Wiener-Landis criterion for Kolmogorov-type operators

We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.


Introduction
Aim of this paper is to establish a necessary and sufficient condition for the regularity of a boundary point for the Dirichlet problem related to a class of hypoelliptic evolution equations of Kolmogorov-type. Our criterion is inspired both to the Evans-Gariepy's Wiener test for the heat equation, and to a criterion by Landis, for the heat equation too, expressed in terms of a series of caloric potentials.
The partial differential operators we are dealing with are of the following type where A = (a i,j ) i,j=1,...,N and B = (b i,j ) i,j=1,...,N are N × N real and constant matrices, z = (x, t) = (x 1 , . . . , x N , t) is the point of R N +1 , ∇ = (∂ x1 , . . . , ∂ xN ), div and , stand for the gradient, the divergence and the inner product in R N , respectively. The matrix A is supposed to be symmetric and positive semidefinite. Moreover, letting E(s) := exp (−sB) , s ∈ R, we assume that the following Kalman condition is satisfied: the matrix is strictly positive definite for every t > 0. As it is quite well known, the condition C(t) > 0 for t > 0 is equivalent to the hypoellipticity of L in (1.1), i.e to the smoothness of u whenever Lu is smooth (see, e.g., [9]). We also assume the operator L to be homogeneous of degree two with respect to a group of dilations in R N +1 . As we will recall in Section 2, this is equivalent to assume A and B taking the blocks form (2.1) and (2.2). Under the above assumptions, one can apply results and techniques from potential theory in abstract Harmonic Spaces, as presented, e.g, in [2]. As a consequence, for every bounded open set Ω ⊆ R N +1 and for every function f ∈ C(∂Ω, R), the Dirichlet problem H Ω f (z) = f (z 0 ) ∀ f ∈ C(∂Ω, R). Aim of this paper is to obtain a characterization of the L-regular boundary points in terms of a serie involving L-potentials of regions in Ω c , the complement of Ω, within different level sets of Γ, the fundamental solution of L. More precisely, if z 0 ∈ ∂Ω and λ ∈]0, 1[ are fixed, we define for k ∈ N Ω c k (z 0 ) = z ∈ Ω c : Then, our main result is the following V Ω c k (z0) (z 0 ) = +∞.
Here and in what follows, if F is a compact subset of R N +1 , V F will denote the Lequilibrium potential of F , and cap (F ) will denote its L-capacity. We refer to Section 3 for the precise definitions.
From Theorem 1.1, one easily obtains a corollary resembling the Wiener test for the classical Laplace and Heat operators.
Let Ω be a bounded open subset of R N +1 and z 0 ∈ ∂Ω. The following statements hold: We can make the sufficient condition for the L-regularity more concrete and more geometrical with the following corollary.
then z 0 is L-regular. In particular, the L-regularity of z 0 is ensured if Ω has the exterior L-cone property at z 0 .
If E is a subset of either R N or R N +1 , |E| stands for the relative Lebesgue measure. Moreover, Q is the homogeneous dimension recalled in Section 2, and the L-cone property will be defined precisely in Section 7. We just mention here that it is a natural adaptation of the parabolic cone condition to the homogeneities of the operator L.
Before proceeding, we would like to comment on Theorem 1.1 and Corollary 1.2. A boundary point regularity test for the heat equation involving infinite sum of (caloric) potentials was showed by Landis in [12]. A similar test for a Kolmogorov equation in R 3 was obtained by Scornazzani in [14]. Our Theorem 1.1 contains, extends, and improves the criterion in [14]. The Wiener test for the heat equation was proved by Evans and Gariepy in [3]. The extension of such a criterion to the Kolmogorov operators (1.1) is an open, and seemingly difficult, problem. Our Corollary 1.2, which is a straightforward consequence of Theorem 1.1, is a Wiener-type test giving necessary and sufficient conditions which look "almost the same". As a matter of fact, in Theorem 1.1 we have considered the L-potentials of the compact sets Ω c k (z 0 ) which are built by the difference of two consecutive super-level sets of Γ(z 0 , ·). These level sets correspond with the sequence of values λ −k log k . The exact analogue of the Evans-Gariepy criterion would have required the sequence with integer exponents λ −k . The presence of the logarithmic term, which makes the growth of the exponents slightly superlinear, is crucial for our proof of Theorem 1.1. Moreover, such presence is also the responsible for the non-equivalence of the necessary and the sufficient condition in Corollary 1.2. To complete our historical comments, we mention that a potential analysis for Kolmogorov operators of the kind (1.1) first appeared in [14], in [4], and in [9]. We also mention that the cone criterion contained in Corollary 1.3 has been recently proved in [6], where such a boundary regularity test has been showed for classes of operators more general than (1.1). For further bibliographical notes concerning Wiener-type tests for both classical and degenerate operators, we refer the reader to [10].
The paper is organized as follows. In Section 2 we show some structural properties of L and fix some notations. Section 3 is devoted to the potential theory for L, while in Section 4 a crucial estimate of the ratio between the fundamental solution Γ at two different poles is proved. In Section 5 the only if part of Theorem 1.1 is proved. The if part, the core of our paper, is proved in Section 6, where the estimates of Section 4 play a crucial rôle. Section 7 is devoted to the proof of Corollary 1.2 and Corollary 1.3.

