Liouville results for fully nonlinear equations modeled on H\"ormander vector fields. I. The Heisenberg group

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the H\"ormander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.


Introduction
In this paper we study Liouville properties for viscosity sub-or supersolutions of fully nonlinear degenerate elliptic equations (1) F (x, u, Du, where F : R d × R × R d × S d → R is at least continuous and proper, i.e., nondecreasing in the second entry and non-increasing in the last entry (with respect to the partial order of symmetric matrices). Our main assumption is the existence of a family X = (X 1 , ..., X m ) of vector fields satisfying the Hörmander bracket generating condition and subunit for F in the following sense: for all i = 1, ..., m (2) sup γ>0 F (x, 0, p, I − γp ⊗ p) > 0 ∀p ∈ R d such that X i (x) · p = 0.
This generalizes the classical definition of subunit vectors for linear operators by Fefferman and Phong [22] and was introduced in our recent paper [8]. Typical examples are subelliptic equations of the form (3) G(x, u, Du, (D 2 X u) * ) = 0 , where (D 2 X u) ij = X i (X j u) is the intrinsic (or horizontal) Hessian associated to X , Y * is the symmetrized matrix of Y , and G is proper and strictly decreasing with respect to the last entry.
Before explaining our results, let us recall some of the many Liouville-type properties for elliptic equations known in the literature, the most related to our work. The classical Liouville theorem for harmonic functions on the whole space states that the only harmonic functions in R d bounded from above or below are constants, and it is a consequence of mean-value formulas or, more generally, of the Harnack inequality. Such result actually holds for classical solutions to more general uniformly elliptic equations, provided the zeroth order coefficient has the appropriate sign for the maximum principle, and the equation is homogeneous, see, e.g., the monograph [26]. For inhomogeneous equations the property is false, for instance ∆(|x| 2 ) = 2d in R d .
The Liouville property holds also in the much larger class of merely subharmonic functions (i.e., subsolutions of −∆u = 0) bounded from above if the space dimension is d = 2, by exploiting the behavior of the fundamental solution log |x| and using the Hadamard Three-Circle Theorem (see, e.g., [38,Theorem 2.29], or Theorem 2.1 below for a different proof). However, this result fails in higher dimensions d ≥ 3: for instance, u 1 (x) := −(1 + |x| 2 ) −1/2 and u 2 (x) := −(1 + |x| 2 ) −1 are nonpositive subharmonic functions in R 3 and, respectively, in R d with d ≥ 4.
For linear degenerate elliptic equations, mean-value properties and Harnack-type inequalities were proved in many cases, typically for vector fields X that generate a stratified Lie group, and Liouville theorems for solutions to such equations were proved, e.g., in [17,12,29,30], see also the references therein. On the other hand, one does not expect the Liouville property for sub-or supersolutions to −∆ X u = 0 when X generates a Carnot group, because the intrinsic dimension of this geometry is larger than 2, which is the maximal one for subharmonic functions. In fact, in Section 4.2 we give simple explicit examples of bounded, non-constant, classical suband supersolutions of the sub-Laplace equation in any Heisenberg group H d .
Liouville theorems for nonlinear elliptic equations were first considered by Gidas and Spruck [25] for semilinear equations and then widely investigated, also in the subelliptic and in the quasilinear settings, see, e.g., [10,16,17,2,13,14] and the references therein.
For fully nonlinear equations as (1), in the simpler form F (x, D 2 u) = 0 and uniformly elliptic, it was proved in [15,Section 4.3 Remark 4] that continuous viscosity solutions either bounded from above or below are constant. We recall that uniform ellipticity with parameters Λ ≥ λ > 0 can be defined by means of Pucci's extremal operators (see their definition in Section 3.3) as (4) M − λ,Λ (M ) ≤ F (x, r, p, M ) − F (x, r, p, 0) ≤ M + λ,Λ (M ) for all symmetric matrices M . The result is a consequence of the Harnack inequality and comparison with Pucci's operators. Further related results for solutions of PDEs of the form F (D 2 u) = 0 can be found in [35,Section 1.7] and [4,Theorem 1.7], and in [36,Theorem 1.5] for equations with F depending also on x and Du.
