NON–TANGENTIALLY ACCESSIBLE DOMAINS FOR VECTOR FIELDS

ROBERTO MONTI AND DANIELE MORBIDELLI

Abstract. We study non–tangentially accessible (NTA) domains for diagonal vector fields. We introduce a geometric notion of “admissible boundary” ensuring the NTA property. For general Hörmander vector fields, we prove that a domain with non characteristic boundary is NTA.

1. Introduction

Analysis of second order elliptic degenerate PDEs has been characterized in recent years by the prominence of metric aspects. Distances associated with second order operators appeared in the late 70s in the work of Nagel and Stein [NaSt], and then in the work of Fefferman and Phong [FP] on subellipticity of operators of the form ∑_{j,k=1}^{n} \frac{∂}{∂x_{j}} (a_{jk} \frac{∂}{∂x_{k}}), and in the work of Franchi and Lanconelli [FL1] on Hölder regularity for weak solutions of equations of the form ∑_{j=1}^{n} λ_{j}^{2} \frac{∂^{2}}{∂x_{j}^{2}} u = 0 in ℝ^{n}. The control distance associated with a system of vector fields X_{1}, …, X_{m} also played a central role in the work by Jerison [J2] on the Poincaré inequality and in the estimates of Sánchez Calle [S] for the fundamental solution of Hörmander operators. Deep structure theorems for such metrics were proved by Nagel, Stein and Wainger [NSW]. Finally, integral curves of the vector fields X_{j} also played a role in Bony’s paper [Bo].

After these seminal papers, the local and global theory of second order PDEs has been intensively studied from a metric point of view. The boundary behavior in Hölder spaces for the Dirichlet problem in the Heisenberg group has been studied in [J1]. Wiener criterion for Hörmander sum of squares has been studied in [NS], [Da] and [HaHu]. After the paper [J2], Poincaré inequalities for vector fields and functional analysis for Sobolev and BV functions have been studied in [L1], [L2], [FGW], [GN1],

[FLW] and [HK]. Jones’ theorems on extension of functions have been generalized to the Carnot–Carathéodory setting in [VG], [GN2] and [G1]. Properties of the trace at the boundary have been studied in [DGN1], [BP], [DGN2], [MM1], [BCX]. Properties of subelliptic harmonic measures have been studied in [CG], [CGN1], [CGN2], [FF1] and [FF2]. Finally, Fatou type theorems for positive subelliptic harmonic functions have been proved in [CG]. Several more references could be enumerated concerning non linear PDEs.

All the difficulties in the analysis at the boundary of a set Ω stem from characteristic points, i.e. points x ∈ ∂Ω where all the given vector fields X_{1}, …, X_{m} are tangent to the boundary. Note that in the Euclidean case (X_{j} = ∂_{j}, j = 1, …, n) there are no such points. If x ∈ ∂Ω is characteristic, then any integral curve of the vector fields starting from x is tangent to the boundary at x. On the other hand, if x is noncharacteristic then there exist integral curves transversal to the surface at x. This difference has a great influence on the size of the control balls and in their interplay with ∂Ω. The quantitative understanding of this phenomenon is the key point in problems at the boundary for degenerate PDEs. In general, nontrivial assumptions are expected to be added to the Euclidean regularity. This is suggested by the work [J2] (see also [MM2]) where examples of smooth sets whose boundary has a “cuspidal behavior” in the control metric are exhibited.

Date: 17th January 2004.

There are several definitions of regular domain which can be formulated in metric spaces, for instance (somehow from the weakest to the strongest) domain with the interior corkscrew property, domain with the twisted cone property (or John domain), (ε, δ)–domain (also called uniform domain), and non–tangentially accessible (briefly NTA) domain. Properties of PDE’s which can be established starting from these notions have been studied in many papers (see the list below). In the framework of vector fields the problem is that only few examples of such regular domains are known, and most of them are in the setting of homogeneous groups. In groups of step 2, bounded open sets with boundary of class C^{2} are known to be NTA (see [MM2], [CG], [CGP]). In groups of step 3, the cone property has been studied in [MM2]. In the specific case of the Heisenberg group, a C^{1} condition does not even guarantee the boundary accessibility through rectifiable curves (see [BM]); Carnot–Carathéodory balls are uniform (see [VG]) but not NTA (see [CG]), cubes centered at the origin are uniform (see [G2]); finally, the uniformity is preserved under quasi–conformal mappings (see [CT]). When no group structure is available no general result is known, except the easy fact that Carnot–Carathéodory balls are John domains. In the case of Grushin vector fields, a class of regular domains (called φ–Harnack domains) has been recently studied in [FF1]. A partial survey on such results can be found in [CGN].

In section 3 we begin our investigation by considering a general system of Hörmander vector fields. We prove the following

Theorem 1.1. A smooth, bounded open domain which is noncharacteristic for a system of Hörmander vector fields is NTA for the control distance.

This result, which answers a question raised in [DGN2], is natural but it was known only for step 2 homogeneous groups (see [CG] and see also [DGN2] for examples of noncharacteristic sets in groups of Heisenberg type). The class of noncharacteristic open sets is believed not to be very rich. On the other hand, it is known that the characteristic set has vanishing surface measure (see [D], [FW], and see also the recent references [Ba] and [Ma]), and regularity properties related to the noncharacteristic part of the boundary have been widely studied by several authors, see [KN], [D],

[BKS] and [FSSC]. In Example 3.4, we give examples of noncharacteristic sets for vector fields of step greater than 2, which naturally arise in the study of solutions of sublaplacians at the boundary of complex domains of the form {(z_{1}, z_{2}) ∈ ℂ^{2} : Im(z_{2}) > |f(z_{1})|^{2}}.

In section 4, which is the central part of the paper, we tackle the problem of characteristic points. We study NTA domains in a class of metric spaces generated by vector fields without any underlying group structure and with an arbitrarily high order of degeneration. We consider a system of diagonal vector fields in ℝ^{n} of the form

X_{1} = λ_{1}(x)∂_{1}, X_{2} = λ_{2}(x)∂_{2}, … X_{n} = λ_{n}(x)∂_{n},

whose control metric, under suitable assumptions on the functions λ_{j}, is known in detail (see [FL]). The basic model case we shall study can be exemplified in ℝ^{3} by the following vector fields

X_{1} = \frac{∂}{∂x_{1}} , X_{2} = x_{1}^{α_{1}} \frac{∂}{∂x_{2}} , X_{3} = x_{1}^{α_{1}}x_{2}^{α_{2}} \frac{∂}{∂x_{3}} , α_{1}, α_{2} ∈ ℕ.  (1.1)

Consider an open set in ℝ^{3} of the form Ω = {x_{3} > φ(x′)}, x′ = (x_{1}, x_{2}) ∈ ℝ^{2}, where φ ∈ C^{1}(ℝ^{2}). By the results of [FL], control balls can be written as Q(x′, r) × ]x_{3} − F_{3}(x, r), x_{3} + F_{3}(x, r)[, where Q(x′, r) are suitable rectangles in the plane and F_{3}(x, r) > 0. We say that the boundary ∂Ω is admissible if for all x′ ∈ ℝ^{2} and r > 0

∑_{i=1,2} osc(X^{i}φ, Q(x′, r)) ≤ C(r ∑_{i=1,2} |X^{i}φ(x′)|^{m} + osc(λ^{3}; Q(x′, r)),  (1.2)

where m is a power suitably dependent on the numbers α_{1} and α_{2} in (1.1). This inequality is a requirement on the oscillation of the derivatives of the function φ along the vector fields X_{1} and X_{2}. The first term in the right hand side vanishes exactly in the characteristic set, while the second one gives an amount of oscillation admitted also at characteristic points. This latter is determined by the oscillation of the function λ_{3}(x) = x_{1}^{α_{1}}x_{2}^{α_{2}}, which is strictly related to the size of control balls in the vertical direction. The balance between the two terms is a very delicate point and it turns out that the correct choice of the power is m = \frac{α_{1}+α_{2}+α_{1}α_{2}−1}{α_{1}+α_{2}+α_{1}α_{2}}. In Definition 4.9, generalizing (1.2), we introduce a class of domains with admissible boundary in the n−dimensional setting. The main result of the paper is the following

Theorem 1.2. Domains with admissible boundary are NTA.