Structural properties of L
In [9, Section 1] it is proved that the operator L is left-translation invariant with respect to the Lie group K whose underlying manifold is R N +1 , endowed with the composition law Furthermore, a fundamental solution for L is given by We assume the operator L to be homogeneous of degree two with respect to a group of dilations. This last assumption, together with the hypoellipticity of L, implies that the matrices A and B take the following form with respect to some basis of R N (see again [9, Section 1]): for some p 0 × p 0 symmetric and positive definite matrix A 0 (p 0 ≤ N ), and where B j is a p j−1 × p j block with rank p j (j = 1, 2, ..., n), p 0 ≥ p 1 ≥ ... ≥ p n ≥ 1 and p 0 + p 1 + ... + p n = N . For such a choice we have trB = 0, and the family of automorphisms of K making L homogeneous of degree two can be taken as x (pi) ∈ R pi , i = 0, . . . , n, r > 0.
We denote by Q + 2 (= p 0 + 3p 1 + ... + (2n + 1)p n + 2) the homogeneous dimension of K with respect to (δ r ) r>0 . We explicitly remark that Q is the homogenous dimension of R N with respect to the dilations Under these notations, the matrix C(t) and the fundamental solution of L with pole at the origin can be written as follows ([9, Proposition 2.3], see also [7]): We observe that Γ is δ r -homogeneous of degree −Q.
Throughout the paper we denote by |·| the Euclidean norms in R N , R p k or R. We also denote, for x ∈ R N , For all x ∈ R N , we have where 4σ 2 C is the smallest eigenvalue of the positive definite matrix E T (1)C −1 (1)E(1). We recall that the homogeneous norm · : R N −→ R + is a D λ -homogeneous function of degree 1 defined as follows We call homogeneous cylinder of radius r > 0 centered at 0 the set , Remark 2.1. The norms · and |·| can be compared as follows On the other hand, for any x = 0, we get 3. Some recalls from Potential Theory for L: L-potentials and L-capacity We briefly collect here some notions and results from Potential Theory applied to the operator L.
For every open set Ω ⊆ R N +1 we denote and we call L-harmonic in Ω the functions in L(Ω).

We say that a bounded open set
(i) u is lower semi-continuous and u < ∞ in a dense subset of Ω; (ii) for every regular set V , V ⊆ Ω, and for every ϕ ∈ C(∂V, R), ϕ ≤ u| ∂V , it follows u ≥ h V ϕ in V. We will denote by L(Ω) the family of the L-superharmonic functions in Ω. Since the operator L endows R N +1 with a structure of β-harmonic space satisfying the Doob convergence property (see [13,2,6]), by the Wiener resolutivity theorem, for every f ∈ C(∂Ω), the Dirichlet problem has a generalized solution in the sense of Perron-Wiener-Bauer-Brelot given by The function H Ω f is C ∞ (Ω) and satisfies Lu = 0 in Ω in the classical sense. However, it is not true, in general, that H Ω f continuously takes the boundary values prescribed by is called L-regular for Ω. For our regularity criteria we still need a few more definitions. We denote by M(R N +1 ) the collection of all nonnegative Radon measure on R N +1 and we call We list some properties of the L-capacities cap. For every F , F 1 and F 2 compact subsets of R N +1 , we have: The properties (i) − (v) are quite standard, and they follow from the features of Γ. We want to spend few words on the last two properties. Property (vi) was proved in [8, Proposizione 5.1] in the case of the heat operator, namely with the capacity build on the Gauss-Weierstrass kernel. It can be proved verbatim proceeding in our situation: the main tools are the facts that Γ has integral 1 over R N , and it reproduces the solutions of the Cauchy problems. Property (vii) appears to be new even in the classical parabolic case (at least to the best of our knowledge), and it can be deduced readily from (vi). As a matter of fact, if a compact The last notions we need are the ones of reduced function and of balayage of 1 on F . They are respectively defined by From general balayage theory we have that V F is less or equal than 1 everywhere, identically 1 in the interior of F , it vanishes at infinity, is a superharmonic function on R N +1 and harmonic on R N +1 \∂F . Furthermore, the following properties will be useful for us. Let F, F 1 , F 2 be compact subsets of R N +1 , and let (F n ) n∈N be a sequence of compact subsets of R N +1 , we have:

