Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators

Nonlinear Analysis 155 (2017) 163–175

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Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators Bin Qian Department of Applied Mathematics, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, PR China

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Article history: Received 30 April 2016 Accepted 25 January 2017 Communicated by Enzo Mitidieri

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities (Theorem 2.1) for the positive solutions to the Schrödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality (Theorem 4.1). Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity (Theorem 5.3). © 2017 Elsevier Ltd. All rights reserved.

MSC: 53C21 58J35 Keywords: Differential Harnack inequalities Perelman entropy Curvature-dimension inequality Subelliptic operators

1. Introduction Let M be a C ∞ connected finite dimensional compact manifold with a smooth measure µ and a secondorder diffusion operator L on M , locally subelliptic, satisfying L1 = 0, ∫ ∫ ∫ f Lgdµ = gLf dµ, f Lf dµ ≤ 0 M

M

M

for every f, g ∈ C ∞ (M ). In the neighborhood of every point x ∈ M , it can be written as m ∑ L=− Xi∗ Xi , i=1

where X1 , X2 , . . . , Xm are Lipschitz continuous, see [16,8]. For any function f, g ∈ C ∞ (M ), the carr´e du champ associated to L is m ∑ 1 Xi f Xi g, (1.1) Γ (f, g) = (L(f g) − f Lg − gLf ) = 2 i=1 and Γ (f ) = Γ (f, f ). E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.na.2017.01.015 0362-546X/© 2017 Elsevier Ltd. All rights reserved.

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In addition, as in [8], we can assume that M is endowed with another smooth symmetric bilinear differential form, indicated with Γ Z , satisfying for f, g ∈ C ∞ (M ), Γ Z (f g, h) = f Γ Z (g, h) + gΓ Z (f, h), and Γ Z (f ) = Γ Z (f, f ). Assume the following assumption (see [8] for detailed explanation) holds throughout the paper: Assumption. For any f ∈ C ∞ (M ), one has Γ (f, Γ Z (f )) = Γ Z (f, Γ (f )). Denote 1 (LΓ (f, g) − Γ (f, Lg) − Γ (g, Lf )) , 2 ) 1( Z LΓ (f, g) − Γ Z (f, Lg) − Γ Z (g, Lf ) , Γ2Z (f, g) = 2

Γ2 (f, g) =

and Γ2 (f ) = Γ2 (f, f ), Γ2Z (f ) = Γ2Z (f, f ). Now we are ready to recall. Definition 1.1 (Due to [8]). We say that L satisfies the generalized curvature-dimension inequality CD(ρ1 , ρ2 , k, d) if there exist constants ρ1 ∈ R, ρ2 > 0, k ≥ 0 and d ∈ [2, ∞] such that the inequality ( ) 1 k Γ2 (f ) + νΓ2Z (f ) ≥ (Lf )2 + ρ1 − Γ (f ) + ρ2 Γ Z (f ), d ν holds for all f ∈ C ∞ (M ) and every ν > 0. The above definition generalizes the curvature dimension inequality CD(K, m) introduced by D. Bakry ´ and M. Emery, see for example [1]. Indeed, we only need to take ρ1 = K, ρ2 = k = 0, d = m. Besides Laplace–Beltrami operators on compact Riemannian manifolds with Ricci curvature bounded below and sub-Laplacian ∆b in a closed pseudohermitian 3-manifold (M 3 , J, θ) with the pseudo-hermitian torsion vanishes, there exists a wide class of examples satisfying the generalized curvature-dimension condition, e.g. see [8]. Denote the heat semigroup Pt = etL , due to the hypo-ellipticity of L, one has the function (t, x) → Pt f (x) ∫ is smooth on M ×(0, ∞) and Pt f (x) = M p(x, y, t)f (y)dµ(y), where p(x, y, t) = p(y, x, t) > 0 is the so-called heat kernel associated to Pt . Suppose the subelliptic operator L satisfies the generalized curvature-dimension inequality CD(ρ1 , ρ2 , k, d), Baudoin and his collaborators ([8,4,6,7,10,5,11] and etc.) have obtained that (1) the heat semigroup Pt is stochastically complete, i.e. Pt 1 = 1; (2) Li–Yau type inequalities: Γ (log Pt f ) + 32 ρ2 tΓ Z (log Pt f ) ≤ ( ) ( ) ( )2 dρ2 2ρ1 dρ1 LPt f 3k 3k d 3k 1 1 + 2ρ − t + t − 1 + + 1 + ; (3) Scale-invariant parabolic Harnack 3 P f 6 2 dρ 2t dρ t 2 2 2 inequality; (4) Off-diagonal Gaussian upper bounds for the heat kernel p(x, y, t); (5) Liouville type property; (6) Bonnet–Myers type theorem, Lichnerowicz–Obata theorem; (7) Volume comparison property, volume doubling property and distance comparison theorem etc.; (8) Functional inequalities such as Poincar´e inequality, Log-Sobolev inequality; · · · · · · , which generalizes the works of S. T. Yau, D. Bakry, M. Ledoux etc., see [19,1,17] and references therein. See also [13,25,26,28] for recent works on gradient estimate and curvature property for subelliptic operators. In the fundamental work [19], P. Li and S. T. Yau established the well known gradient estimates (so-called Li–Yau inequalities) for the positive solutions to the heat equation ∂t u = ∆u on the complete Riemannian