The proof partially relies on some results in [MM3], where we prove that admissible domains are John domains. In the paper [MM3], we also show that the “homogeneous ball” Ω = {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : (|x_{1}|^{2(α_{1}+1)} +x_{2}^{2})^{1+α_{2}} +x_{3}^{2} < 1} has admissible boundary for the vector fields in (1.1) and we give a criterion for checking the admissibility of surfaces of the form x_{3} = g(|x_{1}|^{2(α_{1}+1)} + x_{2}^{2}). Our examples of NTA domains are the first ones in a setting different from homogeneous groups of step 2 and from diagonal vector fields in the plane. In Example 4.15, we also show that Carnot–Carathéodory balls not necessarily are uniform domains.

Notation. If u, v ≥ 0, we write u ≲ v for u ≤ Cv, where C ≥ 1 is an absolute constant. Analogously, u ≃ v stands for u ≲ v and v ≲ u. By d we denote the control metric induced on ℝ^{n} by a system of vector fields. For K ⊂ ℝ^{n} we write diam(K) = sup_{x,y∈K} d(x, y) and dist(x, K) = inf_{y∈K} d(x, y). The Lebesgue measure of a measurable set E ⊂ ℝ^{n} will be denoted by |E|. If γ : [0, 1] → ℝ^{n} is a curve and 0 ≤ a ≤ b ≤ 1, we denote by γ|_{[a,b]} the restriction of γ to the interval [a, b].

2. Basic definitions

In this section we recall all basic definitions and we prove some preliminary propositions that will be used later. We begin with the definition of the control metric associated with a family of vector fields.

Let X = (X_{1}, ..., X_{m}) be a system of vector fields X_{j} = ∑_{i=1}^{n} a_{ij} \frac{∂}{∂x_{i}}, j = 1, ..., m, where the functions a_{ij} are locally Lipschitz continuous in ℝ^{n}. A Lipschitz curve γ : [0, T] → ℝ^{n}, T ≥ 0, is X–subunit if there exists a measurable function h = (h_{1}, ..., h_{m}) : [0, T] → ℝ^{m} such that γ̇(t) = ∑_{j=1}^{m} h_{j}(t)X_{j}(γ(t)) for a.e. t ∈ [0, T] with |h(t)| ≤ 1 a.e. Define d : ℝ^{n} × ℝ^{n} → [0, +∞] by setting

d(x, y) = inf{T ≥ 0 : there exists a subunit curve γ : [0, T] → ℝ^{n}
such that γ(0) = x and γ(T) = y}.

(2.3)

If d(x, y) < +∞ for all x, y ∈ ℝ^{n} then d is a metric on ℝ^{n}, sometimes called control distance (or Carnot–Carathéodory, or sub–Riemannian metric). By Chow theorem, the function d is finite if the vector fields X_{j} ∈ C^{∞}(ℝ^{n}, ℝ^{n}) satisfy the Hörmander condition, see e.g. [NSW]. We denote by B(x, r) the balls in ℝ^{n} defined by the metric d.

Now we introduce domains with the corkscrew property, John domains, (ε, δ)−domains and non–tangentially accessible domains.

Definition 2.1. An open set Ω ⊂ (ℝ^{n}, d) satisfies the interior (exterior) corkscrew condition if there exist r_{0} > 0 and ε > 0 such that for all r ∈ (0, r_{0}) and x ∈ ∂Ω the set B(x, r) ∩ Ω (the set B(x, r) ∩ (ℝ^{n} \ \overline{Ω})) contains a ball of radius εr. An open set Ω satisfies the corkscrew condition if it satisfies both the interior and the exterior corkscrew condition.

Definition 2.2. An open set Ω ⊂ (ℝ^{n}, d) is a John domain (or a domain with the interior cone property) if there exist x_{0} ∈ Ω and σ > 0 such that for all x ∈ Ω there exists a continuous curve γ : [0, 1] → Ω such that γ(0) = x, γ(1) = x_{0} and

B(γ(t), σdiam(γ|_{[0,t]})) ⊂ Ω.  (2.4)

A curve satisfying (2.4) will be called a John curve of parameter σ.

Remark 2.3. If both Ω and ℝ^{n} \ Ω̄ are John domains then Ω satisfies the corkscrew condition.

Definition 2.4. An open set Ω ⊂ (ℝ^{n}, d) is a uniform domain if there exists ε > 0 such that for every x, y ∈ Ω there exists a continuous curve γ : [0, 1] → Ω such that γ(0) = x, γ(1) = y,

diam(γ) ≤ \frac{1}{ε}d(x, y),  (2.5)

and for all t ∈ [0, 1]

dist(γ(t), ∂Ω) ≥ ε min{diam(γ|_{[0,t]}), diam(γ|_{[t,1]})}.  (2.6)

It is known that for bounded domains the uniform property is equivalent to the (ε, δ)–property. Recall that the (ε, δ)–property has been introduced in [J] in the Euclidean case and in [GN2] for vector fields. This property requires that (2.5) and (2.6) hold only for all pairs of points x, y such that d(x, y) ≤ δ, where δ is a suitable positive number.

Remark 2.5. In the definition of John and uniform domain the curves γ are usually required to be rectifiable and the diameter is replaced by their length (see, for instance, [V]). Anyway, in metric spaces of homogeneous (doubling) type and with geodesics, as the metric spaces we are working with, these stronger definitions are equivalent to the weaker ones we are giving here (this is proved in [MS, Theorem 2.7]).

The notion of non–tangentially accessible domain was introduced in the Euclidean case by Jerison and Kenig in [JK], and then generalized to the setting of vector fields in [CG]. Let Ω ⊂ (ℝ^{n}, d) be an open set and α ≥ 1. A sequence of balls B_{0}, B_{1}, ..., B_{k} ⊂ Ω is an α–Harnack chain in Ω if B_{i} ∩ B_{i−1} ≠ ∅ for all i = 1, ..., k, and α^{−1}dist(B_{i}, ∂Ω) ≤ r(B_{i}) ≤ αdist(B_{i}, ∂Ω), where dist(B_{i}, ∂Ω) = inf_{x∈B_{i},y∈∂Ω} d(x, y) and r(B_{i}) is the radius of B_{i}.

Definition 2.6. A bounded open set Ω is a NTA domain in the metric space (ℝ^{n}, d) if the following conditions hold:

(i) there exists α ≥ 1 such that for all η > 0 and for all x, y ∈ Ω such that dist(x, ∂Ω) ≥ η, dist(y, ∂Ω) ≥ η and d(x, y) ≤ Cη for some C > 0, there exists an α−Harnack chain B_{0}, B_{1}, ..., B_{k} ⊂ Ω such that x ∈ B_{0}, y ∈ B_{k} and k depends on C but not on η;

(ii) Ω satisfies the corkscrew condition.

Remark 2.7. If Ω is a uniform domain according to Definition 2.4, then condition (i) in Definition 2.6 is fulfilled (see [CT, Proposition 4.2]).

The following Lemma gives a useful sufficient condition for an open set to be uniform. Roughly speaking, we prove that a domain is uniform if for any pair of points x and y there exist curves γ_{x} and γ_{y} moving far away from the boundary but not from each other.

Lemma 2.8. Let Ω ⊂ (ℝ^{n}, d) be an open set. Assume that there exist constants σ, C_{3}, C_{2} > 0 such that for all x, y ∈ Ω there are John curves γ_{x} : [0, t_{x}] → Ω and γ_{y} : [0, t_{y}] → Ω of parameter σ, with γ_{x}(0) = x and γ_{y}(0) = y and such that

diam(γ_{x}) ≥ C_{3}d(x, y),  (2.7)
d(γ_{x}(t_{x}), γ_{y}(t_{y})) ≤ \frac{σ}{2} C_{3}d(x, y),  (2.8)

and

max {diam(γ_{x}), diam(γ_{y})} ≤ C_{2}d(x, y).  (2.9)

Then Ω is a uniform domain.

Proof. There exists a continuous curve γ̃ joining the point γ_{x}(t_{x}) to the point γ_{y}(t_{y}) and satisfying the condition diam(γ̃) ≤ d(γ_{x}(t_{x}), γ_{y}(t_{y})). Consider the sum path γ = −γ_{y} + γ̃ + γ_{x}, where −γ_{y} stands for a reverse parameterization. We first show condition (2.5):

diam(γ) ≤ diam(γ_{x}) + diam(γ̃) + diam(γ_{y})

≤ C_{2}d(x, y) + \frac{σ}{2} C_{3}d(x, y) + C_{2}d(x, y) ≤ (\frac{σ}{2} C_{3} + 2C_{2})d(x, y).