A crucial estimate
We start by recalling the following identity, whose proof can be found in [9, Remark 2.1] (see also [7]), In what follows we will need the following lemma.
Lemma 4.1. For 0 > t > τ we have the following matrix inequality Proof. Since for symmetric positive definite matrices we have (see [5,Corollary 7.7.4]) and recalling that E −1 (t) = E(−t), it is enough to prove that From the very definition of the matrix C we get which proves (4.2) and the lemma.
A crucial role in the proof of our main theorem will be played by the ratio Γ(z,ζ) Γ(0,ζ) , for z = (x, t) and ζ = (ξ, τ ) with 0 > t > τ . We use the following notations Lemma 4.2. There exists a positive constant C such that, for any z = (x, t), ζ = (ξ, τ ) with 0 > t > τ and µ ≤ min Proof. In our notations we can write Let us deal with the exponential term Lemma 4.1 says in particular that we have Using this in (4.3) we get We are going to bound where A stands for the operator norm of a matrix A (i.e. its biggest eigenvalue for symmetric matrices). By (2.4), for any vector v with |v| = 1 we get Hence, since µ is also less than 1 2 , This gives On the other hand, by the commutation property (4.1), we get Plugging (4.5) and (4.6) in (4.4), we get

Necessary condition for regularity
The characterization in (3.3), together with the following lemma, will give the necessity of (1.3) in Theorem 1.1.
Lemma 5.1. For every fixed p ∈ N, let us split the set G r as follows Proof. From the monotonicity and subadditivity properties of the balayage, we have by (3.2), On the other hand, from the monotonicity and homogeneity properties of the capacities, it follows cap (G * p r ) ≤ cap (C r (z 0 )) = cap (z 0 • δ r (C 1 )) = r Q cap (C 1 (z 0 )). Hence cap (G * p r ) goes to zero as r goes to zero. This proves the lemma. Proof of necessary condition in Theorem 1.
We are going to prove the non regularity of the boundary point z 0 . The assumption implies that for every ε > 0, there exists p ∈ N such that On the other hand, with the notations of the previous lemma, for any positive r Then, from Lemma 5.1, we get lim r→0 V Gr (z 0 ) ≤ ε for every ε > 0, which implies Hence, by (3.3), the boundary point z 0 is not L-regular.

Sufficient condition for regularity
In this section we prove the if part of Theorem 1.1. This is the core of our main result and requires three lemmas.
Suppose also that the following two conditions hold true: Then we have V Gr (z 0 ) ≥ 1 2M 0 for every positive r.
Proof. Let A > 2 M0 , and fix any r > 0. Let us pick m, n ∈ N with m < n such that n k=m F k ⊆ G r and We are going to denote by G m,n = n k=m F k and by W m,n (z) = n k=m V F k (z). We want to estimate W m,n on G m,n . Take z ∈ G m,n . We have then z ∈ F h for some h ∈ {m, . . . , n}. Of course we have If we consider the function v m,n = 1 2+M0 n k=m VF k (z0) W m,n , we thus get v m,n ≤ 1 in O. Moreover, the function v m,n is a nonnegative H-superharmonic in R N +1 , it is H-harmonic in R N +1 G m,n , and it vanishes at the infinity. If we take any function u ∈ Φ Gm,n we have The maximum principle infers that u − v m,n has to be nonnegative in R N +1 G m,n . On the other hand, u ≥ 1 ≥ v m,n in G m,n . Therefore u ≥ v m,n in R N +1 , for every u ∈ Φ Gm,n . This implies that In particular this has to be true at z = z 0 , i.e.
Since the function s → s 2+M0s is increasing, we deduce This concludes the proof since V Gr ≥ V Gm,n .
In order to simplify the notations, from now on we assume z 0 = 0 ∈ ∂Ω. This is not restrictive because of the left-invariance property. We want to choose suitably the compact sets F k of the previous lemma. For any fixed λ ∈ (0, 1), we recall that . Now, we set α(k) = k log k and denote We fix q ∈ N such that and σ C , σ are the constants in (2.3) and (2.4). We also denote by p = 1 + q 2 = 1 + the integer part of q 2 .
So q 2 ≤ p ≤ 1 + q 2 < q − 1. For any k ∈ N we want to consider the sets .
Moreover, we put First we notice that, since kq + p < q(k + 1), F k lies strictly below F k+1 , namely Proof. We are going to prove that F (0) k is contained in a homogeneous cylinder C r k so that This is enough to prove the statement since and by monotonicity and homogeneity we have cap (F (0) k ) ≤ cap (C r k ) = cap (C 1 )r Q k . In order to prove (6.4), we have to find a good bound for r k . Fix z = (x, t) ∈ F (0) k . Since in particular z ∈ Ω c kq (0), we have On the other hand, by (4.1) and (2.3), we get and then Therefore, from (2.4), we deduce Let us remark that from our choice α(k) = k log k we can check that the sequence α(kq + p) − α(kq) is monotone increasing.
This fact and the fact that the functions s → s log β α s Q are increasing in the interval (0, e −β α 1 Q ] allow to bound the term x further. Indeed, having 0 < −t ≤ T * kq , we get Summing up, we have just proved that We are left with verifying (6.4) with this definition of r k . We have thus to prove that The sequences α(kq + 1) − α(kq + p) and α(kq + p) − α(kq) are asymptotically equivalent respectively to (1 − p) log(kq + p) and p log(kq + p). Hence, the series is equivalent to . This proves (6.4), and therefore the lemma. Lemma 6.3. There exists a positive constant M 0 such that Γ(z, ζ) Proof. Fix any h, k ∈ N with h = k. If h ≤ k − 1, then Γ(z, ζ) = 0 and the statement is trivial. Thus, suppose h ≥ k + 1. For every z = (x, t) ∈ F h and ζ = (ξ, τ ) ∈ F k we have .

