Kolmogorov–Fokker–Planck equations: Comparison principles near Lipschitz type boundaries

We prove several new results concerning the boundary behavior of non-negative solutions to the equation K u = 0, where


Introduction
Let N = 2m, where m ≥ 1 is an integer, and let Ω ⊂ R N +1 be a bounded domain, i.e., a bounded, open and connected set. In this paper we establish a number of results concerning the boundary behavior of non-negative solutions to the equation Ku = 0 in Ω, where (1.1) The operator K, referred to as the Kolmogorov or Kolmogorov-Fokker-Planck operator, was introduced and studied by Kolmogorov in 1934, see [19], as an example of a degenerate parabolic operator having strong regularity properties. Kolmogorov proved that K has a fundamental solution Γ = Γ(x, y, t, x, ỹ, t ) which is smooth in the set (x, y, t) = (x, ỹ, t ) . As a consequence, for every distributional solution of Ku = f . Property (1.2) can also be stated as K is hypoelliptic, (1.3) see (1.13) below.
The operator K appears naturally in the context of stochastic processes and in several applications. The fundamental solution Γ(·, ·, ·, x, ỹ, t ) is the density of the stochastic process (X t , Y t ), which solves the Langevin equation dY t = X t dt, Yt =ỹ, (1.4) where W t is a m-dimensional Wiener process. The system in (1.4) describes the density of a system with 2m degrees of freedom. Given z = (x, y) ∈ R 2m , x = (x 1 , . . . , x m ) and y = (y 1 , . . . , y m ) are, respectively, the velocity and the position of the system. (1.4) and (1.1) are of fundamental importance in kinetic theory, they form the basis for Langevin type models for particle dispersion and appear in applications in many different areas including finance [2,23], and vision [10,11]. In [7], [8] and [9], we developed a number of important preliminary estimates concerning the boundary behavior of non-negative solutions to equations of Kolmogorov-Fokker-Planck type in Lipschitz type domains. These papers were the results of our ambition to understand to the extent, and in what sense, scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures, previously established for uniformly parabolic equations with bounded measurable coefficients in Lipschitz type domains, see [13,14,26,12,22,25], can be established for non-negative solutions to the equation Ku = 0 and for more general equations of Kolmogorov-Fokker-Planck type. In this paper we take this program a large step forward by establishing Theorems 1.1, 1.2 and 1.3 stated below. These results are completely new and represent the starting point for far reaching developments concerning operators of Kolmogorov type. Already in the case of uniformly elliptic and parabolic operators this kind of scale and translation invariant estimates are important in the analysis of free boundary problems, see [5], [6] and [1] for instance, and in the harmonic analysis approach to partial differential equations in Lipschitz type domains, see [18,16].

Scalings and translations
The prototype for uniformly parabolic operators in R m+1 is the heat operator (1.5) Considering non-smooth domains, here roughly defined as Lipschitz type domains, the ambition to develop estimates for solutions to Hu = 0 which respect the standard parabolic scalings, and the standard group of translations on R m+1 , naturally leads one to develop estimates for solutions to Hu = 0 in the time-dependent setting of Lip(1, 1/2)-domains. A notion of (local) Lip(1, 1/2)-domains with constants M and r 0 is formulated in the natural way using appropriate local coordinate systems and assuming that in each local chart of size r 0 , the boundary can be represented by a Lip(1, 1/2)-function f with Lip(1, 1/2)-constant M , see [22] for example. Recall that a function f : Compared to the heat operator, the scalings underlying the operator K is different, and the change of variables preserving the equation is more involved. As a consequence the appropriate geometric setting for the equation Ku = 0 becomes more of an issue. In the case of K, the natural family of dilations (δ r ) r>0 on R 2m+1 is defined by δ r (x, y, t) = (rx, r 3 y, r 2 t), (1.7) for every (x, y, t) ∈ R 2m+1 and every positive r. Due to the presence of non-constant coefficients in the drift term of K, the usual Euclidean change of variable does not preserve the Kolmogorov equation. Nevertheless, a Galilean change of variable does. Consider a smooth function u : Ω → R, choose any point (x, ỹ, t ) ∈ R 2m+1 and set w(x, y, t) = u(x + x, ỹ + y − tx, t +t). Then Ku(x, y, t) = f (x, y, t) ⇐⇒ Kw(x, y, t) = f (x + x,ỹ + y − tx, t +t), for every (x, y, t) ∈ Ω. The change of variables used above defines a Lie group in R N +1 with group law (z,t) • (z, t) = (x,ỹ,t) • (x, y, t) = (x + x,ỹ + y − tx,t + t), (1.8) (z, t), (z, t ) ∈ R N +1 . Note that (z, t) −1 = (x, y, t) −1 = (−x, −y − tx, −t), (1.9) and hence (z,t) −1 • (z, t) = (x,ỹ,t) −1 • (x, y, t) = (x −x, y −ỹ + (t −t)x, t −t), (1.10) when (z, t), (z, t ) ∈ R N +1 . Using this notation the operator K is δ r -homogeneous of degree two, i.e., K • δ r = r 2 (δ r • K), for all r > 0. The operator K can be expressed as (1.11) and the vector fields X 1 , . . . , X m and Y are left-invariant with respect to the group law (1.8) in the sense that see [17]. Furthermore, while X i represents a differential operator of order one, ∂ y i acts as a third order operator. This fact is also reflected in the dilations group (δ r ) r>0 defined above. Based on the scalings and group of translations discussed above, writing (x, y, t) = (x 1 , x , y 1 , y , t), (x, ỹ, t ) = (x 1 , x , ỹ 1 , ỹ , t ) ∈ R × R m−1 × R × R m−1 × R, and assuming that x 1 is the dependent variable, it is natural to formulate geometry by using local coordinate charts and expressing the first coordinate x 1 as a function f : (1.14) for some M , where x 1 = f (x , ỹ 1 , ỹ , t ). Here (x, y, t) K = |(x, y)| K + |t| 1 2 , |(x, y)| K = x + y 1/3 (1. 15) whenever (x, y, t) ∈ R m × R m × R = R N +1 , see [8,9]. Note that δ r (x, y, t) K = r (x, y, t) K for every r > 0 and (x, y, t) ∈ R N +1 . Furthermore, as long as f is allowed to depend on the variable y 1 , and x 1 is assumed to be the dependent variable, then the term y 1 −ỹ 1 + (t −t)x 1 has to appear on the right hand side in (1.14) to achieve translation invariance. In line with [8,9], we call a function f satisfying (1.14) a Lip K -function, with Lip K -constant M . From the perspective of scalings and group of translations, Lip K -functions, and associated (local) domains, are the natural replacement in the context of the operator K of the Lip(1, 1/2)-functions and Lip(1, 1/2)-domains considered in the context of H.

