Hindawi Publishing Corporation Boundary Value Problems Volume 2007, Article ID 48232, 16 pages doi:10.1155/2007/48232 Research Article Liouville Theorems for a Class of Linear Second-Order Operators with Nonnegative Characteristic Form Alessia Elisabetta Kogoj and Ermanno Lanconelli Received 1 August 2006; Revised 28 November 2006; Accepted 29 November 2006 Recommended by Vincenzo Vespri We report on some Liouville-type theorems for a class of linear second-order partial dif- ferential e quation with nonnegative characteristic form. The theorems we show improve our previous results. Copyright © 2007 A. E. Kogoj and E. Lanconelli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we survey and improve some Liouville-type theorems for a class of hypoel- liptic second-order oper ators, appeared in the series of papers [1–4]. The operators considered in these papers can be written as follows: ᏸ : = N  i, j=1 ∂ x i  a ij (x) ∂ x j  + N  i=1 b i (x) ∂ x i −∂ t , (1.1) where the coefficients a ij , b i are t-independent and smooth in R N . The matr ix A = (a ij ) i, j=1, ,N is supposed to be symmetric and nonnegative definite at any point of R N . We w ill denote by z = (x,t), x ∈ R N , t ∈ R, the point of R N+1 ,byY the first-order differential operator Y : = N  i=1 b i (x) ∂ x i −∂ t , (1.2) 2 Boundary Value Problems and by ᏸ 0 the stationary counterpart of ᏸ, that is, ᏸ 0 := N  i, j=1 ∂ x i  a ij (x) ∂ x j  + N  i=1 b i (x) ∂ x i . (1.3) We always assume the operator Y to be divergence free, that is,  N i =1 ∂ x i b i (x) = 0atany point x ∈ R N .Moreover,asin[2], we assume the following hypotheses. (H1) ᏸ is homogeneous of degree two with respect to the g roup of dilations (d λ ) λ>0 given by d λ (x, t) =  D λ (x), λ 2 t  , D λ (x) = D λ  x 1 , ,x N  =  λ σ 1 x 1 , ,λ σ N x N  , (1.4) where σ = (σ 1 , ,σ N )isanN-tuple of natural numbers satisfying 1 = σ 1 ≤ σ 2 ≤ ···≤ σ N .Whenwesaythatᏸ is d λ -homogeneous of degree two, we mean that ᏸ  u  d λ (x, t)  = λ 2 (ᏸu)  d λ (x, t)  ∀ u ∈ C ∞  R N+1  . (1.5) (H2) For every (x,t),(y,τ) ∈ R N+1 , t>τ, there exists an ᏸ-admissible path η :[0,T] → R N+1 such that η(0) = (x,t), η(T) =(y,τ). An ᏸ-admissible path is any continuous path η which is the sum of a finite number of diffusion and drift trajectories. A diffusion trajectory is a curve η satisfying, at any points of its domain, the inequality  η  (s),ξ  2 ≤   A  η(s)  ξ,ξ  ∀ ξ ∈ R N . (1.6) Here ·,· denotes the inner product in R N+1 and  A(z) =  A(x,t) =  A(x) stands for the (N +1) ×(N +1)matrix  A =  A 0 00  . (1.7) A drift trajectory is a p ositively oriented integr al curve of Y. Throughout the paper, we will denote by Q the homogeneous dimension of R N+1 with respecttothedilations(1.4), that is, Q = σ 1 + ···+ σ N +2 (1.8) and assume Q ≥ 5. (1.9) Then, the D λ -homogeneous dimension of R N is Q −2 ≥ 3. We explicitly remark that the smoothness of the coefficients of ᏸ and the homo- geneity assumption in (H1) imply that the a ij ’s and the b i ’s are polynomial functions (see [5, Lemma 2]). Moreover, the “oriented” connectivity condition in (H1) implies the A. E. Kogoj and E. Lanconelli 3 hypoellipticity of ᏸ and of ᏸ 0 (see [1, Proposition 10.1]). For any z = (x,t) ∈ R N+1 ,we define the d λ -homogeneous norm |z| by |z|=   (x, t)   :=  | x| 4 + t 2  1/4 , (1.10) where |x|=    x 1 , ,x N    =  N  j=1  x 2 j  σ/σ j  1/2σ , σ = N  j=1 σ j . (1.11) Hypotheses (H1) and (H2) imply