A Carleson-type estimate in Lipschitz type domains for non-negative solutions to Kolmogorov operators

A Carleson-type estimate in Lipschitz type domains for non-negative solutions to Kolmogorov operators

CHIARACINTI, KAJNYSTROM AND¨ SERGIO POLIDORO

Abstract. We prove a Carleson type estimate, in Lipschitz type domains, for non-negative solutions to a class of second order degenerate differential operators of Kolmogorov type of the form

L = Xm i,j=1

ai,j(z)@xixj+ Xm i=1

ai(z)@xi+ XN i,j=1

bi,jxi@_xj @_t,

wherez =(x,t) 2 R^N+1, 1  m  N. Our estimate is scale-invariant and generalizes previous results valid for second order uniformly parabolic equations to the class of operators considered.

Mathematics Subject Classification (2010):35K65 (primary); 35K70, 35H20, 35R03 (secondary).

1. Introduction

In this paper we prove a Carleson type estimate for non-negative solutions to a gen- eral class of hypoelliptic ultraparabolic operators. Specifically, we consider Kol- mogorov operators of the form

L ₌ Xm i,j=1

ai,j(z)@xixj + Xm i=1

ai(z)@xi + XN i,j=1

bi,jxi@_xj @_t, (1.1) wherez =(x,t)2R^N ⇥R^{, 1}m  N, the coefficientsai,j andai are bounded continuous functions and B = (bi,j)_i,j=1,...,N is a matrix of real constants. The main motivation for our research is our long-term goal to establish a regularity theory for the free boundaries occurring in the obstacle problem

(max Lu,' u =0, inR^N⇥]0,T[, u(x,0)='(x,0), for anyx 2R^N,

Received March 14, 2011; accepted August 22, 2011.

considered by Di Francesco, Pascucci and Polidoro in [10]. In [21] and [32] Frentz, Nystr¨om, Pascucci and Polidoro proved optimal regularity properties of the solu- tion to the obstacle problem, for smooth as well as non-smooth obstacles. Up to date, the only known results about boundary regularity for non-negative solutions to Kolmogorov operators have been proved by the authors in [6], see also [7].

Our long-term goal is to establish a boundary regularity theory for Kolmogorov operators. Estimates concerning boundary behaviour are of obvious independent interest from the theoretical point of view. Furthermore, the regularity properties of the Kolmogorov equations onR^N+1depend strongly on a geometric Lie group structure. The extension of the methods used in the Euclidean setting to the Lie group geometry related to Kolmogorov equations is far from trivial. For results con- cerning the boundary behaviour of non-negative solutions and obstacle problems in different, but still subelliptic settings, we refer to the works by Frentz, Garofalo, G¨otmark, Munive and Nystr¨om [20], Capogna, Garofalo and Nhieu [4, 5], Franchi and Ferrari [18, 19], Danielli, Garofalo and Salsa [9], Danielli, Garofalo and Pet- rosyan [8].

We next list our structural assumptions on the operatorL defined in (1.1).

[H.1] The matrix A0(z)= (ai,j(z))i,j=1,...,mis symmetric and uniformly positive definite inR^m: there exists a positive constant such that

1|⇠|² Xm i,j=1

a_i,j(z)⇠_i⇠_j  |⇠|², 8⇠2R^{m, z}2R^N+1. The matrix B=(bi,j)_i,j=1,...,N has real constant entries.

[H.2] Theconstant coefficientsoperator K ₌

Xm i,j=1

ai,j@_xixj + XN i,j=1

bi,jxi@_xj @_t (1.2) is hypoelliptic,i.e.every distributional solution ofKu= f is a smooth classical solution, whenever f is smooth. Here A0 = (ai,j)_i,j=1,...,m is any constant, symmetric and strictly positive matrix.

[H.3] The coefficients ai,j(z) and ai(z) are bounded functions belonging to the H¨older spaceC^0,↵_K (R^N+1),↵2]0,1], defined in (2.6) below.

Note that we can choose A0in[H.2]as them⇥midentity matrix. In our setting, K plays the same role as the heat operator does in the framework of uniformly parabolic pdes. We also note that the operatorK can be written as

K ₌ Xm i=1

X²_i +Y, where

Xi = Xm

j=1

a¯i,j@_xj, i =1, . . . ,m, Y =hx,Bri @_t, (1.3)

and wherea¯i,j’s are the entries of the unique positive matrixA¯0such thatA0= ¯A²₀. Recall that the hypothesis[H.2]is equivalent to the H¨ormander condition [24],

rank Lie(X₁, . . . ,X_m,Y) (z)= N+1, 8z2R^{N+1. (1.4)} It is well-known that the natural framework to study operators satisfying a H¨orman- der condition is the analysis on Lie groups. The relevant Lie group related to the operatorK in (1.2) is defined using the group law

(x,t) (⇠,⌧)=(⇠+exp( ⌧B^T)x,t+⌧), (x,t), (⇠,⌧)2R^{N+1. (1.5)} In particular, the vector fields X1, . . . ,XmandY are left-invariant, with respect to the group law (1.5), in the sense that

X_j(u(⇣ ·))=(X_ju)(⇣ ·), j =1, . . . ,m, Y(u(⇣ ·))=(Y u)(⇣ ·), (1.6) for every⇣ 2R^N+1(henceK(u(⇣ ·))= Ku (⇣ ·)). It is also known that [H.2]is equivalent to the following structural assumption onB[28]:there exists a basis forR^N such that the matrixBhas the form

0 BB BB

@

⇤ B1 0 · · · 0

⇤ ⇤ B2 · · · 0 ... ... ... . .. ...