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manifolds, since then it becomes a powerful tool in differential geometry, PDE, etc. It also plays an important role in the Perelman’s solution to the Poincar´e conjecture. As mentioned in the above section, F. Baudoin and N. Garofalo [8] have obtained the Li–Yau type inequality, but the method (semigroup tools) exploited there (see also [2] for three dimensional subelliptic models) is not adaptable for the diffusion operators with potential LV (Schr¨ odinger operator). The reason is that LV 1 ≤ 0 and PtV is not stochastically complete, hence the parabolic comparison theorem (Proposition 4.5 in [8]) is not clear. While the Li–Yau method (maximum principle) exploited in [19] works for the Schr¨odinger case, hence it would be interesting to find certain method adaptable for the Schr¨ odinger equation. This is the start point of this paper. This paper is devoted to study differential Harnack inequalities (Li–Yau type gradient estimates) for the positive solutions to the heat equation (Schr¨ odinger equation) associated to subelliptic operators with potential, see Section 2 for the statement of Li–Yau type inequalities and Section 3 for the proof. Section 4 is devoted to the Harnack inequalities for positive solutions to the associated Schr¨odinger equation. By applying the differential Harnack inequalities obtained in Section 2, we study the Perelman type entropy and its monotonicity in Section 5. 2. Differential Harnack inequalities In this section, we shall study the Schr¨ odinger equation ∂t u = (L − V )u, where L is the subelliptic diffusion operator as in Section 1 and the potential V = V (x, t) ≥ 0 defined on M × [0, ∞) is C ∞ . Denote LV = L − V . Throughout this section, we assume that L (rather than LV ) satisfies the generalized curvature-dimension inequality CD(ρ1 , ρ2 , k, d). Since M is compact, there exist some positive constants γ1 , γ2 , θ such that Γ (V ) ≤ γ12 , Γ Z (V ) ≤ γ22 , LV ≤ θ.

(2.1) limt→0 aa(t) ′ (t) )2

1

For a given C function a(t) : [0, ∞) → [0, ∞) such that a(0) = 0, ( )2 (∫ ∫ t a2 (s) 2 t a3 (t) 1 a′ 1 ∫s , a(s)ds and for any T > 0, a , ∫ t ds , ′ ′ a (t) a (t) 0 0 2 (

0

a(s)ds)

a(u)du

0

= 0,

a′ a

∫t > 0,

0

a(s)ds a(t)

>0

are continuous and integrable

on the domain (0, T ]. We have the following Li–Yau type inequality for the Schr¨odinger equation: Theorem 2.1. Let u be a positive solution to the Schr¨ odinger equation ∂t u = LV u,

(2.2)

such that u0 := u(·, 0) is C ∞ (M ) and u0 ≥ 0, we have the following differential Harnack inequality: for any ε1 , ε2 ∈ (0, 1), we have Γ (log u) + b(t)Γ Z (log u) − α(t) ((log u)t + V ) ≤ φ(t),

(2.3)

where ) ( ∫t 2(1 − ϵ2 )ρ2 0 a(s)ds d a′ ka(t) b(t) = , η(t) = (1 + ε1 ) + ∫ t − 2ρ1 , a(t) 4 a ρ2 0 a(s)ds and 4 α(t) = d · a(t)

∫ 0

t

1 a(s)η(s)ds, φ(t) = a(t)

∫

t

( a(s)

0

γ12 a(s)|α − 1|2 b2 γ22 2η 2 (s) + + θα(s) + ϵ1 a′ (s) 2ϵ2 ρ2 d

) ds.

