( R x ) , λ ≥ has a zero of infinite order at (and no other zeroes ofinfinite order), and L (operating in x ∈ R n ) and L (operating in y ∈ R m ) are general secondorder PDOs (as in (1.2)) with smooth coefficients and they are assumed to be subelliptic. Itis straightforward to recognize that by subtracting ε to any PDO in (1.5a)-to-(1.5d) we get anoperator of the form ( L − ε ) + λ ( x ) L , where λ has the required features, L is uniformlyelliptic (indeed, a classical Laplacian in all the examples), and L − ε is a uniformly ellipticoperator (cases (1.5a)-to-(1.5c)) or it is a Hörmander operator (case (1.5d)).Before describing the approach to the Harnack Inequality, inspired by the ideas in [8], westate the main needed technical tools on the solvability of the Dirichlet problem for L and forthe perturbed operator L − ε . Lemma 1.7.
Suppose that L is an operator of the form (1.1) , with C ∞ coefficients V > and ( a i,j ) ≥ , and that L satisfies (NTD) . Let ε ≥ be fixed (the case ε = 0 being admissible). Weset L ε := L − ε and we assume that L ε is hypoelliptic on every open subset of R N .Then, there exists a basis for the Euclidean topology of R N , independent of ε , made ofopen and connected sets Ω (with Lipschitz boundary) with the following properties: for every It appears that having some quantitative information on the loss of derivatives may help in facing thisquestion (personal communication by A. Parmeggiani).
ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 7 continuous function f on Ω and for every continuous function ϕ on ∂ Ω , there exists one andonly one solution u ∈ C (Ω , R ) of the Dirichlet problem (cid:26) L ε u = − f on Ω (in the weak sense of distributions), u = ϕ on ∂ Ω (point-wise). (1.8) Furthermore, if f, ϕ ≥ then u ≥ as well. Finally, if f belongs to C ∞ (Ω , R ) ∩ C (Ω , R ) , thenthe same is true of u , and u is a classical solution of (1.8) . We prove this theorem for a considerably larger class of operators than the L ε above; seeTheorem 6.1. We adapt to our context the well established techniques in [8, Section 5] usedfor Hörmander operators. These techniques are perfectly suited to our more general case, sincethey only rely on hypoellipticity and on the Weak Maximum Principle. Since the proof presentsno further difficulties, it is provided in the Appendix, for the sake of completeness only.With the existence of the weak solution of the Dirichlet problem for L ε on a bounded openset Ω , we can define the associated Green operator as usual: Definition 1.8 ( Green operator and Green measure ) . Let ε ≥ be fixed, and let L ε and Ω satisfy, respectively, the hypothesis and the thesis of Lemma 1.7. We consider the operator(depending on L ε and Ω ; we avoid keeping track of the dependency on Ω in the notation)(1.9) G ε : C (Ω , R ) −→ C (Ω , R ) mapping f ∈ C (Ω , R ) into the function G ε ( f ) which is the unique distributional solution u in C (Ω , R ) of the Dirichlet problem (cid:26) L ε u = − f on Ω (in the weak sense of distributions), u = 0 on ∂ Ω (point-wise).(1.10)We call G ε the Green operator related to L ε and to the open set Ω .By the Riesz Representation Theorem (which is applicable thanks to the monotonicity pro-perties in Lemma 1.7 with respect to the function f ), for every x ∈ Ω there exists a (nonnegative)Radon measure λ x,ε on Ω such that G ε ( f )( x ) = Z Ω f ( y ) d λ x,ε ( y ) , for every f ∈ C (Ω , R ) .(1.11)We call λ x,ε the Green measure related to L ε (to the open set Ω and to the point x ) .Let L be as in (1.1); in the rest of the paper, we set once and for all(1.12) d ν ( x ) := V ( x ) d x, that is, ν is the (Radon) measure on R N associated with the (positive) density V in (1.1),absolutely continuous with respect to the Lebesgue measure on R N . It is clear that the measure ν plays the following key rôle:(1.13) Z ϕ L ψ d ν = Z ψ L ϕ d ν, for every ϕ, ψ ∈ C ∞ ( R N , R ) ,thus making L (formally) self-adjoint in the space L ( R N , d ν ) . We observe that (in general)our operators L in (1.1) are not classically self-adjoint; indeed the classical adjoint operator L ∗ of L is related to L by the following identity (a consequence of (1.13))(1.14) L ∗ u = V L ( u/V ) , for every u of class C .The possibility of dealing with non-identically densities V (as in the case of Lie groups, seeExample 1.1-(a)) makes it more convenient to decompose the Green measure λ x,ε with respectto ν in (1.12), rather than w.r.t. Lebesgue measure. Hence we prove the following: Theorem 1.9 ( Green kernel ) . Suppose that L is an operator of the form (1.1) , with C ∞ coefficients V > and ( a i,j ) ≥ , and that L satisfies (NTD) . Let ε ≥ be fixed. We set L ε := L − ε and we assume that L ε is hypoelliptic on every open subset of R N .Let Ω be an open set as in Lemma 1.7. If G ε and λ x,ε are as in Definition 1.8, there existsa function k ε : Ω × Ω → R , smooth and positive out of the diagonal of Ω × Ω , such that thefollowing representation holds true: (1.15) G ε ( f )( x ) = Z Ω f ( y ) k ε ( x, y ) d ν ( y ) , for every x ∈ Ω , ERIKA BATTAGLIA, STEFANO BIAGI, AND ANDREA BONFIGLIOLI and for every f ∈ C (Ω , R ) . We call k ε the Green kernel related to L ε and to the open set Ω .Furthermore, we have the following properties: (i) Symmetry of the Green kernel: (1.16) k ε ( x, y ) = k ε ( y, x ) for every x, y ∈ Ω . (ii) For every fixed x ∈ Ω , the function k ε ( x, · ) is L ε -harmonic in Ω \ { x } ; moreover G ε ( L ε ϕ ) = − ϕ = L ε ( G ε ( ϕ )) for any ϕ ∈ C ∞ (Ω , R ) , that is − ϕ ( x ) = Z Ω L ε ϕ ( y ) k ε ( x, y ) d ν ( y )= L ε (cid:16) Z Ω ϕ ( y ) k ε ( x, y ) d ν ( y ) (cid:17) , for every ϕ ∈ C ∞ (Ω , R ) . (1.17)(iii) For every fixed x ∈ Ω , one has (1.18) lim y → y k ε ( x, y ) = 0 for any y ∈ ∂ Ω . (iv) For every fixed x ∈ Ω , the functions k ε ( x, · ) = k ε ( · , x ) are in L (Ω) , and k ε ∈ L (Ω × Ω) . The key ingredients in the proof of the above results are the following facts: • the hypoellipticity of L ε (as assumed in the hypothesis) which will imply the hypoel-lipticity of the classical adjoint of L ε (see Remark 4.1); • the C ∞ -topology on the space of the L ε -harmonic functions is the same as the L loc -topology, another consequence of the hypoellipticity of L ε (Remark 4.2); • the fact that L is self-adjoint on L ( R N , d ν ) (see (1.13)) so that the same is true of L ε (this will be crucial in proving the symmetry of the Green kernel); • the Strong Maximum Principle for the perturbed operator L ε = L − ε , which we obtainas a consequence of our previous Strong Maximum Principle for L in Theorem 1.3 (seeprecisely Remark 2.2, where nonnegative maxima are considered): this is a key step forthe proof of the positivity of k ε ; • the Schwartz Kernel Theorem (used for the regularity of the Green kernel).The difference with respect to the analogous result given in the framework of the Hörmanderoperators in [8, Théorème 6.1] is the introduction of the relevant measure ν in the integralrepresentation (1.15); indeed, the symmetry property (1.16) of the kernel k ε is connected withthe identity (1.13), which is not true (in general) if we consider Lebesgue measure instead of ν .We are now ready to give the second main result of the paper: Theorem 1.10 ( Strong Harnack Inequality ) . Suppose that L is an operator of the form (1.1) , with C ∞ coefficients V > and ( a i,j ) ≥ , and suppose it satisfies hypotheses (NTD) , (HY) and (HY) ε .Then, for every connected open set O ⊆ R N and every compact subset K of O , there existsa constant M = M ( L , O, K ) ≥ such that (1.19) sup K u ≤ M inf K u, for every nonnegative L -harmonic function u in O .If L is subelliptic or if it has C ω coefficients, then assumption (HY) ε can be dropped. The last assertion follows from Remark 1.6.We now present the spine of the proof of Theorem 1.10.The main step towards the Strong Harnack Inequality is the following Theorem 1.11 fromPotential Theory. A proof of a more general abstract version of this useful result, in theframework of axiomatic harmonic spaces, can be found in the survey notes [9, pp.20–24] byBrelot, where this theorem is attributed to G. Mokobodzki. (See also a further improvement toharmonic spaces which are not necessarily second-countable, by Loeb and Walsh, [32]). Insteadof appealing to an abstract Potential-Theoretic statement, we prefer to formulate the resultunder the following more specific form.
Theorem 1.11.
