Non-divergence form parabolic equations associated with non-commuting vector fields: boundary behavior of nonnegative solutions

Non-divergence form parabolic equations associated with non-commuting vector fields:

boundary behavior of nonnegative solutions

MARIEFRENTZ, NICOLAGAROFALO, ELING ¨OTMARK, ISIDROMUNIVE ANDKAJNYSTROM¨

Abstract. In a cylinder!_T = !×(0,T) ⊂ Rⁿ₊⁺¹we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

H u=

!m i,j=1

a_{i j}(x,t)X_iX_ju−∂_tu=0, (x,t)∈Rⁿ₊⁺¹,

where X = {X1, . . . ,Xm} is a system ofC^∞ vector fields in Rⁿ satisfying H¨ormander’s rank condition (1.2), and!is a non-tangentially accessible domain with respect to the Carnot-Carath´eodory distancedinduced byX. Concerning the matrix-valued functionA= {a_{i j}}, we assume that it is real, symmetric and uni- formly positive definite. Furthermore, we suppose that its entriesa_{i j}are H¨older continuous with respect to the parabolic distance associated withd. Our main results are: 1) a backward Harnack inequality for nonnegative solutions vanish- ing on the lateral boundary (Theorem 1.1); 2) the H¨older continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operatorH(Theorem 1.3). These re- sults generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20, 39]. With one proviso: in those papers the authors assume that the coefficientsai j be only bounded and measur- able, whereas we assume H¨older continuity with respect to the intrinsic parabolic distance.

Mathematics Subject Classification (2010):31C05 (primary); 35C15, 65N99 (secondary).

1. Introduction

Let!⊂Rⁿ be a bounded domain and consider the cylinder!_T =!×(0,T)⊂ Rⁿ₊^{+1, whereT >}0 is fixed. In this paper we establish a number of results concern- Second author supported in part by NSF Grant DMS-07010001.

Fourth author supported in part by the second author’s NSF Grant DMS-07010001.

Received July 21, 2010; accepted January 13, 2011.

ing the boundary behavior of nonnegative solutions in!_Tof second order parabolic equations of the type

H u= Lu−∂_tu=

!m i,j=1

ai j(x,t)XiXju−∂_tu=0. (1.1) Here,X={X1, . . . ,Xm}is a system ofC^∞vector fields inRⁿsatisfying H¨ormander’s rank condition, see [25]:

rank Lie[X1, . . . ,Xm]≡n. (1.2) Concerning the m ×m matrix-valued function A(x,t) = {ai j(x,t)}we assume that it is symmetric, with bounded and measurable entries, and that there exists λ∈[1,∞)such that for every(x,t)∈R^{n+1, andξ} ∈R^m,

λ⁻¹|ξ|²≤

!m i,j=1

ai j(x,t)ξ_iξ_j ≤λ|ξ|². (1.3) Whenm =nand{X₁, . . . ,X_m} = {∂_x1, . . . ,∂_xn}, the operatorHin (1.1) coincides with that studied in [20, 39]. However, in contrast with these papers, in which the coefficients were assumed only bounded and measurable, we will also assume that the entries of the matrix A(x,t)are H¨older continuous with respect to the intrin- sic parabolic distance associated with the system X. More precisely, we indicate with d(x,y) the Carnot-Carath´eodory distance, between x,y ∈ Rⁿ, induced by {X1, . . . ,Xm}. We let

d_p(x,t,y,s)=(d(x,y)²+ |t−s|)^1/2

denote the parabolic distance associated with the metricd. Then, we assume that there existC>0, andσ ∈(0,1), such that for(x,t), (y,s)∈R^n+1,

|ai j(x,t)−ai j(y,s)|≤Cdp(x,t,y,s)^σ, i, j ∈{1, ..,m}. (1.4) The reason for imposing (1.4) will be discussed below.

Concerning the domain ! we will assume that ! is a NTA domain (non- tangentially accessible domain), with parameters M, r0, in the sense of [7, 11], see Definition 2.6 below. Under this assumption we can prove that all points on the parabolic boundary

∂_p!_T =S_T ∪(!×{0}), S_T =∂!×(0,T),

of the cylinder!_Tare regular for the Dirichlet problem for the operator H in (1.1).

In particular, for any f ∈ C(∂p!_T), there exists a unique Perron-Wiener-Brelot- Bauer solutionu=u^!_f^T ∈C(!_T)to the Dirichlet problem

H u=0 in!_T, u= f on∂_p!_T. (1.5)

Moreover, one can conclude that for every(x,t)∈!_T there exists a unique proba- bility measuredω^(x,t)on∂_p!_T such that

u(x,t)=

"

∂p!T

f(y,s)dω^(x,t)(y,s). (1.6)

Henceforth, we refer toω^(x,t)as theH-parabolic measurerelative to(x,t)and!_T. The metric ball centered atx ∈Rⁿwith radiusr >0 will be indicated with

Bd(x,r)= {y∈Rⁿ : d(x,y) <r}.