⇤ ⇤ ⇤ · · · B

⇤ ⇤ ⇤ · · · ⇤ 1 CC CC A

(1.7)

where Bj is amj 1⇥mj matrix of rankmj for j 2{1, . . . ,},1m . . .  m1m0=mandm+m1+. . .+m = N, while⇤represents arbitrary matrices with constant entries. Based on (1.7), we introduce the family of dilations( _r)_r>0 onR^{N+1defined by}

r =(D_r,r²)=diag(r I_m,r³I_m1, . . . ,r^2+1I_m,r²), (1.8) where Ik,k 2N^{, is the}k-dimensional unit matrix. To simplify our presentation, we will also assume the following technical condition.

[H.4] The operatorK in (1.2) is r-homogeneous of degree two,i.e.

K _{r =}r²( _r K), 8r >0.

We explicitly remark that[H.4]is satisfied if (and only if) all the blocks denoted by

⇤in (1.7) are null (see [28]). Moreover we set

q=m+3m1+. . .+(2+1)m,

and we say thatq+2 is thehomogeneous dimensionofR^N+1with respect to the dilations group( _r)_r>0.

Consider the boundary value problem (Lu=0 in,

u=' in@, (1.9)

where is any open subset ofR^{N+1 and'} 2C(@). Using the Perron-Wiener- Brelot method, the existence of a solution to this problem can be established. How- ever, it is well known that the boundary value'is not necessarily attained at every point of@. In the sequel,u'will denote the solution to (1.9), and we set

@_K=n

z2@| lim

w!zu_'(w)='(z)for any'2C(@)o

. (1.10)

We refer to@_Kas theregular boundaryofwith respect to the operatorL. We recall that Manfredini in [30, Proposition 6.1] gives sufficient conditions for the regularity of the boundary points. Recall that a vector⌫2R^{N+1is an}outer normal toatz 2 @if there exists a positiver such that B(z+r⌫,r)\ = ;. Here B(z+r⌫,r)denotes the Euclidean ball inR^N+1with center atz+r⌫ and radius r. Then, in consistency with Fichera’s classification, sufficient conditions for the regularity are expressed in geometric terms, and read as follows. If z 2 @and

⌫ =(⌫₁, ...,⌫_N+1)is an outer normal toatz, then the following holds.

(a)If(⌫₁, . . . ,⌫_m)6=0, thenz2@_K,

(b)if(⌫₁, . . . ,⌫_m)=0 andhY(z),⌫i>0, thenz2@_K, (c)if(⌫₁, . . . ,⌫_m)=0 andhY(z),⌫i<0, thenz62@_K,

(1.11)

whereYis the vector field defined in (1.3). As an example, we consider the simplest Kolmogorov operator@_x²₁+x1@x2 @_t in the set=] 1,1[⇥] 1,1[⇥] 1,0[⇢

R^{3. ThenY(x,t)}=(0,x1, 1), and it is easy to see that

(a)is satisfied by all the points of the set{ 1,1}⇥] 1,1[⇥] 1,0[, (b)is satisfied by all the points of the set] 1,1[⇥] 1,1[⇥{ 1}[

[]0,1[⇥{1}⇥] 1,0[[] 1,0[⇥{ 1}⇥] 1,0[, (c)is satisfied by all the points of the set] 1,1[⇥] 1,1[⇥{0}[

[]0,1[⇥{ 1}⇥] 1,0[[] 1,0[⇥{1}⇥] 1,0[. Condition(a)can be equivalently expressed in terms of the vector fields Xj’s as follows: hXj(z),⌫i 6= 0 for some j = 1, . . . ,m. If this condition holds, in the literaturezis often referred to asa non characteristic point for the operatorL. A more refined sufficient condition for the regularity of the boundary points of@is given in [30, Theorem 6.3] in terms of anexterior cone conditionand that condition will be used here (see Definition 4.1 below).

In [6, Theorem 1.2] we proved a Carleson type estimate for non-negative solu- tionsutoLu=0 assuming thatuvanishes continuously on some open subset6

of@. Furthermore, in [6, Proposition 1.4] we gave an application of [6, Theorem 1.2], assuming that6 is aN-dimensionalC¹-manifold satisfying either condition (a) or condition(b)in (1.11). The purpose of this paper is to improve on these results by relaxing the regularity assumptions on6. Specifically, in this paper we assume that 6is locally a LipK surface, where LipK is a notion of Lipschitz reg- ularity suitably defined in terms of the dilations (1.8) and introduced in the bulk of the paper. We also improve on [6, Theorem 1.2] since our main result, which is stated below, is scale invariantin the sense that (ex,et),C andcdo not depend onr 2]0,r0]. In the following we refer to Definition 2.1 for the meaning of LipK

surfaces, and to (2.9), (2.10) for the definition of the cubeQM,r(x0,t0).