Now we check (2.6). The proof also shows that Ω is arcwise connected. Take a point γ_{x}(t) with t ≤ t_{x}. Since γ_{x} is a John curve of parameter σ we have

dist(γ_{x}(t), ∂Ω) ≥ σdiam(γ_{x}|_{[0,t]}) ≥ σ min{diam(γ_{x}|_{[0,t]}), diam( − γ_{y} + γ̃ + γ_{x}|_{[t,t_{x}]})}.

The same argument works for a point γ_{y}(t), t ≤ t_{y}. Finally, given a point w ∈ γ̃, by the triangle inequality, (2.7) and (2.8) we get

dist(w, ∂Ω) ≥ dist(γ_{x}(t_{x}), ∂Ω) − d(w, γ_{x}(t_{x})) ≥ σdiam(γ_{x}) − \frac{σ}{2} C_{3}d(x, y)
≥ σdiam(γ_{x}) − \frac{σ}{2} diam(γ_{x}) = \frac{σ}{2} diam(γ_{x}).

In order to provide a lower bound for the last term it is enough to note that the hypotheses of the lemma ensure that diam(γ_{x}) ≃ diam(γ) through constants depending on σ, C_{3} and C_{2}. □

3. Non characteristic boundary for Hörmander vector fields

In this section we show that a bounded smooth domain without characteristic points is NTA with respect to the control metric induced by a system of Hörmander vector fields X = (X_{1}, ..., X_{m}). Recall that the system X is of Hörmander type in ℝ^{n} if the vector fields are smooth and for some p ∈ ℕ

span{X_{j_{1}}(x), [X_{j_{1}}, X_{j_{2}}](x), … [X_{j_{1}}, [X_{j_{2}}, …, [X_{j_{p−1}}, X_{j_{p}}]] ⋯](x) : j_{k} = 1, …, m}

has dimension n for any x ∈ ℝ^{n}. Here, [X_{j}, X_{k}] denotes the commutator of X_{j} and X_{k}.

A point x ∈ ∂Ω is characteristic if all the vector fields X_{1}, …, X_{m} are tangent to ∂Ω at x. We say that Ω is non characteristic if all its boundary points are non characteristic.

If x̄ ∈ ∂Ω is non characteristic and ν is a normal vector to ∂Ω at x, then we can find a vector field, say X_{m}, such that ⟨X_{m}(x), ν⟩ ≠ 0. By a standard argument, it can be shown that for a suitable neighborhood U of x̄, there exists a diffeomorphism Φ : U → Φ(U) such that dΦ(x)X_{m}(x) = ∂_{n} for all x ∈ U, Φ(x̄) = 0 and Φ(∂Ω ∩ U) ⊂ {y_{n} = 0}. Therefore, possibly performing such a change of variable, the vector fields can be assumed to be of the form

Y_{j} = b_{j}(y)∂_{y_{n}} + ∑_{i=1}^{n−1} a_{ij}(y)∂_{y_{i}}, j = 1, ..., m − 1, Y_{m} = ∂_{y_{n}}.

and we can consequently assume that Ω = {y_{n} > 0} in a neighborhood of the origin. The vector fields Y_{1}, ..., Y_{m} still satisfy Hörmander condition and induce the control metric d_{Y}. It is now easy to check that the new family of vector fields

X_{j} = ∑_{i=1}^{n−1} a_{ij}(y)∂_{i},  j = 1, ..., m − 1,  X_{m} = ∂_{n}.  (3.10)

still satisfies Hörmander condition. Moreover, if d_{X} is the corresponding control metric, it is not difficult to show (for a proof see [FW]) that there exist two constants c_{1} and c_{2} such that in a neighborhood of the origin we have c_{1}d_{Y} ≤ d_{X} ≤ c_{2}d_{Y}.

We give now an easy lemma.

Lemma 3.1. Let X_{1}, …, X_{m} be Hörmander vector fields of the form (3.10) with a_{ij} ∈ C^{∞}(ℝ^{n}). Then, for all (x′, x_{n}), (y′, y_{n}) ∈ ℝ^{n},

d((x′, x_{n}), (y′, y_{n})) ≥ |y_{n} − x_{n}| = d((x′, x_{n}), (x′, y_{n})).

Proof. Let γ : [0, T] → ℝ^{n} be a subunit curve connecting (x′, x_{n}) = γ(0) and (y′, y_{n}) = γ(T), for some T > 0. Then γ̇(t) = ∑_{j=1}^{m} h_{j}(t)X_{j}(γ(t)) with |h| ≤ 1 almost everywhere. Since all the vector fields X_{1}, …, X_{m−1} lie in ℝ^{n−1}, we have

|y_{n} − x_{n}| = | ∫_{0}^{T} h_{m}(t)dt| ≤ T.

Taking the infimum over all possible such curves we get d((x′, x_{n}), (y′, y_{n})) ≥ |y_{n}−x_{n}|.

Moreover, the subunit curve γ(t) = (x′, x_{n} + t sgn(y_{n} − x_{n})), 0 ≤ t ≤ |y_{n} − x_{n}|, is a geodesic between (x′, x_{n}) and (x′, y_{n}). Thus |y_{n} − x_{n}| = d((x′, x_{n}), (x′, y_{n})). □

Now we recall a deep result due to Nagel, Stein and Wainger. Given a system of Hörmander vector fields X_{1}, …, X_{m} and a compact set K ⊂ ℝ^{n}, denote by Y_{1}, …, Y_{q} a family of commutators which are of maximal rank at every point x ∈ K. Assign to any commutator Y in this family a degree equal to its length, that is write d(Y) = k if Y has length k ≥ 1. Given a n−tuple I = (i_{1}, …, i_{n}) ∈ {1, …, q}^{n}, write

B_{2}(x, r) = { exp (u_{1}Y_{1} + ⋯ + u_{q}Y_{q})(x) : |u_{j}| < r^{d(Y_{j})}}.

Theorem 3.2 ([NSW]). Let d_{2}(x, y) = inf{r > 0 : y ∈ B_{2}(x, r)}. Then d_{2} is locally equivalent to the control distance d.

Now we are ready to prove the main theorem of this section.

Theorem 3.3. A smooth non characteristic bounded domain with respect to a family of Hörmander vector fields is NTA for the control distance.

Proof. We prove that Ω is a uniform domain in the sense of definition 2.4, and by Remark 2.7 condition (i) in Definition 2.6 will be satisfied. Moreover, the proof will show that both Ω and ℝ^{n} \ Ω̄ are John domains and condition (ii) will be satisfied, as well.

By a general result of Väisälä [V, Theorem 4.1], it is sufficient to prove the uniform condition in a neighborhood of a fixed point x ∈ ∂Ω (see also [MM2, Proposition 2.5], where the same localization argument is described). Then, without loss of generality we assume that the vector fields are of the form (3.10) and Ω = {x ∈ ℝ^{n} : x_{n} > 0}. Consider two points x = (x′, x_{n}) and y = (y′, y_{n}) with x_{n}, y_{n} > 0 and assume for instance y_{n} ≥ x_{n}. First of all we define two John curves starting from x and y in the following way

\begin{aligned}γ_{x}(t) & = (x′, x_{n} + t),  0 < t ≤ y_{n} − x_{n} + d(x, y) := t_{x}, \\ γ_{y}(t) & = (y′, y_{n} + t),  0 < t ≤ d(x, y) := t_{y}. \end{aligned}

It is easy to check by Lemma 3.1 that γ_{x} and γ_{y} are John curves of parameter σ = 1.

Denote by x̃ = (x′, y_{n} + d(x, y)) and ỹ = (y′, y_{n} + d(x, y)) the endpoints of γ_{x} and γ_{y}. Let W_{1}, …, W_{q−1}, W_{q} = X_{n} be the family of all the commutators of sufficiently high length in order to apply Theorem 3.2. Note that each commutator W_{j}s, j = 1, 2, …, q − 1, has n−th component equal to zero. Then, by Theorem 3.2, we can write

ỹ = exp (u_{1}W_{1} + ⋯ + u_{q−1}W_{q−1} + u_{n}X_{n})(x̃)

for some u ∈ ℝ^{q} with |u_{j}| ≤ Cd(x̃, ỹ)^{deg(W_{j})}, |u_{q}| ≤ Cd(x̃, ỹ). Since x̃ and ỹ have the same n−th coordinate it must be u_{q} = 0. Define

γ̃(t) = exp (t(u_{1}W_{1} + ⋯ + u_{q−1}W_{q−1}))(x̃), 0 ≤ t ≤ 1.