For(x,t)∈R^{n+1andr >0 we let}

C_r⁻(x,t)=Bd(x,r)×(t −r²,t), Cr(x,t)= Bd(x,r)×(t−r²,t +r²), and we define

'(x,t,r)=S_T ∩C_r(x,t). (1.7) By Definition 2.6 below, if !is a given NTA domain with parameters M andr0, then for anyx0 ∈ ∂!, 0 <r <r0, there exists a non-tangential corkscrew,i.e., a point Ar(x0)∈!, such that

M⁻¹r <d(x₀,A_r(x₀)) <r, and d(A_r(x₀),∂!)≥ M⁻¹r.

In the following we let A_r(x₀,t₀)= (A_r(x₀),t₀)whenever(x₀,t₀)∈ S_T and 0<

r < r0. When we say that a constantcdepends on the operator H we mean that c depends on the dimension n, the number of vector fields m, the vector fields {X1, . . . ,Xm}, the constant λin (1.3) and the parameters C,σ in (1.4). We let diam(!) = sup{d(x,y) | x,y ∈ !}denote the diameter of !. The following theorems represent the main results of this paper.

Theorem 1.1 (Backward Harnack inequality). Let u be a nonnegative solu- tion of H u = 0 in !_T vanishing continuously on ST. Let0 < δ + √

T be a fixed constant, let (x0,t0) ∈ ST, δ² ≤ t0 ≤ T − δ², and assume that r <

min{r0/2,#

(T−t0−δ²)/4,#

(t0−δ²)/4}. Then, there exists a constant c = c(H,M,r0,diam(!),T,δ), 1 ≤ c < ∞, such that for every (x,t) ∈ !_T ∩ Cr/4(x0,t0)one has

u(x,t)≤cu(Ar(x0,t0)).

Theorem 1.2 (Boundary H¨older continuity of quotients of solutions). Letu,vbe nonnegative solutions of H u = 0in!_T. Given(x0,t0) ∈ ST, assume thatr <

min{r₀/2,√

(T −t₀)/4,√

t₀/4}. Ifu, v vanish continuously on'(x₀,t₀,2r), then the quotientv/uis H¨older continuous on the closure of!_T∩C_r⁻(x₀,t₀).

Theorem 1.3 (Doubling property of the H-parabolic measure). Let K ≥ 100 and ν ∈ (0,1) be fixed constants. Let (x0,t0) ∈ ST, and suppose that r <

min{νr₀/2,√

(T−t₀)/4,√

t₀/4}.Then, there exists a constantc=c(H,M,ν,K,r₀), 1 ≤ c < ∞, such that for every (x,t) ∈ !_T, with d(x₀,x) ≤ K|t − t₀|^1/2, t −t0≥16r², one has

ω^(x,t)('(x0,t0,2r))≤cω^(x,t)('(x0,t0,r)).

Concerning Theorems 1.1, 1.2 and 1.3, we note that the study of the type of prob- lems considered in this paper has a long and rich history which, for uniformly parabolic equations inRⁿ⁺¹(i.e., when in (1.1) one hasm=nand{X1, . . . ,Xm} = {∂_x1, . . . ,∂_xn}), culminated with the celebrated papers of Fabes, Safonov and Yuan [19, 20, 39]. In these works the authors proved Theorem 1.1-1.3 for uniformly parabolic equations, both in divergence and non-divergence form, whose coeffi- cients are only bounded and measurable. We remark that, while these authors work in Lipschitz cylinders, one can easily see that their proofs can be generalized to the setting of bounded NTA domains in the sense of [26]. While the works [20, 39]

completed this line of research for parabolic operators in non-divergence form, prior contributions by other researchers are contained in [17, 21, 22, 29]. For the corre- sponding developments for second order parabolic operators in divergence form we refer to [16,19,34]. For the elliptic theory, for both operators in divergence and non- divergence form, we refer to [1, 6, 15, 26]. Finally, and for completion, we also note that second order elliptic and parabolic operators in divergence form with singular lower order terms were studied in [24, 27].