Geometric aspects: Harnack chains
While the outline above gives at hand that Lip K -functions, and associated local Lip K -domains, may serve as good candidates for geometries in which one may attempt to establish more refined boundary comparison principles for solutions to Ku = 0, further considerations are needed. In the corresponding theory for uniformly parabolic operators, the Harnack inequality and a method to connect points and to compare values for non-negative solutions, through Harnack chains in the geometry introduced, are usually very important tools needed to make progress. In this context the progress often builds on the validity of the strong maximum principle, the fact that the spatial variables (z 1 , . . . , z N ) are decoupled from the time variable t, something which naturally also is reflected in the underlying group of translations, and a flexibility in the very formulation of the Harnack inequality. In contrast, this is where things starts to get complicated for the operator K.
The tool used to build Harnack chains is that of K-admissible paths. A path γ : [0, T ] → R N +1 is called K-admissible if it is absolutely continuous and satisfies where ω j ∈ L 2 ([0, T ]), for j = 1, . . . , m, and λ are non-negative measurable functions. We say that γ and γ(T ) = (z, t ). When considering Kolmogorov operators in the domain R N × (T 0 , T 1 ), it is well known that (1.13) implies the existence of a K-admissible path γ for any points (z, t), Here and in the sequel, A (z,t) (Ω) is referred to as the propagation set of the point (z, t) with respect to Ω. The presence of the drift term in K considerably changes the geometric structure of A (z,t) (Ω) and A (z,t) (Ω) compared to the case of uniformly parabolic equations. Indeed, simply consider (z, t) = (x, y, t) ∈ R 3 in which case Consider the domain where R is a given positive constant. In this case 19) and one can prove, see [7], that there exists a non-negative solution u to Ku = 0 in Ω such that u ≡ 0 in A (0,0,0) (Ω) and such that u > 0 in Ω \ A (0,0,0) (Ω). In particular, it is impossible to find a positive constant c such that u(x, y, t) ≤ cu(0, 0, 0) whenever (x, y, t) ∈ Ω \ A (0,0,0) (Ω). Hence, in this sense the Harnack inequality cannot hold in a set greater than A (0,0,0) (Ω) and as a consequence the Harnack inequality we have at our disposal, see Theorem 2.1 stated in the bulk of the paper, is less flexible compared to the corresponding one for uniformly parabolic operators. Naturally this is also related to the Bony maximum principle, see [3]. In this context it is fair to mention that the first proof of the scale invariant Harnack inequality which constitutes one of the building blocks of our paper, can be found in [15]. Furthermore, the introduction of that paper, see p. 776-777 in [15], also contains a discussion of an example showing why a uniform Harnack inequality cannot be expected to hold outside of the propagation set A (z,t) . In [15] the Harnack inequality is expressed in terms of level sets of the fundamental solution, hence depending implicitly on the underlying Lie group structure. This fact was used in [20], where the group law (1.8) was used explicitly and the Harnack inequality, in the form we use it, was proved for the first time.
In general, using (1. 16) we see that if we want to construct a K-admissible path connecting (z, t), (z, t ) ∈ R N +1 , then we have flexibility to define and control the path in the x and t variables by choosing ω j for j = 1, . . . , m, and λ. However, by choosing {ω j } and λ, the path in the y variables becomes determined by these choices. In this sense, any such construction renders a certain lack of control of the path in the y variables and it becomes a difficult task (impossible in some cases) to connect arbitrary points (z, t) = (x, y, t) and (z, t ) = (x, ỹ, t ), in a controlled manner, by K-admissible paths and Harnack chains while taking geometric restrictions into account.
An important contribution of this paper is that we are able to overcome this concrete difficulty by imposing one additional restriction on our Lip K -domains: we consider local Lip K -domains defined by functions f as in (1.14) with the assumption that f does not depend on the variable y 1 . This formulation of the geometry induces an additional degree of freedom which we are able to explore to make progress. (1.20) and we let, with a slight abuse of notation,

Admissible local Lip K -domains
Given positive numbers r 1 , r 2 , we introduce the open cube Given any open set 2 r 1 ,r 2 ⊂ R N −1 × R, we say that a function f , f : 2 r 1 ,r 2 → R, is a Lip K -function, with respect to e 1 = (1, 0, . . . , 0), independent of y 1 and with constant M ≥ 0, if whenever (x , y , t), (x , ỹ , t ) ∈ 2 r 1 ,r 2 . In addition, given positive numbers r 1 , r 2 , r 3 , we let For positive M and r, we let Q M,r = Q r, √ 2r,4Mr . Finally, given f as above with f (0, 0, 0) = 0 and M, r > 0, we define Definition 1. Let f be a Lip K -function, with respect to e 1 = (1, 0, . . . , 0), independent of y 1 and with constant M ≥ 0. Let Ω f,r and Δ f,r be defined as above. Given M , r 0 , we say that Ω f,2r 0 is an admissible local Lip K -domain, with Lip K -constants M , r 0 . Similar we refer to Δ f,2r 0 as an admissible local Lip K -surface with Lip K -constants M , r 0 . Remark 1.1. Our results, see Theorems 1.1, 1.2 and 1.3 below, are established near an admissible local Lip K -surface Δ f,2r 0 . The surface Δ f,2r 0 is contained in the non-characteristic part of the boundary of Ω f,2r 0 . Recall that a vector ν ∈ R N +1 is an outer normal to Ω f,2r 0 at (z 0 , t 0 ) ∈ Δ f,2r 0 if there exists a positive r such that B((z 0 , t 0 ) + rν, r) ∩ Ω f,2r 0 = ∅. Here B((z 0 , t 0 ) + rν, r) denotes the (standard) Euclidean ball in R N +1 with center at (z 0 , t 0 ) + rν and radius r. Now X j (z 0 , t 0 ), ν = 0, for some j = 1, . . . , m, whenever (z 0 , t 0 ) ∈ Δ f,2r 0 . Hence, by definition all points (z 0 , t 0 ) ∈ Δ f,2r 0 are non-characteristic points for the operator K. For a more thorough discussion of this, regular points for the Dirichlet problem, and Fichera's classification, we refer to subsection 2.4, see (2.15) in particular. Remark 1.2. We emphasize that an admissible local Lip K -surface Δ f,2r 0 is defined through a function f which is independent of the y 1 variable. This formulation of the geometry induces an additional degree of freedom which we are able to explore to make progress. In particular, as discussed, due to the lack of flexibility when constructing K-admissible paths and Harnack chains, it is difficult to connect arbitrary points (z, t) = (x, y, t) and (z, t ) = (x, ỹ, t ), in a controlled manner, while taking geometric restrictions into account. However, using that Δ f,2r 0 is independent y 1 , and as our equation is invariant under translations in the y 1 variable, we are able to explore this independence in the proof of our main results in a manner similar to how t independence is explored in [12]. We refer to subsection 1.5 below for a more thorough discussion, see also Remark 1.5 below.

