Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators

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1 Harnack inequaliies and Gaussian esimaes for a class of hypoellipic operaors Sergio Polidoro Diparimeno di Maemaica, Universià di Bologna Absrac We announce some resuls obained in a recen sudy [14], concerning a general class of hypoellipic evoluion operaors in R N+1. A Gaussian lower bound for he fundamenal soluion and a global Harnack inequaliy are given. 1 Inroducion In his noe we will discuss a resul obained in a recen sudy by Andrea Pascucci and myself [14]. Le us consider he linear second order operaor in R N+1 of he form L = m Xp + X p=1 In 1.1 z = x, denoes he poin in R N+1 and he X p s are smooh vecor fields on R N, i.e. X p x = N a p j x x j, p = 0,..., m, j=1 where any a p j is a C funcion. In he sequel we also consider he X p s as vecor fields in R N+1, moreover we denoe Y = X 0, and λ X λ 1 X λ m X m, 1. for λ = λ 1,..., λ m R m. We say ha a curve γ : [0, T ] R N+1 is L-admissible if i is absoluely coninuous and saisfies γ s = λs Xγs + Y γs, a.e. in [0, T ], for suiable piecewise consan real funcions λ 1,..., λ m. We nex sae our main assumpions: AMS Subjec Classificaion: 5K57, 5K65, 5K70. Piazza di Pora S. Donao 5, 4016 Bologna Ialy. polidoro@dm.unibo.i 1

2 [H.1] here exiss a homogeneous Lie group G = R N+1,, δ λ such ha i X 1,..., X m, Y are lef ranslaion invarian on G; ii X 1,..., X m are δ λ -homogeneous of degree one and Y is δ λ -homogeneous of degree wo; [H.] for every x,, ξ, τ R N+1 wih > τ, here exiss an L-admissible pah γ : [0, T ] R N+1 such ha γ0 = x,, γt = ξ, τ. Le us recall ha a Lie group G = R N+1, is called homogeneous if a family of dilaions δ λ λ>0 exiss on G. Hypoheses [H.1]-[H.] imply ha R N has a direc sum decomposiion R N = V 1 V n such ha, if x = x x n wih x k V k, hen he dilaions are δ λ x x n, = λx λ n x n, λ, for any x, R N+1 and λ > 0. The naural number Q = + n k=1 k dimv k is usually called he homogeneous dimension of G wih respec o δ λ λ>0. We also inroduce he following δ λ -homogeneous norm on R N : { x k } x G = max 1 k k = 1,..., n, i = 1,..., m k. i Operaors of he form 1.1, verifying assumpions [H.1]-[H.], have been inroduced by Kogoj and Lanconelli in [5] and [6]. Under hese hypoheses he Hörmander condiion holds: rank Lie{X 1,..., X m, Y }z = N + 1, z R N+1 ; 1. hence L in 1.1 is hypoellipic i.e. every disribuional soluion o Lu = 0 is smooh; see, for insance, Proposiion 10.1 in [5] and has a fundamenal soluion Γ which is smooh ou of he pole and δ λ -homogeneous of degree Q: Γ δ λ z = λ Q Γz, λ > Hence operaor 1.1 belongs o he general class of hypoellipic operaors on homogeneous groups firs sudied by Folland [], Rohschild and Sein [17], Nagel, Sein and Wainger [11]. The main resul in [5] is an invarian local Harnack inequaliy for L; one-side Liouville heorems are given in [6]. In he paper [14] a non-local Harnack inequaliy has been proved see Theorem 5.1 below. Moreover in [14] i is proved a lower bound for he fundamenal soluion Γ of he operaor L, under he assumpion ha he group G has sep hree, i.e. Q R N = V 1 V V. Proposiion 1.1 Le L be he operaor in 1.1 on a group of sep hree and Γ is fundamenal soluion. There exiss a posiive consan C such ha Γx, C exp C x 6 G, x, R N R