Let L be a second order linear PDO in R N with smooth coefficients. Supposethe following conditions are satisfied. ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 9 (Regularity):
There exists a basis B for the Euclidean topology of R N (consisting ofbounded open sets) such that, for every Ω ∈ B \ { ∅ } and for every ϕ ∈ C ( ∂ Ω , R ) , thereexists a unique L -harmonic function H Ω ϕ ∈ C (Ω) ∩ C (Ω) solving the Dirichlet problem (cid:26) Lu = 0 in Ω u = ϕ on ∂ Ω ,and satisfying H Ω ϕ ≥ whenever ϕ ≥ . (Weak Harnack Inequality): For every connected open set O ⊆ R N , every compactsubset K of O and every y ∈ O , there exists a constant C ( y ) = C ( L, O, K, y ) > such that sup K u ≤ C ( y ) u ( y ) , for every nonnegative L -harmonic function u in O .Then, the following Strong Harnack Inequality for L holds: for every connected open set O andevery compact subset K of O there exists a constant M = M ( L, O, K ) ≥ such that (1.20) sup K u ≤ M inf K u, for every nonnegative L -harmonic function u in O . See also Remark 5.2 for some equivalent assumptions that can replace the above (WeakHarnack Inequality) to get the Strong Harnack Inequality. The proof of Theorem 1.11 is givenin Section 5, starting from a result by Mokobodzki and Brelot in [9, Chapter I]: in the latter itis shown that if the axioms (Regularity) and (Weak Harnack Inequality) are fulfilled then, forany connected open set O ⊆ R N and any x ∈ O , the set(1.21) Φ x := n h ∈ H L ( O ) : h ≥ , h ( x ) = 1 o is equicontinuous at x . The proof of this fact rests on some deep results of Functional Analysisconcerning the family of the so-called harmonic measures { µ Ω x } x ∈ ∂ Ω related to L (and to aregular set Ω for the Dirichlet problem), jointly with some basic properties of the harmonicsheaf associated with the operator L .As observed by Bony in [8, Remarque 7.1, p.300], the Strong Harnack Inequality classicallyrelies on two-sided estimates of the ratios h ( x , · ) /h ( x , · ) , where h ( x, y ) is the relevant Poissonkernel; these estimates were unavailable in the setting considered in [8], as they are (to thebest of our knowledge) in our setting too. However, like in [8], the unavailability of theseestimates can be overcome by the use of the Green kernel for the perturbed operator L − ε andby the Strong Maximum Principle, as they jointly lead to the Weak Harnack Inequality. It isinteresting to observe that, once the Weak Harnack Inequality is available, the equicontinuity of(1.21) (an equivalent version of the Strong Harnack Inequality) is derived by Mokobodzki andBrelot by the comparison (in the sense of measures) µ Ω x ≤ M µ Ω x for harmonic measures: thiscomparison seems to be the core substitute for the mentioned pointwise estimates with Poissonkernels centered at different points x , x .Due to Theorem 1.11, the focus on the Strong Harnack Inequality is now shifted to theWeak Harnack Inequality, which is easier to establish. As already anticipated, the latter isbased on the lower bound (1.7) as we now briefly describe. First, we remark that the proof of(1.7) is a two-line comparison argument: it suffices to apply L − ε on both sides of (1.7) to seethat they produce the same result, namely − ε u ; then one uses the Weak Maximum Principle,since the right-hand side is null on ∂ Ω whereas the left-hand side is nonnegative. Secondly,with inequality (1.7) at hands and the strict positivity of k ε (a consequence of the SMP), it isnot difficult to prove that u ( x ) dominates the L loc -norm of u , on suitable compact sets. Then,due to the equivalence of the L loc and C ∞ topologies on the space of the L -harmonic functions(this fact deriving from (HY)), one can infer the following: Theorem 1.12 (Weak Harnack inequality for derivatives) . Let L satisfy (NTD) , (HY) and (HY) ε . Then, for every connected open set O ⊆ R N , every compact subset K of O , every m ∈ N ∪ { } and every y ∈ O , there exists a positive C ( y ) = C ( L , ε, O, K, m, y ) such that (1.22) X | α |≤ m sup x ∈ K (cid:12)(cid:12)(cid:12) ∂ α u ( x ) ∂x α (cid:12)(cid:12)(cid:12) ≤ C ( y ) u ( y ) , for every nonnegative L -harmonic function u in O . We remark that topological properties similar to those mentioned above for the space of the L -harmonic functions are also valid when L in (1.1) is not necessarily hypoelliptic , provided thatit possesses a positive global fundamental solution: see e.g., [3] by the first and third namedauthors, where Montel-type results are proved (in the sense of [34]), jointly with the equivalenceof the topologies induced on H L (Ω) by L loc and by L ∞ loc , under no hypoellipticity assumptions. Acknowledgements.
We wish to thank Alberto Parmeggiani for many helpful discussionson hypoellipticty, leading to an improvement of the manuscript.2.
The Strong Maximum Principle for L The aim of this section is to prove the Strong Maximum Principle for L in Theorem 1.3.Clearly, a fundamental step is played by a suitable Hopf-type lemma, furnished in Lemma 2.1.(For a recent interesting survey on maximum principles and Hopf-type results for uniformlyelliptic operators, see López-Gómez [33].)First the relevant definition and notation: given an open set Ω ⊆ R N and a relatively closedset F in Ω , we say that ν is externally orthogonal to F at y , and we write(2.1) ν ⊥ F at y ,if: y ∈ Ω ∩ ∂F ; ν ∈ R N \ { } ; B ( y + ν, | ν | ) is contained in Ω and it intersects F only at y . Hereand throughout B ( x , r ) is the Euclidean ball in R N of centre x and radius r > ; moreover | · | will denote the Euclidean norm on R N . The notation (2.1) does not explicitly refer toexternality, but this will not create any confusion in the sequel. It is not difficult to recognizethat if Ω is connected and if F is a proper (relatively closed) subset of Ω , then there alwaysexist couples ( y, ν ) such that ν ⊥ F at y .Finally, throughout the paper we write ∂ i for ∂∂x i . Lemma 2.1 ( of Hopf-type for L ) . Suppose that L is an operator of the form (1.1) with C coefficients V > and a i,j , and let us set A ( x ) := ( a i,j ( x )) i,j . (We recall that A ( x ) ≥ forevery x ∈ R N .) Let Ω ⊆ R N be a connected open set. Then, the following facts hold. (1) Let u ∈ C (Ω , R ) be such that L u ≥ on Ω ; let us suppose that (2.2) F ( u ) := n x ∈ Ω : u ( x ) = max Ω u o is a proper subset of Ω . Then (2.3) h A ( y ) ν, ν i = 0 whenever ν ⊥ F ( u ) at y. (2) Suppose c ∈ C ( R N , R ) is nonnegative on R N , and let us set L c := L − c . Let u ∈ C (Ω , R ) be such that L c u ≥ on Ω ; let us suppose that F ( u ) in (2.2) is a propersubset of Ω and that max Ω u ≥ . Then (2.3) holds true.Proof. We begin by proving part (1) in the statement of the Lemma, from which we alsoinherit the notation and hypotheses on u and F ( u ) . Notice that the assumption ensures that M := max Ω u ∈ R . To this aim, let us assume by contradiction that(2.4) h A ( y ) ν, ν i > , for some ν ⊥ F ( u ) at y .We define a smooth function w : R N −→ R as follows w ( x ) := e − λ | x − ( y + ν ) | − e − λ | ν | , where λ is a positive real number chosen in a moment. We set b j := P Ni =1 ∂ i ( V a i,j ) /V , so that L = P i,j a i,j ∂ i,j + P j b j ∂ j . A simple computation shows that(2.5) L w ( y ) = λ e − λ | ν | h A ( y ) ν, ν i − λ N X j =1 (cid:0) a j,j ( y ) − b j ( y ) ν j (cid:1) , and thus, by (2.4), it is possible to choose λ > in such a way that L w ( y ) > . By the continuity of L w , we can then find a positive real number δ such that V := B ( y, δ ) is compactly contained in Ω and L w > on V . We now define, for ε > , a function v ε : V → R by setting v ε := u + ε w . Clearly, v ε ∈ C ( V, R ) ∩ C ( V , R ) , and we claim that the maximum of v ε on V is attained in V . ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 11
Indeed, let us consider the splitting of ∂V given by the two sets K := ∂V ∩ B ( y + ν, | ν | ) and K := ∂V \ K . For every x ∈ K , one has v ε ( x ) = u ( x ) + εw ( x ) < u ( x ) ≤ M. On the other hand, for all x ∈ K , we have v ε ( x ) ≤ max K u + ε max K w, and since max K u < M (observe that u < M outside F ( u ) and that K is a compact setcontained in Ω \ F ( u ) ), it is possible to choose ε > so small that v ε < M on K . By gatheringtogether these facts we see that, for every x ∈ ∂V (note that y ∈ F ( u ) and w ( y ) = 0 ) v ε ( x ) < M = u ( y ) = v ε ( y ) ≤ max V v ε , and this proves the claim. From L v ε = L u + ε L w ≥ ε L w (and the latter is > on V ) thefunction v ε is a strictly L -subharmonic function on V , that is, L v ε > on V , admitting amaximum point on the open set V , say p . Then we have (recall that A ( p ) ≥ and noticethat ∇ v ε ( p ) = 0 and H ( p ) := ( ∂ i,j v ε ( p )) i,j ≤ ) < L v ε ( p ) = X i,j a i,j ( p ) ∂ i,j v ε ( p ) = trace (cid:0) A ( p ) · H ( p ) (cid:1) ≤ , (2.6)which is clearly a contradiction.Part (2) in the statement of the Lemma can be proved in a totally analogous way: wereplace L with L c and we notice that w ( y ) = 0 so that L c w ( y ) = L w ( y ) , and (2.5) is leftunchanged. Arguing as above, we let again p ∈ V be such that v ε ( p ) = max V v ε . This gives v ε ( p ) ≥ v ε ( y ) = u ( y ) = M . Hence (2.6) becomes < L c v ε ( p ) = trace (cid:0) A ( p ) · H ( p ) (cid:1) − c ( p ) v ε ( p ) ≤ − c ( p ) M, where in the last inequality we used the assumption c ≥ and the fact that v ε ( p ) ≥ M . Bythe assumption M ≥ (and again by the assumption on the sign of c ), we have − c ( p ) M ≤ ,and we obtain another contradiction. (cid:3) We are now in a position to provide the
Proof (of Theorem 1.3).