Statement of the main results
Let Ω f,2r 0 be an admissible local Lip K -domain in the sense of Definition 1, with Lip K -constants M , r 0 . The topological boundary is denoted by ∂Ω f,2r 0 . As discussed in the bulk of the paper, all points on Δ f,2r 0 are regular for the Dirichlet problem for the operator K in Ω f,2r 0 . For every (z, t) ∈ Ω f,2r 0 , there exists a unique probability measure ω K (z, t, ·) on ∂Ω f,2r 0 such that the Perron-Wiener-Brelot solution to Ku = 0 in Ω f,2r 0 , with boundary data ϕ on ∂Ω f,2r 0 , equals (1.25) We refer to ω K (z, t, ·) as the Kolmogorov measure relative to (z, t) and Ω f,2r 0 . To formulate our results we also have to introduce certain reference points.
Definition 2. Given > 0 and Λ > 0 we let In Theorem 1.2 below we use the notation

27)
and assume that m − > 0. Then there exist constants and assume that m −  [8,9], in greater generality, developed a number of important preliminary results concerning the boundary behavior of non-negative solutions like, for example, the Carleson estimate. This paper can be seen as a rather far reaching continuation of these papers.
Remark 1.5. In Theorem 1.1, Theorem 1.2 and Theorem 1.3, as well as in the generalizations stated in Theorem 7.1 and Theorem 7.2 below, the underlying function f defining the local domain is assumed to be independent of a set of properly chosen variables. It is fair to pose the question if this is really necessary for the validity of this type of results. Though our argument relies heavily on independence, we believe that the answer to this question likely is no. We believe that the results established in this paper can serve as a starting point for the development of the corresponding results under weaker assumptions. We here leave this problem for future research.

Brief discussion of the proof and organization of the paper
Section 2 is of preliminary nature and we here state facts about the fundamental solution associated to K, we state the Harnack inequality, we discuss the Dirichlet problem and we introduce the Kolmogorov measure and the Green function. In Section 3 we elaborate on the Harnack inequality, K-admissible paths and Harnack chains under geometric restrictions. Some of the material in this section builds on results established in [8,9]. In Section 4 we establish an important relation between the Kolmogorov measure and the Green function. In Section 5 we first prove Lemma 5.1 which gives a weak comparison principle at the boundary. Using Lemma 5.1 we in Section 5 then prove an important lemma: Lemma 5.3. In fact, it is Lemma 5.3 which enables us to, in the end, complete the proofs of Theorems 1.1, 1.2 and 1.3. In the context of admissible local Lip K -domains, Lemma 5.3 states that there exist constants , (1.29) whenever (x 1 , x , y 1 , y , t) ∈ Ω f, 1 /c 3 . I.e., for (x 1 , x , y , t) fixed and up to the boundary, all values of the function are comparable to u(x 1 , x , 0, y , t), uniformly in (x 1 , x , y , t), but with constants depending on the (acceptable) quotient u(A + 0 ,Λ )/u(A − 0 ,Λ ). Using this result we have a crucial additional degree of freedom at our disposal when building Harnack chains to connect points: we can freely connect points in the x 1 variable, taking geometric restriction into account, accepting that the path in the y 1 variable will most probably not end up in 'the right spot'. In the proof of Lemma 5.3 we use the fact that by the very definition of an admissible local Lip K -domain, the surface Δ f,2r 0 is independent of y 1 , hence we are able to translate with respect to this variable. Section 6 is devoted to the proof of Theorem 1.1, Theorem 1.2 and Theorem 1.3. Section 7 is devoted to a discussion of to what extent Theorems 1.1, 1.2 and 1.3 can be extended to more general operators of Kolmogorov type.

Preliminaries
In general we will establish our estimates in an admissible local Lip K -domain Ω f,2r 0 ⊂ R N +1 , with Lip K -constants M , r 0 . Therefore, throughout the paper c will in general denote a positive constant c ≥ 1, not necessarily the same at each occurrence, depending at most on N and M . Naturally c = c(a 1 , . . . , a l ) denotes a positive constant c ≥ 1 which may depend only on a 1 , . . . , a l and which is not necessarily the same at each occurrence. Two quantities A and B are said to be comparable, or

Notation
Recall the definition of |(x, y)| K , (x, y) ∈ R N , in (1.15) and that δ r (x, y, t) K = r (x, y, t) K for every r > 0 and (x, y, t) ∈ R N +1 . We recall the following pseudo-triangular inequality: there exists a positive constant c such that whenever (x, y, t), (x, ỹ, t ) ∈ R N +1 . We define the quasi-distance d K by setting and we introduce the ball Note that from (2.1) it follows directly that , (x,ỹ,t))), (2.4) whenever (x, y, t), (x, ŷ, t ), (x, ỹ, t ) ∈ R N +1 . For any (x, y, t) ∈ R N +1 and H ⊂ R N +1 , we define (2.5) Using this notation we say that a function f : We let Note that, if O is any bounded subset of R N +1 , then every u ∈ C 0,α K (O) is Hölder continuous in the usual sense as

Fundamental solution
Following [19] and [20] it is well known that an explicit fundamental solution, Γ, associated to K can be written down. Let for s ∈ R, where I m , 0, represent the identity matrix and the zero matrix in R m , respectively. * denotes the transpose. Furthermore, let whenever t ∈ R. Note that det C(t) = t 4m /12 and that Using this notation we have that Here ·, · denotes the standard inner product on R N . We also note that where q = 4m and c = c(N ). Often q + 2 is referred to as the homogeneous dimension of R N +1 with respect to the dilations group (δ r ) r>0 .