The main result proved in this paper reads as follows.

Theorem 1.1. Let be an open subset ofR^{N+1, and let6} ⇢ @be a compact LipK surface with constantsMandr0. Then there exist two positive constantsC,c, depending only onL andM, such that

sup

QM,cr(x0,t0)\

u(x,t)C u A_r⁺(x₀,t₀) , A⁺_r(x₀,t₀)=(x₀,t₀) _r(ex,et),

for any(x0,t0)26, for every non-negative solutionutoLu =0invanishing continuously on Q_M,r(x₀,t₀)\@, and for everyr 2]0,r₀]. Here(ex,et)2R^N+1 is such that

k(ex,et)kK C, dK A⁺_r(x0,t0),@ cr, for everyr 2]0,r0[.

To put Theorem 1.1 in the context of the existing literature devoted to boundary estimates for second order elliptic and parabolic operators, we here briefly discuss previous results valid for uniformly parabolic operators of the form

L= XN i,j=1

ai j(x,t)@_xixj @_t, (x,t)2R^{N+1, (1.12)} where the matrix(a_i,j(x,t))satisfies[H.1]withm = N. The study of the bound- ary behavior of non-negative solutions to non-divergence form uniformly parabolic equations Lu = 0, as well as the associatedL-parabolic measure, has a long his- tory. We quote the papers by Fabes and Kenig [14], Fabes and Stroock [17], Garo- falo [22], Krylov and Safonov [27], leading up to the results of Fabes, Safonov and Yuan in [16] and [33]. The corresponding developments for second order parabolic operators in divergence form are treated by Fabes, Garofalo and Salsa [13], Fabes and Safonov [15], Nystr¨om [31]. We refer to Bauman [1], Caffarelli, Fabes, Mor- tola and Salsa [2], Fabes, Garofalo, Marin-Malave and Salsa [12], and Jerison and Kenig [25] for both divergence and non-divergence form elliptic operators. Finally, we also note that second order elliptic and parabolic operators in divergence form with singular lower order terms have been studied by Kenig and Pipher [26] and by

Hofmann and Lewis [23]. Today the boundary regularity theory in the setting of uniformly elliptic and parabolic operators has reached a quite advanced level.

A natural geometrical setting for the uniformly parabolic operators in (1.12) is that of Lip(1,1/2)domains which we next introduce. We fix a j 2 1, . . . ,N , we letx⁰ =xj and we letx⁰⁰denote the remainingN 1 coordinates of anyx 2R^N. We also lete⁰ be the j-th vector of the canonical basis of R^N, and we say that a function f :R^{N 1}⇥R!R^{is Lip(1,}1/2), with respect toxj, if

|f(x⁰⁰,t) f(⇠⁰⁰,⌧)|M |x⁰⁰ ⇠⁰⁰| + |t ⌧|¹² , (1.13) for any(x⁰⁰,t), (⇠⁰⁰,⌧) 2 R^{N 1}⇥R, and for some positive constant M. For any (x,t)2R^{N+1andr > 0 we setCr(x,t)}= B(x,r)⇥]t r²,t +r²[. LetT be a positive number, let⇢R^N+1be a bounded domain. Let_T =\(R^N⇥[0,T]), ST =@\(R^N⇥]0,T[), and1(x,t,r)=ST\Cr(x,t). Letz =(x⁰,x⁰⁰,t)2ST, with x⁰ = xj for some j 2 {1, . . . ,N}. If there existrz > 0 and a Lip(1,1/2) function f with constantMz, such that

_T\Crz(z)= (⇠,⌧)2R^N⇥[0,T] |⇠⁰> f(⇠⁰⁰,⌧) \Crz(z),

ST\Crz(z)= (⇠,⌧)2R^N⇥[0,T] |⇠⁰= f(⇠⁰⁰,⌧) \Crz(z), (1.14) then we say thatST\Crz(z)is Lip(1,1/2)surface, with constantMz. IfSTis covered by a finite set of cylinders C_rzi(z_i) , withz_i2S_T,r_zi>0, we setM=max_i M_zi and r₀=min_ir_zi, and we say that_T is a Lip(1,1/2)domain with constants Mandr₀.

Let _T be a Lip(1,1/2) domain with constants M andr₀, let(x₀,t₀)be any point on ST. For every r 2]0,r0[ with t0 +r² < T, let A⁺_r(x0,t0) = (x0+ 2Mre⁰,t0+r²)2. Then

M ¹r d_P A⁺_r(x₀,t₀),S_T , and d_P A⁺_r(x₀,t₀), (x₀,t₀) <r,

where dP denotes the standard parabolic distance function, dP((x,t), (y,s)) =

|x y| + |t s|¹² for any(x,t), (y,s)2R^N+1.