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Remarks 2.2. (i) For the elliptic operators (possibly with potential) satisfying the curvature dimension inequality CD(ρ, m) introduced in [1], the corresponding result has been obtained in [27]. Moreover, the local differential Harnack inequalities have been derived there. (ii) For diffusion operator LV = L (i.e. V ≡ 0, see Proposition 2.6), a similar result has been obtained by Baudoin and Garofalo [8] by different method, see Corollary 4.3 in [8]. The method in this paper is motivated by the one in [18,27], it works not only for the diffusion operators but also works for the Schr¨odinger operators (i.e. diffusion operators with potential), see Section 3. While the one in [8] does not work for the Schr¨odinger operator LV . But due to the validity of the parabolic comparison theorem (or maximum principle), we assume M is compact here. (iii) In the case of sub-Laplacian ∆b in a closed pseudohermitian 3-manifold (M 3 , J, θ) with pseudohermitian torsion vanishes, the Li–Yau type inequality has been shown in [14] (Theorem 1.1) by Chang et al. It is easy to see that ∆b satisfies CD(k, 21 , 1, 2) where k is the lower bound of Tanaka–Webster curvature, hence the above result generalizes the corresponding one in [14]. A possible function a(t) is a(t) = tγ with γ > 1. In this case, Theorem 2.1 reduces to Corollary 2.3. Under the assumption of Theorem 2.1, there exist positive constants Ci (i = 1, 2, 3, 4) depend on ρ1 , ρ2 , k, d, γ1 , γ2 , γ, θ, we have for all t > 0, ( ) 2 ρ2 3 (1 + γ)k d(1.5γρ2 + (1 + γ)k) 2ρ1 Z Γ (log u) + tΓ (log u) ≤ + ((log u)t + V ) + − 1+γ 2 γρ2 1+γ 8(γ − 1)ρ22 t dρ1 (1.5γρ2 + (1 + γ)k) − + C1 t + C2 t2 + C3 t3 + C4 t4 . (2.4) 2γρ2 2

Another possible function a(t) is a(t) = eγρ1 t (eγρ1 t − 1) with γ · sgn(ρ1 ) > 0, taking ε1 = ε2 = Theorem 2.1 reduces to

1 2,

Corollary 2.4. Under the assumption of Theorem 2.1, there exist positive constants C1 , C2 depend on ρ1 , ρ2 , k, d, γ1 , γ2 , γ, θ, we have for all t > 0, Γ (log u) +

) ρ2 ( γρ1 t C2 e − 1 e−γρ1 t Γ Z (log u) ≤ α(t) ((log u)t + V ) + C1 + γρ t , 3γρ1 e 1 −1

(2.5)

where 3 k 2 α(t) = + − + 2 ρ2 3γ

(

2 k + 3γ 2ρ2

)

e−γρ1 t .

Comparing with Corollary 2.3, (2.5) works for both small time and large time t, while (2.4) works for the finite time t. As an application of the above corollary, we have Corollary 2.5. Assume L satisfies curvature inequality CD(ρ1 , ρ2 , k, d) with ρ1 ̸= 0. Let u be a positive solution to the equation LV u = 0 with V defined on M , there exists a positive constant C such that ( ) ρ2 Z k Γ (log u) + Γ (log u) ≤ 2 + V + C. (2.6) 4|ρ1 | ρ2

∫t 0

In particular, for a given C 1 function a(t) : [0, ∞) → [0, ∞) such that a(0) = 0, limt→0 aa(t) ′ (t) = 0, a(s)ds a(t)

> 0 and for any T > 0,

′2

a a

3

, ∫ ta (

0

(t)

a(s)ds)2

a′ a

> 0,

are continuous and integrable on the domain (0, T ], we

have Li–Yau type differential Harnack inequalities for subelliptic diffusion operator L (i.e. V ≡ 0).

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Proposition 2.6. Let u be a positive solution to the heat equation ∂t u = Lu such that u0 := u(·, 0) ∈ C ∞ (M ) with u0 ≥ 0, we have the following differential Harnack inequality: ∫t 2ρ2 0 a(s)ds Z Γ (log u) − α(t)(log u)t ≤ φ(t), (2.7) Γ (log u) + a(t) where (

a′ ka(t) + ∫t − 2ρ1 a ρ2 0 a(s)ds

)

Taking a(t) = tγ with γ > 1, then η(t) =

d 4

d η(t) = 4

, α(t) = (

4

∫t 0

2 a(s)η(s)ds , φ(t) = d · a(t)

∫t 0

a(s)η 2 (s) ds . d · a(t)