3 In he above saemen Γ denoes he fundamenal soluion of L wih pole a he origin. Due o he lef -invariance of Γ, we have ha Γz, ζ = Γζ 1 z and a lower bound analogous o 1.5 also holds for Γ, ζ. The above esimae looks raher rough, since i is naural o expec x G in he exponen in 1.5. Indeed he following Gaussian upper bound has been proved by by Kogoj and Lanconelli in [5]: Γx, C Q exp x G, x R N, > 0, 1.6 C being C a posiive consan. However i is known ha he fundamenal soluion of he Kolmogorov operaor x 1 + x 1 x is Γx 1, x, = see.4 below so ha, in paricular, Γ0, x, = On he oher hand, we have Γx 1, 0, = π exp x 1 x 1x x, x 1, x R, > π exp x = π exp x 1 = π exp π exp 0, x 6 G. 1.8 x 1, 0 G, so ha neiher 1.5 nor 1.6 are sharp. However we can hope o sharpen 1.5 a leas in some componen of x. The following example shows ha furher hypoheses on he operaor L are needed o obain such a resul. Consider he operaor L = X + Y in R, where X = x1 + x 1 x, and Y = x 1 x. I is sraighforward o verify [H.1], [H.] for L he dilaions are δ λ x 1, x, = λx 1, λ x, λ. The fundamenal soluion is Γx 1, x, = Γx 1, x x 1, wih Γ in 1.7, hen Γx 1, 0, = π exp x 1 x 1 4 x 1 6, x 1, R R In Secions,, and 4 we give some examples of operaors ha moivae our sudy. In hese paricular cases we will give sharp esimaes for he case of a Lie algebra of sep hree see Proposiions.1,.1 and 4.1 below. The resuls saed in Secions and agree wih he known resuls for he same kind of operaors, he resuls saed in Secion 4 are new. Las secion conains an ouline of he proof.

4 Hea operaors on Carno groups Consider he operaor L in 1.1 under assumpions [H.1] and rank Lie { X 1,..., X m } x = N, x R N..1 In his case G = R N,, δ λ is a Carno or sraified group see, for insance, [] and [19]. When X 0 0 in 1.1, we have L = G,. where as usual G denoes he canonical sub-laplacian on G: G = m Xp. p=1 We recall he well-known Gaussian upper and lower bounds for hea kernels due o Jerison and Sánchez-Calle [4], Kusuoka and Sroock [8], Varopoulos, Saloff-Cose and Coulhon [19]. These resuls apply o Lie groups which are no necessarily homogeneous. We also quoe he more recen and accurae esimaes by Saloff-Cose and Sroock [18], Bonfiglioli, Lanconelli and Uguzzoni []. More generally condiion.1 is saisfied by operaors 1.1 of he form L = G + X 0,. wih X 0 Lie { X 1,..., X m }. Operaors of his kind have been considered by Alexopoulos in [1]. The following saemen conains he global lower bound and he global Harnack inequaliy for a parabolic operaor on a Carno group of sep hree proved in [14]. The esimaes are in accord wih he classical ones given in [4], [8] and in [1]. Proposiion.1 Le L be a parabolic operaor on a Carno group of sep hree and le z 0 = x 0, 0 R N+1, T > 0. Then: There exiss a posiive consan C such ha Γx, C Q exp C x G, x, R N R +..4 There exis wo consans c > 0 and C > 1, only dependen on L, such ha, if u is a non-negaive soluion o Lu = 0 in R N ] 0 ct, 0 + T ], hen for every z = x, R N ]0, T ]. uz 0 exp C 1 + x G uz 0 z,.5 4

5 Kolmogorov ype operaors Assume X p = p, p = 1,..., m, and he coefficiens of X 0 are linear funcions of x R N : for a consan N N marix B. Then L = X 0 = x, B m p + X 0..1 p=1 This kind of operaor has been exensively sudied see [10] and [9] for a comprehensive bibliography. I is known ha [H.1]-[H.] for L are equivalen o he following hypohesis: [H.] he marix B akes he form 0 B B 0 B = B n for some basis of R N, where B k is a d k d k+1 marix of rank d k, k = 1,,..., n wih m = d 1 d d n+1 1 and d d n+1 = N. The equivalence of [H.] and he couple of hypoheses [H.1]- 1. has been proved in [10]. As said before [H.1]-[H.] yield [H.1]-1., on he oher hand in [16] i is proved he converse implicaion for Kolmogorov operaors. Under assumpion [H.], he dilaions are δ λ = diagλi d1, λ I d,..., λ n+1 I dn+1, λ, λ > 0,. where I dk is he d k d k ideniy marix. Moreover he fundamenal soluion of L in 1.1 is explicily known: 1 Γz = 4π N de C exp 1 4 C 1 x, x,.4 for > 0, and Γz = 0 for 0. In.4, we denoe E = exp B T and C = 0 EsAE T sds,.5 where B T is he ranspose marix of B. We remark ha condiion [H.] ensures ha C > 0 for any > 0 cf. Proposiion A.1 in [10], see also [7]. In his case he group law is x, ξ, τ = ξ + Eτx, + τ, x,, ξ, τ R N+1..6 In he sequel we call K R N+1, a Kolmogorov group. 5