In the subelliptic setting of the present paper, i.e., when m < n and X = {X1, . . . ,Xm} is assumed to satisfy (1.2), much less is known. Several delicate new issues arise in connection with the intricate (sub-Riemannian) geometry as- sociated with the vector fields, and the interplay of such geometry with the so- called characteristic points on the boundary of the relevant domain. In addition, the derivatives along the vector fields do not commute, and the commutators are effectively derivatives of higher order. For all these aspects we refer the reader to the works [7, 10, 11, 13, 14, 30–32, 35], but this only represents a partial list of references.

In the stationary case, and for operators in divergence form, results similar to those in the present paper have been obtained in [7, 10, 11], see also [8, 9], whereas for parabolic operators in divergence form the reader is referred to the recent pa- per by one of us [33]. The methods in [33], however, extensively exploit the di- vergence structure of the operator and do not apply to the setting of the present paper.

We stress that for non-divergence form operators such as those treated in this paper, results such as Theorems 1.1-1.3 are new even for the case of stationary equations such as

Lu=

!m i,j=1

ai j(x)XiXju=0.

In view of these considerations our paper provides a novel contribution to the under- standing of the boundary behavior of solutions to parabolic equations arising from a system of non-commuting vector fields.

Concerning the proofs of Theorems 1.1-1.3 our approach is modeled on the ideas developed by Fabes, Safonov and Yuan in [20, 39]. In fact, the ideas in those papers have provided an important guiding line for our work. Yet, the arguments in [20,39] use mainly elementary principles like comparison principles, interior reg- ularity theory, the (interior) Harnack inequality, H¨older continuity type estimates and decay estimates at the lateral boundary, for solutions which vanish on a portion of the lateral boundary, as well as estimates for the Cauchy problem and the funda- mental solution associated to the operator at hand. In this connection it is important that the reader keep in mind that when the matrix A(x,t)= {ai j(x,t)}in (1.1) has entries which are just bounded and measurable, then most of these results presently represent in our settingterra incognita. More specifically, the counterparts of the Harnack inequality of Krylov and Safonov [29] and the Alexandrov-Bakel’man- Pucci type maximum principle due to Krylov [28] presently constitute fundamental open questions.

With this being said, our work uses heavily the recent important results of Bramanti, Brandolini, Lanconelli and Uguzzoni [5], see also [4], concerning the (interior) Harnack inequality, the Cauchy problem and the existence and Gaussian estimates for fundamental solutions for the non-divergence form operators H de- fined in (1.1). In fact, we assume (1.4) precisely in order to be able to use results from [5]. We want to stress, however, that we have strived throughout the whole paper to provide proofs which are “purely metrical”. By this we mean that, should the above mentioned counterpart of the results in [28, 29] become available, then our proofs would carry to the more general setting of bounded and measurable co- efficients in (1.1) with minor changes.

In closing we mention that the rest of the paper is organized as follows. Section 2 is of a preliminary nature. In it we collect some notation and results concerning basic underlying principles, and we also introduce the notion of NTA domains fol- lowing [7]. In Section 3 we prove a number of basic estimates concerning the boundary behavior of nonnegative solutions of (1.1). In addition we prove a num- ber of technical lemmas which allow us to present the proofs of Theorems 1.1-1.3 in a quite condensed manner. Finally, the proofs of Theorems 1.1 and 1.2 will be presented in Section 4, whereas that of Theorem 1.3 will be given in Section 5.

2. Preliminaries

In this section we introduce some notation and state a number of preliminary results for the operator H defined in (1.1). Specifically, we will discuss the Cauchy prob- lem and Gaussian estimates for the fundamental solution, the Harnack inequality and comparison principle, and the Dirichlet problem in bounded domains. In par- ticular, we also justify the notion of H-parabolic measure and introduce the notion of NTA domain.

2.1. Notation

InR^{n, withn} ≥3, we consider a system X = {X₁, . . . ,X_m}ofC^∞ vector fields satisfying H¨ormander’s rank condition (1.2). As in [18], a piecewise C¹ curve γ : [0,+]→Rⁿis called subunitary if at everyt ∈[0,+]at whichγ^.(t)exists one has for everyξ∈Rⁿ

<γ^.(t),ξ>²≤

!m j=1

< X_j(γ(t)),ξ >².

We note explicitly that the above inequality forcesγ^.(t)to belong to the span of {X1(γ(t)), . . . ,Xm(γ(t))}. The subunit length ofγ is by definitionls(γ)= +. If we fix an open set!⊂Rⁿ, then givenx,y ∈!, denote byS!(x,y)the collection of all subunitary γ : [0,+] → !which joinx to y. The accessibility theorem of Chow and Rashevsky, [12,37], states that, if!is connected, then for everyx,y∈! there existsγ ∈S!(x,y). As a consequence, if we define

d!(x,y)=inf{ls(γ)|γ ∈S!(x,y)},

we obtain a distance on!, called the Carnot-Carath´eodory distance, associated with the systemX. When!=R^{n, we writed(x,y)instead ofd}Rⁿ(x,y). It is clear that d(x,y)≤d!(x,y),x,y∈!, for every connected open set!⊂Rⁿ. In [36] it was proved that, given!⊂⊂Rⁿ, there existC,,>0 such that

C|x−y|≤d!(x,y)≤C⁻¹|x−y|^,, x,y∈!. (2.1) This givesd(x,y) ≤C⁻¹|x−y|^,,x,y∈!, and therefore

i :(R^n,| · |)→(R^n,d) is continuous.