The first results for mere sub-or supersolutions of F (x, D 2 u) = 0 are due to A. Cutrì and F. Leoni [20]. They proved that if u ∈ C(R d ) is either bounded above and satisfying (5) M + λ,Λ (D 2 u) ≤ 0 in R d in viscosity sense, or bounded below and satisfying  (5) and (6) with D 2 u replaced by (D 2 H d u) * . Here the condition d ≤ Λ λ + 1 is replaced by Q ≤ Λ λ + 1, Q = 2d + 2 standing for the homogeneous dimension of H d . An example of classical subsolution violating the Liouville property when Q > Λ λ + 1 is in Section 4.2. This is consistent with the aforementioned failure of Liouville properties for subharmonic functions in the Heisenberg group.
In [20] the authors also prove Liouville results for sub-and supersolutions of F (x, D 2 u)+ u p = 0 with F uniformly elliptic, F (x, 0) = 0 and p in a suitable range. This was recently extended to Carnot groups of Heisenberg type in [27]. See also [3] and [31] for related results.
Liouville properties for PDEs involving gradient terms of the form were first investigated by Capuzzo Dolcetta and Cutrì [18]. They assume that g is bounded and such that for |x| large, and use suitable extensions of the Hadamard three-sphere theorem. Note that this is a smallness condition at infinity on the first order terms of the PDE. See also [27] for similar recent results on the Heisenberg group. Related papers for fully nonlinear PDEs with gradient dependence are [40], [36] and [19].
A new approach to Liouville properties for sub-and supersolutions of Hamilton-Jacobi-Bellman elliptic equations involving operators of Ornstein-Uhlenbeck type was introduced in [6], based on the strong maximum principle and the existence of a sort of Lyapunov function for the equation. It was applied in [5] to fully nonlinear uniformly elliptic equations of the form (1) and to some quasilinear hypoelliptic equations, under assumptions on the sign of the coefficients of the first and zero-th order terms, and on their size. Here these terms must be large for large |x|, contrary to the results quoted above. In the case of Pucci's operators the results of [5] are different from those in [20] and fit better the treatment of uniformly elliptic equations via the inequalities (4). In Section 4.2 we give examples showing their optimality. The paper [33] treats a linear equation on the Heisenberg group in the same spirit.
In the present paper we study Liouville properties for viscosity sub-and supersolutions of equations of the form (1) under the condition (2) for a Hörmander family X . Our main motivation are equations of the form (3), uniformly subelliptic in the sense that G satisfies the inequalities (4) with Pucci operators M ± λ,Λ acting on m-dimensional instead of d-dimensional symmetric matrices. In the first part we prove a general result under two additional assumptions, that for subsolutions are: a subadditivity condition ((S1) of Section 3.1) which for linear equations corresponds to the homogeneity, and the existence of a Lyapunov-like function w such that lim |x|→∞ w(x) = +∞ and supersolution of (1) out of a large ball (symmetric assumptions are made for supersolutions). Here we adapt the approach of [5] to degenerate equations by means of the new strong maximum and minimum principles obtained by the authors in the recent paper [8] using the generalized subunit vectors for fully nonlinear equations (2).
In the second part of the paper we find more explicit sufficient conditions for the Liouville properties in the case of the Heisenberg group H d by taking w = log ρ as Lyapunov function, where ρ is a norm 1-homogeneous with respect to the dilations of the group H d . As must be expected from the results quoted before, these assumptions concern the sign and the strength of either the first or the zero-th order terms in the equation (or both).
They are related to recurrence conditions in the probabilistic literature, and are a form of dissipativity (cfr., e.g., [37]). An example of our results, for the uniformly subelliptic equation H d u are the horizontal gradient and Hessian in H d , is the following: if with c α ≥ 0, we prove the Liouville property for subsolutions under the condition .., d, and Q = 2d + 2 is the homogeneous dimension of H d . This condition is satisfied if either c α > 0 or b α (x) · η < 0 for x large, and under suitable growth conditions at infinity of the data. In Section 4.2 we use again the norm ρ to discuss the sharpness of this condition.
In our companion paper [7] we apply the general results of Section 3 to other classical families of Hörmander vector fields, namely, the generators of free step 2 Carnot groups, and Grushin-type fields, whose associated geometry has not a group structure.
It is well known that Liouville properties have many applications to various issues. We are motivated in particular by their consequences in ergodic problems, large time stabilization in parabolic equations, and singular perturbation problems, as in, e.g., [5,6,33,34]. For other forms of Liouville-type theorems for different equations let us also mention [9] in the Heisenberg group, the recent paper [32] for PDEs arising in conformal geometry, and [11] for versions of Pucci's extremal equations with different degeneracies than in our work.