By Theorem 3.2, we have diam(γ) ≤ Cd(x̃, ỹ). Moreover, by the triangle inequality and Lemma 3.1 d(x̃, ỹ) ≤ d(x̃, x) + d(x, y) + d(y, ỹ) ≲ d(x, y). Thus, condition (2.5) is satisfied. Finally, again by Lemma 3.1 we have dist(γ̃(t), ∂Ω) ≥ y_{n} + d(x, y). The path γ = −γ_{y} + γ̃ + γ_{x} satisfies all requirements of Definition 2.4. □

Example 3.4. Consider in ℝ^{3} the vector fields

X = \frac{∂}{∂x} + 2k|z|^{2k−2}y \frac{∂}{∂t},  Y = \frac{∂}{∂y} − 2k|z|^{2k−2}x\frac{∂}{∂t},  (3.11)

where (z, t) = (x, y, t) ∈ ℝ^{3}. The vector fields X and Y naturally arise in the analysis of the sublaplacian of the boundary of a domain in ℂ^{2}. Moreover, X and Y satisfy for any k ∈ ℕ Hörmander condition. When k = 1 we have the Heisenberg vector fields.

The open set

Ω = {(z, t) ∈ ℂ × ℝ : (|z|^{k} − 2)^{2} + t^{2} < 1}

is bounded and has boundary of class C^{∞}. We show that Ω is non characteristic for X and Y (see [DGN2] for the same example in the setting of the Heisenberg group k = 1). Thus, by Theorem 3.3 Ω is a NTA domain in the associated metric space. A defining function Φ = 0 for the boundary of Ω is Φ(z, t) = (|z|^{k} − 2)^{2} + t^{2} − 1. Since

\begin{aligned}XΦ(z, t) & = 2kx (|z|^{k} − 2) |z|^{k−2} + 2ky|z|^{2k−2}t \\ Y Φ(z, t) & = 2ky (|z|^{k} − 2) |z|^{k−2} − 2kx|z|^{2k−2}t, \end{aligned}

we find

|XΦ(z, t)|^{2} + |Y Φ(z, t)|^{2} = 4k^{2}|z|^{2k−2} {(|z|^{k} − 2)^{2} + |z|^{2k}t^{2}} .

The last expression never vanishes when (z, t) ∈ ∂Ω.

The vector fields X and Y appear in subelliptic analysis as follows. Let f(z) = z^{k}, where k ∈ ℕ is a fixed integer and z ∈ ℂ. We write (z_{1}, z_{2}) = (x_{1} +iy_{1}, x_{2} +iy_{2}) ∈ ℂ^{2}. Consider the domain D = {(z_{1}, z_{2}) ∈ ℂ^{2} : Im(z_{2}) > |f(z_{1})|^{2}} ⊂ ℂ^{2}. The holomorphic tangent vector field to the boundary of D is Z = \frac{∂}{∂z_{1}} + 2if′(z_{1})\overline{f(z_{1})}\frac{∂}{∂z_{2}} , where \frac{∂}{∂z_{k}} = \frac{1}{2}(\frac{∂}{∂x_{k}} − i\frac{∂}{∂y_{k}} ). In the tangential coordinates z = z_{1} and t = Re(z_{2}) we have

Z = \frac{∂}{∂z} + if′(z)\overline{f(z)}\frac{∂}{∂t}.

Writing Z = \frac{1}{2} (X − iY) we get the vector fields in (3.11). The subelliptic Laplacian arising from this situation has been studied in [GS] and [BGG].

4. Non–tangentially accessible domains for diagonal vector fields

In this section we describe the geometry of diagonal vector fields, we introduce a class of admissible domains and we show that they are NTA for the related control metric. Consider

X_{j} = λ_{j}(x)\frac{∂}{∂x_{j}} ,  j = 1, …, n,  (4.1)

where

λ_{1}(x) = 1 and λ_{j}(x) = ∏_{i=1}^{j−1} |x_{i}|^{α_{i}}, j = 2, ..., n.  (4.2)

We assume that

α_{i} = 0 or α_{i} ∈ [1, ∞[ , i = 1, …, n.  (4.3)

This condition ensures that the functions λ_{j}, and thus the vector fields X_{j}, are locally Lipschitz continuous.

Remark 4.1. If the numbers α_{i} are integers then the functions λ_{j} in (4.2) can be defined writing x_{i}^{α_{i}} in place of |x_{i}|^{α_{i}}. In this case the vector fields X_{1}, ..., X_{n} are smooth and satisfy the Hörmander condition. In this smooth case, all the definitions that follow remain unchanged and all the results still hold.

According to the definition in (2.3), the vector fields X_{1}, …, X_{n} define a control metric d in ℝ^{n}. Thanks to the special form (4.2) of the functions λ_{j}, the metric balls B(x, r) can be described rather explicitly. Following [FL], for all j = 1, ..., n define inductively the functions F_{j} : ℝ^{n} × [0, +∞) → [0, +∞) by

\begin{matrix}F_{1}(x, r) = r,  F_{2}(x, r) = rλ_{2}(|x_{1}| + F_{1}(x, r)), … \\ F_{j}(x, r) = rλ_{j}(|x_{1}| + r, |x_{2}| + F_{2}(x, r), …, |x_{j−1}| + F_{j−1}(x, r)). \end{matrix}  (4.4)

Equivalently, the definition can be also written in the following recursive way

F_{j+1}(x, r) = F_{j}(x, r)(|x_{j}| + F_{j}(x, r))^{α_{j}}.  (4.5)

Note that F_{j}(x, r) actually depends only on x_{1}, …, x_{j−1}. It is easy to check that r ↦ F_{j}(x, r) satisfies the following doubling property

F_{j}(x, 2r) ≤ CF_{j}(x, r), x ∈ ℝ^{n}, 0 < r < +∞,  (4.6)

for all j = 1, ..., n. Here and in the sequel C > 0 is an absolute constant. Moreover, an inspection of the form (4.4) of the functions F_{j} shows that

\begin{aligned}F_{j}(x, ϱr) & ≤ ϱF_{j}(x, r),  ϱ ≤ 1, r > 0 \\ (1 + η)F_{j}(x, r) & ≤ F_{j}(x, (1 + η)r),  η ≥ 0. \end{aligned}  (4.7)

Finally, since for any fixed x ∈ ℝ^{n} the function F_{j}(x, ·) is strictly increasing from [0, ∞[ onto itself, we denote its inverse by G_{j}(x, ·) = F_{j}(x, ·)^{−1}.

The following theorem proved in [FL] shows that the structure of the control balls B(x, r) can be described by means of the following boxes

Box(x, r) = {x + h : |h_{j}| < F_{j}(x, r), j = 1, ..., n}.  (4.8)

Theorem 4.2 ([FL]). There exists a constant C > 0 such that:

\begin{matrix}Box(x, C^{−1}r) ⊂ B(x, r) ⊂ Box(x, Cr),  x ∈ ℝ^{n}, r ∈ ]0, +∞[ , \\ C^{−1}d(x, y) ≤ ∑_{j=1}^{n} G_{j}(x, |y_{j} − x_{j}|) ≤ Cd(x, y),  x, y ∈ ℝ^{n}. \end{matrix}  (4.9)

Remark 4.3. Looking at the form of the vector fields, is easy to check that diam{x+ se_{k} : 0 ≤ s ≤ t} = d(x, x + te_{k}) = d((x_{1}, …, x_{k−1}, 0, …, 0), (x_{1}, …, x_{k−1}, t_{,}0 …, 0)) for all x ∈ ℝ^{n}, t ≥ 0, k = 1, …, n.

Before proceeding we introduce the following convention. If j = 1, ..., n and x = (x_{1}, ..., x_{n}) ∈ ℝ^{n} we write x̂_{j} = (x_{1}, ..., x_{j−1}, 0, x_{j+1}, ..., x_{n}) and we identify it with (x_{1}, ..., x_{j−1}, x_{j+1}, ..., x_{n}) ∈ ℝ^{n−1}.