the existence of a fundamental s olution Γ(z,ζ)ofᏸ with the following proper ties (see [2, page 308]): (i) Γ is smooth in {(z, ζ) ∈ R N+1 ×R N+1 | z = ζ}, (ii) Γ( ·,ζ) ∈ L 1 loc (R N+1 )andᏸΓ(·,ζ) =−δ ζ for every ζ ∈R N+1 , (iii) Γ(z, ·) ∈ L 1 loc (R N+1 )andᏸ ∗ Γ(z, ·) =−δ z for every z ∈ R N+1 , (iv) limsup ζ→z Γ(z, ζ) =∞for every z ∈ R N+1 , (v) Γ(0,ζ) → 0asζ →∞, Γ(0,d λ (ζ)) = λ −Q+2 Γ(0,ζ), (vi) Γ((x,t),(ξ,τ)) ≥ 0, > 0ifandonlyift>τ, (vii) Γ((x,t),(ξ,τ)) = Γ((x,0),(ξ,τ −t)). In (iii) ᏸ ∗ denotes the formal adjoint of ᏸ. These properties of Γ allow to obtain a mean value formula at z = 0 for the entire solutions to ᏸu = 0. We then use this formula to prove a scaling invariant Harnack in- equality for the nonnegative solutions ᏸu = f in R N+1 . Our first Liouville-type theorems will follow from this Harnack inequality. All t hese results will be showed in Section 2. In Section 3, we show some asymptotic Liouville theorem for nonnegative solution to ᏸu = 0 in the halfspace R N ×] −∞,0[ assuming that ᏸ, together with (H1) and (H2), is left invariant with respect to some Lie groups in R N+1 . Finally, in Section 4 some examples of operators to which our results apply are showed. 2. Polynomial Liouville theorems Throughout this section, we will assume that ᏸ in (1.1) satisfies hypotheses (H1) and (H2). Let Γ be the fundamental solution of ᏸ with pole at the origin. With a standard procedure based on the Green identity for ᏸ and by using the properties of Γ recalled in the introduction, one obtains a mean value formula at z = 0forthesolutiontoᏸu = 0. Precisely, for every point (0,T) ∈ R N+1 and r>0, define the ᏸ-ball centered at (0,T)and with radius r,asfollows: Ω r (0,T):=  ζ ∈ R N+1 : Γ  (0,T),ζ  >  1 r  Q−2  . (2.1) Then, if ᏸu = 0inR N+1 ,onehas u(0,T) =  1 r  Q−2  Ω r (0,T) K(T,ζ)u(ζ)dζ, (2.2) 4 Boundary Value Problems where K(T,ζ) =  A(ξ)∇ ξ Γ,∇ ξ Γ  Γ 2 , ζ = (ξ,τ), (2.3) and Γ stands for Γ((0, T),(ξ,τ)). Moreover, ·,· denotes the inner product in R N and ∇ ξ is the gradient operator (∂ ξ 1 , ,∂ ξ N ). Formula (2.2) is just one of the numerous extensions of the classical Gauss mean value theorem for harmonic functions. For a proof of it, we directly refer to [6, Theorem 1.5]. We would like to stress that in this proof one uses our assumption div Y = 0. The kernel K(T, ·) is strictly positive in a dense open subset of Ω r (0,T)foreveryT,r> 0 (see [2, Lemma 2.3]). This property of K(T, ·), together with the d λ -homogeneity of ᏸ, leads to the following Harnack-type inequality for entire solutions to ᏸu = 0. Theorem 2.1. Let u : R N+1 → R be a nonnegative solution to ᏸu = 0 in R N+1 . Then, there exist two positive constants C = C(ᏸ) and θ = θ(ᏸ) such that sup C θr u ≤ Cu(0,r 2 ) ∀r>0, (2.4) where, for ρ>0, C ρ denotes the d λ -symme tric ball C ρ :=  z ∈ R N+1 ||z| <ρ  . (2.5) The proof of this theorem is contained in [2, page 310]. By using inequality (2.4) together with some basic properties of the fundamental solu- tion Γ, one easily gets the following a priori estimates for the positive solution to ᏸu = f in R N+1 . Corollary 2.2. Let f beasmoothfunctionin R N+1 and let u be a nonnegative solution to ᏸu = f in R N+1 . (2.6) Then there exists a positive constant C independent of u and f such that u(z) ≤ Cu  0,  |z| θ  2  + |z| 2 sup |ζ|≤|z|/θ 2   f (ζ)   , (2.7) where θ is the constant in Theorem 2.1. This result allows to use the Liouville-type theorem of Luo [5] to obtain our main result in this section. Theorem 2.3. Let u : R N+1 → R be a smooth function such that ᏸu = p in R N+1 , u ≥ q in R N+1 , (2.8) A. E. Kogoj and E. Lanconelli 5 where p and q are polynomial function. Assume u(0,t) = O  t m  as t −→ ∞ . (2.9) Then, u is a polynomial function. Proof. We split the proof into two steps. Step 1. There exists n>0suchthat u(z) = O  | z| n  as z −→ ∞. (2.10) Indeed, letting v : = u −q,wehave ᏸv = p −ᏸq in R N+1 , v ≥ 0inR N+1 , (2.11) and v(0,t) = u(0,t) −q(0,t) = O(t n 1 )ast →∞,forasuitablen 1 > 0. Moreover, since p and ᏸq are polynomial functions, (p −ᏸq)(z) = O(|z| m 1 )asz →∞for a suitable m 1 > 0. Then, by the previous corollary, there exists m 2 > 0suchthat v(z) = O  | z| m 2  as z −→ ∞. (2.12) From this estimate, since v = u + q,andq is a polynomial function, the assertion (2.10) follows. Step 2. Since p is a polynomial function and ᏸ is d λ -homogeneous, there exists m ∈ N such that ᏸ (m) p ≡ 0, (2.13) where ᏸ (m) = ᏸ ◦···◦ᏸ is the mth iterated of ᏸ. It follows that ᏸ (m+1) u = 0inR N+1 . (2.14) Moreover, since ᏸ is d λ -homogeneous and hy poelliptic, the same properties hold for ᏸ (m+1) . On the other hand, by Step 1 , u(z) = O(z m )asz →∞,sothatu is a tempered distribution.Then,byLuo’spaper[5,Theorem1],u is a polynomial function.  Remark 2.4. It is well known that hypothesis (2.9) in the previous theorem cannot be removed. Indeed, if ᏸ = Δ −∂ t is the classical heat operator and u(x,t) = exp(x 1 + ···+ x N + Nt), x = (x 1 , ,x N ) ∈ R N and t ∈R,wehave ᏸu = 0inR N+1 , u ≥0, (2.15) and u is not a polynomial function. In the previous theorem, the degree of the polynomial function u can be estimated in terms of the ones of p and q. For this, we need some more notation. If α =(α 1 , ,α N ,α N+1 ) is a multi-index with (N + 1) nonnegative integer components, we let |α| d λ :=σ 1 α 1 + ···+ σ N α N +2α N+1 , (2.16) 6 Boundary Value Problems and, if z = (x,t) = (x 1 , ,x N ,t) ∈ R N+1 , z α := x α 1 1 ···x α N N t α N+1 . (2.17) As a consequence, we can write every polynomial function p in R N+1 ,asfollows: p(z) =  |α| d λ ≤m c α z α (2.18) with m ∈ Z, m ≥ 0, and c α ∈ R for every multi-index α.If  |α| d λ =m c α z α ≡ 0inR N+1 , (2.19) then we set m = deg d λ p. (2.20) If p is independent of t, that is, if p is a polynomial function in R N , we denote by deg D λ p (2.21) the degree of p withrespecttothedilations(D λ ) λ>0 . Obviously, in this case, deg D λ p = deg d λ p. Proposition 2.5. Let u, p : R N+1 → R be polynomial functions such that ᏸu = p in R N+1 . (2.22) Assume u ≥ 0. Thus, the following statements hold. (i) If p ≡ 0, then u = constant. (ii) If p ≡ 0, then deg d λ u = 2+deg d λ p. (2.23) This proposition is a consequence of the fol lowing lemma. Lemma 2.6. Let u : R N+1 → R be a nonnegative polynomial function d λ -homogeneous of degree m>0. Then ᏸu ≡ 0 in R N+1 . Proof. We argue by contradiction and assume ᏸu = 0. Since u is nonnegative and d λ - homogeneous of strictly positive degree, we have u(0,0) = 0 = min R N+1 u. (2.24) Let us now denote by ᏼ the ᏸ-propagation set of (0,0) in R N+1 , that is, the set ᏼ : =  z ∈ R N+1 : there exists an ᏸ-admissible path η :[0,T] −→ R N+1 , s.t. η(0) = (0,0), η(T) = z  . (2.25) A. E. Kogoj and E. Lanconelli 7 From hypotheses (H2), we obtain ᏼ = R N ×] −∞,0] so that, since (0, 0) is a minimum point of u and the minimum spread all over ᏼ (see [7]), we have u(z) = u(0,0) = 0 ∀z ∈ R N ×] −∞,0]. (2.26) Then, being u a polynomial function, u ≡ 0inR N+1 . This contradicts the assumption deg d λ u>0, and completes the proof.  