Furthermore, it is easy to see that also the continuity of the opposite inclusion holds [23], and therefore the metric and the Euclidean topologies are equivalent.

For x ∈ R^{n andr > 0, we let B}d(x,r) = {y ∈ Rⁿ | d(x,y) < r}. The basic properties of these balls were established by Nagel, Stein and Wainger in their seminal paper [36]. These authors proved in particular that, given bounded open setU ⊂ Rⁿ, there exist constantsC,R0 > 0 such that, for anyx ∈ U, and 0<r ≤R0,

C≤ Bd(x,r)

-(x,r) ≤C⁻¹, where -(x,r) = $

I|aI(x)|r^dI is a polynomial function with continuous coeffi- cients. As a consequence, one has withC1>0,

|Bd(x,2r)|≤C1|Bd(x,r)| for every x ∈U and 0<r ≤ R0. (2.2)

In what follows, given β ∈ (0,1), we let/^β(!_T)denote the space of functions u:!_T →R^{such that}

||u||/^β(!T): =sup

!T

|u|

+ sup

(x,t),(x^.,t^.)∈!T, (x,t)/=(x^.,t^.)

|u(x,t)−u(x^.,t^.)| d_p(x,t,x^.,t^.)^β <∞.

(2.3)

We say thatu has a Lie derivative along Xj, at(x,t) ∈ !_T, ifu◦γ is differen- tiable at 0, whereγ is the integral curve ofXj such thatγ(0)=(x,t). Moreover, we indicate with/^2+β(!_T)the space of functionsu ∈ /^β(!_T)which admit Lie derivatives up to second order along X1, . . . ,Xm, and up to order one with respect to t, in /^β(!_T). Ifu ∈ /^2+β(!_T)then we let||u||_/²+β(!T) denote the naturally defined norm of u. Furthermore, u ∈ /_loc^β (!_T) ifu ∈ /^β(D) for any compact subset D of!_T. The space/_loc^2+β(!_T)is defined analogously. Finally, ifβ = 0 then we simply write/²(!_T)for/²⁺⁰(!_T). Throughout the paper we will use the following notation:

Cr(x,t)= Bd(x,r)×(t−r²,t+r²), C_r⁺(x,t)= Bd(x,r)×(t,t+r²), C_r⁻(x,t)= Bd(x,r)×(t−r²,t), Cr1,r2(x,t)= Bd(x,r1)×(t−r₂²,t +r₂²), C_r⁺_1,r2(x,t)= Bd(x,r1)×(t,t+r₂²), C_r⁻_1,r2(x,t)= Bd(x,r1)×(t−r₂²,t),

(2.4)

for(x,t)∈R^{n+1andr,r1,r2>} 0. Furthermore, if!⊂Rⁿ is a bounded domain, andT >0 andδ>0 are given, then we let

!^δ = {x ∈!|d(x,∂!) >δ}, !^δ_T =!^δ×(0,T). (2.5) 2.2. The Cauchy problem

Let H be defined as in (1.1), with the hypothesis (1.2), (1.3) and (1.4) in place.

These assumptions allow us to use some basic results established in [5]. In partic- ular, for what concerns the existence of a fundamental solution of the operator H, and Gaussian estimates, we will henceforth suppose, as it is done in [5], that the sub-Laplacian$m

i=1X²_i associated with X coincides with the standard Laplacian '=$_n

j=1∂_x²_j inRⁿoutside of a fixed compact set inR^n.