The Harnack inequality
To formulate the Harnack inequality we first need to introduce some additional notation. We let, for where e 1 is the unit vector pointing in the direction of x 1 and B( 1 2 e 1 , 1) and B(− 1 2 e 1 , 1) are standard Euclidean balls of radius 1, centered at 1 2 e 1 and − 1 2 e 1 , respectively. Similarly, we let In the following we formulate two versions of the Harnack inequality. Recall, given a domain Ω ⊂ R N +1 and a point (z, t) ∈ Ω, the sets A (z,t) (Ω) and A (z,t) (Ω) = A (z,t) (Ω) defined in the introduction.
Theorem 2.1. There exist constants c > 1 and α, β, γ, θ ∈ (0, 1), with 0 < α < β < γ < θ 2 , such that the following is true. Assume u is a non-negative solution to . Then there exists a positive constant c K , depending only on Ω and K, such that for every non-negative solution u of Ku = 0 in Ω.
Remark 2.1. We emphasize, and this is different compared to the case of uniform parabolic equations, that the constants α, β, γ, θ in Theorem 2.1 cannot be arbitrarily chosen. In particular, according to Theorem 2.2, the cylinder Q − r (z 0 , t 0 ) has to be contained in the interior of the propagation set A (z 0 ,t 0 ) (Q − r (z 0 , t 0 )).

The Dirichlet problem
Let Ω ⊂ R N +1 be a bounded domain with topological boundary ∂Ω. Given ϕ ∈ C(∂Ω) we consider here the well posedness of the boundary value problem The existence of a solution to this problem can be established by using the Perron-Wiener-Brelot method and, in the sequel, u ϕ will denote this solution to (2.13). In the following we first introduce what we refer to as the Kolmogorov boundary of Ω, denoted ∂ K Ω. The notion of the Kolmogorov boundary replaces the notion of the parabolic boundary used in the context of uniformly parabolic equations.
Definition 3. The Kolmogorov boundary of Ω, denoted ∂ K Ω, is defined as By Definition 3, ∂ K Ω ⊂ ∂Ω is the set of all points on the topological boundary of Ω which is contained in the closure of the propagation of at least one interior point in Ω. The importance of the Kolmogorov boundary of Ω is highlighted by the following lemma.
Proof. The lemma is a consequence of the Bony maximum principle, see [3]. 2 ∂ K Ω is the largest subset of the topological boundary of Ω on which we can attempt to impose boundary data if we want to construct non-trivial solutions. Hence, also the notion of regular points for the Dirichlet problem only makes sense for points on the Kolmogorov boundary and we let ∂ R Ω be the set of all (z 0 , t 0 ) ∈ (2.14) We refer to ∂ R Ω as the regular boundary of Ω with respect to the operator K. By definition ∂ R Ω ⊆ ∂ K Ω. Given a bounded domain Ω ⊂ R N +1 , in [21, Proposition 6.1] Manfredini gives sufficient conditions for regularity of boundary points. Recall that a vector ν ∈ R N +1 is an outer normal to Ω at (z 0 , t 0 ) ∈ ∂Ω if there exists a positive r such that B((z 0 , t 0 ) + rν, r) ∩ Ω = ∅. Here B((z 0 , t 0 ) + rν, r) denotes the (standard) Euclidean ball in R N +1 with center at (z 0 , t 0 ) + rν and radius r. In consistency with Fichera's classification, sufficient conditions for the regularity can be expressed in geometric terms as follows. If (z 0 , t 0 ) ∈ ∂Ω and ν = (ν 1 , . . . , ν N +1 ) is an outer normal to Ω at (z 0 , t 0 ), then the following holds: where Y is the vector field defined in (1.11). Condition (a) can be equivalently expressed in terms of the vector fields X j 's as follows: X j (z 0 , t 0 ), ν = 0 for some j = 1, . . . , m. If this condition holds, then in the literature (z 0 , t 0 ) is often referred to as a non-characteristic point for the operator K. A more refined sufficient condition for the regularity of the boundary points of ∂Ω is given in [21,Theorem 6.3] in terms of an exterior cone condition.
i.e., all points on the Kolmogorov boundary are regular for the operator K.
Proof. First, using Lemma 3.6 below and the sufficient condition for the regularity of the boundary points in terms of the existence of exterior cones referred to above, see [21, Theorem 6.3], we have that follows, as discussed above, also by using the results in [21]. 2 (2. 16) In the case of the adjoint operator K * we denote the associated Kolmogorov boundary of Ω f,2r 0 by ∂ * K Ω f,2r 0 . The above discussion and lemmas then apply to K * subject to natural modifications.
, to the Dirichlet problem in (2.13) and to the corresponding Dirichlet problem for K * , respectively. Furthermore, u is continuous up to the boundary at all boundary points contained in ∂ K Ω and u * is continuous up to the boundary at all boundary points contained in Proof. The lemma is an immediate consequence of Lemma 2.1 and Lemma 2.2. 2 is referred to as the Kolmogorov measure relative to (z, t) and Ω = Ω f,2r 0 , and ω * K (z, t, ·) is referred to as the adjoint Kolmogorov measure relative to (z, t) and Ω = Ω f,2r 0 .