Define

Box_{n}(x̂_{n}, r) = {x̂_{n} + ĥ_{n} : |h_{i}| < F_{i}(x̂_{n}, r), i = 1, …, n − 1}  (4.10)

and let Λ_{n}(x̂_{n}, r) = sup_{ŷ_{n}∈Box_{n}(x̂_{n},r)} |λ_{n}(ŷ_{n}) − λ_{n}(x̂_{n})|. For each j = 1, ..., n define induc-

tively the real number d_{j} by

d_{1} = 1, d_{2} = 1 + α_{1}, …, d_{j} = 1 + ∑_{i=1}^{j−1} d_{i}α_{i} = (1 + α_{1}) · . . . · (1 + α_{j−1}). (4.11)

We say that d_{j} is the degree of the variable x_{j}. Note that F_{j}(0, r) = r^{d_{j}}. The following proposition is proved in [MM3].

Proposition 4.4. There is η > 0 such that for all r > 0 and ϱ ∈ ]0, 1]

Λ_{n}(x̂_{n}, ϱr) ≤ h(ϱ)Λ_{n}(x̂_{n}, r), where h(ϱ) = \frac{ϱ}{ϱ + η(1 − ϱ)}.  (4.12)

Moreover, there exists a constant C > 0 such that Λ_{n}(x̂_{n}, r) ≤ (C/r)F_{n}(x, r) and Λ_{n}(x̂_{n}, r) ≥ r^{d_{n}−1}.

Denote in the following by c_{ϱ} any positive constant depending on ϱ > 0 such that c_{ϱ} → 0, as ϱ ↓ 0. The following Lemma holds.

Lemma 4.5. Box(y, r) ⊂ Box(x, (1 + c_{ϱ})r) for all x, y, r satisfying d(x, y) ≤ ϱr.

Proof. By definition, z ∈ Box(y, r) if and only if |z_{j}−y_{j}| ≤ F_{j}(y, r) for all j = 1, …, n. We need to prove

|z_{j} − x_{j}| ≤ F_{j}(x, (1 + c_{ϱ})r),  j = 1, …, n.  (4.13)

The assumptions of the lemma, Theorem 4.2 and the first inequality in (4.7) give

\begin{aligned}|z_{j} − x_{j}| & ≤ |z_{j} − y_{j}| + |y_{j} − x_{j}| \\ & ≤ F_{j}(y, r) + F_{j}(x, Cd(x, y)) ≤ F_{j}(y, r) + c_{ϱ}F_{j}(x, r). \end{aligned}  (4.14)

We claim that

F_{k}(y, r) ≤ F_{k}(x, (1 + c_{ϱ})r) for all k = 1, …, n.  (4.15)

If the claim is proved, then inserting (4.15) in (4.14) we conclude

|z_{j} − x_{j}| ≤ F_{j}(x, (1 + c_{ϱ})r) + c_{ϱ}F_{j}(x, r) ≤ (1 + c_{ϱ})F_{j}(x, (1 + c_{ϱ})r) ≤ F_{j}(x, (1 + c_{ϱ})^{2}r),

by (4.7) (in our notations (1 + c_{ϱ})^{2} = 1 + c_{ϱ}). Then the lemma is proved.

In order to show (4.15) we use induction on k. The statement is trivial for k = 1. If (4.15) holds for some k then by (4.5)

\begin{aligned}F_{k+1}(y, r) & = F_{k}(y, r)(|y_{k}| + F_{k}(y, r))^{α_{k}} \\ & ≤ F_{k}(x, (1 + c_{ϱ})r)(|x_{k}| + |y_{k} − x_{k}| + F_{k}(x, (1 + c_{ϱ})r))^{α_{k}}. \end{aligned}  (4.16)

Recall that, by Theorem 4.2, |y_{k} − x_{k}| ≤ F_{k}(x, Cd(x, y)) ≤ c_{ϱ}F_{k}(x, r), and

c_{ϱ}F_{k}(x, r) + F_{k}(x, (1 + c_{ϱ})r) ≤ (1 + c_{ϱ})F_{k}(x, (1 + c_{ϱ})r) ≤ F_{k}(x, (1 + c_{ϱ})^{2}r),

by (4.7). Inserting the last inequality into the second line of (4.16) we immediately conclude the proof of (4.15). □

Now we introduce our definition of admissible surface with respect to the vector fields X_{1}, …, X_{n} in (4.1) for surfaces of the type {x_{n} = φ(x̂_{n})}. We proceed as follows. First of all we give the definition of “admissible surface” for a graph of the form x_{n} = φ(x̂_{n}). This is the most degenerate case and contains all the difficulty of the problem. Then, we will show that a graph of the form x_{j} = φ(x̂_{j}), with j ≠ n, can be studied reducing to the previous case. Finally, in Definition 4.9 we introduce the notion of open set with admissible boundary.

Definition 4.6. Let φ ∈ C^{1}(ℝ^{n−1}). The surface {x_{n} = φ(x̂_{n})} is said to be admissible if there exist C > 0 and r_{0} > 0 such that, for all x̂_{n} ∈ ℝ^{n−1} and r ∈ ]0, r_{0}],

∑_{i≠n} osc(X^{i}φ, Box^{n}(x̂_{n}, r)) ≤ C(r ∑_{i≠n} |X^{i}φ(x̂_{n})|^{\frac{dn−2}{dn−1}} + Λ^{n}(x̂_{n}, r)).  (4.17)

Note that the exponent \frac{d_{n}−2}{d_{n}−1} is nonnegative as soon as at least one of the numbers α_{i} is non zero (otherwise we are in the Euclidean case).

In order to define admissible surfaces of the type {x_{j} = φ(x̂_{j})} when j ≠ n, we start with the following euristic remark. The variables x_{j+1}, …, x_{n} are “more degenerate” than x_{j}: the size of the balls in their direction is greater than the size in the j−th direction. This suggests that the behavior of the function φ with respect to the mentioned variables does not need to be controlled in a careful way.

To implement this idea, consider the new functions and vector fields

λ̃_{i}(x) = \begin{cases}λ_{i}(x) & if i ≤ j \\ λ_{j}(x) & if i ≥ j, \end{cases}  and  X̃_{i} = λ̃_{i}∂_{i}, i = 1, …, n.  (4.18)

The functions F̃_{j} and Λ̃_{j} are defined exactly as above using X̃_{1}, …, X̃_{n}. Define the boxes B̃ox_{j}(x̂_{j}, r) = {x̂_{j} + ĥ_{j} : |h_{i}| < F̃_{i}(x̂_{j}, r), i ≠ j} and denote by d̃ the metric constructed as in (2.3) using subunit curves with respect to the vector fields X̃ = (X̃_{1}, ..., X̃_{n}), and let B̃(x, r) be the corresponding balls. The new vector fields have the advantage that the variable x_{j} can be thought of as the n−th variable. In [MM3] we prove the following proposition describing some relations between d and d̃:

Proposition 4.7. The following properties hold:

(i) for any C_{1} > 0 there is C_{2} > 0 such that if |x_{j}|, |y_{j}|, r < C_{1} then B(x, r) ⊂ B̃(x, C_{2}r) and d̃(x, y) ≤ C_{2}d(x, y);

(ii) writing x′ = (x_{1}, ..., x_{j}) and x′′ = (x_{j+1}, ..., x_{n}), we have d((x′, x′′), (y′, x′′)) ≃ d̃((x′, x′′), (y′, x′′)).

Definition 4.8. Let φ ∈ C^{1}(ℝ^{n−1}). The surface {x_{j} = φ(x̂_{j})} is said to be admissible if there exist C > 0 and r_{0} > 0 such that for all x̂_{j} ∈ ℝ^{n−1} and r ∈ ]0, r_{0}]

∑_{i≠j} osc(X̃_{i}φ, B̃ox^{j}(x̂_{j}, r)) ≤ C(r ∑_{i≠j} |X̃_{i}φ(x̂_{j})|^{\frac{d_{j} −2}{d_{j} −1}} + Λ̃_{j}(x̂_{j}, r)).  (4.19)

Definitions 4.6 and 4.8 can be stated also for a bounded graph x_{j} = φ(x̂_{j}), where φ is defined on a bounded open set of ℝ^{n−1}.

Definition 4.9. A bounded open set Ω ⊂ ℝ^{n} is said to be with admissible boundary with respect to X if it is of class C^{1} and for all x ∈ ∂Ω there exists a neighborhood U of x such that ∂Ω ∩ U is an admissible surface according to Definitions 4.6 or 4.8.