The paper is organised as follows. In section 2 we explain the approach to Liouville properties based on Lyapunov functions and strong maximum principles in the simple case of classical subsolutions of linear equations, for the reader's convenience, and discuss some related literature. Section 3 presents an abstract result and its various applications to nonlinear equations with general Hörmander vector fields. In Section 4 we study PDEs involving the generators of the Heisenberg group H d , in the form (7) where only the horizontal gradient appears, as well as in the form (3) involving the Euclidean gradient. Section 4.2 makes a detailed comparison with the literature, in the cases of R d and H d , and discusses by means of explicit examples the optimality of the sufficient conditions for Liouville properties.

A glimpse on the method of proof for linear equations
Before showing our main results, we present the proof of a Liouville-type theorem for classical C 2 subsolutions to linear uniformly elliptic equations in the Euclidean framework, which serves as a guideline for our proof in the nonlinear and subelliptic setting. It uses only classical arguments such as strong maximum and comparison principles, but not Harnack inequalities.
Let also u ∈ C 2 (R d ) be such that Lu ≤ 0 and 0 ≤ u(x) ≤ C in R d . Then u is constant.
Remark 2.2. This result is essentially a special case, e.g., of [5, Theorem 2.1] and applies to the case of the Laplacian (i.e. a ij = δ ij and b = 0) when d ≤ 2; therefore it gives a different proof of the Liouville theorem for subsolutions, see e.g. [38,Theorem 2.29]. Indeed, the function w := log |x| fulfills the above assumptions, giving thus that every subharmonic function bounded from above is constant. However, as pointed out in the introduction, this is not the case when d ≥ 3, where w is no longer a classical supersolution of −∆u = 0. A more general result in the context of Riemannian manifolds can be found in [28,Corollary 7.7] under the same sufficient conditions (i)-(ii). It applies to subsolutions of −∆u + b(x) · Du = 0 in any space dimension d under assumptions on the drift b implying the existence of a Lyapunov-like function w (cfr. [5]).
Lv ζ = Lu − ζLw ≤ 0 for every x such that |x| >R . Since On one hand, letting ζ → 0 we conclude On the other hand, owing to the weak maximum principle in the set B(0,R) we obtain u, for all |y| <R .
Combining the above inequalities one concludes Hence, u attains its nonnegative maximum at some point of ∂B(0,R) and then the conclusion follows by the strong maximum principle for classical linear uniformly elliptic equations [26, Theorem 3.5].
Remark 2.3. The same result remains true if L is replaced by a degenerate elliptic operator provided the vector fields X satisfy the Hörmander condition and b : R m → R n , n ≤ m is smooth, the proof being exactly the same thanks to Bony strong maximum principle for subelliptic equations. An example of such result is [33, Proposition 3.1]. Note also that the assumption u ≤ C can be replaced by lim sup |x|→∞ u(x)/w(x) ≤ 0, whereas the sign condition u ≥ 0 can be dropped if c ≡ 0.
We further remark that, when b ≡ 0, L reduces to a Schrödinger-type operator. When u is a solution of the equation −∆u+cu = 0, the Liouville property is proved in [28,Corollary 13.7] under the same sufficient conditions (i)-(ii) on Riemannian manifolds, and it is connected to recurrence and non-explosive properties of Brownian motions on Riemannian manifolds, see [28, Theorem 5.1 and Section 13.2]. Our result is more general in that it allows u to be merely a subsolution to the equation. We also refer to [37] for a control theoretic interpretation of the Liouville property for Ornstein-Uhlenbeck operators.
3. The general case 3.1. An abstract result. In this Section we consider a general equation of the form We will denote F [u] := F (x, u, Du, D 2 u) and make the following assumptions and F (x, r, 0, 0) ≥ 0 for every x ∈ Ω and r ≥ 0. (ii) F satisfies the comparison principle in any bounded open set Ω, namely, if u and v are, respectively, a viscosity sub-and supersolution of (8) such that u ≤ v on ∂Ω, then u ≤ v in Ω. (iii) There exists R o ≥ 0 and w ∈ LSC(R d ) viscosity supersolution of (8) for |x| > R o and satisfying lim |x|→∞ w(x) = +∞. (iv) F satisfies the strong maximum principle, namely, any viscosity subsolution of (8) that attains an non-negative maximum must be constant.