Let L be as in the statement of Theorem 1.3; suppose that Ω ⊆ R N is a connected open set and that u ∈ C (Ω , R ) satisfies L u ≥ on Ω and u attains a maximumin Ω . We set F ( u ) := n x ∈ Ω : u ( x ) = max Ω u o . By assumption F ( u ) = ∅ , say ξ ∈ F ( u ) . We show that F ( u ) = Ω .To this aim, let us rewrite L as follows: L = 1 V X i,j ∂ i (cid:16) V a i,j ∂ j (cid:17) = 1 V X i,j V ∂ i ( a i,j ∂ j ) + X i,j ∂ i VV a i,j ∂ j = X i,j ∂ i ( a i,j ∂ j ) + X j b j ∂ j , where b j := V P Ni =1 ∂ i V a i,j (for j = 1 , . . . , N ). Let us consider the vector fields X i := N X j =1 a i,j ∂ j , i = 1 , . . . , N, X := N X j =1 b j ∂ j . (2.7)We explicitly remak the following useful fact: X is a linear combination (with smooth-coefficients) of X , . . . , X N ; indeed(2.8) X = N X j =1 b j ∂ j = N X j =1 V N X i =1 ∂ i V a i,j ∂ j = N X i =1 ∂ i VV N X j =1 a i,j ∂ j = N X i =1 ∂ i VV X i . Summing up, we have written L as follows L u = N X i =1 ∂ i ( X i u ) + N X i =1 ∂ i VV X i u, ∀ u ∈ C . Thanks to the assumption (NTD) of non-total degeneracy of L and due to the smoothness ofits coefficients, we are entitled to use a notable result [2, Theorem 2] by Amano, which statesthat the hypoellipticity assumption (HY) ensures the controllability of the ODE system(2.9) ˙ γ = ξ X ( γ ) + N X i =1 ξ i X i ( γ ) , ( ξ , ξ , . . . , ξ N ) ∈ R N , on every open and connected subset of R N (see e.g., [26, Chapter 3] for the notion of control-lability). Since Ω is open and connected, this implies that any point of Ω can be joined to ξ by a continuous curve γ contained in Ω which is piecewise an integral curve of a vector fieldbelonging to V := span R { X , X , . . . , X N } . It then suffices to prove that if γ is an integral curveof a vector field X ∈ V starting at a point of F ( u ) (which is non-empty), then γ ( t ) remains in F ( u ) for every admissible time t . In this case we say that F ( u ) is X -invariant .By a result of Bony, [8, Théorème 2.1], the X -invariance of F ( u ) is equivalent to the tan-gentiality of X to F ( u ) : this latter condition means that(2.10) h X ( y ) , ν i = 0 whenever ν ⊥ F ( u ) at y. Hence, by all the above arguments, the proof of the SMP is complete if we show that (2.10) isfulfilled by any X ∈ V . Since X is a linear combination of X , X , . . . , X N and due to (2.8),it suffices to prove this identity when X is replaced by any element of { X , . . . , X N } . Dueto identity (2.3) in the Hopf-type Lemma 2.1, it is therefore sufficient to show that for every i ∈ { , . . . , N } and every x ∈ R N , there exists λ i ( x ) > such that(2.11) h X i ( x ) , ν i ≤ λ i ( x ) h A ( x ) ν, ν i for every ν ∈ R N . Indeed, (2.11) together with (2.3) implies that the left-hand side of (2.11) is null whenever ν ⊥ F ( u ) at y , which is precisely (2.10) for X ∈ { X , . . . , X N } . Due to the very definition of X i , inequality (2.11) boils down to proving that, given a real symmetric positive semidefinitematrix A = ( a i,j ) , for every i there exists λ i > such that (cid:16) X j a i,j ν j (cid:17) ≤ λ i X i,j a i,j ν i ν j for every ν ∈ R N , which is a consequence of the Cauchy-Schwarz inequality and the characterization (for a sym-metric A ≥ ) of ker ( A ) as { x ∈ R N : h Ax, x i = 0 } .This proves part (1) of Theorem 1.3. As for part (2), let c be smooth and nonnegative on R N , and let us set L c := L − c . Suppose u ∈ C (Ω , R ) satisfies L c u ≥ on Ω and that itattains a nonnegative maximum in Ω . For F ( u ) = ∅ as above, we show again that F ( u ) = Ω .The hypoellipticity and non-total degeneracy of L ensure again (by Amano’s cited result for L ) the controllability of system (2.9). This again grants a connectivity property of Ω by meansof continuous curves, piecewise integral curves of elements in the above vector space V . ByBony’s quoted result on invariance/tangentiality, the needed identity F ( u ) = Ω follows if weshow again that (2.10) is fulfilled when X is replaced by X i , for i = 1 , . . . , N (the case i = 0 deriving as above from (2.8)).Now, by part (2) of Lemma 2.1, it is at our disposal a Hopf-type Lemma for operators ofthe form L c , and for functions u such that L c u ≥ and attaining a nonnegative maximum. Inother words, we know that (2.3) holds true, again as in the previous case (1). The validity of(2.11) allows us to end the proof, as in the previous part. (cid:3) A close inspection to the above proof shows that we have indeed demonstrated the followingresult as well (replacing the hypothesis of hypoellipticity of L by that of L − c ), since Amano’sresults on hypoellipticity/controllobility are independent of the presence of a zero-order term: Remark . Suppose that L is an operator of the form (1.1) , with C ∞ coefficients V > and ( a i,j ) ≥ , and that it satisfies (NTD) . Let c ∈ C ∞ ( R N , R ) be nonnegative and suppose that theoperator L c := L − c is hypoelliptic on every open subset of R N .If Ω ⊆ R N is a connected open set, then any function u ∈ C (Ω , R ) satisfying L c u ≥ on Ω and attaining a nonnegative maximum in Ω is constant throughout Ω . As a Corollary of Theorem 1.3 we immediately get the following result.
ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 13
Corollary 2.3 ( Weak Maximum Principle for L ) . Suppose that L is an operator of theform (1.1) , with C ∞ coefficients V > and ( a i,j ) ≥ , and that it satisfies (NTD) and (HY) .Suppose also that c ∈ C ∞ ( R N , R ) is nonnegative on R N (the case c ≡ is allowed), and let usset L c := L − c . Then, L c satisfies the Weak Maximum Principle on every bounded open set Ω ⊆ R N , that is: (2.12) u ∈ C (Ω , R ) L c u ≥ on Ωlim sup x → x u ( x ) ≤ for every x ∈ ∂ Ω = ⇒ u ≤ on Ω .As a consequence, if Ω ⊆ R N is bounded, and if u ∈ C (Ω) ∩ C (Ω) is nonnegative and such that L c u ≥ on Ω , then one has sup Ω u = sup ∂ Ω u. Proof.
Suppose that the open set Ω ⊂ R N is bounded and u is as in the left-hand side of (2.12).Let x ∈ Ω be such that(2.13) lim sup x → x u ( x ) = sup Ω u. If x ∈ ∂ Ω , then (2.12) ensures that lim sup x → x u ( x ) ≤ , so that (due to (2.13)) sup Ω u ≤ ,proving the right-hand side of (2.12). If x ∈ Ω , then (2.13) gives u ( x ) = max Ω u . If u ( x ) < ,we conclude as above that max Ω u = u ( x ) < . If u ( x ) ≥ , we consider Ω ⊆ Ω the connectedcomponent of Ω containing x , and, thanks to part (2) of the Strong Maximum Principle inTheorem 1.3, the existence of an interior maximum point of u on Ω ⊇ Ω (and the fact that u ( x ) ≥ ) ensures that u ≡ u ( x ) on Ω . Let us take any ξ ∈ ∂ Ω ; we have max Ω u = u ( x ) = lim sup Ω ∋ x → ξ u ( x ) ≤ lim sup Ω ∋ x → ξ u ( x ) ≤ , where the last inequality follows from ∂ Ω ⊆ ∂ Ω and from the assumption in (2.12).We remark that when c ≡ the proof is slightly simpler, as an interior maximum of u propagates up to the boundary, regardless of the sign of this maximum.Finally we prove the last assertion of the corollary. Let Ω ⊆ R N be bounded and let u ∈ C (Ω) ∩ C (Ω) be nonnegative on Ω and satisfying L c u ≥ on Ω ; then we set M := sup ∂ Ω u and we observe that M ≥ since this is true of u . We have (recall that c ≥ ) L c ( u − M ) = L c u − L c M ≥ − L c M = − ( L − c ) M = cM ≥ . Since (by definition of M ) we have u − M ≤ on ∂ Ω (and u − M is continuous up to ∂ Ω ), wecan apply (2.12) to get u − M ≤ , that is u ≤ sup ∂ Ω u on Ω . This clearly proves the needed sup Ω u = sup ∂ Ω u . (cid:3) Arguing as in the previous proof (and exploiting Remark 2.2 instead of Theorem 1.3-(2))we also get the following result, where we alternatively replace the hypothesis of hypoellipticityof L by that of L − c : Remark . Suppose that L is an operator of the form (1.1) , with C ∞ coefficients V > and ( a i,j ) ≥ , and that it satisfies (NTD) . Let c ∈ C ∞ ( R N , R ) be nonnegative and suppose that theoperator L c := L − c is hypoelliptic on every open subset of R N .Then L c satisfies the Weak Maximum Principle on every bounded open set Ω ⊆ R N .As a consequence, if Ω ⊆ R N is bounded, and if u ∈ C (Ω) ∩ C (Ω) is nonnegative and suchthat L c u ≥ on Ω , then one has sup Ω u = sup ∂ Ω u. Analytic coefficients: A Unique Continuation result for L In this short section, by means of the ideas of controllability/propagation introduced in theprevious section, we prove the following result.