Example 4.10. Consider in ℝ^{3} the vector fields

X_{1} = ∂_{1}, X_{2} = |x_{1}|^{α_{1}}∂_{2}, X_{3} = |x_{1}|^{α_{1}}|x_{2}|^{α_{2}}∂_{3},  (4.20)

with α_{1}, α_{2} ≥ 1. In [MM3] the open set

Ω = {(x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} : (|x_{1}|^{2(α_{1}+1)} + x_{2}^{2})^{1+α_{2}} + x_{3}^{2} < 1}

is proved to have admissible boundary with respect to X = (X_{1}, X_{2}, X_{3}) according to Definition 4.9. A key tool in the proof is the following proposition.

Proposition 4.11 ([MM3]). Let N(x_{1}, x_{2}) = |x_{1}|^{2(α_{1}+1)}+x_{2}^{2} and consider the function φ(x_{1}, x_{2}) = g(N(x_{1}, x_{2})), where g ∈ C^{2}(0, +∞) is such that for some C > 0

0 ≤ g′(t) ≤ Ct^{\frac{α_{2}−1}{2}} , |g′′(t)| ≤ C \frac{g′(t)}{t}, g′(2t) ≤ Cg′(t), t > 0.  (4.21)

Then the surface {x_{3} = φ(x_{1}, x_{2})} is admissible according to Definition 4.6.

Now we recall how to construct a John curve starting from a point in an open set with admissible boundary. The construction is taken from [MM3] and it relies on (4.17). Here, we study the uniform property, which is stronger than the cone condition. We need to deduce from (4.17) some deeper information describing how the John curve starting from a point x changes when the point x moves. This is done in Lemma 4.13.

Consider again an open set of the form Ω = {x_{n} > φ(x̂_{n})}, take a point x = x̂_{n} + x_{n}e_{n} ∈ Ω and introduce the following notation

ν_{i} = ν_{i}(x̂_{n}) = −∂_{i}φ(x̂_{n}) and N_{i} = \frac{ν_{i}}{|ν_{i}|}, if ν_{i} ≠ 0, i ≠ n.  (4.22)

In order to construct a John curve γ_{x} : [0, 1] → Ω starting from x, two different situations need to be distinguished:

max_{i<n} |X_{i}φ(x̂_{n})| ≤ λ_{n}(x̂_{n}) (Case 1),
max_{i<n} |X_{i}φ(x̂_{n})| > λ_{n}(x̂_{n}) (Case 2). (4.23)

In Case 1, the characteristic case, define the curve

γ_{x}(t) = x + te_{n} = x̂_{n} + (x_{n} + t)e_{n}, t ≥ 0.  (4.24)

In Case 2, the curve γ_{x} is defined in two steps. First of all, take any k = 1, …, n − 1 such that |X_{k}φ(x̂_{n})| is “maximal” in the following sense (this choice is not unique)

|X_{k}φ(x̂_{n})| ≥ \frac{1}{2} max_{i<n} |X_{i}φ(x̂_{n})| > \frac{1}{4}λ_{n}(x̂_{n}),  (4.25)

and let δ_{k}(x) be the solution of the following equation in the variable δ

Λ_{n}(x̂_{n}, δ) = ε_{0}|X_{k}φ(x̂_{n})|,  (4.26)

(the solution is unique because Λ_{n}(x̂_{n}, ·) is strictly increasing; here, ε_{0} > 0 is a suitable constant which depends on the surface and whose choice is discussed in [MM3]). Finally, define the positive time t(x) = t_{k}(x) by

t_{k}(x) = F_{k}(x, δ_{k}(x)).  (4.27)

The first piece of γ_{x} is defined for t ∈ [0, t_{k}(x)] by letting

γ_{x}(t) = x + tN_{k}e_{k}.  (4.28)

Here, N_{k} = N_{k}(x) depends on x. The number δ_{k}(x) essentially represents the diameter of the first piece of the path. The second piece is

γ(t) = x + t_{k}(x)N_{k}e_{k} + (t − t_{k}(x))e_{n},  t ≥ t_{k}(x).  (4.29)

The following theorem has been proved in [MM3].

Theorem 4.12. Assume that φ ∈ C^{1}(ℝ^{n−1}) satisfies (4.17). Let Ω = {x_{n} > φ(x̂_{n})}. Then there exists a constant σ > 0 such that: if x ∈ Ω and Case 1 holds, then the curve γ_{x} defined as in (4.24) is a John curve of parameter σ; if x ∈ Ω and Case 2 holds, then for any k such that (4.25) holds, the curve γ_{x} defined in (4.28)–(4.29) is a John curve of parameter σ.

Now we start the core of our discussion. For any x ∈ Ω for which Case 2 in (4.23) holds, fix a k = k(x) ∈ {1, …, n − 1} such that |X_{k}φ(x̂_{n})| = max_{i<n} |X_{i}φ(x̂_{n})|. Introduce now the parameter Δ(x) (equivalent to the diameter of the first piece of the path γ_{x} starting from x) as follows:

Δ(x) = \begin{cases}0 & if x satisfies (4.23), Case 1, \\ δ_{k}(x) & if x satisfies (4.23), Case 2, \end{cases}

where, if x satisfies Case 2, δ_{k}(x) is given by (4.26).

Let ϱ > 0 be a constant that will be fixed later. Given a pair of points x and y ∈ Ω, we distinguish two cases. The first case is

d(x, y) > ϱ max{Δ(x), Δ(y)} (Case A).

If Case A does not hold, assuming for instance Δ(x) ≥ Δ(y), it should be d(x, y) ≤ ϱΔ(x). Moreover, if k = k(x) is the number selected above, we can write Δ(x) = δ_{k}(x). Then the second case is

\{ \begin{matrix}|X_{k}φ(x̂_{n})| = max_{i≠n} |X_{i}φ(x̂_{n})| > λ_{n}(x̂_{n}) \\ d(x, y) ≤ ϱδ_{k}(x). \end{matrix}  (Case B)

Case B is the more delicate one. The problem here is that if the points x and y are very near and we want to connect them by a curve with total diameter comparable with d(x, y), we have to use only the first piece of the paths γ_{x} and γ_{y} starting from x and y. The following Lemma provides the suitable tools to prove that if y is near x (in other words, if we are in Case B and ϱ is small) then we can choose a John curve γ_{y} from y which starts in the same direction of the curve γ_{x} starting from x. This Lemma gives the correct bound on the oscillation of the horizontal derivatives X_{i}φ near characteristic points. The properties established in this lemma are crucial.

Lemma 4.13. Let φ ∈ C^{1}(ℝ^{n−1}) satisfy (4.17). There is a constant ϱ_{0} > 0 and a function ϱ ↦ c_{ϱ} from (0, ϱ_{0}) to ℝ^{+}, with lim_{ϱ↓0} c_{ϱ} = 0 and such that, if Case B holds for a pair of points x, y ∈ {x_{n} > φ(x̂_{n})} and for a number k = 1, …, n − 1, then we have

|X_{i}φ(x̂_{n}) − X_{i}φ(ŷ_{n})| ≤ c_{ϱ}|X_{k}φ(x̂_{n})| ∀ i = 1, …, n − 1,  (4.30)
|X_{k}φ(ŷ_{n})| ≥ (1 − c_{ϱ})λ_{n}(ŷ_{n}),  (4.31)

and, denoting by δ_{k}(y) the solution of (4.26) with ŷ_{n} instead of x̂_{n},

δ_{k}(y) ≥ \frac{1}{2}δ_{k}(x).  (4.32)

Using Lemma 4.13, whose proof will be given later, we can prove the main theorem of this section.

Theorem 4.14. If Ω ⊂ ℝ^{n} is an admissible domain for X_{1}, …, X_{n}, then it is a NTA domain in the metric space (ℝ^{n}, d).

Proof. We show that Ω is a uniform domain in the sense of Definition 2.4, and this will prove condition (i) in Definition 2.6. Condition (ii) is a direct consequence of Theorem 4.12.