) (γρ2 + (γ + 1)k) ρ12 t − 2ρ1 , Proposition 2.6 reduces to

Corollary 2.7. Let u be a positive solution to the heat equation ∂t u = Lu such that u0 = u(·, 0) ∈ C ∞ (M ) with u0 ≥ 0, we have for γ > 1, ( ) ( ) (1 + γ)k 2ρ1 dρ21 t dρ1 (1 + γ)k 2ρ2 Z Γ (log u) + tΓ (log u) ≤ 1 + − t (log u)t + − 1+ 1+γ γρ2 1+γ 2(1 + γ) 2 γρ2 ( )2 2 (1 + γ)k dγ 1+ + . (2.8) 8(γ − 1)t γρ2 If ρ1 ≥ 0, it is easy to see (2.8) reduces to ( ) ( )2 (1 + γ)k dγ 2 (1 + γ)k 2ρ2 Z tΓ (log u) ≤ 1 + (log u)t + 1+ . Γ (log u) + 1+γ γρ2 8(γ − 1)t γρ2

(2.9)

Remark 2.8. Choose γ = 2, Corollary 2.4 reduces to

Γ (log u) +

2ρ2 Z tΓ (log u) ≤ 3

( 1+

3k 2ρ1 t − 2ρ2 3

)

)2 ( ( ) d 1 + 3k 2ρ2 dρ1 3k (log u)t + t− 1+ , + 6 2 2ρ2 2t dρ21

which is nothing less than Theorem 5.1 in [8]. In particular, if ρ1 ≥ 0, we have )2 ( 3k ( ) d 1 + 2ρ 3k 2ρ2 Z 2 tΓ (log u) ≤ 1 + . Γ (log u) + (log u)t + 3 2ρ2 2t

(2.10)

As a consequence, we can get the Liouville property if ρ1 ≥ 0, i.e. there is no positive constant solution to Lu = 0. β

Taking a(t) = eγρ1 t (1 − eγρ1 t ) ∫t 0

a(s)ds

= reduces to a(t)

1 γρ1 t (1+β)γρ1 (e

with β > 1 even, γ ≥

− 1)e−γρ1 t and η(t) =

2ρ2 (ρ2 +k)(1+β) ,

then

a′ a

dρ1 (γ−2)ρ2 −(γ(ρ2 +k)(1+β)−2ρ2 )eγρ1 t . 4ρ2 1−eγρ1 t

=

γρ1 ((1+β)eγρ1 t −1) , eγρ1 t −1

Hence Proposition 2.6

Corollary 2.9. Let u be a positive solution to the heat equation ∂t u = Lu such that u0 := u(·, 0) ∈ C ∞ (M ) 2ρ2 with u0 ≥ 0, we have for β > 1 even, γ ≥ (ρ2 +k)(1+β) , Γ (log u) −

( ) 2ρ2 1 − eγρ1 t e−γρ1 t Γ Z (log u) ≤ α(t)(log u)t + φ(t), (1 + β)γρ1

where α(t) = 1 +

(1 + β)k −γρ1 t e + βρ2

(

k 2 − (1 + β)γ ρ2

)

(1 − eγρ1 t )e−γρ1 t ,

(2.11)

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and 2

φ(t) =

) dρ1 γ(ρ2 β + k + kβ)2 e−γρ1 t dρ1 (γ(ρ2 + k)(1 + β) − 2ρ2 ) ( 1 − e−γρ1 t + 2 8γ(1 + β)ρ2 8ρ22 (β − 1) eγρ1 t − 1 dρ1 γ(γ(ρ2 + k)(1 + β) − 2ρ2 )(ρ2 β + k + kβ) −γρ1 t + e . 4βρ22

In particular, let γ =

2ρ2 (ρ2 +k)(1+β)

( β > 1 even), we have ) ( 2ρ1 ρ2 t 1 ρ2 t ρ2 + k k + βk + βρ2 − (ρ 2ρ − (ρ +k)(1+β) Γ Z (log u) ≤ Γ (log u) + 1−e 2 e 2 +k)(1+β) (log u)t ρ1 βρ2 −

2ρ1 ρ2 t

e (ρ2 +k)(1+β) dρ1 (βρ2 + k + kβ)2 . + 1 ρ2 t 4ρ2 (ρ2 + k)(β + 1)(β − 1) (ρ 2ρ +k)(1+β) 2 e −1

(2.12)

It yields the following lower bound for (log u)t : (log u)t ≥ −

d(βρ2 + k + kβ) 4(ρ2 + k)(β + 1)(β − 1)

ρ1 e

2ρ1 ρ2 t (ρ2 +k)(1+β)

. −1

Comparing with Corollary 2.7, (2.11) works for both small time and large time t while (2.8) only works for finite time t in the case of ρ1 ≥ 0. 3. Proof of Theorem 2.1 In this section, we give the proof of Theorem 2.1. The method of the proof is inspired by [27,18]. Assume u is a positive solution to the Schr¨ odinger equation ∂t u = LV u. Let f = log u, we have Lf + Γ (f ) = ft + V,