Theorem 3.1 ( Unique Continuation for L ) . Suppose that L is an operator of the form (1.1) satisfying assumptions (NTD) and (HY) . Suppose that L has C ω coefficients a i,j and V .Let Ω ⊆ R N be a connected open set. Then any L -harmonic function on Ω vanishing onsome non-empty open subset of Ω is identically zero on Ω .Proof. Let u ∈ H L (Ω) be vanishing on the open set U ⊆ Ω , U = ∅ . Let F ⊆ Ω be the supportof u . We argue by contradiction, by assuming that F = ∅ . Let us fix any y ∈ ∂F ∩ Ω and any ν ∈ R N \ { } such that ν ⊥ F at y (see the notion of exterior orthogonality at the beginning of Section 2; the assumption F = ∅ ensures the existence of such a couple ( y, ν ) ). We considerthe Euclidean open ball B := B ( y + ν, | ν | ) which is completely contained in Ω \ F , so that u ≡ on B . We observe that B is the sub-level set { f ( x ) < } , where f ( x ) = | x − ( y + ν ) | − | ν | . There are only two cases:(a) The boundary of B is non-characteristic for L at y , that is, h A ( y ) ∇ f ( y ) , ∇ f ( y ) i 6 = 0 .Due to the C ω assumption we are allowed to use the classical Holmgren’s Theorem (see e.g.,[24, Theorem 8.6.5]), ensuring that u vanishes in a neighborhood of y , so that y ∈ Ω \ F . Since F is relatively closed in Ω , this is in contradiction with y ∈ ∂F ∩ Ω . Hence it is true that:(b) The boundary of B is characteristic for L at y , that is, h A ( y ) ∇ f ( y ) , ∇ f ( y ) i = 0 . Since ∇ f ( x ) = 2 ( x − y − ν ) , this condition boils down to h A ( y ) ν, ν i = 0 . Let X , X , . . . , X N be thevector fields introduced in (2.7). The same Linear Algebra argument leading to (2.11) showsthat h A ( y ) ν, ν i = 0 implies h X i ( y ) , ν i = 0 for every i = 1 , . . . , N .Identity (2.8) guarantees that the same holds for i = 0 as well. Therefore, one has h X ( y ) , ν i = 0 for every X ∈ V := span { X , X , . . . , X N } . Arguing as in Section 2, by means of the result byBony [8, Théorème 2.1], this geometric condition (holding true for arbitrary ν ⊥ F at y ) impliesthat the closed set F is X -invariant for any X ∈ V (that is F contains the trajectories of theintegral curves of X touching F ).On the other hand, the hypoellipticity assumption (HY) on L ensures (due to the recalledresult by Amano [2, Theorem 2]) that any pair of points of the connected open set Ω can bejoined by a continuous curve which is piecewise an integral curve of some vector fields X in V .Gathering together all the mentioned results, the fact that F = ∅ implies that any point of Ω belongs to F , contradicting the assumption that U ⊆ Ω \ F . (cid:3) The Green function and the Green kernel for L − ε The aim of this section is to prove Theorem 1.9. In the first part of the proof (Steps I–III)we follow the classical scheme by Bony (see [8, Théorème 6.1]), hence we skip many details;it is instead in Step IV that a slight difference is presented, in that we exploit the measure d ν ( x ) = V ( x ) d x in order to obtain the symmetry property of the Green kernel even when ouroperator L is not (classically) self-adjoint. The problem of the behavior of the Green kernelalong the diagonal is more subtle, as it is shown by Fabes, Jerison and Kenig in [14] whoproved that, for divergence-form operators as in (1.1) (when V ≡ and, roughly put, when thedegeneracy of A ( x ) is controlled by a suitable weight) the limit of the Green kernel along thediagonal need not be infinite; we plan to investigate this behavior in a future study, since ourassumption (NTD) prevents the existence of any vanishing Muckenhoupt-type weight.Throughout this section, we fix an operator L of the form (1.1), with C ∞ coefficients V > and ( a i,j ) ≥ , and we assume that L satisfies (NTD). Moreover, we also fix ε ≥ (note thatthe case ε = 0 is allowed) and we set L ε := L − ε ; we assume that L ε is hypoelliptic on everyopen subset of R N . Finally, Ω is a fixed open set as in Lemma 1.7, such that the Dirichletproblem (1.8) is (uniquely) solvable.From Lemma 1.7, we know that there exists a monotone operator G ε (which we called theGreen operator related to L ε and Ω ); since ε ≥ is fixed, in all this section we drop the subscript ε in G ε , k ε , λ x,ε and we simply write G, k, λ x . Hence we are given the monotone operator G : C (Ω , R ) −→ C (Ω , R ) mapping f ∈ C (Ω , R ) into the unique function G ( f ) ∈ C (Ω , R ) satisfying (cid:26) L ε ( G ( f )) = − f on Ω (in the weak sense of distributions), G ( f ) = 0 on ∂ Ω (point-wise).(4.1)We also know that the (Riesz) representation G ( f )( x ) = Z Ω f ( y ) d λ x ( y ) for every f ∈ C (Ω , R ) and every x ∈ Ω (4.2)holds true, with a unique Radon measure λ x defined on Ω (which we called the Green measurerelated to L ε , Ω and x ). ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 15
Finally, we set d ν ( x ) := V ( x ) d x and we observe that (as in (1.13))(4.3) Z ϕ L ε ψ d ν = Z ψ L ε ϕ d ν, for every ϕ, ψ ∈ C ∞ ( R N , R ) . Step I.
We fix x ∈ Ω . We begin by proving that λ x is absolutely continuous with respectto the Lebesgue measure on Ω . To this end, let ϕ ∈ C ∞ (Ω , R ) ; by (4.1) it is clear that G ( L ε ϕ ) = − ϕ , so that (see (4.2)) − ϕ ( x ) = Z Ω L ε ϕ ( y ) d λ x ( y ) , for every ϕ ∈ C ∞ (Ω , R ) .If we consider λ x as a distribution on Ω in the standard way, this identity boils down to(4.4) ( L ε ) ∗ λ x = − Dir x in D ′ (Ω) , where Dir x denotes the Dirac mass at x , and ( L ε ) ∗ is the classical adjoint operator of L ε . It isnoteworthy to observe that, in general, ( L ε ) ∗ is neither equal to L ε nor of the form e L − ε forany e L a divergence operator as in (1.1).However, the following crucial property of ( L ε ) ∗ is fulfilled: Remark . The operator ( L ε ) ∗ is hypoelliptic on every open subset of R N . Indeed, let U ⊆ W be open sets and let u ∈ D ′ ( W ) be such that ( L ε ) ∗ u = h in D ′ ( U ) , where h ∈ C ∞ ( U, R ) . This gives the following chain of identities (here ψ ∈ C ∞ ( U, R ) is arbitrary) Z h ψ = h u, L ε ψ i = h u, L ψ − εψ i (1.14) = D u, L ∗ ( V ψ ) V − εψ E = D uV , L ∗ ( V ψ ) − εψ V E = D uV , ( L ε ) ∗ ( V ψ ) E . If we write R h ψ = R hV ( ψ V ) , and if we observe that C ∞ ( U, R ) = { ψ V : ψ ∈ C ∞ ( U, R ) } , theabove computation shows that L ε ( u/V ) = h/V in D ′ ( U ) . The hypoellipticity of L ε now gives u/V ∈ C ∞ ( U, R ) whence u ∈ C ∞ ( U, R ) , as V is smooth and positive. (cid:3) Identity (4.4) gives in particular ( L ε ) ∗ λ x = 0 in D ′ (Ω \ { x } ) ; thanks to Remark 4.1, thisensures the existence of g x ∈ C ∞ (Ω \ { x } , R ) such that the distribution λ x restricted to Ω \ { x } is the function-type distribution associated with the function g x ; equivalently(4.5) Z ϕ ( y ) d λ x ( y ) = Z ϕ ( y ) g x ( y ) d y, for every ϕ ∈ C ∞ (Ω \ { x } , R ) .Clearly g x ≥ on Ω \ { x } and ( L ε ) ∗ g x = 0 in Ω \ { x } . This temporarily proves that λ x coincideswith g x ( y ) d y on Ω \ { x } . We claim that this is also true throughout Ω . This will follow if weshow that C := λ x ( { x } ) = 0 . Clearly, by the definition of C , on Ω we have λ x = C Dir x + ( λ x ) | Ω \{ x } = C Dir x + g x ( y ) d y. Treating this as an identity between distributions on Ω , we apply the operator ( L ε ) ∗ to get C ( L ε ) ∗ Dir x = − Dir x − ( L ε ) ∗ ( g x ( y ) d y ) . Here we used (4.4). We now proceed as follows:- we multiply both sides by a C ∞ function χ compactly supported in Ω and χ ≡ near x ;- we compute the Fourier transform of the tempered distributions obtained as above;- on the left-hand side we obtain a function-type distribution associated with function y C e − i h x,y i (cid:16) − X i,j a i,j ( x ) y i y j + { polynomial in y of degree ≤ } (cid:17) , where ( a i,j ) is the principal matrix of L ;- on the right-hand side we obtain a function-type distribution associated with a function whichis the sum of y
7→ − e − i h x,y i with a function of the form y
7→ − X i,j α i,j ( x, y ) y i y j + { polynomial in y of degree ≤ } , where α i,j ( x, y ) = − Z g x ( ξ ) χ ( ξ ) a i,j ( ξ ) e − i h ξ,y i d ξ. By the Riemann-Lebesgue Theorem one has α i,j ( x, y ) −→ as | y | → ∞ . This implies that C = 0 , since at least one of the entries of ( a i,j ( x )) is non-vanishing, due to the (NTD)hypothesis on L .We have therefore proved that, for any x ∈ Ω ,(4.6) d λ x ( y ) = g x ( y ) d y on Ω .Since λ x is a finite measure (recalling that Ω is compact), from (4.6) we get g x ∈ L (Ω) forevery x ∈ Ω . Step II.