It will be enough to consider the case Ω = {x_{n} > φ(x̂_{n})} where φ ∈ C^{1}(ℝ^{n−1}) is a function satisfying (4.17). We start the discussion with Case B. Let x, y ∈ Ω and k ∈ {1, …, n−1} be as in Case B for some ϱ > 0. The estimates provided by Lemma 4.13 and a choice of ϱ small enough easily imply

|X_{k}φ(ŷ_{n})| ≥ \frac{1}{2}|X_{i}φ(ŷ_{n})|, for all i ≠ n,  (4.33)
|X_{k}φ(ŷ_{n})| > \frac{1}{2}λ_{n}(ŷ_{n}).  (4.34)

By Theorem 4.12 and (4.25) there are two John curves γ_{x} and γ_{y} of parameter σ > 0, starting respectively from x and y, which are of the form (compare (4.28))

γ_{x}(t) = x + tN_{k}e_{k}, t ≤ t_{k}(x) and  γ_{y}(t) = y + tN_{k}e_{k}, t ≤ t_{k}(y). (4.35)

The numbers t_{k}(x) and t_{k}(y) are respectively defined by t_{k}(x) = F_{k}(x, δ_{k}(x)) and t_{k}(y) = F_{k}(y, δ_{k}(y)), where δ_{k}(x) and δ_{k}(y) are solutions of equation (4.26) written in x and y, respectively. Moreover, note that γ_{x} and γ_{y} are parallel. This is a consequence of the fact that (4.33) and (4.34) give (4.25) with y instead of x. In addition, X_{k}φ(x̂_{n}) and X_{k}φ(ŷ_{n}) must have the same sign by (4.30) and thus N_{k}(x) = N_{k}(y) (recall (4.22)). We denoted both by N_{k}.

We claim that if ϱ > 0 is small enough, there exist constants C_{2}, C_{3} > 0 (independent of x and y) and times t_{x} ≤ t_{k}(x) and t_{y} ≤ t_{k}(y) such that the curves γ_{x} and γ_{y} satisfy assumptions (2.7)–(2.9) of Lemma 2.8. This will show that Ω is a uniform domain.

Define the numbers

δ^{∗} = \frac{1}{2ϱ}d(x, y) and t^{∗} = F_{k}(x, δ^{∗}).  (4.36)

Since we are in Case B, we trivially have δ^{∗} ≤ \frac{δ_{k}(x)}{2} and by (4.32) δ^{∗} ≤ δ_{k}(y). It follows that t^{∗} ≤ t_{k}(x), t_{k}(y). We would like to apply Lemma 2.8 for the times t_{x} = t_{y} = t^{∗}. This would require the estimate (2.8), i.e. d(γ_{x}(t^{∗}), γ_{y}(t^{∗})) ≤ \frac{σC_{3}}{2} d(x, y). Unfortunately, it may happen that γ_{x}(t^{∗}) belongs (or is very near) to the plane {x_{k} = 0}. In this case the size of the boxes may become too small (this can be seen letting x_{k} = 0 in (4.4)) and estimate (2.8) does not seem to hold. To overcome this problem we operate as follows.

Consider the projection of x onto the k’th coordinate plane x_{k} = 0 and denote it by π(x) = ∑_{i≠k} x_{i}e_{i}. We distinguish the following two cases:

d(x + t^{∗}N_{k}e_{k}, π(x)) ≥ \frac{1}{4}d(x, π(x)), and  (4.37)
d(x + t^{∗}N_{k}e_{k}, π(x)) < \frac{1}{4}d(x, π(x)).  (4.38)

We first study case (4.37). Case (4.38) can be reduced to the first one (this is discussed after equation (4.44)). Choose t_{x} = t_{y} = t^{∗} and let γ_{x} : [0, t^{∗}] → Ω and γ_{y} : [0, t^{∗}] → Ω be as in (4.35). We first check (2.7), which is easier. By Theorem 4.2

diam(γ_{x}) ≥ C_{0}δ^{∗} = C_{0} \frac{d(x, y)}{2ϱ},  (4.39)

where C_{0} < 1 is an absolute constant. Then (2.7) holds with

C_{3} = \frac{C_{0}}{2ϱ} .  (4.40)

Now we have to check (2.8), which is

d(γ_{x}(t^{∗}), γ_{y}(t^{∗})) = d(x + t^{∗}N_{k}e_{k}, y + t^{∗}N_{k}e_{k}) ≤ \frac{σC_{0}}{4ϱ} d(x, y).  (4.41)

We claim that there exists a constant C_{4} > 0 independent of ϱ, x, y such that

d(γ_{x}(t^{∗}), γ_{y}(t^{∗})) ≤ C_{4}d(x, y),  (4.42)

whenever x satisfies (4.37). Then (4.41) follows choosing ϱ small enough to ensure C_{4} ≤ \frac{σC_{0}}{4ϱ}.

To prove (4.42), first of all notice that by Theorem 4.2 condition (4.37) implies G_{k}(π(x), |x_{k} + t^{∗}N_{k}e_{k}|) ≥ CG_{k}(π(x), |x_{k}|) and thus

|x_{k} + t^{∗}N_{k}e_{k}| ≥ F_{k}(π(x), CG_{k}(π(x), |x_{k}|)) ≥ CF_{k}(π(x), G_{k}(π(x), |x_{k}|)) = C|x_{k}|,

for some absolute (small) constant C. This estimate together with the explicit form (4.2) and (4.4) of the vector fields also implies

F_{i}(x + t^{∗}N_{k}e_{k}, s) ≥ ε_{1}F_{i}(x, s), ∀s > 0, i = 1, …, n,  (4.43)

where ε_{1} > 0 is a new absolute small constant. Then

|y_{i} − x_{i}| = F_{i}(x, G_{i}(x, |y_{i} − x_{i}|)) ≤ F_{i}(x, Cd(x, y)) ≤ ε_{1}^{−1}F_{i}(x + t^{∗}N_{k}e_{k}, Cd(x, y)).

This is equivalent to saying that y + t^{∗}N_{k}e_{k} ∈ Box(x + t^{∗}N_{k}e_{k}, Cd(x, y)), which gives (4.42) (by Theorem 4.2) provided C_{4} is large enough. Note that all such estimates do not depend on ϱ. This proves the claim (4.41).

We have proved hypotheses (2.7) and (2.8) of Lemma 2.8 under condition (4.37). We discuss later the turning condition (2.9).

Now we study case (4.38). We shall show that it can be essentially reduced to case (4.37). By continuity there is t^{∗∗} < t^{∗} such that

d(x + t^{∗∗}N_{k}e_{k}, π(x)) = \frac{1}{4}d(x, π(x)).  (4.44)

In this case we choose t_{x} = t_{y} = t^{∗∗} and we define define δ^{∗∗} by t^{∗∗} = F_{k}(x, δ^{∗∗}).

Now we are using shorter paths. We have to make sure that their diameter is large enough to ensure that (2.7) continue to hold. In order to check (2.7), notice that the triangle inequality and (4.38) give

d(x, π(x)) ≥ d(x, γ_{x}(t^{∗})) − d(π(x), γ_{x}(t^{∗})) > d(x, γ_{x}(t^{∗})) − \frac{1}{4}d(x, π(x)),

which yields d(x, π(x)) ≥ \frac{4}{5}d(x, γ_{x}(t^{∗})). Thus, by (4.44)

\begin{aligned}diam(γ_{x}|_{[0,t^{∗∗}]}) & ≥ d(x, π(x)) − d(γ_{x}(t^{∗∗}), π(x)) = \frac{3}{4}d(x, π(x)) \\ & ≥ \frac{3}{5}d(x, γ_{x}(t^{∗})) ≥ \frac{3}{5}C_{3}d(x, y), \end{aligned}

where C_{3} is given by (4.40). In other words, changing δ^{∗} with δ^{∗∗} does not give any problem in checking (2.7). We just have to modify slightly the constant C_{3} in (4.40).

Moreover, since (4.44) holds, we can prove (4.42) and ultimately (4.41) with t^{∗∗} instead of t^{∗}. This shows that (2.8) holds in case (4.38), as well.

In order to finish the proof of the theorem in Case B, we have to check condition (2.9). We check the upper bound for t^{∗}, which is greater than t^{∗∗}. The estimate diam(γ_{x}|_{[0,t^{∗}]}) ≲ C \frac{d(x,y)}{ϱ} follows from the definition of δ^{∗}. It remains to estimate the diameter of γ_{y}. Since by Theorem 4.2 diam(γ_{y}|_{[0,t^{∗}]}) ≃ G_{k}(y, t^{∗}), the proof is concluded as soon as we show that

G_{k}(y, t^{∗}) ≤ 2G_{k}(x, t^{∗}).

Since t^{∗} = F_{k}(x, δ^{∗}), the claim is equivalent to

G_{k}(y, F_{k}(x, δ^{∗})) ≤ 2δ^{∗} ⇔ F_{k}(x, δ^{∗}) ≤ F_{k}(y, 2δ^{∗}),

which holds (also with 1 + c_{ϱ} instead of 2) in force of (4.15) (in the statement of Lemma 4.5 x and y can be interchanged). The proof of Case B is concluded.