6 Non-local Harnack inequaliies for his kind of operaor are proved in [1], moreover Gaussian esimaes for he fundamenal soluion are given in [15], [16] and [1] in he case of non-consan coefficiens of he second order derivaives. The following saemen conains he global lower bound and he global Harnack inequaliy for operaors on a Kolmogorov group of sep hree proved in [14]. Also in his case he esimaes are in accord wih he ones given in [10]. Proposiion.1 Le L be a Kolmogorov ype operaor on a group K of sep hree and le z 0 = x 0, 0 R N+1, T > 0. Then: There exiss a posiive consan C such ha Γx, C x 1 exp C K + Q x 6 K, x, R N R +..7 There exis wo consans c > 0 and C > 1, only dependen on L, such ha, if u is a non-negaive soluion o Lu = 0 in R N ] 0 ct, 0 + T ], hen x 1 x 6 K K uz 0 exp C uz 0 z,.8 for every z = x, R N ]0, T ]. 4 Operaors on linked groups Le L = G K be he linked group of a Carno group G on R m R n and a Kolmogorov group K on R m R r R, as defined by Kogoj and Lanconelli in [5] Sec. 10. We consider he operaor L = G + Y. 4.1 For reader s convenience, we recall here he definiion of link of Carno and Kolmogorov groups. Consider a Carno group G = R m R n,, δλ G, where x, y denoes he poin in R m R n and assume ha X p = p + a p x, y y, p = 1,..., m. 4. Hence he dilaions and he group law ake he following form: δ G λ x, y = λx, ρ G λ y, x, y x, y = x + x, Qx, y, x, y. Moreover he Kolmogorov group is 1 K = R m R r R,, δλ K, 1 We use he same noaion for he composiion law in differen groups; he conex will avoid ambiguiy. 6

7 where we denoe x, w, he poin in R m R r R. We assume ha Y = X 0 x, w = x, w, B x,w. 4. The dilaions. and he group law.6 will be denoed by: δ K λ x, w, = λx, ρ K λ w, λ, x, w, x, w, = x + x, Rx, w,, x, w,, +. The link L = G K is defined as follows: L = R m R n R r R,, δλ L, where and δ L λ x, y, w, = λx, ρ G λ y, ρk λ w, λ x, y, w, x, y, w, = x + x, Qx, y, x, y, Rx, w,, x, w,, I urns ou ha L is a homogeneous group, he X p s and Y considered as vecor fields on R m R n R r R saisfy [H.1]-[H.] see Proposiions 10.4 and 10.5 in [5]. Le explicily noe ha he operaions defined in L exend he ones in G and K. In paricular we have x, y, 0, 0 x, y, 0, 0 = x, y x, y, 0, The following saemen conains he global lower bound and he global Harnack inequaliy for operaors on a linked group of sep hree proved in [14]. Proposiion 4.1 Le L be he operaor in 4.1 on a linked group L = G K of sep hree. Le z 0 = ξ 0, η 0, ω 0, 0 R N+1 R m R n R r R and T > 0. Then: There exiss a posiive consan C such ha Γx, y, w, C x, y exp C L + Q w 6 L, x, y, w, R N R There exis wo consans c > 0 and C > 1, only dependen on L, such ha, if u is a non-negaive soluion o Lu = 0 in R N ] 0 ct, 0 + T ], hen uz 0 exp for every z = x, y, w, R N ]0, T ]. C x, y L w 6 uz 0 z, 4.7 L 7

8 5 Ouline of he proof The main ool in he proof of Proposiions.1,.1 and 4.1 is he following non-local Harnack inequaliy given in [14], Theorem 1.1: Theorem 5.1 Le z 0 = x 0, 0 R N+1 and s > 0. There exis wo consans c, C > 1, only dependen on L, such ha uexpsλ X + Y z 0 C 1+s λ uz 0, 5.1 for every non-negaive soluion u o Lu = 0 in R N ] 0 cs, 0 ], λ R m. In he above saemen we denoed as usual expsxz = γs, where γ is he unique and globally defined soluion o he Cauchy problem γ = Xγ; γ0 = z. In he sequel we also use he following noaion e X = expx0 and recall ha expxz = z e X. The above Harnack inequaliy has been proved by using repeaedly he invarian local Harnack inequaliy by Kogoj and Lanconelli [5], Theorem 7.1. In order o sae he Harnack inequaliy in [5], we se some noaions. Given r > 0, ε ]0, 1[ and z 0 R N+1, we pu C r z 0 = z 0 δ r C 1, S r ε z 0 = z 0 δ r S ε 1, where C 1 = {z = x, R N+1 z G 1, 0}, S ε 1 = {z = x, ε z C 1 }. Theorem 5. Le O be an open se in R N+1 conaining C r z 0 for some z 0 R N+1 and r > 0. Given ε ]0, 1[, here exis wo posiive consans θ = θl, ε and C = CL, ε such ha for every non-negaive soluion u of L in O. sup u Cuz 0, 5. S ε θr z 0 The conneciviy assumpion [H.] and Theorem 5.1 direcly yield a global Harnack inequaliy for posiive soluions o Lu = 0 of he form: ux, Hx,, ξ, τ uξ, τ, x,, ξ, τr N+1, < τ. 5. When we are able o find explicily an L-admissible pah γ connecing x, o ξ, τ, hen we can express explicily Hx,, ξ, τ and obain a more useful esimae. Aiming o ake ino accoun of he homogeneous srucure of he Lie group, we consruc such a γ by considering separaely he commuaors of differen homogeneiy of X 1,..., X m, Y. We remark ha hese commuaors can be convenienly approximaed by L-admissible pahs: for insance, he direcion of he commuaor [X p, X q ] can be obained by using he inegral curves of X p, X q, X p, X q. To be more specific, by using he Campbell-Hausdorff formula, we have e Xp+Y e Xq+Y e Xp+Y e Xq+Y = e 4Y +[Xp,Xq]+R, 8