We next show that λ x ( ∂ Ω) = 0 for any x ∈ Ω . For small δ > , we let D δ denotethe closed δ -neighborhood of ∂ Ω of the points in R N having distance from ∂ Ω less than or equalto δ ; we then choose a function F ∈ C ( R N , [0 , which is identically on ∂ Ω and is supportedin the interior of D δ . We denote by f the restriction of F to Ω . From (4.2) we have ≤ G ( f )( x ) = Z Ω f ( y ) d λ x ( y ) ≤ Z Ω d λ x ( y ) = G (1)( x ) , for every x ∈ Ω .(4.7)For any x ∈ Ω we have λ x ( ∂ Ω) = Z ∂ Ω d λ x ( y ) = Z ∂ Ω f ( y ) d λ x ( y ) ≤ Z Ω f ( y ) d λ x ( y ) = G ( f )( x ) ≤ sup Ω G ( f ) = max (cid:26) sup Ω ∩ D δ G ( f ) , sup Ω \ D δ G ( f ) (cid:27) =: max { I , II } . We claim that I and II in the above right-hand side are bounded from above by sup Ω ∩ D δ G (1) .This is true of I, due to (4.7); as for II we invoke the last assertion in Remark 2.4 applied to:- the hypoelliptic operator L ε = L − ε ,- the bounded open set Ω := Ω \ D δ ,- the nonnegative function G ( f ) , which satisfies L ε G ( f ) = − f = 0 on Ω both weakly andstrongly due to the hypoellipticity of L ε .The mentioned Remark 2.4 then ensures that the values of G ( f ) on Ω \ D δ are bounded fromabove by the values of G ( f ) on the boundary of this set, so that II ≤ I. Summing up, λ x ( ∂ Ω) ≤ max { I , II } ≤ sup Ω ∩ D δ G (1) . As δ goes to , the right-hand side tends to sup ∂ Ω G (1) = 0 by (4.1). This gives the desired λ x ( ∂ Ω) = 0 , for any x ∈ Ω . By collecting together (4.6) and λ x ( ∂ Ω) = 0 , we infer that (forevery f ∈ C (Ω , R ) and x ∈ Ω ) G ( f )( x ) (4.2) = Z Ω f ( y ) d λ x ( y ) = Z Ω f ( y ) d λ x ( y ) (4.6) = Z Ω f ( y ) g x ( y ) d y. This proves the identity G ( f )( x ) = Z Ω f ( y ) g x ( y ) d y, for every f ∈ C (Ω , R ) and every x ∈ Ω .(4.8)If ϕ ∈ C ∞ (Ω , R ) , since we know that G ( L ε ϕ ) = − ϕ , we get − ϕ ( x ) = Z Ω L ε ϕ ( y ) g x ( y ) d y, for every x ∈ Ω .(4.9)This is equivalent to ( L ε ) ∗ g x = − Dir x for every x ∈ Ω .(4.10) Step III. If g x is as in Step I, we are ready to set g : Ω × Ω −→ [0 , ∞ ] , g ( x, y ) := (cid:26) g x ( y ) if x = y ∞ if x = y .Hence the representation (4.8) becomes G ( f )( x ) = Z Ω f ( y ) g ( x, y ) d y, for every f ∈ C (Ω , R ) and every x ∈ Ω .(4.11)We aim to prove that g is smooth outside the diagonal of Ω × Ω . ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 17
Remark . Let O be any open subset of R N . The hypoellipticity of a general PDO L as in (1.2) ensures the equality of the topologies on H L ( O ) inherited by the Fréchet spaces C ∞ ( O ) and L ( O ) . Indeed, let X and Y denote respectively the topological space H L ( O ) with the topologiesinherited by C ∞ ( O ) and L ( O ) . Then X and Y are Fréchet spaces, since, if a sequence u n ∈ H L ( O ) converges to u uniformly on the compact sets of Ω or, more generally in L loc , Z u n L ∗ ϕ n →∞ −−−−→ Z u L ∗ ϕ, ∀ ϕ ∈ C ∞ ( O, R ) . Now, the identity map ι : X → Y is trivially linear, bijective and continuous, whence, by theOpen Mapping Theorem, ι is a homeomorphism, whence the mentioned topologies coincide. (cid:3) We next resume our main proof. The set { g x } x ∈ Ω is bounded in L (Ω) , since ≤ Z Ω g x ( y ) d y = G (1)( x ) ≤ max Ω G (1) . A fortiori, the set { g x } x ∈ Ω is also bounded in the topological vector space L (Ω) . We next fixtwo disjoint open sets U, W with closures contained in Ω . The family of the restrictions n ( g x ) (cid:12)(cid:12) U o x ∈ W is contained in the space of the ( L ε ) ∗ -harmonic functions on U . By Remark 4.2, the set G isalso bounded in the topological vector space H ( L ε ) ∗ ( U ) , endowed with the C ∞ -topology.This means that, for every compact set K ⊂ U and for every m ∈ N , there exists a constant C ( K, m ) > such that(4.12) sup | α |≤ m sup y ∈ K (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ∂∂y (cid:17) α g ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( K, m ) , uniformly for x ∈ W .Following Bony [8, Section 6], we introduce the operator F transforming any distribution T compactly supported in U into the function on W defined by F ( T ) : W −→ R , F ( T )( x ) := h T, g x i ( x ∈ W ) . The definition is well-posed since g x ∈ C ∞ ( U, R ) (and T is compactly supported in U ). Weclaim that F ( T ) ∈ C ∞ ( W, R ) . Once this is proved, by the Schwartz Kernel Theorem (see e.g.,[13, Section 11] or [40, Chapter 50]), we can conclude that g ( x, y ) is smooth on W × U . Bythe arbitrariness of the disjoint open sets U, W this proves that g ( x, y ) is smooth out of thediagonal of Ω × Ω , as desired.As for the proof of the claimed F ( T ) ∈ C ∞ ( W, R ) , we can take (say, by some appropriateconvolution) a sequence of continuous functions f n , supported in U , converging to T in theweak sense of distributions; due to the compactness of the supports (of the f n and of T ), lim n →∞ Z U f n ϕ = h T, ϕ i , for every ϕ ∈ C ∞ ( U, R ) .We are hence entitled to take ϕ = g x (for any fixed x ∈ W ). From (4.11) we get(4.13) lim n →∞ G ( f n )( x ) = h T, g x i = F ( T )( x ) , for any x ∈ W .We now prove that F ( T ) ∈ L ∞ ( W ) ; this follows from the next calculation (here C > and m ∈ N are constants depending on T and on the compact set U ) k F ( T ) k L ∞ = sup x ∈ W |h T, g x i| ≤ sup x ∈ W C X | α |≤ m sup y ∈ U (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ∂∂y (cid:17) α g ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) (4.12) ≤ e C ( U , m ) < ∞ . We finally prove that L ε ( F ( T )) = 0 in the weak sense of distributions on W ; by the hypoellip-ticity of L ε this will yield the smoothness of F ( T ) on W . We aim to show that, Z W F ( T )( x ) ( L ε ) ∗ ϕ ( x ) d x = 0 for any ϕ ∈ C ∞ ( W ) . Now, the left-hand side is (by (4.13)) Z lim n →∞ G ( f n )( x ) ( L ε ) ∗ ϕ ( x ) d x. If a dominated convergence can be applied, this is equal to lim n →∞ Z W G ( f n )( x ) ( L ε ) ∗ ϕ ( x ) d x (4.1) = − lim n →∞ Z W f n ( x ) ϕ ( x ) d x = 0 , the last equality descending from the fact that the f n are supported in U for every n . Weare then left with showing that the dominated convergence is fulfilled: this is a consequenceof (4.12), of the boundedness of F ( T ) on W , and of the fact that the convergence in (4.13) isindeed uniform w.r.t. x ∈ W (a general result of distribution theory: the uniform convergencefor sequences of distributions on bounded sets). Step IV.