Case A is the easy part. Denote by x̃ e ỹ the endpoints of the paths γ_{x} e γ_{y} at the end of their first piece, i.e. x̃ = x + t_{k(x)}(x)N_{k(x)}e_{k(x)} and ỹ = y + t_{k(y)}(y)N_{k(y)}e_{k(y)}. Here k(x) may be different from k(y). This does not matter because the points are not too near. It could also be x̃ = x or ỹ = y if one or both of the points belong to Case 1 in (4.23). At any rate we have

d(x, x̃) ≤ Δ(x) ≤ \frac{1}{ϱ}d(x, y).

The same estimate holds for d(y, ỹ) (we are assuming Δ(x) ≥ Δ(y)). Here ϱ is small but has been fixed in the proof of Case B. We have the paths

γ_{x}(s) = x̃ + se_{n} and γ_{y}(s) = ỹ + se_{n},

with s ≥ 0. The proof of Case A can be concluded noting that by invariance with respect to translations along the n−th direction we have, independently of s,

d(x̃ + se_{n}, ỹ + se_{n}) = d(x̃, ỹ) ≤ d(x̃, x) + d(x, y) + d(ỹ, y) ≤ (\frac{1}{ϱ} + 1 + \frac{1}{ϱ})d(x, y).
□

Proof of Lemma 4.13. Fix k ∈ {1, …, n−1} such that |X_{k}φ(x̂_{n})| = max_{i=1,...,n−1}|X_{i}φ(x̂_{n})|. Then (4.17) gives

|X_{i}φ(x̂_{n}) − X_{i}φ(ŷ_{n})| ≤ osc (X_{i}φ, Box_{n}(x̂_{n}, d(x, y)))
≤ C(d(x, y)|X_{k}φ(x̂_{n})|^{\frac{dn−2}{dn−1}} + Λ_{n}(x̂_{n}, d(x, y)))
≤ C(ϱδ_{k}(x)|X_{k}φ(x̂_{n})|^{\frac{dn−2}{dn−1}} + CϱΛ_{n}(x̂_{n}, δ_{k}(x))),

by Case B and Proposition 4.4. Now, in order to estimate the right hand side note that by (4.26) Λ_{n}(x̂_{n}, δ_{k}(x)) = ε_{0}|X_{k}φ(x̂_{n})|. Moreover, by Proposition 4.4

δ_{k}(x) ≤ Λ_{n}(x̂_{n}, δ_{k}(x))^{1/(d_{n}−1)} = (ε_{0}|X_{k}φ(x̂_{n})|)^{1/(d_{n}−1)}.

Then (4.30) is proved. Letting i = k in (4.30) we get

|X_{k}φ(ŷ_{n})| ≥ (1 − c_{ϱ})|X_{k}φ(x̂_{n})|.  (4.45)

We are now ready to prove (4.31). By the definition of Λ_{n} we have

\begin{aligned}λ_{n}(ŷ_{n}) & ≤ λ_{n}(x̂_{n}) + Λ_{n}(x̂_{n}, d(x, y)) ≤ λ_{n}(x̂_{n}) + Λ_{n}(x̂_{n}, ϱδ_{k}(x)) \\ & ≤ λ_{n}(x̂_{n}) + c_{ϱ}ε_{0}|X_{k}φ(x̂_{n})| ≤ (1 + c_{ϱ})|X_{k}φ(ŷ_{n})|, \end{aligned}

where we used Case B to estimate λ_{n}(x̂_{n}) and (4.45). Then (4.31) is proved. We prove (4.32). By (4.45) and by the definition of δ_{k}, we have

\begin{aligned}Λ_{n}(ŷ_{n}, δ_{k}(y)) & = ε_{0}|X_{k}φ(ŷ_{n})| ≥ ε_{0}(1 − c_{ϱ})|X_{k}φ(x̂_{n})| \\ & ≥ (1 − c_{ϱ})Λ_{n}(x̂_{n}, δ_{k}(x)). \end{aligned}  (4.46)

Assume by contradiction that δ_{k}(y) < \frac{1}{2}δ_{k}(x). Then, we have

Box_{n}(ŷ_{n}, δ_{k}(y)) ⊂ Box_{n}(ŷ_{n}, \frac{1}{2}δ_{k}(x)) ⊂ Box_{n}(x̂_{n}, \frac{1}{2}(1 + c_{ϱ})δ_{k}(x)),

by Lemma 4.5 (recall that d(x, y) ≤ ϱδ_{k}(x), by Case B). Then

Λ_{n}(ŷ_{n}, δ_{k}(y)) = sup_{ẑ_{n}∈Box_{n}(ŷ_{n},δ_{k}(y))} |λ_{n}(ẑ_{n}) − λ_{n}(ŷ_{n})|

≤ Λ_{n}(x̂_{n}, \frac{1}{2}(1 + c_{ϱ})δ_{k}(x)) + |λ_{n}(x̂_{n}) − λ_{n}(ŷ_{n})|

≤ Λ_{n}(x̂_{n}, \frac{1}{2}(1 + c_{ϱ})δ_{k}(x)) + Λ_{n}(x̂_{n}, ϱδ_{k}(x))

≤ (h(\frac{1}{2}(1 + c_{ϱ})) + h(ϱ))Λ_{n}(x̂_{n}, δ_{k}(x)),

where h is the function introduced in Proposition 4.4. By the properties of h, we immediately see that the last chain of inequalities is in contradiction with (4.46), if ϱ is small enough. This finishes the proof of Lemma 4.13. □

Example 4.15. It is known that control balls defined by vector fields are always John domains (see [FGW] and [GN1]). We show that they not necessarily are uniform domains. Consider in ℝ^{2} the vector fields X_{1} = ∂_{1} and X_{2} = x_{1}∂_{2} and let (ℝ^{2}, d) be the metric space with metric defined as in (2.3). Applying Theorem 4.2 to this special case it is not difficult to see that

d((x_{1}, x_{2}), (0, y_{2})) ≃ |x_{1}| + |x_{2} − y_{2}|^{\frac{1}{2}} .  (4.47)

The ball B = B(0, 1) is a symmetric set with respect to x_{1} and x_{2} and can be computed explicitly (see for instance [F] and [Be]). Precisely,

∂B ∩ \{(x_{1}, x_{2}) : x_{1}, x_{2} ≥ 0\} = \{(x_{1}(ϑ), x_{2}(ϑ)) = (\frac{sin ϑ}{ϑ}, \frac{2ϑ − sin 2ϑ}{4ϑ^{2}}) : 0 ≤ ϑ ≤ π\}.

Notice that

(x_{1}(π), x_{2}(π)) = (0, \frac{1}{2π} ), (x′_{1}(π), x′_{2}(π)) = ( − \frac{1}{π} , −\frac{1}{π^{2}} ).

Then, all the points of the set \{x : x_{2} = \frac{1}{2π}(1 + |x_{1}|)\} belong to B, if |x_{1}| is small enough.

Take the points x^{+} = (x_{1}, \frac{1}{2π} (1 + |x_{1}|)) and x^{−} = ( − x_{1}, \frac{1}{2π} (1 + |x_{1}|)), where x_{1} > 0 is small. If γ : [0, 1] → B is a continuous curve joining the point x^{+} to the point x^{−}, then it must intersect the x_{2} axis. Call (0, y_{2}) this intersection point. It must be |y_{2}| < \frac{1}{2π}. Then by (4.47)

diam(γ) ≥ d((x_{1}, \frac{1}{2π} (1 + |x_{1}|)), (0, y_{2})) ≃ |x_{1}| + (\frac{1}{2π} (1 + |x_{1}|) − y_{2})^{\frac{1}{2}} ≥ (\frac{1}{2π} |x_{1}|)^{\frac{1}{2}} .

On the other hand, d(x^{+}, x^{−}) = 2|x_{1}|, and we find

diam(γ) ≥ \frac{C}{|x_{1}|^{1/2}} d((x_{1}, \frac{1}{2π} (1 + |x_{1}|)), ( − x_{1}, \frac{1}{2π} (1 + |x_{1}|)))

for some absolute constant C > 0. Letting x_{1} → 0 we see that condition (2.5) can not hold uniformly.

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Roberto Monti: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland

  E-mail address: roberto.monti@math-stat.unibe.ch

Daniele Morbidelli: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40127 Bologna, Italy

  E-mail address: morbidel@dm.unibo.it