In [5] it is proved that, under such hypothesis, there exists a fundamental solu- tion,/, for H, with a number of important properties. In particular,/is a continu- ous function away from the diagonal ofRⁿ⁺¹×R^{n+1and/(x,t,ξ,τ)}=0 fort ≤

τ. Moreover,/(·,·,ξ,τ) ∈/²_loc^+α(Rⁿ⁺¹\ {(ξ,τ)})for every fixed(ξ,τ)∈ Rⁿ⁺¹ andH(/(·,·,ξ,τ))=0 inRⁿ⁺¹\ {(ξ,τ)}. For everyψ ∈C₀^∞(Rⁿ⁺¹⁾the function

w(x,t)=

"

Rⁿ⁺¹/(x,t,ξ,τ)ψ(ξ,τ)dξdτ

belongs to/²_loc^+α(Rⁿ⁺¹⁾and we haveHw=ψinRⁿ⁺¹. Furthermore, letµ≥0 and T2>T1be such that(T2−T1)µis small enough, let 0<β≤α, letg∈C^0,β(Rⁿ× [T1,T2])and f ∈C(Rⁿ⁾be such that|g(x,t)|,|f(x)|≤cexp(µd(x,0)²)for some constantc>0. Then, forx ∈R^n,t ∈(T1,T2], the function

u(x,t)=

"

Rⁿ

/(x,t,ξ,T1)f(ξ)dξ+

"t

T1

"

Rⁿ

/(x,t,ξ,τ)g(ξ,τ)dξdτ, (2.6)

belongs to the class/²_loc^+β(Rⁿ×(T1,T2))∩C(Rⁿ×[T1,T2]). Moreover,usolves the Cauchy problem

H u=ginR^{n×(T1,T2), u(·,T1)= f(·)in}R^{n. (2.7)} One also has the following Gaussian bounds.

Lemma 2.1. There exist a positive constant C and, for every T > 0, a positive constantc=c(T)such that, if0<t −τ ≤T,x,ξ ∈R^{n, then}

c⁻¹e^{−Cd(x,ξ)2/(t−τ)}

|B(x,√

t −τ)| ≤/(x,t,ξ,τ)≤ce^{−C−1d(x,ξ)2/(t−τ)}

|B(x,√

t−τ)| . (2.8) Furthermore, one also has

|Xi/(·,t,ξ,τ)(x)|≤c(t−τ)^−1/2e^{−C−1d(x,ξ)2/(t−τ)}

|B(x,√

t−τ)| , (2.9) and

|XiXj/(·,t,ξ,τ)(x)|+|∂_t/(x,·,ξ,τ)(t)|≤c(t−τ)⁻¹ce^{−C−1d(x,ξ)2/(t−τ)}

|B(x,√

t −τ)| . (2.10) 2.3. The Harnack inequality and strong maximum principle

We next state the Harnack inequality and the strong maximum principle for the operator H, see [4] and also [5].

Theorem 2.2. Let R > 0,0 < h1 < h2 < 1andγ ∈ (0,1). Then, there exists a positive constant C = C(h1,h2,γ,R)such that the following holds for every (ξ,τ)∈R^n+1,r ∈(0,R]. If

u∈/²(C_r⁻(ξ,τ))∩C(C_r⁻(ξ,τ)) satisfiesH u=0,u≥0, inC_r⁻(ξ,τ), then

u(x,t)≤Cu(ξ,τ)whenever(x,t)∈C_γ⁻_r,h2r(ξ,τ)\C_γr,h⁻ _1r(ξ,τ).

Theorem 2.3. Let!⊂Rⁿ be a connected, bounded open set, and letT > 0. Let u ∈/²(!_T)and assume thatLu≥0,u≤0in!_T. Assume thatu(x0,t0)=0for some(x0,t0)∈!_T. Thenu(x,t)≡0whenever(x,t)∈!_T∩{t :t ≤t0}.

2.4. The Dirichlet problem

In the following we let D be any bounded open subset ofRⁿ⁺¹ and we study the Dirichlet problem

H u=0 inD,u= f on∂_pD, (2.11)

with f ∈C(∂pD). Here,∂_pDdenotes the parabolic boundary ofD. Ifu: D→R is a smooth function satisfyingH u =0 inD, then we say thatuisH-parabolic in D. We denote byP(D)the linear space of functions which are H-parabolic inD.

We say that D is H-regular if for any f ∈ C(∂pD) there exists a unique function H^D_f ∈ P(D) such that lim(x,t)→(x0,t0)H^D_f (x,t) = f(x0,t0) for every (x₀,t₀) ∈ ∂_pD. Following the arguments in [30], see in particular Theorems 6.5 and 10.1, we can easily construct a basis for the Euclidean topology ofR^n+1which is made of cylindrical H-regular sets. Furthermore, ifDisH-regular, then in view of Theorem 2.3 (one actually only needs the weak maximum principle) for every fixed (x,t) ∈ D the map f 1→ H_f^D(x,t)defines a positive linear functional on C(∂pD). By the Riesz representation theorem there exists a unique Borel measure ω=ω_D, supported in∂_pD, such that

H^D_f (x,t)=

"

∂pD

f(y,s)dω^(x,t)(y,s), for every f ∈C(∂_pD). (2.12) We will refer toω^(x,t)=ω^(x_D^,t)as theH-parabolic measurerelative toDand(x,t).