We are finally ready to introduce our kernel(4.14) k : Ω × Ω −→ [0 , ∞ ) , k ( x, y ) := g ( x, y ) V ( y ) . Clearly, from (4.11) and (1.13) we immediately have G ( f )( x ) = Z Ω f ( y ) k ( x, y ) d ν ( y ) , for every f ∈ C (Ω , R ) and every x ∈ Ω .(4.15)This gives the representation (1.15) whilst (1.17) follows from (4.9).The integrability of k ( x, · ) in Ω is a consequence of g x ∈ L (Ω) (and the positivity of thecontinuous function V on R N ). Moreover, k is smooth on Ω × Ω deprived of the diagonal byStep III. Also, the nonnegative function k is integrable on Ω × Ω as this computation shows: ≤ Z Ω × Ω k ( x, y ) d x d y = Z Ω (cid:16) Z Ω V ( y ) k ( x, y ) d ν ( y ) (cid:17) d x (4.15) = Z Ω G (1 /V )( x ) d x < ∞ , the last inequality following from the continuity of G (1 /V ) on the compact set Ω .For fixed x ∈ Ω , the L ε -harmonicity of the function k ( x, · ) in Ω \ { x } is a consequence ofthe following computation (4.10) = ( L ε ) ∗ g x (1.14) = V L ε (cid:16) g x V (cid:17) (4.14) = V L ε ( k ( x, · )) . The fact that V is positive then gives L ε ( k ( x, · )) = 0 in Ω \{ x } . From the SMP for L ε = L − ε inRemark 2.2, we deduce that the nonnegative function k ( x, · ) (which is L ε -harmonic in Ω \ { x } )cannot attain the (minimal) value ; therefore k ( x, · ) > on the connected open set Ω \ { x } .A crucial step consists in proving the symmetry property (1.16). We take any nonnegative ϕ ∈ C ∞ (Ω , R ) and we set (note the reverse order of x and y , if compared to G ( ϕ ) ) Φ( x ) = Z Ω ϕ ( y ) k ( y, x ) d ν ( y ) , x ∈ Ω . We claim that Φ ≥ G ( ϕ ) on Ω ; once the claim is proved, from (4.15) we infer that Z Ω ϕ ( y ) k ( x, y ) d ν ( y ) ≤ Z Ω ϕ ( y ) k ( y, x ) d ν ( y ) , x ∈ Ω . The arbitrariness of ϕ will then give k ( x, y ) ≤ k ( y, x ) (recalling that d ν = V ( y ) d y with positive V ) for every y ∈ Ω ; since x, y ∈ Ω are arbitrary, we get k ( x, y ) = k ( y, x ) on Ω × Ω . We provethe claim. We observe that Φ is continuous on Ω and that L ε Φ = − ϕ in D ′ (Ω) , as the followingcomputation shows ( ψ ∈ C ∞ (Ω , R ) is arbitrary): Z Ω Φ( x ) ( L ε ) ∗ ψ ( x ) d x = Z Ω ϕ ( y ) (cid:16) Z Ω k ( y, x ) ( L ε ) ∗ ψ ( x ) d x (cid:17) d ν ( y )= Z Ω ϕ ( y ) (cid:16) Z Ω k ( y, x ) ( L ε ) ∗ ψ ( x ) V ( x ) d ν ( x ) (cid:17) d ν ( y ) (1.14) = Z Ω ϕ ( y ) (cid:16) Z Ω k ( y, x ) L ε (cid:16) ψ ( x ) V ( x ) (cid:17) d ν ( x ) (cid:17) d ν ( y ) (1.17) = − Z Ω ϕ ( y ) ψ ( y ) V ( y ) d ν ( y ) = − Z Ω ϕ ( y ) ψ ( y ) d y. ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 19
From the hypoellipticity of L ε we get Φ ∈ C ∞ (Ω , R ) and L ε Φ = − ϕ point-wise. We now applythe WMP in Remark 2.4 to the operator L ε = L − ε and to the function G ( ϕ ) − Φ : this functionis smooth and L ε -harmonic on Ω , and G ( ϕ ) − Φ ≤ G ( ϕ ) on Ω (since Φ is nonnegative), so that lim sup x → x ( G ( ϕ ) − Φ)( x ) ≤ lim sup x → x G ( ϕ )( x ) = 0 for every x ∈ ∂ Ω . Therefore G ( ϕ ) − Φ ≤ on Ω as claimed.We finally prove (1.18). Due to the symmetry property of k , (1.18) will follow if we showthat, given x ∈ Ω and y ∈ ∂ Ω , one has(4.16) lim n →∞ k ( y n , x ) = 0 , for every sequence y n in Ω converging to y . To this end, we fix an open set Ω ′ containing x and with closure contained in Ω , and it is non-restrictive to suppose that y n / ∈ Ω ′ for every n .The functions k n : Ω ′ −→ R , k n ( x ) := k ( y n , x ) , x ∈ Ω ′ are smooth and L ε -harmonic in Ω ′ . We also have k n −→ in L (Ω ′ ) , as it follows from ≤ Z Ω ′ k n ( x ) d x ≤ Z Ω k ( y n , x ) d x = Z Ω g ( y n , x ) V ( x ) d x ≤ sup Ω V Z Ω g ( y n , x ) d x = sup Ω V G (1)( y n ) n →∞ −−−−→ . From Remark 4.2 we get that k n −→ in the Fréchet space H L ε (Ω ′ ) with the C ∞ -topology, sothat k n −→ uniformly on the compact sets of Ω ′ and in particular point-wise on Ω ′ . (cid:3) The Harnack inequality
We begin by proving the next crucial lemma. This is the first time that, broadly speaking,the PDOs L and the perturbed L − ε clearly interact. Lemma 5.1.
Let L be as in (1.1) and let it satisfy (NTD) and (HY) ε . Let Ω be an open set in R N as in the thesis of Lemma 1.7, and let Ω ′ be an open set containing Ω . Finally, we denoteby k ε the Green kernel related to L ε and to the set Ω (as in Theorem 1.9).Then we have the estimate (5.1) u ( x ) ≥ ε Z Ω u ( y ) k ε ( x, y ) d ν ( y ) , ∀ x ∈ Ω , holding true for every smooth nonnegative L -harmonic function u in Ω ′ .Proof. We consider the function v ( x ) = R Ω u ( y ) k ε ( x, y ) d ν ( y ) on Ω . From (1.15) (and thedefinition of Green operator) we know that v = G ε ( u ) , where G ε is the Green operator relatedto L ε (and to the open set Ω ); moreover, since u is smooth (by assumption) on Ω , we knowfrom Lemma 1.7 (and the hypoellipticity of L ε ) that v ∈ C ∞ (Ω) ∩ C (Ω) is the solution of (cid:26) L ε v = − u on Ω ,v = 0 on ∂ Ω .(5.2)This gives L ε ( ε v − u ) = − ε u − ( L − ε ) u = − ε u + ε u = 0 on Ω ; moreover, on ∂ Ω , ε v − u = − u ≤ ,by the nonnegativity of u . By the WMP in Remark 2.4, we get ε v − u ≤ on Ω which isequivalent to (5.1). (cid:3) We are ready for the proof of the Weak Harnack Inequality (for higher order derivatives).
Proof (of the Weak Harnack Inequality for derivatives, Theorem 1.12).
We distinguish two ca-ses: y / ∈ K and y ∈ K . The second case can be reduced to the former. Indeed, let usassume we have already proved the theorem in the former case, and let y ∈ K . If we take any y ′ ∈ O \ K , and we consider the inequality u ( y ′ ) ≤ C ′ u ( y ) , resulting from (1.22) by considering m = 0 and the compact set { y ′ } , we get X | α |≤ m sup x ∈ K (cid:12)(cid:12)(cid:12) ∂ α u ( x ) ∂x α (cid:12)(cid:12)(cid:12) (1.22) ≤ C u ( y ′ ) ≤ C C ′ u ( y ) . We are therefore entitled to assume that y / ∈ K . By the aid of a classical argument (with a chainof suitable small open sets { Ω n } pn =1 covering a connected compact set containing K ∪ { y } ), itis not restrictive to assume that K ∪ { y } ⊂ Ω ⊂ Ω ⊂ O , where Ω is one of the basis open setsconstructed in Lemma 1.7.Let x ∈ K be arbitrarily fixed. The function k ε ( x , · ) (the Green kernel related to L ε and Ω ) is strictly positive in Ω \ { x } (this is a consequence of the SMP applied to the L ε -harmonicfunction k ε ( x , · ) ; see Theorem 1.9). In particular, since y / ∈ K , we infer that k ε ( x , y ) > .Hence, there exist a neighborhood W of x (contained in Ω ) and a constant c = c ( ε, y , x ) > such that(5.3) inf z ∈ W k ε ( z, y ) ≥ c > . Our assumptions allow us to apply Lemma 5.1: hence, for every nonnegative u ∈ H L ( O ) , wehave the following chain of inequalities u ( y ) (5.1) ≥ ε Z Ω u ( z ) k ε ( y , z ) d ν ( z ) ≥ ε Z W u ( z ) k ε ( y , z ) d ν ( z ) (1.16) = ε Z W u ( z ) k ε ( z, y ) d ν ( z ) (5.3) ≥ ε c Z W u ( z ) d ν ( z ) ≥ ε c inf W V Z W u ( z ) d z. Summing up, for every x ∈ K there exist a neighborhood W of x and a constant c > (alsodepending on x but independent of u ) such that(5.4) u ( y ) ≥ c Z W u ( z ) d z, for every nonnegative u ∈ H L ( O ) .Next, from Remark 4.2, we know that the hypothesis (HY) for L ensures the equality ofthe topologies on H L ( W ) inherited by the Fréchet spaces C ∞ ( W ) and L ( W ) . In particular,to any chosen open neighborhood U of x (with U ⊂ W ) we are given a positive constant c = c ( U, W, m ) such that(5.5) X | α |≤ m sup x ∈ U (cid:12)(cid:12)(cid:12) ∂ α u ( x ) ∂x α (cid:12)(cid:12)(cid:12) ≤ c Z W u ( z ) d z, for every nonnegative u ∈ H L ( O ) . Gathering together (5.4) and (5.5), we infer that, for every x ∈ K there exist a neighborhood U of x and a constant c > (again depending on x butindependent of u ) such that u ( y ) ≥ c X | α |≤ m sup x ∈ U (cid:12)(cid:12)(cid:12) ∂ α u ( x ) ∂x α (cid:12)(cid:12)(cid:12) , for every nonnegative u ∈ H L ( O ) . The compactness of K allows us to derive (1.22) from thelatter inequality, and a covering argument. (cid:3) We now present a proof of Theorem 1.11, crucially based on [9, Chapter I].
Proof (of Theorem 1.11).