The following theorem is essentially due to Salsa, see [34, Theorem 3.1].

Let_T ⇢R^{N+1be aLip(1,1/2)}domain with constantsM andr0, let(x0,t0)2ST

and assume that r < min{r0/2,p

(T t0)/4,p

t0/4}. Let u be a non-negative solution toLu=0in_T\C2r(x0,t0)and assume thatuvanishes continuously on 1(x0,t0,2r). Then there exists a constantc=c(L,M,r0),1c<1, such that

u(x,t)c u(A_r⁺(x0,t0)) whenever(x,t)2_T \Cr/c(x0,t0).

While Salsa proved this theorem in the setting of time-independent Lipschitz cylinders, the proof goes through essentially unchanged in the more general set- ting of Lip(1,1/2) domains. This estimate is often referred to as a Carleson-type

estimate, since this type of estimate first occurs in a paper by Carleson [3] on Fatou-type theorems for harmonic functions. Note that Theorem 1.1 is a gener- alization of this theorem. Indeed, in the Euclidean case we have(x₀,t₀) _r(ex,et)= (x₀+rex,t₀+r²et), and we can choose(ex,et)=(2Me⁰,1)to recover the statement above. Furthermore, in the Euclidean case the notion of LipK surfaces introduced in Definition 2.1 below coincides with the notion of Lip(1,1/2)surfaces.

Returning to operators of Kolmogorov typeL, we note that if6is a smooth subset of @ which is non-characteristic with respect to L according to (a) in Fichera’s classification (1.11), then6 is LipK. If we require less regularity on6, we may consider a surface 6 ⇢ @which locally agrees with the graph{x_j = f(x⁰⁰,t)}for some Lip(1,1/2)function f. In Proposition 2.2 below we prove that in this case 6 is also a Lip_K surface. Hence Proposition 2.2 provides us with a simple sufficient condition for the LipK regularity. We also note that sinceL may be strongly degenerate, it is possible that a wide part6of the boundary of a given open setis characteristic and in this case(b)in (1.11) is of interest. Indeed, when (b)is satisfied in some neighborhoodW\6of a pointz02@, thenW\6can be characterized ast =g(x), wheregdoes not depend on(x1, . . . ,xm). In this case, we in Proposition 2.4 below give a statement analogous to Theorem 1.1. Note that, in the case of uniformly parabolic operators, it turns out that(b)is satisfied only by the lower basis of a cylinder.

The rest of the paper is organized as follows. Section 2 contains the definitions of LipK functions and LipK surfaces, and the statement of the key propositions es- tablished in order to prove our main result, Theorem 1.1. Section 3 is of preliminary nature and we here collect some notions and we state a few results concerning the interior Harnack inequality. In Section 4 we prove a few basic boundary estimates for non-negative solutions toLu =0 near LipK surfaces. Section 5 is devoted to the proof of Theorem 1.1, and to the proof of the propositions stated in Section 2.

ACKNOWLEDGEMENTS. We thank E. Lanconelli, F. Ferrari and S. Salsa for their interest in our work.

2. LipK surfaces and statement of key propositions

In the sequel we will write the dilation (1.8) in the following form

r =diag(r^↵1, . . . ,r^↵N,r²) (2.1) where we set↵₁=. . .=↵_m=1, and↵_{m+m1+···+mj 1+1}=. . .=↵_{m+m1+···+mj+1}= 2j+1 for j =1, . . . ,. According to (1.8), we split the coordinatex 2R^{N as}

x = x⁽⁰⁾,x⁽¹⁾, . . . ,x^() , x⁽⁰⁾2R^{m, x(j)}2R^{mj, j} 2{1, . . . ,}, (2.2)

and we define

|x|K = X

j=0

x^{(j) 2j1+1}, k(x,t)kK = |x|K+ |t|¹².

Note thatk rzkK =rkzkK for everyr >0 andz2R^N+1. We recall the following pseudo-triangular inequality: there exists a positive constantcsuch that

kz ¹kK ckzkK, kz ⇣kK c(kzkK +k⇣kK), z,⇣ 2R^{N+1. (2.3)} We also define thequasi-distancedK by setting

dK(z,⇣):=k⇣ ¹ zkK, z,⇣ 2R^{N+1, (2.4)} and theball

BK(z0,r):= {z2R^N+1|dK(z,z0) <r}. (2.5) Note that from (2.3) it directly follows

dK(z,⇣)c(dK(z, w)+dK(w,⇣)), z,⇣, w2R^N+1. For anyz2R^N+1andH ⇢R^N+1, we define

d_K(z,H):=inf{d_K(z,⇣)|⇣ 2H}.