Harnack chains under geometric restrictions
In this section we discuss the construction of Harnack chains in domains Ω ⊂ R N +1 and we derive some important lemmas. The following lemma gives the general connection between appropriate K-admissible paths and the possibility to compare values of non-negative solutions to Ku = 0 in Ω.
for some ∈ (0, 1) small enough to ensure that Ω = ∅. Consider (z, t), (z, t ) ∈ Ω , t < t. Then the following is true for every non-negative solution u of Then there exists a positive constant c, depending only on N , such that if we define c(γ, ) through Remark 3.1. The problem when attempting to apply Lemma 3.1 is that, in general, we have no method at our disposal based on which we, in concrete situations, can construct a K-admissible path (γ(τ ), } is a Harnack chain in Ω connecting (z, t ) to (z, t) and let c be the constant appearing in Theorem 2.1. Then, using Theorem 2.1, we see that and hence, Next we recall the following lemmas, Lemma 3.2 and Lemma 3.3. Lemma 3.2 is Lemma 2.2 in [4]. Then Let h and β be as in Lemma 3.2 and define {τ j } as follows. Let τ 0 = 0, and define τ j , for j ≥ 1, recursively as follows: Proof. This lemma is essentially proved in [4]. In particular, that (γ(τ ), t − τ ) : [0, t −t] → R N +1 is a K-admissible path, and that (3.7) holds, follow by a direct computation. Similarly, We now apply Lemma 3.2 to the path in (3.6). Let {{(z j , t j )} k j=1 , {r j } k j=1 } be constructed as in the statement of Lemma 3.3. Then, using Lemma 3.2, and the assumption in (3.9), it follows that is a Harnack chain in R N +1 connecting (z, t ) to (z, t). Furthermore, the length of the chain, k, can be estimated and This completes the proof of the lemma. 2 Remark 3.2. The crucial assumption to be verified when applying Lemma 3.3 is (3.9), i.e., we have to ensure This condition is trivially satisfied when Ω = R N × (T 0 , T 1 ) for some T 0 < τ − r 2 < t < T 1 . In this case, the path constructed in Lemma 3.3 is the solution of an optimal control problem giving the K-admissible path connecting (z, t), (z, t ), t < t, which minimizes the energy This path is constructed without reference to any geometric restrictions and it is not a straight line. Clearly, this introduces new difficulties when we impose some geometric restrictions on the domain Ω as it is, in Lemma 3.3, the path which imposes restrictions on Ω. In reality we want the opposite: we want to construct a path subject to the geometric restrictions imposed by Ω. Finally, following [4] we can also conclude that Lemma 3.3 holds for much more general operators of Kolmogorov type.
By a straightforward computation we see that where A ij are bounded functions defined on the interval [0, 1] and A ij (0) = 0. Note also that Furthermore, simply using the short notation z = (x, y), z = (x, ỹ), A ij = A ij (τ /δ), we get, after some computations, that where Remark 3.5. Consider Lemma 3.3 and let δ = t −t. Then, by similarly considerations as in Remark 3.3 we see that Remark 3.6. Inequality (3.10) in Lemma 3.3 gives the sharp bound for a non-negative solution in R N . The exponent appearing in (3.10) is found by solving an optimal control problem as briefly discussed in Remark 3.2. However, in the context of the equation Ku = 0 it is also possible to give a more intuitive construction of Harnack chains, a construction that gives a non-sharp, but equivalent, exponent. In the following we show how to construct such a K-admissible path connecting (x, y, t) ∈ R N × R + to (0, 0, 0).
We now let, for suitable vectors ω, ω ∈ R m to be chosen, 3 4 t, t . Specifically, we choose ω so that x t 2 = 0. A direct computation shows that and if we choose ω = − 2 t x, then x t 2 = 0 and y t 2 = y + t 4 x. In particular, for τ ∈ t 2 , t and (x(t), (y(t)) = (0, 0) if we choose ω = − 16 t 2 y + t 4 x . Based on this construction we now use Lemma 3.2 to give an estimate for the constant k in (3.4). Indeed, let k 0 be the positive integer which satisfies By Lemma 3.2, the points z j = γ t 2βj , 1 ≤ j ≤ k 0 , form a Harnack chain of length k 0 . Analogously, we let k 1 be the positive integer which satisfies and we form a Harnack chain of length k 1 . The construction made in the interval [ t 2 , 3 4 t) gives a Harnack chain also for 3 4 t, t . We eventually obtain a Harnack chain of length k = k 0 + 2k 1 + 3. Put together, the above two inequalities imply that u(0, Proof. Note that by definition Hence, by a direct computation In particular, where
, be defined as above. Then
For the details we refer to Lemma 4.3 in [9]. 2

Additional estimates based on the Harnack inequality
Let Ω f,2r 0 be an admissible local Lip K -domain, with Lip K -constants M , r 0 . Recall that given f with f (0, 0, 0) = 0 and M, r > 0, we defined where Q M,r = Q r, √ 2r,4Mr was introduced below (1.24). Let Λ, c 0 , η, 0 , 1 , be in accordance with Remark 3.7 and consider (z 0 , t 0 ) ∈ Δ f, 1 , 0 < < 1 . Let Q M,r (z 0 , t 0 ) = (z 0 , t 0 ) • Q M,r and consider the sets Ω f,2r 0 ∩ Q M,r 0 /2 (z 0 , t 0 ) and Ω f,2r 0 ∩ Q M, (z 0 , t 0 ). Then, by a change of variables, for a new function f , f (0, 0, 0) = 0, having the same properties as f . Keeping this in mind we will in the following, with a slight abuse of notation, simply use the following notation: Proof. We just give the proof in case (z 0 , t 0 ) = (0, 0) as our estimates will only depend on N and the Lip K -constant of f , and as we may, by construction and as by discussed above, after a redefinition f →f , also reduce the general case (z 0 , t 0 ) ∈ Δ f, 1 to this situation (z 0 , t 0 ) = (0, 0). By Lemma 3.7 we see that there exist, given (z, t) ∈ Ω f, and 0 < < 1 , points (z ± 0 , t ± 0 ) ∈ Δ f,c 0 and ± such that for some c = c(N, M ), 1 ≤ c < ∞. Hence, it suffices to prove the lemma with (z, t) replaced with A ± ± ,Λ (z ± 0 , t ± 0 ) as above. In the following we let δ, 0 < δ 1, δ , 0 <δ 1, δ ≤ δ, be fixed degrees of freedom to be chosen. Based on δ, δ we impose the restriction that (z, t) ∈ Ω f,δ and we let ¯ = δ . Then, using Lemma 3.9 we see that Keeping δ fixed we choose δ =δ(N, M, δ) such that, in the above construction, we have where c is the constant appearing in Lemma 3.8. Then, using Lemma 3.8 we can conclude that 1 , be in accordance with Remark 3.7. Let ε ∈ (0, 1) be given. Then there exists c = c(N, M, ε), 1 < c < ∞, such that following holds. Assume (z 0 , t 0 ) ∈ Δ f, 1 , 0 < < 1 , and that u is a non-negative solution to Ku = 0 in Ω f,2 (z 0 , t 0 ), vanishing continuously on Δ f,2 (z 0 , t 0 ). Then Proof. This lemma can be proved by a straightforward barrier argument. We refer to Lemma 3.1 in [8] and Lemma 4.5 in [9] for the details. 2 Assume that u is a non-negative solution to Ku = 0 in Ω f,2 0 , vanishing continuously in Δ f,r 0 , and that (z 0 , t 0 ) ∈ Δ f, 1 . Then Proof. This is essentially Theorem 1.1 in [9]. 2 Remark 3.8. Let Ω f,2r 0 be an admissible local Lip K -domain, with Lip K -constants M , r 0 . Based on the above lemmas, from now on we will let Λ, c 0 , η, 0 , 1 , be in accordance with Remark 3.7 and we recall that 1 0 . In this work we then prove estimates related to a scale satisfying 0 < < 1 .