To prove the analogous results for viscosity supersolutions we need to replace (i) and (iii)-(iv) above by The next result extends the proof of Theorem 2.1 for linear equations to the fully nonlinear degenerate setting. Its proof is essentially the same as the one done in [5] for HJB equations, so we only outline it for the reader's convenience.
for w as in (iii). If u ≥ 0, then u is constant.
As a consequence, we have Then, for all ζ > 0 there exists R ζ > R o such that The main step is proving that u ζ is a viscosity subsolution of r). Indeed, takex ∈ B(x, r) and ψ smooth such that ζw + ϕ − k − ψ has a minimum atx. Using that w is a viscosity supersolution to (8), F proper, (S1), and (13) we get Then and hence u attains its maximumx over R d . Since u ≥ 0 the SMP gives the desired conclusion.
Remark 3.2. Note that if u is bounded above, then (9) is satisfied.
The next result says that the assumption u ≥ 0 can be dropped provided r → F (x, r, p, X) is constant: this will be the case for some HJB operators we discuss in the next sections. Corollary 3.3. Assume (i)-(iv). Let u ∈ USC(R d ) be a viscosity subsolution to (8) satisfying (9) for w as in (iii). Assume r → F (x, r, p, X) is constant for all x, p, X and F (x, r, 0, 0) = 0 for every x ∈ Ω. Then u is constant.
Proof. The proof goes along the same lines as Proposition 3.1. It is sufficient to note that since r → F (x, r, p, X) is constant for all x, p, X, u + |u(x)|,x standing for the maximum point in Proposition 3.1, is again a subsolution, and one concludes.
A symmetric result holds for the case of supersolutions to (8), see [5] for the details of the proof.
Then v is constant.

3.2.
Equations with Hörmander vector fields. In this section we discuss Liouville properties for PDEs over Hörmander vector fields. We recall that the vector fields Z 1 , ..., Z m satisfy the Hörmander's rank condition if (H) the vector fields are smooth and the Lie algebra generated by them has full rank d at each point.
The classical smoothness requirement on Z i is C ∞ , but it can be reduced to C k for a suitable k, and considerably more if the Lie brackets are interpreted in a generalized sense, see [23] and the references therein. Before stating the main result for subsolutions, we recall a crucial scaling assumption for the validity of the strong maximum principle for fully nonlinear subelliptic equations together with the concept of generalized subunit vector field.
The proof is a simple consequence of Proposition 3.4, and the strong minimum principle Corollary 2.12 in [8].
Let v ∈ LSC(R d ) be a viscosity supersolution to (8) satisfying (14) for W as in (iii'). Assume either v ≤ 0, or r → F (x, r, p, X) is constant for all x, p, X and F (x, r, 0, 0) = 0 for every x ∈ Ω.
Then v is constant.
Next we apply the last two theorems to subelliptic equations of the form ..., X m ) are C 1,1 vector fields on R d satisfying the Hörmander condition (H), D X u := (X 1 u, ..., X m u), (D 2 X u) ij := X i (X j u), and Y * is the symmetrized matrix of Y . Here G : R d × R × R m × S m → R is at least continuous, proper, satisfying (S1), and it is elliptic for any x and p fixed in the following sense: After choosing a basis in Euclidean space we write X j = σ j · D, with σ j : R d → R d , and σ = σ(x) = [σ 1 (x), ..., σ m (x)] ∈ R d×m . Then Therefore, for u ∈ C 2 , and we can rewrite the equation (15) in Euclidean coordinates, i.e., in the form F (x, u, Du, D 2 u) = 0, by taking Du)).
The ellipticity condition (16) implies that the vector fields X j = σ j · D are subunit for F (cfr. [8,Lemma 3.1]). Moreover, if G satisfies (SC) or (SC'), also F does. Assume finally that F given by (17) satisfies property (ii) about the weak comparison principle. Then we have the following. Now we prove the Liouville property for subsolutions of the equation and for supersolutions of Here A is a set of indices such that H i and H s are finite. We assume that b α : R d → R m is locally Lipschitz in x uniformly in α, i.e., for all R > 0 there exists K R > 0 such that  The comparison principle (ii) holds for both equations in view of [8,Example 4.6]. Finally, observe that when c α ≡ 0, then G(x, r, 0, 0) = 0 for every x ∈ Ω, r ∈ R, and r → G(x, r, p, X) is constant for every x, p, X. Corollary 3.9 concerns only operators that are either convex or concave with respect to the derivatives of the solution. However, we will use it in the next Section 3.4 to study general fully nonlinear uniformly subelliptic equations.