Proof of Proposition 2.5. Obviously, if u = constant, we have nothing to prove. If we as- sume m : = deg d λ u>0andprovethat m ≥ 2, p ≡0, deg d λ p = m −2, (2.27) then it would complete the proof. Let us write u as follows: u = u 0 + u 1 + ···+ u m , (2.28) where u j is a polynomial function d λ -homogeneous of degree j, j =0, ,m,andu m ≡ 0 in R N+1 . Then p = ᏸu = ᏸu 0 + ᏸu 1 + ···+ ᏸu m , (2.29) and, since ᏸ is d λ -homogeneous of degree two,  ᏸu j  d λ (x)  = λ j−2 ᏸu j (x) (2.30) so that ᏸu 0 = ᏸu 1 ≡ 0anddeg d λ ᏸu j = j −2ifandonlyifᏸu j ≡ 0. On the other hand, the hypothesis u ≥ 0 implies u m ≥ 0sothat,beingu m ≡ 0andd λ - homogeneous of degree m>0, by Lemma 2.6,wegetᏸu m ≡ 0. Hence m ≥ 2. Moreover, by (2.29), p = ᏸu ≡ 0and deg d λ p = deg d λ ᏸu m = m −2. (2.31)  This proposition al l ows us to make more precise the conclusion of Theorem 2.3.In- deed, we have the following. Proposition 2.7. Let u, p,q : R N+1 → R be polynomial functions such that ᏸu = p in R N+1 , u ≥ q in R N+1 . (2.32) Then deg d λ u ≤ max  2+deg d λ p,deg d λ q  . (2.33) In particular, and more precisely, if q = 0,thatis,ifu ≥0, then deg d λ u = 2+deg d λ p if p ≡ 0, u = constant if p ≡ 0. (2.34) 8 Boundary Value Problems Proof. If q ≡ 0, the assertion is the one of Proposition 2.5.Supposeq ≡ 0. By letting v := u −q,wehave ᏸv = p −ᏸq, v ≥ 0. (2.35) By Proposition 2.5 ,wehave deg d λ v ≤ 2+deg d λ (p −ᏸq) ≤ 2+max  deg d λ p,deg d λ q −2  = max  2+deg d λ p,deg d λ q  (2.36) and (2.33)follows.  Proposition 2.7, together with Theorem 2.3, extends and improves the Liouville-type theorems contained in [2, 4] (precisely [2, Theorem 1.1] and [4, Theorem 1.2]). From Theorem 2.3 and Proposition 2.7, we straightforwardly get the following poly- nomial Liouville theorem for the stationary operator ᏸ 0 in (1.3). Theorem 2.8. Let P,Q : R N → R be polynomial functions and let U : R N → R be a smooth function such that ᏸ 0 U = P, U ≥Q, in R N . (2.37) Then, U is a polynomial function and deg D λ U ≤ max  2+deg D λ P,deg D λ Q  . (2.38) In particular, and more precisely, if Q ≡ 0,thatis,ifU ≥0, then deg D λ U = 2+deg D λ P if P ≡ 0, U = constant if P ≡ 0. (2.39) Proof. Let us define u(x,t) = U(x), p(x,t) = P(x), q(x,t) = Q(x). (2.40) Then p, q are polynomial functions in R N+1 and u is a smooth solution to the equation ᏸu = p in R N+1 , (2.41) such that u ≥ q.Moreover, u(0,t) = U(0) = O(1) as t −→ ∞. (2.42) Then, by Theorem 2.3, u is a polynomial function in R N+1 . This obviously implies that U is a polynomial in R N . The second part of the theorem immediately follows from Proposition 2.5.  A. E. Kogoj and E. Lanconelli 9 Remark 2.9. The class of our stationary operators ᏸ 0 also contains “parabolic”-typ e op- erators like, for example, t he following “forward-backward” heat operator ᏸ 0 := ∂ 2 x 1 + x 1 ∂ x 2 in R 2 . (2.43) Nevertheless, in Theorem 2.8, we do not require any a priori behavior at infinity, like condition (2.9)inTheorem 2.3. 