A lower semi-continuous function u : D →] − ∞,∞] is said to be H- superparabolic inDifu <∞in a dense subset ofDand if

u(x,t)≥

"

∂Vu(y,s)dω^(x,t_V ⁾(y,s),

for every open H-regular setV ⊂ V ⊂ D and for every(x,t) ∈ V. We denote by S(D)the set ofH-superparabolic functions in D, and byS⁺(D)the set of the

functions in S(D)which are nonnegative. A functionv : D → [−∞,∞[is said to be H-subparabolic in D if−v ∈ S(D) and we write S(D) := −S(D). As the collection of H-regular sets is a basis for the Euclidean topology, it follows that S(D)∩S(D) = P(D). Finally, we recall that H^D_f can be realized as the generalized solution in the sense of Perron-Wiener-Brelot-Bauer to the problem in (2.11). In particular,

inf U^Df =supU^Df = H^D_f , (2.13) where we have indicated with U^Df the collection of allu∈ S(D)such that infDu>

−∞, and

lim inf

(x,t)→(x0,t0)u(x,t)≥ f(x0,t0),∀(x0,t0)∈∂_pD, and withU^Df the collection of allu∈S(D)for which sup_Du<∞, and

lim sup

(x,t)→(x0,t0)

u(x,t)≤ f(x0,t0), ∀(x0,t0)∈∂_pD.

Lemma 2.4. Let D ⊂Rⁿ⁺¹be a bounded open set, let f ∈C(∂pD)and letube the generalized Perron-Wiener-Brelot-Bauer solution to the problem in(2.11), i.e., u= H^D_f whereH_f^Dbe defined as in(2.13). Thenu∈/²(D).

Proof. This follows from Theorem 1.1 in [40].

In the following we are concerned with the issue of regular boundary points and we note, concerning the solvability of the Dirichlet problem for the operator H, that in [40] Uguzzoni developes what he refers to as a “cone criterion” for non- divergence equations modeled on H¨ormander vector fields. This is a generalization of the well-known positive density condition of classical potential theory. We next describe his result in the setting of domains of the form!_T =!×(0,T), where

!⊂Rⁿ is assumed to be a bounded domain. In [40] a bounded open set!is said to haveouter positived-density atx0∈∂!if there existr0,θ>0 such that

|Bd(x0,r)\ ¯!|≥θ|Bd(x0,r)|, for allr ∈(0,r0). (2.14) Furthermore, ifr0andθ can be chosen independently ofx0 then one says that! satisfies the outer positive d-density condition. The following lemma is a special case of [40, Theorem 4.1].

Lemma 2.5. Assume that!satisfies the outer positived-density condition. Given f ∈ C(∂_p!_T)and g ∈ /^β(!_T) for some0 < β ≤ σ, whereσ is the H¨older exponent in(1.4), there exists a unique solutionu ∈/^2+β(!_T)∩C(!_T ∪∂_p!_T) to the problem

H u=gin!_T, u= f on∂_p!_T. In particular,!_T isH-regular for the Dirichlet problem(2.11).

2.5. NTA domains

In this section we recall the notion of NTA domain with respect to the control dis- tance d(x,y)induced by the system X = {X₁, . . . ,X_m}. We recall that, when d(x,y) = |x− y|, the notion of NTA domain was introduced in [26] in connec- tion with the study of the boundary behavior of nonnegative harmonic functions.

The first study of NTA domains in a sub-Riemannian context was conducted in [7], where a large effort was devoted to the nontrivial question of the construction of examples. In that paper the relevant Fatou theory was also developed and, in par- ticular, the doubling condition for harmonic measure, and the comparison theorem for quotients of nonnegative solutions of sub-Laplacians. Subsequently, in the pa- pers [10, 11] the notion of NTA domain was combined with an intrinsic outer ball condition to obtain the complete solvability of the Dirichlet problem.

Given a bounded open set! ⊂ Rⁿ, we recall that a ball Bd(x,r)is M-non- tangential in!(with respect to the metricd) if

M⁻¹r <d(Bd(x,r),∂!) < Mr.

Furthermore, given x,y ∈ ! a sequence of M-non-tangential balls in !, Bd(x1,r1),. . . , Bd(xp,rp), is called a Harnack chain of length p joining x to y if x ∈ Bd(x1,r1), y ∈ Bd(xp,rp), and Bd(xi,ri)∩ Bd(xi+1,ri+1) /= ∅ ^for i ∈ {1, . . . ,p−1}. We note that in this definition consecutive balls have com- parable radii.