Kolmogorov measure and the Green function: relations
Let Ω f,2r 0 be an admissible local Lip K -domain, with Lip K -constants M , r 0 . Let (z, t) ∈ Ω f,2r 0 and recall the notion of the Kolmogorov measure relative to (z, t) and Ω f,2r 0 , ω K (z, t, ·), introduced in Definition 4 and Lemma 2.3. The purpose of this section is to prove the following lemma.
Proof. Let in the following (z, t) ∈ Ω f,2 0 . We first prove statement (i). By definition 2.18 we have Obviously, we have that (2.12). Recalling that the t-coordinate of the point A + ,Λ is ρ 2 we introduce the sets Using (2.10) and (4.2) we see that Next, using a simple argument based on Lemma 3.11 we see that there exists c = c(N, M ), 1 ≤ c < ∞, such that and v(z, t) = 1 in Δ f, . Hence the function u(z, t) = 1 − v(z, t) satisfies the assumptions of Lemma 3.11 and (4.5) follows. Next, we note that if we choose δ sufficiently small, then S 2 ⊂ B K (A + ,Λ , /c), where the constant c is the one appearing in (3.34) of Lemma 3.8. In particular, we can conclude that we can choose δ = δ(N, M ), 0 < δ 1, use (4.5) and apply inequality (i) of (3.34) to the function v(z, t) = ω K (z, t, Δ f, ), to conclude that for some c =c(N, M ), 1 ≤c < ∞. Note that G(z, t, A + ,Λ ) = 0 if (z, t) ∈ S 1 . Hence, from (4.4), (4.6), and from the maximum principle, it follows that This completes the proof of (i). We next prove statement (ii). Let (z, t) ∈ Ω f,2r 0 ∩ {(z, t) : t ≥ 8 2 } and let δ, 0 < δ 1, be a degree of freedom to be chosen. Recall that Based on this we in the following let (4.8) Using this notation, and given δ, we let θ ∈ C ∞ (R N +1 ) be such that θ ≡ 1 on the set Q δ /2 and θ ≡ 0 on the complement of Q 3δ /4 . Such a function θ can be constructed so that |Kθ(z, t)| ≤ c(δ ) −2 , whenever (z, t) ∈ R N +1 . Using θ we immediately see that By the representation formula in (2.20) we have that Next, using the adjoint version of Lemma 3.12 and (4.11) we see that we can choose δ = δ(N, M ), 0 < δ 1, so that (4.12) for some constant c = c(N, M ), 1 ≤ c < ∞. This completes the proof of (ii). 2 and Ω f,2r 0 and let G(A + 0 ,Λ , ·) be the adjoint Green function for Ω f,2r 0 with pole at A + 0 ,Λ . Then there exists c = c(N, M ), 1 ≤ c < ∞, such that Proof. The lemma is an immediate consequence of Lemma 4.1. 2 Remark 4.1. Following the arguments used in the proof of Lemma 4.1 we can prove the This inequality will be useful in the sequel.

A weak comparison principle and its consequences
The main purpose of this section is to prove Lemma 5.1 and Lemma 5.3 stated below.
Proof. Let in the following ε = ε(N, M ), 0 < ε 1, be a degree of freedom to be chosen. Consider the set Δ f,6ε \ Δ f,4ε . We claim that there exist δ = δ(N, M ), 0 < δ 1, and a set of points and such that for some k only depending on the diameter of the cylinder Q M,1 and on the constant c appearing in the triangular inequality (2.4). Furthermore, the construction can be made so that for some c = c (N, M, δ(N, M ) The claim is a direct consequence of a Vitali covering argument and the method used in the proof of (4.5).
Using the claim we introduce the auxiliary function where k 1 is a large degree of freedom to be chosen below, and we let whenever (z, t) ∈ Γ 2 . Furthermore, we claim that, if k is big enough, then by elementary estimates and the Harnack inequality. To give a more detailed proof of this claim, recall the notation introduced in (3.27) and (4.8). Let Ω = A − kε ,Λ • Q 4ε and let G denote the Green function for the set Ω. Using the dilation invariance of the fundamental solution Γ, and of the cone C − ρ,η,Λ (0, 0), we see that we can use (2.18) to prove that for some η = η(N, M ), 0 < η 1. Using this, we see that by the comparison principle. (5.11) now follows from (5.13) and as, by the Harnack inequality, To proceed with the proof of Lemma 5.1 we next note, combining (5.8)-(5.11), and using the maximum principle, we can conclude that there exist k = k (N, M ) and c = c(N, M ) whenever (z, t) ∈ Ω f,5ε . To continue, having estimated v from above we next want to estimate u from below. To start the estimate we introduce the sets ). (5.16) and, by arguing as in Lemma 4.1, we see that holds whenever (z, t) ∈ Ω f,5ε . Then, by using the continuity of u, choosing δ sufficiently small and also using the maximum principle, we find that there exist k = k (N, M ) and whenever (z, t) ∈ Ω f,5ε . We now claim that there exists c = c(N, M ), 1 ≤ c < ∞ such that whenever (z, t) ∈ ∂ K Ω f,ε . Assuming (5.19) it follows from (5.18), (5.19), and the maximum principle, that exist k = k(N, M ) and c = c(N, M ), 1 ≤ c < ∞, such that whenever (z, t) ∈ Ω f,ε and hence the proof of the lemma is complete once we define ε through the relation kε = 1. Finally, to prove (5.19) it follows, by construction, that we only have to prove that and that u and v vanish continuously on Δ f,2 (z 0 , t 0 ). Then Proof. Note that Lemma 5.2 is a localized version of Lemma 5.1. In fact, analyzing the proof of Lemma 5.1, using appropriate localized versions of Lemma 3.8, Lemma 3.9, Lemma 3.10 and Lemma 3.12, localized in the sense that u does not have to be a solution in all of Ω f,2r 0 or Ω f,2 0 , we see that the conclusion of Lemma 5.2 is true. We omit further details.  = c(N, M ), 1 ≤ c < ∞, such that the following is true. Assume that u is a non-negative solution to Ku = 0 in Ω f,2r 0 and that u vanishes continuously on Δ f,2r 0 . Then Proof. Consider u = u(x, y, t) = u(x 1 , x , y 1 , y , t) as in the statement of the lemma and let v = v(x, y, t) = v(x 1 , x , y 1 , y , t) = u(x 1 , x , y 1 ±δ, y , t) for some δ > 0 small. Let r 0 = (r 0 −δ)/4. Then Kv = 0 in Ω f,2r 0 and v vanishes continuously on Δ f,2r 0 since we are assuming that the function defining Δ f,2r 0 is independent of the y 1 -coordinate. We can now apply Lemma 5.1 to the functions v and u, with r 0 , 0 , 1 replaced by r 0 , ˜ 0 , ˜ 1 , and conclude that whenever (x, y, t) ∈ Ω f, /c and 0 <˜ 0 ≤˜ 1 . We now fix ˜ 0 , ˜ 1 as above, and we claim that there exists c = c(N, M ), 1 ≤ c < ∞, such that whenever (x 1 , x , y 1 , y , t) ∈ Ω f,˜ 1 /c . To prove this we first make the trivial observations that, for any degree of freedom ε = ε(N, M ), 0 < ε 1, and assume that m − > 0. Then there exist constants Proof. Assuming that m − > 0 we see that Lemma 3.10 implies that m + > 0. By Lemma 5.3 we have 1 , and recall that We now consider the path which is a K-admissible such that By construction, the definition of the points A − ,Λ (z 0 , t 0 ), A ,Λ (z 0 , t 0 ), and the fact that the function defining Δ f,2r 0 is independent of the y 1 -coordinate, the path γ is contained in Ω f,2r 0 . Thus we can construct a Harnack chain connecting A ,Λ (z 0 , t 0 ) and γ( 2 ), based on which we can conclude that for some c = c(N, M ), 1 ≤ c < ∞. Note that the coordinates A − ,Λ (z 0 , t 0 ) and γ( 2 ) only differ in the y 1 -coordinate. In particular, using (5.29) we have The other inequality is proved analogously. 2