We say that a function f : !Ris H¨older continuous of exponent↵2]0,1], in short f 2C^0,↵_K (), if there exists a positive constantCsuch that

|f(z) f(⇣)|C dK(z,⇣)^↵, for every z,⇣ 2. (2.6) We next define LipK functions and LipK surfaces. For any givenx2R^{N we set}

x⁰ =x₁ and x⁰⁰=(x₂, . . . ,x_N) . (2.7) Moreover, using the notation in (2.2) we also letx₀⁽⁰⁾= x1 =x⁰, and we letx₀₀⁽⁰⁾ = (x2, . . . ,xm). Using this notation, we define the norm

k(x⁰⁰,t)k⁰⁰K = x₀₀⁽⁰⁾ + X

j=1

x^{(j) 2j1+1} + |t|¹² (2.8)

inR^{N 1}⇥R. Recalling (1.8), we let

D_r⁰⁰ =diag r Im 1,r³Im1, . . . ,r^2+1Im ,

so that D_r⁰⁰x⁰⁰ = (Drx)⁰⁰, while (Drx)⁰ = r x⁰. Note that k(D_r⁰⁰x⁰⁰,r²t)k⁰⁰K = rk(x⁰⁰,t)k⁰⁰_K for everyr >0 and(x⁰⁰,t) 2R^{N 1}⇥R. Using the notation in (2.1) we set, for any positiver1,r2,r3,

Qr1,r2,r3 = {(x,t)2R^N+1| |x⁰|r1,|xi|r₂^↵ifor anyi =2, . . . ,N, |t|r₃²}, Q_r⁰⁰_2,r3 = {(x⁰⁰,t)2R^{N 1}⇥R| |xi|r₂^↵i for anyi =2, . . . ,N, |t|r₃²},

(2.9) and, for any positive Mandr and any arbitrary pointz02R^N+1, we define

QM,r = Q_4Mr,r,^p_2r, Q⁰⁰_r = Q⁰⁰_r,^p_2r, QM,r(z0)=z0 QM,r. (2.10) Note that Q_M,r = rQ_M,1 and Q⁰⁰_r = D⁰⁰_rQ⁰⁰₁ for every r > 0. Moreover, the continuity of the group law (1.5) implies that there exists" = "(L,M) 2]0,1[ such that

(x,t) (⇠,⌧)2 QM,r 8(⇠,⌧)2QM,"r, 8(x,t)2Q_M,^r₂. (2.11) Furthermore, for every positiveM, there exist two positive constantsc⁰_M,c⁰⁰_M such that

BK(z0,c⁰_Mr)✓ QM,r(z0)✓BK(z0,c⁰⁰_Mr), (2.12) for everyz02R^{N+1andr >0.}

Lete⁰ 2 R^m be such thatke⁰k = 1. Since the form of the matrix Bin (1.7) remains invariant under rotation in the firstmvariables it is not restrictive to assume e⁰=e1=(1,0, . . . ,0). Given any open set⁰⁰⇢R^{N 1}⇥R, we say that f :⁰⁰! R^{is a Lip}K function with respect toe⁰, ifx⁰=x1and

f x+exp( t B^T)x₀ ⁰⁰,t +t₀ f(x₀⁰⁰,t₀) Mk(x⁰⁰,t)k⁰⁰K, (2.13) for every(x₀⁰⁰,t0)2⁰⁰, and(x⁰⁰,t)2R^{N 1}⇥R^{such that x}+exp( t B^T)x0 00,t+ t₀ 2 ⁰⁰, x⁰₀ = f(x₀⁰⁰,t₀). Note that we can assume, without loss of generality, that (x₀,t₀) = (0,0)and f(0,0) = 0. Indeed, if necessary, it is enough to set g(x⁰⁰,t)= f x+exp( t B^T)x₀ ⁰⁰,t+t₀ f(x₀⁰⁰,t₀). Equivalently, f :⁰⁰!R is Lip_K if

f(x⁰⁰,t) f(⇠⁰⁰,⌧) M x exp((⌧ t)B^T)⇠ ⁰⁰,t ⌧ ⁰⁰_K, (2.14) for every (x⁰⁰,t), (⇠⁰⁰,⌧)2 ⁰⁰,x⁰ = f(x⁰⁰,t),⇠⁰ = f(⇠⁰⁰,⌧). Given f as above with f(0,0)=0 and M,r >0, we define

_f,r= {(x,t)2QM,r | f(x⁰⁰,t) <x⁰}, 1_f,r= {(x,t)2 QM,r | f(x⁰⁰,t)=x⁰}. (2.15) Note that, according to the dilations randD_r⁰⁰, we have

_f,r = r_fr,1, 1_f,r = r1_fr,1,

where fr(x⁰⁰,t) = r ¹f(D_r⁰⁰x⁰⁰,r²t). We point out that if f is a LipK function, then so is fr. Indeed, as the normk·k⁰⁰_K is(D_r⁰⁰,r²)-homogeneous, we have

f_r(y⁰⁰,s) f_r(x₀⁰⁰,t₀) M y exp((t₀ s)B^T)x₀ ⁰⁰,s t₀ ⁰⁰_K, for every(y⁰⁰,s), (x₀⁰⁰,t0)2(D_r⁰⁰,r²) ¹⁰⁰ andr >0. Finally, we let

_f,r(z0)=z0 _f,r, 1_f,r(z0)=z0 1_f,r, z02R^N+1.