Definition 2.6. We say that a connected, bounded open set ! ⊂ R^{n is a non-} tangentially accessible domainwith respect to the systemX= {X₁, . . . ,X_m}(NTA domain, hereafter) if there existM,r₀>0 for which:

(i) (Interior corkscrew condition) For any x0 ∈ ∂! and r ≤ r0 there exists Ar(x0) ∈ ! such that M⁻¹r < d(Ar(x0),x0) ≤ r andd(Ar(x0),∂!) >

M⁻¹r. (This implies thatBd(Ar(x0), (2M)⁻¹r)is(3M)-nontangential.) (ii) (Exterior corkscrew condition)!^c=Rⁿ\!satisfies property (i).

(iii) (Harnack chain condition) There existsC(M) > 0 such that for any, > 0 andx,y∈!such thatd(x,∂!) >,,d(y,∂!) >,, andd(x,y) <C,, there exists a Harnack chain joiningxtoywhose length depends onCbut not on,.

We observe that the Chow-Rashevski accessibility theorem implies that the metric space(R^n,d)be locally compact, see [23]. Furthermore, for any bounded set!⊂ Rⁿ there exists R0 = R0(!) > 0 such that the closure of balls B(x0,R) with x0 ∈!and 0< R< R0are compact. We stress that metric balls of large radii fail to be compact in general, see [23]. In view of these observations, for a given NTA domain!⊂Rⁿwith constantMandr0we will always assume, following [7], that the constantr0has been adjusted in such a way that the closure of balls B(x0,R), withx0∈!and 0< R<r0, be compact.

We note the following lemma which will prove useful in the sequel and which follows directly from Lemma 2.5 and Definition 2.6. In its statement the numberσ denotes the H¨older exponent in (1.4).

Lemma 2.7. Let! ⊂ R^{n beNTA}domain, then there exist constantsC,R1, de- pending on the NTA parameters of !, such that for every y ∈ ∂! and every 0<r < R₁one has,

C|Bd(y,r)|≤min{|!∩Bd(y,r)|,|!^c∩Bd(y,r)|}≤C⁻¹|Bd(y,r)|. In particular, everyNTAdomain has outer positived-density and therefore, in view of Lemma2.5, given f ∈C(∂p!_T), there exists a unique solutionu∈/^2+σ(!_T)∩ C(!T∪∂_p!_T)to the Dirichlet problem(2.11). In particular,!_T is H-regular.

Assume that!⊂ Rⁿ is a non-tangentially accessible domain with respect to the system X = {X₁, . . . ,X_m}and with parametersM,r₀. LetT > 0 and define

!_T = !×(0,T). Based on Definition 2.6, for every(x₀,t₀) ∈ ST, 0< r <r₀, we introduce the following points of reference whenever

A_r⁺(x0,t0)=(Ar(x0),t0+2r²), A_r⁻(x0,t0)=(Ar(x0),t0−2r²),

Ar(x0,t0)=(Ar(x0),t0).

(2.15)

We note here that according to [30, Lemma 6.4],Rⁿ\Bd(x0,R)satisfies condition (ii) in Definition 2.6, and thus it also satisfies the uniform outer positived-density condition, and one can solve the Dirichlet problem there. Also note that the same is true of the intersection of two sets that satisfy condition (ii) in Definition 2.6. This is used to prove the following lemma ( [30, Theorem 6.5]) which states that one can approximate any bounded open set with a set where one can solve the Dirichlet problem (2.11).

Lemma 2.8. Let D ⊂ Rⁿ be a bounded open set. Then, for everyδ > 0 there exists a set Dδ such that{x ∈ D: d(x,∂D) >δ}⊂ Dδ ⊂ D, and Dδsatisfies the uniform outer positived-density condition.

To apply the Harnack inequality to the equation (1.1) in a cylinder !_T, we will need to connect two points of!_T with a suitable Harnack chain of parabolic cylinders. We thus introduce the relevant geometric definition.