A different, although similar, class of extremal operators was introduced by C. Pucci in the seminal paper [39] (for the Euclidean Hessian). Here we consider them on the Hessian associated to a the vector fields X . Consider for α > 0 the class of matrices As pointed out in [39] (see also [26,Chapter 17]), these operators can be rewritten in terms of the ordered eigenvalues of the matrix M ∈ S m as follows The Liouville properties of Corollary 3.9 hold under the same assumptions for the equations (19) and (21) with the operators M − λ,Λ ((D 2 X u) * ) and M + λ,Λ ((D 2 X u) * ) replaced, respectively, by P − α ((D 2 X u) * ) and P + α ((D 2 X u) * ). In Section 4.1 we will give some explicit results for extremal equations involving P ± on the Heisenberg group.
3.4. Fully nonlinear uniformly subelliptic equations. In this section we consider the general fully nonlinear subelliptic equation , and, as a consequence, by setting H(x, r, p) := G(x, r, p, 0), one can infer Liouville results for viscosity subsolutions and supersolutions to (28) by comparison with the equation (19) and (21) of the previous section. For this purpose we assume either (30) G(x, r, p, for a concave Hamiltonian of the form (20), or for a convex H s as in (22).
(a) Let (30) hold and u ∈ USC(R d ) be a viscosity subsolution to (28) satisfying (9) for w as in (iii). If either u ≥ 0 or c α (x) ≡ 0, then u is constant. (b) Let (31) hold and v ∈ LSC(R d ) be a viscosity supersolution to (28) Proof. It is sufficient to observe that u and v satisfies the differential inequalities The same kind of result holds for equations of the form where the dependence is on the Euclidean gradient Du instead of the horizontal one D X u. We assume G satisfies (29) and either (33) G for a concave Hamiltonian of the form (20), but with b α : R d → R d a vector field in R d , or for a convex Hamiltonian of the form (22) with b α : R d → R d . On b α and c α we make the same assumptions (23), (24), and the fields X are C 1,1 and satisfy (H). The arguments leading to Corollaries 3.9 and 3.10 give the following.

3.5.
Normalized p-Laplacian. The result of the last section encompasses degenerate equations of the form where div X is the intrinsic divergence over the fields of the family X . In fact the operator E(D X u, (D 2 X u) * ) := −|D X u| 2−p div X (|D X u| p−2 D X u), called normalized or game-theoretic p-Laplacian, can be rewritten as −Tr In other words, showing that E is uniformly subelliptic for p ∈ (1, ∞). Therefore such nonlinear operators can be compared with Pucci's extremal operators M ± λ,Λ with λ = min{1, p − 1} and Λ = max{1, p − 1} over Hörmander vector fields. They were studied recently by several authors, see e.g. [1] for the case of Carnot groups, and the references therein. The game theoretic p-Laplace operator on X is the sublaplacian if p = 2, whereas for p = 1 it drives the evolutive equation describing the motion of level sets by sub-Riemannian mean curvature.

The Heisenberg vector fields
The aim of this section is to specialize the results obtained in the previous one to viscosity subsolutions of (28) fulfilling (29) over Heisenberg vector fields. We briefly recall some standard facts on the Heisenberg group. For further details we refer the reader to the monograph [12]. The Heisenberg group H d can be identified with (R 2d+1 , •), where 2d + 1 stands for the topological dimension and the group law • is defined by We denote with x a point of R 2d+1 and set The d-dimensional Heisenberg algebra is the Lie algebra spanned by the m = 2d vector fields and are 1-homogeneous with respect to the family of (anisotropic) dilations Following [12, Definition 5.1.1], it is useful to consider the following homogeneous norm defined via the stratification property of H d (35) ρ which is 1-homogeneous with respect to the previous group of dilations. We emphasize that this norm is easier to compute than the Carnot-Carathéodory norm.

4.1.