Definition 2.1. Let ⇢ R^N+1be a bounded domain. We say that6 ⇢ @is a Lip_K surface with constantsM =max{M₁, . . . ,M_k}andr₀= min{r₁, . . . ,r_k}if, for any j =1, . . . ,k, there exists a pointz_j 26, and a Lip_K function f_j : Q⁰⁰_2rj! Rdefined with respect to a suitablee⁰_j 2R^mand with Lipschitz constantMj, such that

6⇢ [k j=1

1_fj,rj(z_j),

6\QMj,2rj(zj)=1_fj,2rj(zj),

\QMj,2rj(zj)=_fj,2rj(zj).

Recall the definition of Lip(1,1/2)functions given in (1.13). For the proof of the following proposition we refer to Section 5.

Proposition 2.2. Let⇢R^N+1be a bounded domain. If6 ⇢@locally agrees with the graph{xj = f(x⁰⁰,t)}of someLip(1,1/2)function f, j 2 {1, . . . ,m}, then6is aLipK surface.

The following proposition plays a key role in the proof of Theorem 1.1 and will be proved in Section 5.

Proposition 2.3. Let Q_r⁰⁰ be as in(2.10)withr 2]0,1]. Let f : Q⁰⁰_r ! R ^{be a} LipK function such that f(0,0)= 0. Then there exist a point(ex,et) 2R^N+1and two positive constantsC,c, depending only onL andM, such that

sup

f,cr(x0,t0)

u(x,t)C u A_r⁺(x₀,t₀) , A⁺_r(x₀,t₀)=(x₀,t₀) _r(ex,et),

for every positive solutionutoLu =0in_f,r(x0,t0)vanishing continuously on 1_f,r(x0,t0).

We next formulate two versions of Proposition 2.3 which apply to characteris- tic boundary points with respect toL. In particular, Proposition 2.4 below applies to a cylinder e_g,r(z0) = z0 e_g,r obtained by using the operation “ ”, while in Proposition 2.5 the operation “ ” does not appear. Note that, in the latter case, the Lipschitz constantMof the functiongdepends on the cylinder. To proceed, let

Qer =n⇣

x⁽¹⁾, . . . ,x^()⌘

2R^{N m} | |x^(j)|r^2j+11 , j =1, . . . ,o

. (2.16)

Let g : Qer ! Rbe a Lipschitz function in the classical sense, with Lipschitz constant M,i.e.

|g⇣

x⁽¹⁾, . . . ,x^()⌘ g⇣

y⁽¹⁾, . . . ,y^()⌘

|M(|x⁽¹⁾ y⁽¹⁾| +....+ |x^() y^()|) (2.17) whenever x⁽¹⁾, . . . ,x^() , y⁽¹⁾, . . . ,y^() 2Qer. Furthermore, assumeg(0)=0.

Then, for positiver andz₀2R^N+1, we define e

_g,r =n

(x,t)2 Qr,r,Mr |t >g⇣

x⁽¹⁾, . . . ,x^()⌘o , e

1_g,r =n

(x,t)2Qr,r,Mr |t =g⇣

x⁽¹⁾, . . . ,x^()⌘o , e

_g,r(z0)=z0 e_g,r, e1_g,r(z0)=z0 1e_g,r.

(2.18)

Proposition 2.4. Let Qer be as in(2.16)withr 2]0,1]. Let g : Qer ! R^{be a} Lipschitz function with Lipschitz constant M such thatg(0) =0. Then there exist three positive constantsC,candet, depending only onL andM, such that

sup

e

g,cr(x0,t0)

u(x,t)C u A_r⁺(x0,t0) , A⁺_r(x0,t0)=(x0,t0) (0,r²et),

for every positive solutionutoLu =0ine_g,r(x0,t0)vanishing continuously on e

1_g,r(x₀,t₀).

Let Q be a bounded set ofR^{N, and letg} : Q ! Rbe a Lipschitz function, which does not depend onx1, ...,xm, with constantM,i.e.g(x)=g x⁽¹⁾, ...,x^() . Lettingg(x₀)=t₀for somex₀2Q, andc_Q=sup_x2Q|x x₀|, we define

e

_g=n

(x,t)2R^N+1|x2 Q, g(x) <t t0+McQ

o , 1e_g =n

(x,t)2R^N+1|x 2Q, t =g(x)o .

Proposition 2.5. Let x0 2 R^{N, let Q} be a bounded neighborhood of x0, and let M be a positive constant such thatMsup_x2QkB^Txk < 1. Letg : Q ! R^{be a} Lipschitz function, which does not depend onx1, . . . ,xm, with constant M and let t0=g(x0). Then there exist three positive constantsC,r andet, only depending on L andQ, such that

sup

e

g\BK((x0,t0),r)

u(x,t)C u A⁺_r (x0,t0) , A_r⁺(x0,t0)=(x0,t0) (0,r²et),

for every positive solutionutoLu=0ine_gvanishing continuously one1_g. The proofs of Proposition 2.4 and Proposition 2.5 are given in Section 5.