3. Asymptotic Liouville theorems in halfspaces The operator ᏸ in our class do not satisfy the usual Liouville property. Precisely, if u is a nonnegative solution to ᏸu = 0inR N+1 , (3.1) then we cannot conclude that u ≡ constant without asking an extra condition on the solution u (see Theorem 2.3 and Remark 2.4). However, if we also assume that ᏸ is left translation invariant with respect to the com- position law of some Lie group in R N+1 , then we can show that ever y nonnegative solution of (3.1) is constant at t =−∞. To be precise, let us fix the new hypothesis on ᏸ and give the definition of ᏸ-parabolic trajectory. Suppose ᏸ satisfies (H2) of the introduction and, instead of (H1), the follow ing con- dition (H1) ∗ There exists a homogeneous Lie group in R N+1 , L =  R N+1 ,◦,d λ  (3.2) such that ᏸ is left translation invariant on L and d λ -homogeneous of degree two. We assume the composition law ◦ is Euclidean in the time variable, that is, (x, t) ◦(x  ,t  ) =  c(x,t,x  ,t  ),t + t   , (3.3) where c(x,t, x  ,t  ) denotes a suitable function of (x,t)and(x  ,t  ). It is a standard matter to prove the existence of a p ositive constant C such that |z ◦ζ|≤C  | z|+ |ζ|  ∀z, ζ ∈ R N+1 . (3.4) Let γ :[0, ∞[→R N be a continuous function such that limsup s→∞   γ(s)   2 s < ∞ (3.5) (here |·|denotes the D λ -homogeneous norm (1.11)). 10 Boundary Value Problems Then, the path s −→ η(s) =  γ(s),T −s  , T ∈ R, (3.6) will be called an ᏸ-parabolic trajectory. Obviously, the curve s −→ η(s) =(α,T −s), α ∈R N , T ∈R (3.7) is an ᏸ-parabolic trajectory. It can be proved that every integral curve of the vector fields Y in (1.2)alsoisanᏸ-parabolic trajectory (see [3, Lemma 3]). Our first asymptotic Liouville theorem is the following one. Theorem 3.1. Let ᏸ satisfy hypotheses (H1) ∗ and (H2), and let u be a nonnegative solution to the equation ᏸu = 0 (3.8) in the halfspace S = R N ×] −∞,0[. (3.9) Then, for every ᏸ-parabolic trajectory η, lim s→∞ u  η(s)  = inf S u. (3.10) In particular lim t→−∞ u(x,t) = inf S u ∀x ∈ R N . (3.11) The proof of this theorem relies on a left translation and scaling invariant Harnack inequality for nonnegative solutions to ᏸu = 0. For every z 0 ∈ R N+1 and M>0, let us put P z 0 (M):= z 0 ◦P(M), (3.12) where P(M): =  (x, t) ∈ R N+1 : |x| 2 ≤−Mt  . (3.13) Then, the following theorem holds. Theorem 3.2 (left and scaling invariant Harnack inequality). Let u be a nonnegative so- lution to ᏸu = 0 in R N ×] −∞,0[. (3.14) [...]... Kogoj and E Lanconelli, “An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations,” Mediterranean Journal of Mathematics, vol 1, no 1, pp 51–80, 2004 [2] A E Kogoj and E Lanconelli, “One-side Liouville theorems for a class of hypoelliptic ultraparabolic equations,” in Geometric Analysis of PDE and Several Complex Variables, vol 368 of Contemporary Math., pp 305–312, American... Journal of Functional Analysis, vol 216, no 2, pp 455–490, 2004 [13] A E Kogoj and E Lanconelli, “Link of groups and applications to PDE’s,” to appear in Proceedings of the American Mathematical Society Alessia Elisabetta Kogoj: Dipartimento di Matematica, Universit` di Bologna, a Piazza di Porta San Donato 5, 40126 Bologna, Italy Email address: kogoj@dm.unibo.it Ermanno Lanconelli: Dipartimento di Matematica,... American Mathematical Society, Providence, RI, USA, 2005 16 Boundary Value Problems [3] A E Kogoj and E Lanconelli, Liouville theorems in halfspaces for parabolic hypoelliptic equations,” Ricerche di Matematica, vol 55, no 2, pp 267–282, 2006 [4] E Lanconelli, A polynomial one-side Liouville theorems for a class of real second order hypoelliptic operators, ” Rendiconti della Accademia Nazionale delle... 