As anticipated in the Introduction, the proof is based in an essentialway on the ideas by Mokobodzki-Brelot in [9, Chapter I], ensuring the equivalence of the StrongHarnack Inequality with a series of properties comprising the Weak Harnack Inequality, providedsome assumptions are fulfilled. We furnish some details in order to be oriented through theseequivalent properties.We denote by H L the harmonic sheaf on R N defined by O H L ( O ) (here O ⊆ R N isany open set). Under the assumptions of (Regularity) and (Weak Harnack Inequality), Brelotproves that (see [9, pp.22–24]), for any connected open set O ⊆ R N , and any x ∈ O , the set(5.6) Φ x := n h ∈ H L ( O ) : h ≥ , h ( x ) = 1 o is equicontinuous at x . The proof of this fact rests on some results of Functional Analysisrelated to the family of the so-called harmonic measures { µ Ω x } x ∈ ∂ Ω associated with L (and onbasic properties of the harmonic sheaf H L ). Next, we show how to prove (1.20) starting fromthe equicontinuity of Φ x at x . Indeed, let K ⊂ O , where K is compact and O is an open andconnected subset of R N . By possibly enlarging K , we can suppose that K is connected as well.Let u ∈ H L ( O ) be nonnegative. If u ≡ then (1.20) is trivial; if u is not identically zero then ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 21 (from the Weak Harnack Inequality) one has u > on O . For every x ∈ K , the equicontinuityof Φ x ensures the existence of δ ( x ) > such that (with the choice h = u/u ( x ) in (5.6))(5.7) u ( x ) ≤ u ( ξ ) ≤ u ( x ) , for all ξ ∈ B x := B ( x, δ ( x )) .From the open cover { B x } x ∈ K we can extract a finite subcover B x , . . . , B x p of K . It is alsonon-restrictive (since K is connected) to assume that the elements of this subcover are chosenin such a way that B x ∩ B x = ∅ , ( B x ∪ B x ) ∩ B x = ∅ , . . . ( B x ∪ · · · ∪ B x p − ) ∩ B x p = ∅ . From (5.7) it follows (1.20) with K replaced by B x (with M = 3 ); since B x intersects B x ,one can use again (5.7) in order to prove (1.20) with K replaced by B x ∪ B x (with M = 3 );by proceeding in an inductive way, one can prove (1.20) with K replaced by B x ∪ · · · ∪ B x p (and M = 3 p ), and this finally proves (1.20), since B x ∪ · · · ∪ B x p covers K . (cid:3) Remark . Following Brelot [9, pp.14–17], it being understood that axiom (Regularity) inTheorem 1.11 holds true, the axiom (Weak Harnack Inequality) can be replaced by any of thefollowing equivalent assumptions (see also Constantinescu and Cornea [11]): (Brelot Axiom):
For every connected open set O ⊆ R N , if F is an up-directed familyof L -harmonic functions in O , then sup u ∈ F u is either + ∞ or it is L -harmonic in O . (Harnack Principle): For every connected open set O ⊆ R N , if { u n } n is a non-decrea-sing sequence of L -harmonic functions in O , then lim n →∞ u n is either + ∞ or it is an L -harmonic function in O .We are ready to derive our main result for this section: due to all our preliminary results, theproof is now a few lines argument. Proof (of Harnack Inequality, Theorem 1.10).
Due to Theorem 1.11, it suffices to prove thatour operator L as in the statement of Theorem 1.10 satisfies the properties named (Regularity)and (Weak Harnack Inequality) in Theorem 1.11: the former is a consequence of Lemma 1.7(with f = 0 ), whilst the latter follows from Theorem 1.12. (cid:3) Appendix: The Dirichlet problem for L The aim of this appendix is to prove Lemma 1.7 under the following more general form inTheorem 6.1: our slightly more general framework (we indeed deal with general hypoellipticoperators which are non-totally degenerate at every point) compared to the one considered byBony in [8] (where Hörmander operators are concerned) does not present much more difficultiesthan the one in [8, Section 5], and the proof is given for the sake of completeness only.
Theorem 6.1.
Suppose that L is an operator on R N of the form (6.1) L = N X i,j =1 α i,j ∂ ∂x i ∂x j + N X i =1 β i ∂∂x i + γ, with α i,j , β i , γ ∈ C ∞ ( R N , R ) , with ( α i,j ) symmetric and positive semi-definite. We assume that L is non-totally degenerate at every x ∈ R N and that L is C ∞ -hypoelliptic in every open set.Then there exists a basis for the Euclidean topology of R N made of open sets Ω with thefollowing properties: for every continuous function f on Ω and for every continuous function ϕ on ∂ Ω , there exists one and only one solution u ∈ C (Ω , R ) of the Dirichlet problem (cid:26) Lu = − f on Ω (in the weak sense of distributions), u = ϕ on ∂ Ω (point-wise). (6.2) Furthermore, if f, ϕ ≥ then u ≥ as well. Finally, if f belongs to C ∞ (Ω , R ) ∩ C (Ω , R ) , thenthe same is true of u , and u is a classical solution of (6.2) .Finally, if the zero-order term γ of L is non-positive on R , the above basis { Ω } does notdepend on γ . If γ < , the basis { Ω } only depends on the principal matrix ( α i,j ) of L . The key step is to construct a basis for the Euclidean topology of R N as follows: F is said to be up-directed if for any u, v ∈ F there exists w ∈ F such that max { u, v } ≤ w . Lemma 6.2.
Let A ( x ) = ( a i,j ( x )) be a matrix with real-valued continuous entries on R N ,which is symmetric, positive semi-definite and non-vanishing at a point x ∈ R N .Then, there exists a basis of connected open neighborhoods B x of x such that any Ω ∈ B x satisfies the following property: for every y ∈ ∂ Ω there exists ν ∈ R N \{ } such that B ( y + ν, | ν | ) intersects Ω at y only, and such that (6.3) h A ( y ) ν, ν i > . Proof.
By the assumptions on A ( x ) there exists a unit vector h such that(6.4) h A ( x ) h , h i > . Following the idea of Bony [8], we choose the neighborhood basis B x = { Ω( ε ) } as follows: Ω( ε ) := B ( x + ε − h , ε − + ε ) ∩ B ( x − ε − h , ε − + ε ) . It suffices to show that there exists ε > such that every Ω( ε ) with < ε ≤ ε satisfies therequirement of the lemma. Now, the set Ω( ε ) (which is trivially an open neighborhood of x )shrinks to { x } as ε shrinks to . Moreover, every y ∈ ∂ Ω( ε ) belongs to one at least of thespheres ∂B ( x ± ε − h , ε − + ε ) ; accordingly, we choose ν = ν ε ( y ) := y − ( x ± ε − h ) ε − + ε to get the geometric condition B ( y + ν, | ν | ) ∩ Ω( ε ) = { y } . It obviously holds that ν ε ( y ) tends to h ( x ) as ε → (uniformly for bounded x , y, h ), so that (6.3) follows from (6.4) by continuityarguments, for any ≤ ε ≤ ε , with ε conveniently small. (cid:3) We proceed with the proof of Theorem 6.1 by constructing, for any given x ∈ R N , a basis ofneighborhoods of x as required. The crucial step is to reduce L to some equivalent operator e L with zero-order term e L (1) which is strictly negative around x . We observe that this procedureis not necessary if γ = L (1) is already known to be negative on R N . In general, we let e Lu := w L ( w u ) , where w ( x ) = 1 − M | x − x | , with M ≫ to be chosen. Let us denote by B ( x ) the Euclidean ball of centre x and radius / √ M . It is readily seen that the second order parts of L and e L are equal, modulo the factor w . This shows that e L is non-totally degenerate at any point of B ( x ) and that the principalmatrix of e L is symmetric and positive semi-definite at any point of B ( x ) . Since e L (1)( x ) = w ( x ) γ ( x ) − M trace ( A ( x )) − M N X i =1 β i ( x ) ( x − x ) i , if we choose M so large that M > γ ( x ) / (2 trace ( A ( x ))) (we recall that trace ( A ( x )) > atany x since L is non-totally degenerate at any point), then e L (1)( x ) < . By continuity, thereexists r > small enough such that B ′ ( x ) := B ( x , r ) ⊆ B ( x ) and such that e L (1) < on theclosure of B ′ ( x ) . We explicitly remark (and this will prove the final statement of the theorem)that the condition γ ≤ allows us to take M = 1 for all x and to use the bound e L (1)( x ) ≤ − trace ( A ( x )) − N X i =1 β i ( x ) ( x − x ) i , in order to chose r independently of γ . Remark . Classical arguments, [31], show that, due to the strict negativity of e L (1) on B ′ ( x ) ,the operator e L satisfies the Weak Maximum Principle on every open subset of B ′ ( x ) , that is:(6.5) Ω ⊂ B ′ ( x ) , u ∈ C (Ω , R ) e Lu ≥ on Ωlim sup x → y u ( x ) ≤ for every y ∈ ∂ Ω = ⇒ u ≤ on Ω .The rest of the proof consists in demonstrating the following statement: (S): there exists a basis B x of neighborhoods Ω of x all contained in B ′ ( x ) with theproperties required in Theorem 6.1 relative to e L (in place of L ) . ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 23