9 where he error erm R conains commuaors δ λ homogeneous of order greaer han wo. This fac is well-known and has been used by many auhors in he sudy of he regulariy of ellipic and parabolic operaors of he form m p=1 X p and m Xp, 5.4 p=1 respecively. However he sudy of operaor 1.1 involves commuaors of he form [X p, Y ] ha do no occur in he examples 5.4. In his case, we have o use a differen combinaion of vecor fields, namely e X p+y e X p+y = e Y +[X p,y ]+R, where R is an error erm of order hree. The above argumen can be adaped o commuaors of higher lengh and leads o explici esimaes of H in 5. which are saed in Proposiions.1,.1 and 4.1. We omi here he deails of he proof, ha are conained in he paper [14]. References [1] G. K. Alexopoulos, Sub-Laplacians wih drif on Lie groups of polynomial volume growh, Mem. Amer. Mah. Soc., , pp. x+101. [] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Uniform Gaussian esimaes of he fundamenal soluions for hea operaors on Carno groups., Adv. Differ. Equ., 7 00, pp [] G. B. Folland, Subellipic esimaes and funcion spaces on nilpoen Lie groups, Ark. Ma., , pp [4] D. S. Jerison and A. Sánchez-Calle, Esimaes for he hea kernel for a sum of squares of vecor fields, Indiana Univ. Mah. J., , pp [5] A. E. Kogoj and E. Lanconelli, An invarian Harnack inequaliy for a class of hypoellipic ulraparabolic equaions, Medierr. J. Mah., 1 004, pp [6], One-side Liouville heorems for a class of hypoellipic ulraparabolic equaions, Geomeric Analysis of PDE and Several Complex Variables Conemporary Mahemaics Proceedings, 004. [7] L. P. Kupcov, The fundamenal soluions of a cerain class of ellipic-parabolic second order equaions, Differencial nye Uravnenija, 8 197, pp , [8] S. Kusuoka and D. Sroock, Applicaions of he Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sec. IA Mah., , pp [9] E. Lanconelli, A. Pascucci, and S. Polidoro, Linear and nonlinear ulraparabolic equaions of Kolmogorov ype arising in diffusion heory and in finance, in Nonlinear problems in mahemaical physics and relaed opics, II, vol. of In. Mah. Ser. N. Y., Kluwer/Plenum, New York, 00, pp

10 [10] E. Lanconelli and S. Polidoro, On a class of hypoellipic evoluion operaors, Rend. Sem. Ma. Univ. Poliec. Torino, , pp Parial differenial equaions, II Turin, 199. [11] A. Nagel, E. M. Sein, and S. Wainger, Balls and merics defined by vecor fields. I. Basic properies, Aca Mah., , pp [1] A. Pascucci and S. Polidoro, A Gaussian upper bound for he fundamenal soluions of a class of ulraparabolic equaions, J. Mah. Anal. Appl., 8 00, pp [1], On he Harnack inequaliy for a class of hypoellipic evoluion equaions, Trans. Amer. Mah. Soc., , pp elecronic. [14], Harnack inequaliies and Gaussian esimaes for a class of hypoellipic operaors, o appear on Trans. Amer. Mah. Soc., 006. [15] S. Polidoro, On a class of ulraparabolic operaors of Kolmogorov-Fokker-Planck ype, Maemaiche Caania, , pp [16], A global lower bound for he fundamenal soluion of Kolmogorov-Fokker-Planck equaions, Arch. Raional Mech. Anal., , pp [17] L. P. Rohschild and E. M. Sein, Hypoellipic differenial operaors and nilpoen groups, Aca Mah., , pp [18] L. Saloff-Cose and D. W. Sroock, Opéraeurs uniformémen sous-ellipiques sur les groupes de Lie, J. Func. Anal., , pp [19] N. T. Varopoulos, L. Saloff-Cose, and T. Coulhon, Analysis and geomery on groups, vol. 100 of Cambridge Tracs in Mahemaics, Cambridge Universiy Press, Cambridge,