Proof of the main results
In this section we prove Theorem 1.1, Theorem 1.2 and Theorem 1.3. The proofs rely heavily on Lemma 5.3. We prove the theorems based on the set up concluded in Remark 3.8. Using a by now familiar argument it suffices to prove Theorem 1.1, Theorem 1.2 and Theorem 1.3 in the case (z 0 , t 0 ) = (0, 0) only. Thus, throughout this section we will assume (z 0 , t 0 ) = (0, 0). Furthermore, we again note that Lemma 3.10 implies, assuming m − > 0 in Theorem 1.1 and m − 1 > 0, m − 2 > 0 in Theorem 1.2, that m + > 0 and m + 1 > 0, m + 2 > 0.

Proof of Theorem 1.1
Assume that u is a non-negative solution to Ku = 0 in Ω f,2r 0 and that u vanishes continuously on Δ f,2r 0 . In the sequel, the constants Λ, c 0 , η, 0 , 1 will be chosen in accordance with Remark 3.8. Hence, to prove Theorem 1.1 we have to show that there exist constants whenever (z, t) ∈ Ω f, /c 1 and 0 < < 1 . Based on this we from now on consider 0 and , 0 < < 1 , as fixed. To start the proof we introduce where γ is the constant appearing in Lemma 3.9. Furthermore, we let By the definition of ˜ in (6.2) we see that Furthermore, using Lemma 3.9 we see that In the following we prove that there exists a constant c =c(N, M, m + /m − ), 1 ≤c < ∞, such that for this particular choice of ˜ . In fact, assuming (6.5) we first see, combining Lemma 3.12, (6.3), (6.4) and (6.5), that To prove (6.5) we let K 1 be an other degree of freedom based on which we divide the proof into two cases.
The case 0 /(8K) <˜ . In this case we immediately obtain from Lemma 3.9 that and the conclusion follows immediately.
The case ˜ ≤ 0 /(8K). In this case we first note, by the definition of ˜ , that <˜ < 0 and that h(2K˜ ) < h(˜ ), i.e., Using Lemma 3.12 we see that the above inequality implies that for some c = c(N, M ), 1 ≤ c < ∞. In the following we can, without loss of generality, assume that ˜ = 1. Based on this we let K = K/c and we introduce where TC stands for Thin Cylinder. Using this notation, (6.7) implies that u, (6.9) again for some c =c(N, M ), 1 ≤c < ∞. We emphasize that K is a degree of freedom which remains to be chosen. Furthermore, we can, by a redefinition of u, and without loss of generality, assume that sup TC f,2K u = 1. (6.10) Hence (6.9) becomes We now let We now use the following lemma, the proof of which we postpone to the next subsection.
Lemma 6.1. Let c and γ be as in (6.11). Then there exists K =K(N, M ), K 1, such that Using Lemma 6.1 and (6.11) we see that We can therefore conclude that In particular, using Lemma 3.12 we see that there exists ε, 0 < ε 1, depending on N and M , such that for every (z 1 , t 1 ) ∈ ΓK ,B ∩ Δ f,2r 0 . In the above inequality c is the constant appearing in Lemma 3.12. Then, using also Lemma 3.5, we can conclude that u(A + 1,Λ ) ≤ 2c u(z,t), (6.16) for some (z, t ) ∈Γ εK ,B , wherẽ To complete the proof we now use the following lemma, the proof of which we also postpone to the next subsection. Using Lemma 6.2 and (6.16) we can conclude that (6.5) also holds in this case. This completes the proof of Theorem 1.1 modulo the proofs of Lemma 6.1 and Lemma 6.2 given below. 2