Fully nonlinear PDEs on the Heisenberg group. In the next result we provide sufficient conditions for the validity of the Liouville property for viscosity subsolutions to (19). We search the Lyapunov functions of property (iii) and (iii') among the radial ones (e.g. w = log ρ and W = − log ρ). Here and in the next examples we exploit a classical chain rule to compute the horizontal gradient and Hessian of a "radial" function with respect to the homogeneous norm ρ. Indeed, for a sufficiently smooth radial function f = f (ρ) and given a system of vector fields In this section we denote the Heisenberg horizontal gradient and symmetrized Hessian by D H d and (D 2 H d ) * . We premise the following auxiliary result taken from [21, Lemma 3.1 and Lemma 3.2]. Lemma 4.1. Let ρ be defined in (35). Then, for |x H | = 0, where η ∈ R 2d is defined by for i = 1, ..., d. Moreover where the matrices B = (b ij ) and C = (c ij ) are defined as follows for i, j = 1, ..., d (in particular B = B T and C = −C T ). In addition, for a radial function f = f (ρ) we have and its eigenvalues are , which are simple, and f ′ (ρ) with multiplicity 2d − 2.
for ρ sufficiently large and |x H | = 0, where Q = 2d + 2 is the homogeneous dimension of H d , b α (x) takes values in R 2d , and η = (η i , η i+d ) is defined by (36).
(A) Let u ∈ USC(R 2d+1 ) be a viscosity subsolution of (19) such that If either c α (x) ≡ 0 or u ≥ 0, then u is a constant. (B) Let v ∈ LSC(R 2d+1 ) be a viscosity supersolution of (21) such that If either c α (x) ≡ 0 or v ≤ 0, then v is a constant. Proof. We only have to check property (iii) for the Lyapunov function w(x) = log ρ(x). Note that lim |x|→∞ w(x) = ∞ because ρ → ∞ as |x| → ∞. By Lemma 4.1 applied to the radial function w the eigenvalues of (D 2 H d w) * are − |x H | 2 ρ 4 and 3 |x H | 2 ρ 4 , which are simple, and |x H | 2 ρ 4 with multiplicity 2d − 2 when |D H d ρ| = 0. Otherwise all the eigenvalues vanish identically and w is trivially a supersolution to (19) because M − λ,Λ ((D 2 H d w) * ) = 0 and c α u ≥ 0. Hence, we are able to compute the Pucci's minimal operator at points where |x H | = 0 owing to Lemma 4.1 as Thus, w is a supersolution at all points where because D H d w = η/ρ 4 by Lemma 4.1. In particular, this inequality holds when ρ is sufficiently large under condition (37) by recalling that Q = 2d + 2. Similarly, one can check that (37) implies that the function W (ρ) = − log ρ is a subsolution to (21) for |x| sufficiently large at points where |D H d ρ| = 0. Therefore Corollary 3.9 gives the conclusion.
Remark 4.4. Condition (37) is comparable to that obtained in [5, condition (2.17)], but here typical quantities of Carnot groups appear. One may think that the ratio plays exactly the same role as |x| 2 in [5, condition (2.17)], while the dimension d of the Euclidean setting is precisely replaced by its sub-Riemannian counterpart Q, as expected.
Remark 4.5. A simple condition that implies (37), and therefore the Liouville property, is lim sup since c ≥ 0. Compare the above condition to that in [5,Remark 2.4]: Q replaces the dimension d of the Euclidean case and x ∈ R d is replaced by the vector η/|x H | 2 ∈ R 2d , where η is defined by (36).
Proof. As in the proof of Theorem 4.2 we must only check that w = log ρ is a supersolution. Observe that −|D H d w| = −|σ T Dw| = min |α|=1 {−α · σ T Dw}. Hence we can write the right-hand side of the inequality (40) with p = D H d w as a results that appears to be new even in the linear case λ = Λ.
We end this subsection with a result on the following equations driven by the Pucci's extremal operators P ± λ defined by (25) (26) and (27) for P ± we can get optimal sufficient conditions. Corollary 4.10. Let X = {X 1 , ...., X 2d } be the system of vector fields generating the Heisenberg group H d . Assume (23), (24), and for ρ sufficiently large and |D H d ρ| = 0. Then the same conclusions as in Theorem 4.2 hold for subsolutions of (43) and supersolutions of (44).