Once this is proved, given any Ω ∈ B x , any f ∈ C (Ω , R ) and any ϕ ∈ C ( ∂ Ω , R ) , we obtain thesolution e u of the problem (cid:26) e L e u = − w f on Ω (in the weak sense of distributions), e u = ϕ/w on ∂ Ω (point-wise);(6.6)then we set u := w e u , and a simple verification shows that u solves (6.2), so that existence isproved. As for uniqueness, it suffices to observe that for any fixed Ω ∈ B x , to any solution u of (6.2) on Ω , there corresponds a solution e u = u/w of (6.6) (which is unique, as it is claimedin (S)). Finally all the other requirements on u in the statement of Theorem 6.1 are satisfied,since w is positive and smooth on Ω ⊆ B ( x ) . Remark . We remark that the operator e L is C ∞ -hypoelliptic on every open subset of B ( x ) . Indeed, for any open sets
V, V ′ such that V ⊆ V ′ ⊆ B ( x ) , a distribution u ∈ D ′ ( V ′ ) suchthat e Lu = f ∈ C ∞ ( V, R ) satisfies L ( w u ) = f /w ∈ C ∞ ( V, R ) ; thus, by the hypoellipticity of L ,we infer that w u ∈ C ∞ ( V, R ) so that u ∈ C ∞ ( V, R ) (recalling that w = 0 on B ( x ) ).We are then left to prove statement (S). From now on we choose a neighborhood basis B x of x consisting of open sets (contained in B ′ ( x ) ) as in Lemma 6.2 relative to the principal matrix e A of the operator e L (the matrix e A ( x ) is symmetric, positive semi-definite and non vanishing,as already discussed). We will show that any Ω ∈ B x has the requirements in statement (S).For the uniqueness part, it suffices to use in a standard way the WMP in Remark 6.3 jointlywith the hypoellipticity condition in Remark 6.4. As for existence, we split the proof in severalsteps and, to simplify the notation, we write P instead of e L .(I): f smooth and ϕ ≡ . We fix Ω as above, f ∈ C ∞ (Ω , R ) ∩ C (Ω , R ) and ϕ ≡ . We use astandard elliptic approximation argument. For every n ∈ N we set P n := P + 1 n N X j =1 (cid:16) ∂∂x j (cid:17) . We observe that:- P n is uniformly elliptic on R N ;- the zero-order term P n (1) = P (1) (= e L (1)) is (strictly) negative on Ω ;- Ω satisfies an exterior ball condition, due to Lemma 6.2;- f ∈ C ∞ (Ω , R ) .These conditions imply the existence (see e.g., Gilbarg and Trudinger [21]) of a classical solution u n ∈ C ∞ (Ω , R ) ∩ C (Ω , R ) of the Dirichlet problem (cid:26) P n u n = − f on Ω u n = 0 on ∂ Ω .Let c > be such that P (1) < − c on the closure of B ′ ( x ) . With this choice, we observe that(setting k f k ∞ = sup Ω | f | ) P n (cid:16) ± u n − k f k ∞ c (cid:17) = ∓ f − k f k ∞ c P (1) ≥ ∓ f + k f k ∞ c c ≥ on Ω ± u n − k f k ∞ c = − k f k ∞ c ≤ on ∂ Ω .Arguing as in Remark 6.3, the Weak Maximum Principle for P n proves that(6.7) k u n k ∞ = sup x ∈ Ω | u n ( x ) | ≤ k f k ∞ c uniformly for every n ∈ N . This provides us with a subsequence of u n (still denoted by u n ) and a function u ∈ L ∞ (Ω) suchthat u n tends to u in the weak ∗ topology, that is(6.8) lim n →∞ Z Ω u n h = Z Ω u h, for all h ∈ L (Ω) .Moreover one knows that(6.9) k u k L ∞ ( U ) ≤ lim sup n →∞ k u n k L ∞ ( U ) , for all U ⊆ Ω . From (6.8) it easily follows that Z Ω u P ∗ ψ = − Z Ω f ψ, for all ψ ∈ C ∞ (Ω , R ) .This means that P u = − f in the weak sense of distributions. As P is hypoelliptic on everyopen set (Remark 6.4), we infer that u can be modified on a null set in such a way that u ∈ C ∞ (Ω , R ) . Thus P u = − f in the classical sense on Ω . We aim to prove that u can becontinuously prolonged to on ∂ Ω . To this end, given any y ∈ ∂ Ω , in view of Lemma 6.2 (andthe choice of Ω ), there exists ν ∈ R N \ { } such that B ( y + ν, | ν | ) intersects Ω at y only, andsuch that (see (6.3))(6.10) h e A ( y ) ν, ν i > . As in the Hopf-type Lemma 2.1, we consider the function w ( x ) := e − λ | x − ( y + ν ) | − e − λ | ν | , where λ is a positive real number chosen in a moment. For every n and for every x one has P n w ( x ) = P w ( x ) + 1 n e − λ | x − ( y + ν ) | (cid:16) λ | x − ( y + ν ) | − λN (cid:17) ≥ P w ( x ) − λN e − λ | x − ( y + ν ) | . (6.11)If we set P = P i,j e a i,j ∂ i,j + P j e b j ∂ j + e c , a simple computation (similar to (2.5)) shows that (cid:16) P w ( x ) − λN e − λ | x − ( y + ν ) | (cid:17)(cid:12)(cid:12)(cid:12) x = y = e − λ | ν | (cid:18) λ h e A ( y ) ν, ν i − λ N X j =1 (cid:0)e a j,j ( y ) − e b j ( y ) ν j (cid:1) − λ N (cid:19) . Thanks to (6.10), there exists λ ≫ such that the above right-hand side is strictly positive.Therefore, due to (6.11) there exist ε > and an open ball V = B ( y, δ ) (with ε and δ independentof n ) such that(6.12) P n w ( x ) ≥ ε for every x ∈ V and every n ∈ N .We are willing to apply the Weak Maximum Principle for the operator P n on the open set Ω ∩ V , and for the functions M w ± u n , where M ≫ is chosen as follows. First we have P n ( M w ± u n ) = M P n w ± P n u n = M P n w ∓ f ≥ M ε ∓ f ≥ M ε − k f k ∞ , in Ω ∩ V .
Consequently we first chose
M > k f k ∞ /ε . Then we study the behavior of M w ± u n on ∂ (Ω ∩ V ) = [ V ∩ ∂ Ω] ∪ [Ω ∩ ∂V ] =: Γ ∪ Γ . Firstly, on Γ we have M w ± u n = M w ≤ since Γ ⊆ R N \ B ( y + ν, | ν | ) . Secondly, on Γ , M w ± u n ≤ M max Γ w + k u n k ∞ (6.7) ≤ M max Γ w + k f k ∞ c . Since Γ is a compact set on which w is strictly negative, we have max Γ w < and the furtherchoice M ≥ −k f k ∞ / ( c max Γ w ) yields M w ± u n ≤ on Γ . Summing up, (cid:26) P n ( M w ± u n ) ≥ on Ω ∩ VM w ± u n ≤ on ∂ (Ω ∩ V ) .The Weak Maximum Principle yields M w ± u n ≤ on Ω ∩ V , that is (since w < on Ω ) | u n ( x ) | ≤ M | w ( x ) | for every x ∈ Ω ∩ V and for every n ∈ N .Since w ( y ) = 0 , for every σ > there exists an open neighborhood W ⊂ V of y such that k w k L ∞ ( W ) < σ ; the above inequality then gives k u n k L ∞ ( W ∩ Ω) ≤ M σ . Jointly with (6.9) wededuce that k u k L ∞ ( W ∩ Ω) ≤ M σ , so that lim Ω ∋ x → y u ( x ) = 0 . From the arbitrariness of y , weobtain that u prolongs to be on ∂ Ω with continuity.In order to complete the proof of (S), we are left to show that if f ∈ C ∞ (Ω , R ) ∩ C (Ω , R ) isnonnegative, then the unique solution u ∈ C (Ω , R ) of (cid:26) P u = − f on Ω (in the weak sense of distributions) u = 0 on ∂ Ω (point-wise) ARNACK INEQUALITY FOR SUBELLIPTIC OPERATORS 25 is nonnegative as well. From the hypoellipticity of P (see Remark 6.4), we already know that u ∈ C ∞ (Ω , R ) , and we can apply the WMP to − u (see Remark 6.3) to get − u ≤ .(II): f and ϕ smooth . We fix Ω as above, and f is in C ∞ (Ω , R ) ∩ C (Ω , R ) and ϕ is therestriction to ∂ Ω of some function Φ which is smooth and defined on an open neighborhood of Ω . As in Step (I), we consider the unique solution v ∈ C ∞ (Ω , R ) ∩ C (Ω , R ) of (cid:26) P v = − f − P Φ on Ω v = 0 on ∂ Ω ,and we observe that u = v + Φ is the (unique) classical solution of (cid:26) P u = − f on Ω u = Φ | ∂ Ω = ϕ on ∂ Ω .If furthermore f, ϕ ≥ , the nonnegativity of u is a consequence of the WMP as in Step (I).(III): f and ϕ continuous . Finally we consider f ∈ C (Ω , R ) and ϕ ∈ C ( ∂ Ω , R ) . By theStone-Weierstrass Theorem, there exist polynomial functions f n , ϕ n uniformly converging to f, ϕ respectively on Ω , ∂ Ω as n → ∞ . As in Step (II), for every n ∈ N we consider the uniqueclassical solution u n of (cid:26) P u n = − f n on Ω u n = ϕ n on ∂ Ω .From the fact that − c := max Ω P (1) < , we can argue as in Step (I), obtaining the estimate k u n − u m k C (Ω) ≤ max (cid:26) c k f n − f m k C (Ω) , k ϕ n − ϕ m k C ( ∂ Ω) (cid:27) . This proves that there exists the uniform limit u := lim n →∞ u n in C (Ω , R ) . Clearly one has: u = ϕ point-wise on ∂ Ω and P u = − f in the weak sense of distributions on Ω . From thehypoellipticity of P (Remark 6.4) we infer that f smooth implies u smooth. Finally, supposethat f, ϕ ≥ . By the Tietze Extension Theorem, we prolong f out of Ω to a continuous function F on R N ; we consider a mollifying sequence F n ∈ C ∞ ( R N , R ) uniformly converging to F on the compact sets of R N . Since mollification preserves the sign, the fact that F | Ω ≡ f ≥ on Ω gives that F n ≥ on Ω . As above in this Step, we solve the problem (cid:26) P U n = − F n on Ω U n = ϕ on ∂ Ω , with U n ∈ C ∞ (Ω , R ) ∩ C (Ω , R ) , and we get that U n uniformly converges on Ω to the unique continuous solution u of (cid:26) P u = − f in D ′ (Ω) u = ϕ on ∂ Ω .From the WMP for − U n (recalling that F n ≥ and ϕ ≥ ), we derive U n ≥ on Ω ; this gives u ( x ) = lim n →∞ U n ( x ) ≥ for all x ∈ Ω . This completes the proof. (cid:3) References [1] Abbondanza, B., Bonfiglioli, A.:
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