3. Interior Harnack inequalities

We say that a path : [0,T]!R^N+1isL-admissibleif it is absolutely continu- ous and satisfies

0(s)=

Xm j=1

!_j(s)Xj( (s))+ (s)Y( (s)), for a.e. s 2[0,T], (3.1) where!=(!₁, . . . ,!_m)2L²([0,T],R^{m), and} is a strictly positive measurable function. We say that steers z0 toz if (0) = z0 and (T) = z. Concerning the problem of theexistenceof admissible paths, we recall that it is acontrollability problem, and that[H.2]is equivalent to the followingKalman condition:

rank⇣

A B¯ ^TA¯ · · · B^{T N 1}A¯⌘

=N. (3.2)

HereA¯is theN⇥N matrix defined by

✓A¯0 0 0 0

◆

and A¯0 is them ⇥m constant matrix introduced in (1.3). We recall that (3.2) is a sufficient condition for theglobal controllabilityof (3.1),i.e.the property that any pointz0=(x0,t0)2R^N+1can be connected to anyz=(x,t)2R^{N+1witht <t}0

by anL-admissible path (see [29], Theorem 5, page 81). In the sequel we let Az0()= z2|there exists anL-admissible : [0,T]!connectingz0toz ,

(3.3) and we defineA_{z0 =} A_z0()= Az0()as the closure (inR^{N+1) of A}z0(). We will refer to the setA_z0as theattainable set.

We next recall a Harnack type inequality which is stated in terms ofL-admis- sible paths and attainable sets.

Theorem 3.1. (THEOREM 2.4 IN [6]) LetL be an operator in the form(1.1), satisfying assumptions[H.1-3]. Letbe an open subset ofR^{N+1and letz}0 2.

For every compact set H ✓ Int A_z0 , there exists a positive constant CH, only dependent on,z0,H and on the operatorL, such that

sup

H

uCHu(z0), for every non-negative solutionuofLu=0in.

The following Proposition 3.2 is a consequence of [11, Theorem 1.2] and [11, Lemma 6.2]. We give here a simple proof based on our Theorem 3.1. For any positiver, and(x0,t0)2R^N+1, according to notation (2.9) we set

Q_r = Qr,r,r\ (x,t)2R^N+1 |t <0 , Q_r(x0,t0)=(x0,t0) Q_r , K r= Q r, r,r\ (x,t)2R^N+1|t = r²/2 , K r(x0,t0)=(x0,t0) K r.

(3.4)

Proposition 3.2. Let be anL-admissible path satisfying (0)=(x0,t0). There exist two positive constantshandCsuch that

Z s

0 |!(⌧)|²d⌧ h ) u( (s))C u(x0,t0),

for every non-negative solutionuofLu=0inQ₁(x0,t0)ands2]0,1/2]. Proof.We first claim that there exists 2]0,1[such thatK r(x0,t0)is contained in Int A_(x0,t0)(Q_r(x0,t0)) . To check the above statement, it is sufficient to consider the case(x0,t0)=(0,0)andr =1. We first note that, as!₁⌘0, . . . ,!_m⌘0, we have (s)=(0, s), then(0, 1/2)2 A(0,0)(Q₁). To prove that(0, 1/2)is an interior point ofA_(0,0)(Q₁), we recall that the system

0(s)=

Xm j=1

!_j(s)Xj( (s))+Y( (s)), (0)=(0,0), (3.5) is globally controllable, by (3.2). Then, as!varies in a neighborhood of the vector 0 2 L²([0,T],R^m), the image of the map! 7! (T)covers a neighborhood of (0, 1/2), and (s)2Q₁ for everys2[0,T]. This proves the claim.

Let be as above. By Theorem 3.1, there exists a positive constantC such that

sup

K r(x0,t0)

uC u(x0,t0), (3.6) for every non-negative solutionuofLu =0 inQ₁(x0,t0)and for anyr 2]0,1]. Finally, a plain application of the H¨older inequality shows that

(s)2K r(x0,t0) whenever s2]0,1/2] satisfies Z s

0 |!(⌧)|²d⌧ h, (3.7) for a suitable constanth. We refer to [11, Lemma 6.2] for more details. The con- clusion of the proof easily follows from (3.7) and (3.6). ⇤

4. Basic boundary estimates

We first introduce a family of cones defined in terms of the dilation and the translation “ ”. For any given z₀ 2 R^N+1, x¯ 2 R^{N, t¯} 2 R⁺, we consider an open neighborhoodU ⇢R^{N of}x¯, and we denote byZ_x¯,¯t,U(z₀)andZ⁺_x,¯¯t,U(z₀)the followingtusk-shapedsets

Z_x,¯¯t,U(z₀)= z_{0 s}(x, t¯)|x 2U,0<s 1 ,

Z⁺_x,¯¯t,U(z0)= z0 s(x,t)¯ |x 2U,0<s1 . (4.1)

In the sequel, aiming to simplify the notations, we will often write Z^±(z0)instead ofZ^±_x,¯¯t,U(z0)if the choice ofx¯,t,¯ Uis clear from the context. Note thatZ (z0)and Z⁺(z0)are cones with the same vertex atz0=(x0,t0), while the basis ofZ (z0)is at the time levelt₀ t¯<t₀, and the basis ofZ⁺(z₀)is at the time levelt₀+ ¯t >t₀. Definition 4.1. Letbe an open subset ofR^{N+1and let6}⇢@.