Proof of Lemma 6.1 and Lemma 6.2
We here prove Lemma 6.1 and Lemma 6.2. We note that Lemma 6.2, together with Lemma 5.3, represent the main (novel) technical components of the paper.
Proof of Lemma 6.1. Using the normalization in (6.10) we see that Recall the sets Q · introduced in (4.8), and let λ, 1 ≤ λ K , be an additional degree of freedom. Let θ be Then θ is a (smooth) approximation of the characteristic function for the set ( defined by the function θ. Given K 1 we claim that there exist λ ≥ 1 and a constant c, both just depending on N , and hence independent of K , such that where the fundamental solution associated to K, Γ, is defined in (2.8). Using (6.21) we see that the bound from below in (6.20) follows from elementary estimates. Next, using (6.18), (6.20), and the maximum principle, we see that Note that (6.23) and that, by (2.8) and (2.9), we have (6.24) whenever θ(z) = 0 and for some harmless constant c, 1 ≤ c < ∞. In particular, combining the above we see that (6.25) and hence Lemma 6.1 follows for K large enough. 2 Proof of Lemma 6.2. Consider an arbitrary point (z, t ) ∈Γ εK ,B where Γ εK ,B is the set defined in (6.17). We want to prove that there exists a constant c, depending at most on N , M , and ε, such that u(z,t) ≤cu(A − 1,Λ ). (6.26) To do this we will construct a K-admissible path (γ(τ ), such that (γ(0), −1) = A − 1,Λ = (Λ, 0, 2 3 Λ, 0, −1) = (x 1 , x , y 1 , y , −1) =: (z, t), and an associated Harnack chain, targeting (z, t ) = (x 1 , x , ỹ 1 , ỹ , t ). Note that 3 ≥ −1 −t ≥ 3 − ε and hence (z, t) and (z, t ) are well separated in time. In the following we let δ := −1 −t. As the first step in the construction we construct a path γ (τ ) := (γ x (τ ), γ y (τ )) in R N −2 connecting z := (0, 0) to z := (x , ỹ ). Indeed we simply let γ (τ ) be the path in (3.6), i.e., we consider (γ (τ ), We now first note, using Remark 3.4 and the fact that (z, t ) ∈Γ εK ,B , that Furthermore, using (6.29), and the Lip K -character of f , we can conclude that there exists δ = δ (N, K, ε) = δ (N, M, ε), 0 < δ δ, such that In particular, using that (0, 0, 0) ∈ Δ f,2r 0 , (6.32), and the Lip K -character of f , we can conclude that there exists a constant c =c(N, (6.39) whenever 0 ≤ a ≤ b ≤ δ. Using (6.39) we will construct a finite sequence of real numbers {r j } k j=1 , and a sequence of points {(z j , t j )} k j=1 , such that (z 1 , t 1 ) = (z, t) and such that To start the construction we note, see (6.37), that we can in the following use that there exists ε =ε(N, M, ε), 0 <ε 1, such that and we will build a Harnack chain with r j =ε for all j. We construct {(z j , t j )} k j=1 inductively as follows. Let (z 1 , t 1 ) = (z, t) and assume that (z j , t j ) = (γ(τ j ), −1 −τ j ) has been constructed for some j ≥ 1. If τ j = δ, then the construction is stopped and we let k = j. If τ j < δ then we construct (z j+1 , t j+1 ) = (γ(τ j+1 ), −1 − τ j+1 ) by arguing as follows. There are two options, either (i) τ j +ε 2 β < δ or (ii) τ j +ε 2 β ≥ δ, (6.42) where β is the constant appearing in Lemma 2.1 and hence in the definition of the sets { Q − r k (z k , t k )}. We consider (i) first and we note that there are now two additional options: either If (i ) is true, then we set τ j+1 = τ j +ε 2 β, z j+1 = γ(τ j+1 ). If (ii ) is true, then we set We next consider (ii). In this case τ j ≥ δ −ε 2 β. Assume first that, in addition, In this case we set τ j+1 = δ, z j+1 = γ(τ j+1 ), and we can again conclude that (6.45) holds. If, on the contrary, (6.46) does not hold, then we set , and we again see that (6.45) holds. We note that by this construction there will be a first j such that τ j = δ and we then set k = j. The next step is to estimate k and we note that 0 < τ j+1 − τ j ≤ε 2 β for all j. Let I 1 denote the set of all index j for which either (i) + (ii ) or (ii), and the scenario leading up to (6.47), occur. Let I 2 denote the set of all index j for which either (i) + (i ) or (ii) + (i ), occur. Note the union of the sets I 1 and I 2 is the set of all indices occurring in the construction. Now, by continuity of ω(τ ) = (ω 1 (τ ), ω (τ )) = (ω 1 (τ ), . . . , ω m (τ )) we first see that In particular, Furthermore, we easily see that In particular, Hence, using (6.51), Lemma 3.3, Remark 3.5, the fact that 3 ≥ δ ≥ 3 − ε, and the explicit construction in In particular, combining (6.52) and (6.53) we see that the proof of Lemma 6.2 is complete. 2

Proof of Theorem 1.2
Assume that u and v are non-negative solutions to Ku = 0 in Ω f,2r 0 and that v and u vanish continuously on Δ f,2r 0 . Relying on the set up concluded in Remark 3.8 we introduce m ± 1 , m ± 2 , as in (1.28). As previously noted, the assumption min{m − 1 , m − 2 } > 0 implies min{m + 1 , m + 2 } > 0. We intend to prove that there exist constants c 1 = c 1 (N, M ) whenever (z, t), (z, t ) ∈ Ω f, /c 1 and 0 < < 1 . The proof is based on interior Hölder continuity estimates, Lemma 5.1, Lemma 5.2, Theorem 1.1 and its proof, see (6.5) in particular. To start the proof, let whenever (z, t) and ˜ are such that Q M,˜ (z, t) is contained in the closure of the set Ω f, 1 /(100c 1 ) , where c 1 are as in the statement of Theorem 1.1. Using Lemma 5.1, and the assumptions on m ± 1 , m ± 2 , we first see that O v,u (0, 0, 1 /c 1 ) < ∞. Let now be fixed and let ¯ = δ for some degree of freedom δ = δ(N, M ), 0 < δ 1, to be chosen. Consider 0 <˜ ≤¯ , pick (z, t) ∈ Ω f,¯ and let d = d K (z, t, Δ f,2r 0 ). Given ˜ , (z, t), d, we consider two cases: ˜ ≤ d (interior case) and ˜ > d (boundary case).

Further results: generalizations and extensions
In this section we briefly discuss, without giving the complete proofs, to the extent one can generalize Theorems 1.1, 1.2 and 1.3 to the context of a subset of the more general operators of Kolmogorov type considered in [7], [8] and [9]. In [7], [8] and [9] we considered Kolmogorov operators of the form where (z, t) ∈ R N × R, 1 ≤ m ≤ N . The coefficients a i,j and a i are bounded continuous functions and B = (b i,j ) i,j=1,...,N is a matrix of real constants. Following [7], [8] and [9] we here impose the structural assumptions a i,j (z, t)ξ i ξ j ≤ λ|ξ| 2 , ∀ ξ ∈ R m , (z, t) ∈ R N +1 .
[H.3] The coefficients a i,j (z, t) and a i (z, t) are bounded functions belonging to the Hölder space C 0,α K (R N +1 ), α ∈ (0, 1], defined with respect to the appropriate metric associated to L.
Following [20] we have that [H.4] is satisfied if (and only if) all the blocks denoted by * in (7.6) are null. Building on L we next construct a new operator L of Kolmogorov type by adding variables. Let m = κ, where κ ≥ 1 is an integer, and let N = N +m + 1. We now add the variables z = (z 1 , . . . , zm +1 ) and form the operatorL = ∂z 1z1 +m i=1z i ∂z i+1 + L (7.10) which we consider in Rm +1 × R N × R = RN × R. We emphasize that the operator L is independent of the variables (z 1 , . . . , zm +1 ). Furthermore, both L and L are operators of Kolmogorov type in the sense outlined above satisfying the structural assumptions [H.1]-[H.4].
Next, given > 0 and Λ > 0 we define the points z Λ,+ , z Λ,− ∈ Rm +1 as follows. We let Using this notation we introduce the following reference points.