Proof. The proof is the same as Theorem 4.2 using the Lyapunov function w(ρ) = log ρ and the formulas (26) and (27). By the expression of the eigenvalues of (D 2 H d w) * in the proof of Theorem 4.2 one finds Similarly, one uses W = − log ρ as Lyapunov function for the maximal operator P + λ .

4.2.
Comparison with the literature and sharpness of the conditions. In this section we make a comparison with the results in the literature, showing the sharpness of our conditions and those of [5,20,21] the former for viscosity subsolutions bounded above, and the second for supersolutions bounded from below, when d ≤ λ Λ + 1. This complements the result of [20] on (5) and (6) recalled in the Introduction, but with a more restrictive condidion on d, which is, however, still sharp for the Lapalcian (λ = Λ). The next counterexample shows that the inequalities can have nonconstant solutions when d > λ Λ + 1.
Since it is radial, the eigenvalues of the Hessian matrix can be computed by [20, Lemma 3.1] and one checks that it is a classical solution to M + λ,Λ (D 2 u 2 ) ≥ 0 in R d . Moreover it is bounded and not constant if β > 2, which is equivalent to d > λ/Λ + 1, so the Liouville property for supersolutions to M + (D 2 u) = 0 is false in this case. Similarly, v 2 = −u 2 gives a counterexample for solutions to Remark 4.13. The paper [20] studies a similar but different problem with respect to [5], namely, the Liouville property for viscosity supersolutions to M − λ,Λ (D 2 u) = 0 in R d and for subsolutions to M + λ,Λ (D 2 u) = 0. They prove it under the less restrictive condition d ≤ Λ λ + 1 [20, Theorem 3.2]. Note, however, that their theorem cannot be applied to general uniformly elliptic operators via the inequalities (29), whereas the results in [5] allow such application. The next example shows that also the condition in [20] is optimal.
Counterexample 4.14 (From [20]). Set α := λ Λ (d − 1) + 1 and consider the function which is a classical solution to M + λ,Λ (D 2 u 3 ) ≤ 0 in R d . Moreover it is bounded and not constant if α > 2, which is equivalent to d > Λ/λ + 1. Similarly, v 3 = −u 3 yields a counterexample for the corresponding property for the minimal operator. 4.2.2. The Heisenberg case: sublaplacians. Liouville's theorem for classical harmonic functions on the Heisenberg group is a consequence of the Harnack inequality, see, e.g., [12,Theorem 8.5.1], or mean-value formulas [17,Theorem 3.1]. However, the Liouville property for classical subsolutions (resp., supersolutions) bounded from above (resp., below) of is false for all dimensions d, as the next example shows. We recall that Q := 2d + 2 is the homogeneous dimension of H d and ρ(x) is the homogeneous norm defined in (35).
G(x, u, Du, (D 2 H d u) * ) = 0 in R d , with G : R 2d+1 × R × R 2d+1 × S 2d → R, so they involve the Heisenberg Hessian D 2 H d u and the Euclidean gradient Du. As at the end of Section 3.4 we assume G is uniformly subelliptic and its first order part is bounded from below by a concave Hamiltonian H i or from above by a convex one H s . Then Corollary 3.11 gives one of the Liouville properties if we find a suitable super-or subsolution out of a big ball. The next result gives an explicit sufficient condition saying that the vector fields b α in the drift part of H i and H s point toward the origin for |x| large enough, as in the Ornstein-Uhlenbeck operators. It involves the homogeneous norm ρ of the Heisenberg group defined by (35).
Proof. We check that w = log ρ is a supersolution. Let C 1 := Λ(2d + 1) − λ > 0. As in the proof of Theorem 4.2, w is a supersolution at all points where Since Dρ = (2|x H | 2 x H , x 2d+1 )/(2ρ 3 ), we get from (47) that the left hand side is larger than for ρ large enough, by taking either |x H | 2 > C 1 /γ o , or |x H | 2 ≤ C 1 /γ o and x 2 2d+1 > 2C 2 1 /γ 2 o . The last result is based on a condition of positivity of the coefficients c α at infinity similar to Example 4.8. Remark 4.20. Corollary 4.18 generalizes to fully nonlinear equations the Liouville properties for linear Ornstein-Uhlenbeck operators with Heisenberg sub-laplacian proved in [33]. The condition (49) in Corollary 4.19 obviously holds if c α (x) ≥ c o > 0 for |x| large enough, and in such case the condition (51) can be weakened to sup α∈A |b α (x)| = o(ρ log ρ).