Definition 2.9. Let (y1,s1), (y2,s2) ∈ !_T, withs2 > s1 . Suppose that (s2− s1)^1/2 ≥η⁻¹d(y1,y2)for someη > 1, and thatd(y1,∂!) > ,,d(y2,∂!) > ,, (T −s₂) > ,²,s₁ > ,² andd_p((y₁,s₁), (y₂,s₂)) < c, for some, > 0. We say that{Cr_ˆi,ρˆi(yˆi,sˆi)}⁺_i=1is aparabolic Harnack chain of length+connecting(y1,s1) to(y2,s2),ifrˆi,ρˆ_i,yˆi,sˆi satisfy the following:

(i) c(η)⁻¹≤ ^ρ_rˆ^ˆ_iⁱ ≤c(η)fori =1,2, . . . ,+, (ii) sˆi+1−sˆi ≥c(η)⁻¹rˆ_i²,fori =1,2, . . . ,+−1,

(iii) Bd(yˆi,rˆi)is M-nontangential in!fori =1,2, . . . ,+, (iv) (y1,s1)∈C_rˆ1,ρˆ1(yˆ1,sˆ1), (y2,s2)∈C_rˆ+,ρˆ+(yˆ+,sˆ+),

(v) C_rˆi+1,ρˆ_i+1(yˆi+1,sˆi+1)∩C_rˆi,ρˆi(yˆi,sˆi)/=∅^fori =1,2, . . . ,+−1.

Lemma 2.10. Let!⊂R^{nbe a}NTA-domain. GivenT >0and(y1,s1),(y2,s2)∈

!_T, suppose that s2 > s1, (s2 −s1)^1/2 ≥ η⁻¹d(y1,y2) for some η > 1, that d(y₁,∂!) >,,d(y₂,∂!) >,,(T−s₂) >,²,s₁>,²and thatd_p((y₁,s₁), (y₂,s₂)) <

c,for some,>0. Then, there exists a parabolic Harnack chain{Cr_ˆi,ρˆi(yˆi,sˆi)}⁺_i=1, connecting(y1,s1)to(y2,s2)in the sense of Definition2.9. Furthermore, the length +of the chain can be chosen to depend only onηandc, but not on,.

Proof. Since!is a NTA domain and sinced(y1,∂!) >,,d(y2,∂!) >,, d(y₁,y₂)≤d_p((y₁,s₁), (y₂,s₂)) <c,,

it follows that we can use Definition 2.6 to conclude the existence of a Harnack chain of length+ˆ = ˆ+(c),{Bd(yˆi,rˆi)}⁺_i^ˆ₌₁, connecting y1and y2. In the following we letβbe a degree of freedom to be fixed below. Usingβwe defineρˆ_i =βrˆi, we letsˆi =s1+¹_β$_i

j=1rˆ²_j fori ∈{1, ..,+ˆ}, and we consider the sequence of cylinders {Cr_ˆi,ρˆi(yˆi,sˆi)}⁺_i^ˆ₌₁.

If we now choose β > 1, and if we assume thatβ is chosen as a function ofη, then (i), (ii), (iii), (v) and the first part of (iv) in Definition 2.9 are satisfied. In particular, it only remains to ensure that the second part of (iv) in Definition 2.9 is satisfied. To do this we first note that we can assume, without loss of generality, that

ˆ

ri ≤d(y1,y2)for alli ∈{1, . . . ,+ˆ}. Hence,$lˆ

1rˆ_i²≤+ˆ·d(y1,y2)². Furthermore, sinced(y₁,y₂)²≤η²(s₂−s₁), we have

ˆ

s_+ˆ−s1= 1 β

ˆ

!+ 1

ˆ r²_j ≤ +ˆ

βd(y1,y2)²≤ +ˆ

βη²(s2−s1). (2.16) We now letβ = ˆ+·η²and we can conclude thatsˆ_+ˆ ≤s2. Ifsˆ_+ˆ =s2we are done.

Otherwise, we only step up in time with cylindersCj = {Cr_ˆ+ˆ,rˆ_+ˆ(y2,sˆ_+ˆ+ jrˆ_+ˆ)}until we reach (y2,s2). The time that is left depends onη, and, in particular, we have thats2−sˆ_+ˆ ≤ c²,². Furthermore, sincerˆ+ ≤ c,, the number of steps we need to reach(y2,s2)only depends onc. In particular, it is clear that the length of the entire parabolic Harnack chain only depends oncandη.

Lemma 2.11. Let u be a nonnegative solution to the equation H u = 0 in !_T. Furthermore, let (y1,s1), (y2,s2) ∈ !_T, suppose that s2 > s1, (s2 −s1)^1/2 ≥ η⁻¹d(y1,y2)for someη>1, thatd(y1,∂!) >,,d(y2,∂!) >,,(T−s2) >,², s1 > ,²and thatdp((y1,s1), (y2,s2)) < c, for some, > 0. Then, there exists a constantcˆ= ˆc(H,η,c,r0),1≤cˆ<∞, such that

u(y1,s1)≤cu(yˆ 2,s2).

Proof. To prove the lemma we simply use the parabolic Harnack chain from Lem- ma 2.10 and apply Theorem 2.2 in each cylinder. Note that the dependence of constant cˆ on r₀ enters through the size parameter R in the statement of Theo- rem 2.2.