By monotonicity we have
. By our choice of q (6.1) we have then This fact allows us to exploit Lemma 4.2 and get for some structural positive constant C. To prove the statement we need to show that the term is uniformly bounded for z ∈ F h and ζ ∈ F k . By estimating as in (6.5) we have and analogously In order to bound µM 2 (z)M 2 (ζ) we are thus going to estimate the term Since p < q − 1 and α(n + s) − α(n) is asymptotically equivalent to s log(n + s) as n goes to ∞, it is easy to check that the sequences A k and B h are convergent to 0. Therefore they are a fortiori bounded. This proves the lemma.
We can assume without loss of generality that i = 0, i.e.
Let us split the sets Ω c kq (0) as in (6.2). In this way we have defined the sequence of compact sets F k . We want to check that such a sequence satisfies the hypotheses of Lemma 6.1. First of all, from (6.3), we have that the F k 's are disjoint. Moreover, since F k ⊂ Ω c kq (0), it is easy to see that the sets converge from below to the point 0 (e.g., using that Γ(0, ·) is δ r -homogeneous of degree −Q). Lemma 6.3 provide the existence of a positive constant M 0 for which condition (ii) in Lemma 6.1 holds true. The last assumption we have to verify is the condition (i). To do this, we recall that the subadditivity of the equilibrium potentials implies that which is condition (i). Then, we can apply Lemma 6.1 and infer that V Gr (0) ≥ 1 2M0 for all positive r. The regularity of the point 0 is thus ensured by the characterization in (3.3).

The Wiener-type test, and the cone condition
In this section we want prove Corollary 1.2, and Corollary 1.3. First, we want to show how one can deduce the Wiener-type test of Corollary 1.2 from Theorem 1.1: it follows easily from the representation of the potentials (3.2).
The assertions (i) and (ii) directly follow from these inequalities, and from Theorem 1.1.
The main statement in Corollary 1.3 follows from the sufficient condition (i) we have just proved, and from (3.1). In fact, we have where we recall that T Q 2 k = c N λ k log k . Finally, we have to deal with the proof of the cone condition. To this aim, we need some definitions. We call L-cone of vertex 0 ∈ R N +1 a set of the form  We can now complete the proof.
Proof of Corollary 1.3. As we said, from (7.1) we get |Ω c k (z 0 )| λ Q+2 Q k log k and the first part of the proof follows. If we suppose that Ω has the exterior L-cone property at z 0 , we want to prove that the series on the r.h.s. is divergent. In particular, we are going to prove that the terms of that series are uniformly bigger than a positive constant, for k big enough. Without loss of generality, we can assume z 0 = 0. Denote F θ r := z ∈ R N +1 : for r > 0, and for θ > 1, and let r k = λ k log k . For any θ > 1 there existsk such that we have Ω c k (z 0 ) ⊇ F θ r k ∩ K R (B) ∀k ≥k.
On the other hand