(i) We say that6satisfies anuniform exterior cone conditionif there existx¯ 2 R^N,t¯>0 and an open neighborhoodU ✓R^{N of}x¯such that

Z (z0)\=; for every z026, where Z (z0)= Z_x,¯¯t,U(z0).

(ii) We say that6satisfies an uniform interior cone conditionif there existx¯ 2 R^N,t¯>0 and an open neighborhoodU ✓R^{N of}x¯such that

Z⁺(z0)⇢ for every z026, where Z⁺(z0)= Z⁺_x,¯¯t,U(z0).

(iii) Assume that6satisfies an uniform interior cone condition in the sense stated above. We then say that the cones{Z⁺(z0)} = {Z⁺_x,¯¯t,U(z0)},z0 26, satisfy a strong Harnack connectivity condition if the functions7!(x0,t0) _{1 s}(x,¯ t¯) is anL-admissible path.

We point out that thestrong Harnack connectivity conditionis more restrictive than theHarnack connectivity conditionused in our previous work [6]. Nevertheless, we are able to prove the validity of this condition near Lip_K surfaces.

We next show how to find a pointx32R^N such that the paths 7! 1 s(x3,1) isL-admissible. To this aim, we recall notations (1.7) and (2.2).

Lemma 4.2. For any positive3we define the pointx32R^N as follows:

x₃⁽⁰⁾ =3e⁰, x₃^(j) = 2

2j+1 B^T_j x₃^{(j 1)}, j =1, . . . ,. (4.2) Heree⁰ is the unit vector ofR^m pointing towards thex⁰-direction. Then, the path [0,1]3s! (s)= 1 s(x3,1)isL-admissible.

Proof. We show that satisfies (3.1), namely

0(s)= ¯A0!(s)+ (s) B^T (s) @_t a.e. in[0,1], (4.3) for a!2L²([0,1],R^m)and a positive measurable function . By a direct compu- tation,

(s)=

✓

(1 s)x₃⁽⁰⁾, 2

3(1 s)³B₁^Tx⁽⁰⁾₃ , . . . , ( 2)^

(2+1)!!(1 s)^2+1B_^T· · ·B₁^Tx₃⁽⁰⁾, (1 s)²

◆ ,

so that we find

0(s)= x⁽⁰⁾₃ ,0, . . . ,0 +2(1 s) B^T (s) @_t , s2[0,1]. This proves (4.3) with!= A¯₀¹x₃⁽⁰⁾and (s)=2(1 s). ⇤

Our next result improves on the analogous one in [6, Proposition 3.2]. Indeed, as noticed in [6, Remark 4.2], it fails under the weakerHarnack connectivity con- ditionassumed in [6].

Lemma 4.3. LetZ_x⁺_¯,¯t,U(0,0)be a cone satisfying the strong Harnack connectivity condition(iii)in Definition4.1. Then there exist two positive constantsC₁ and , which only depend on Z⁺and on the operatorL, such that

u( _s(x¯,t¯)) C₁

k s(x¯,t¯)k_K u(x¯,t¯) 0<s <1, for every non-negative solutionuofLu=0inZ⁺.

Proof. We first show that there exist a positive constantCeands02]0,1[such that u( (x¯,t))¯ C u(e x,¯ t¯), for every 2[1 s0,1[. (4.4) To this aim, we note that there exists⇢2]0,1]such that Q_⇢(x¯,t¯)⇢ Z⁺_x¯,¯t,U(0,0).

Moreover, the path (s)= 1 s(x,¯ t¯)isL-admissible by the strong Harnack con- nectivity condition. Since !₁, . . . ,!_m 2 L²([0,1]), there existss0 2]0,1[ such

that Z s0

0 |!(⌧)|²d⌧ h,

where his the positive constant appearing in Proposition 3.2. Then (4.4) directly follows from Proposition 3.2.

We next conclude the proof by applying several times (4.4). For a givens 2 ]0,1 s0[, we seteZ⁺(0,0)= (1 s0)/s Z⁺(0,0) . Note that the function

us :eZ⁺(0,0)!R^{, us}=u _{s/(1 s0})(·) is a non-negative solution toL_sus=0, where

L_s= Xm i,j=1

ai,j s/(1 s0)(z) @_xixj+ Xm i=1

s

(1 s0)ai s/(1 s0)(z) @_xi

+ XN i,j=1

bi,jxi@_xj @_t.