1986 [10] A Bonfiglioli and E Lanconelli, Liouville- type theorems for real sub-Laplacians,” Manuscripta Mathematica, vol 105, no 1, pp 111–124, 2001 [11] E Lanconelli and S Polidoro, “On a class of hypoelliptic evolution operators, ” Rendiconti Seminario Matematico Universit` e Politecnico di Torino, vol 52, no 1, pp 29–63, 1994 a [12] E Priola and J Zabczyk, Liouville theorems for non-local operators, ”... degenerate elliptic-parabolic operators, ” Indiana University Mathematics Journal, vol 28, no 4, pp 545–557, 1979 [8] R Ja Glagoleva, Liouville theorems for the solution of a second order linear parabolic equation with discontinuous coefficients,” Matematicheskie Zametki, vol 5, no 5, pp 599–606, 1969 [9] H S Bear, Liouville theorems for heat functions,” Communications in Partial Differential Equations, vol 11,.. .A E Kogoj and E Lanconelli 11 Then, for every z0 ∈ RN ×] − ∞,0[ and M > 0, there exists a positive constant C = C(M), independent of z0 and u, such that sup u ≤ Cu z0 (3.15) Pz0 (M) Proof It follows from Theorem 2.1 and the left translation invariance of ᏸ The details are contained in [3, Proof of Theorem 3] From this theorem we obtain the proof of Theorem 3.1 Proof of Theorem 3.1 We may assume... detta dei XL, vol 29, pp 243–256, 2005 [5] X Luo, Liouville s theorem for homogeneous differential operators, ” Communications in Partial Differential Equations, vol 22, no 11-12, pp 1837–1848, 1997 [6] E Lanconelli and A Pascucci, “Superparabolic functions related to second order hypoelliptic operators, ” Potential Analysis, vol 11, no 3, pp 303–323, 1999 [7] K Amano, “Maximum principles for degenerate... Value Problems Theorem 3.1 is contained in [3, Theorem 1] The idea of our proof is taken from Glagoleva’s paper [8], in which classical parabolic operators of Cordes-type are considered For the heat equation, a stronger version of Theorem 3.1 was proved by Bear [9] The following theorem improves Theorem 3.1 Theorem 3.3 Let ᏸ and u as in Theorem 3.1 For every M > 0 and t < 0, define M(u,t) = sup u(x,t)... explicitly remark that the homogeneous dimension of L is Q := Q0 + 2 In [1, page 70], it is proved that ᏸ also satisfies (H2) 14 Boundary Value Problems Remark 4.2 The stationary part of the operator ᏸ in (4.7) is the sub-Laplacian ΔG For this kind of operator, the polynomial Liouville theorem in Theorem 2.8 was first proved in [10, Theorem 1.4] Example 4.3 (B-Kolmogorov operators) Let us split RN as follows:... usual Laplace operator in R p , ·, · is the inner product in RN , and D = (∂x1 , ,∂xN ) In this case, we have Y = Bx,D − ∂t (4.18) The operator ᏷ satisfies (H1)∗ and (H2), and it is left translation invariant on K (see [1, 11]) A E Kogoj and E Lanconelli 15 Remark 4.4 The matrix E(t) in (4.13) takes the following triangular form: E(t) = Ip 0 , E1 (t) Ir (4.19) where I p and Ir are the identity matrix . invariant the op erator ᏸ. References [1] A. E. Kogoj and E. Lanconelli, “An invariant Harnack inequality for a class of hypoelliptic ultra- parabolic equations,” Mediterranean Journal of Mathematics,. Hindawi Publishing Corporation Boundary Value Problems Volume 2007, Article ID 48232, 16 pages doi:10.1155/2007/48232 Research Article Liouville Theorems for a Class of Linear Second-Order Operators with. 2004. [2] A. E. Kogoj and E. Lanconelli, “One-side Liouville theorems for a class of hypoelliptic ultra- parabolic equations,” in Geometric Analysis of PDE and Several Complex Variables, vol. 368 of Contemporary