(3.1)

where Γ is induced by L rather than LV . For some positive function b(t) and some functions α(t), φ(t), define the parameter function: F = Γ (f ) + b(t)Γ Z (f ) − α(t)(ft + V ) − φ(t). Through calculation, we obtain ( ) (L − ∂t )F = LΓ (f ) − 2Γ (f, ft ) + b(t) LΓ Z (f ) − 2Γ Z (f, ft ) − α(t)(Lft − ftt ) − α(t)(LV − Vt ) − b′ (t)Γ Z (f ) + α′ (t)(ft + V ) + φ′ (t). Applying (3.1) and Assumption, we have (L − ∂t )F = 2Γ2 (f ) + 2b(t)Γ2Z (f ) − 2Γ (f, F ) − 2(α(t) − 1)Γ (f, V ) + 2b(t)Γ Z (f, V ) − α(t)LV − b′ (t)Γ Z (f ) + α′ (t)ft + α′ (t)V + φ′ (t). Thanks to the generalized curvature dimension inequality CD(ρ1 , ρ2 , k, d), ( ) 2 2k (L − ∂t )F ≥ −2Γ (f, F ) + (Lf )2 + 2ρ1 − Γ (f ) + (2ρ2 − b′ (t)) Γ Z (f ) d b(t) − 2(α(t) − 1)Γ (f, V ) + 2b(t)Γ Z (f, V ) − α(t)LV + α′ (t)(V + ft ) + φ′ (t) ( ) 4η(t) 2η 2 (t) 2k ≥ −2Γ (f, F ) − Lf − + 2ρ1 − Γ (f ) + (2ρ2 − b′ (t)) Γ Z (f ) d d b(t) − 2(α(t) − 1)Γ (f, V ) + 2b(t)Γ Z (f, V ) − α(t)LV + α′ (t)(V + ft ) + φ′ (t),

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where η = η(t) is some function to be determined later. It follows by (3.1), ( ) ( ) 4η 2k 4η (L − ∂t )F ≥ −2Γ (f, F ) + + 2ρ1 − Γ (f ) + (2ρ2 − b′ (t))Γ Z (f ) − − α′ (t) (ft + V ) d b(t) d 2 2η (t) + φ′ (t). − 2(α(t) − 1)Γ (f, V ) + 2b(t)Γ Z (f, V ) − α(t)LV − d 2

By (2.1) and the basic fact 2ax ≤ aε + εx2 , ∀a, ε ∈ R+ , we have for any positive functions ϵ1 (t), ϵ2 (t), ) ( ) ( 2k 4η 4η ′ Z ′ + 2ρ1 − Γ (f ) + (2ρ2 − b (t)) Γ (f ) − − α (t) (ft + V ) (L − ∂t )F + 2Γ (f, F ) ≥ d b(t) d 2η 2 (t) − 2|α(t) − 1|γ1 (Γ (f ))1/2 − 2b(t)γ2 (Γ Z (f ))1/2 − α(t)θ − + φ′ (t) d ) ( 2k 4η + 2ρ1 − − ϵ1 (t) Γ (f ) + (2ρ2 − b′ (t) − ϵ2 (t)) Γ Z (f ) ≥ d b(t) ( ) 4η |α(t) − 1|2 γ12 b2 (t)γ22 2η 2 (t) ′ − − α (t) (ft + V ) − − − α(t)θ − + φ′ (t). d ϵ1 (t) ϵ2 (t) d For any ϵ1 , ϵ2 ∈ (0, 1), now we take ϵ1 a′ , a ∫t 2(1 − ϵ2 )ρ2 0 a(s)ds b(t) = , a(t) ∫ t 4 a(s)η(s)ds, α(t) = d · a(t) 0 ϵ1 (t) =

ϵ2 (t) = 2ϵ2 ρ2 , ( ) d a′ 2k η(t) = (1 + ϵ1 ) + − 2ρ1 , (3.2) 4 a b(t) ( 2 ) ∫ t 1 γ1 a(s)|α − 1|2 b2 γ22 2η 2 (s) φ(t) = a(s) + ds + θα(s) + a(t) 0 ϵ1 a′ (s) 2ϵ2 ρ2 d

so that 4η 2k a′ + 2ρ1 − = (1 + ϵ1 ) , d b(t) a ′ a b 2(1 − ϵ2 )ρ2 − b′ (t) = , a a′ α 4η − α′ (t) = , d a |α − 1|2 γ12 a′ b2 γ22 2η 2 φ′ (t) − = −φ . − − αθ − ϵ1 ϵ2 d a Indeed, the above equations display the reason for our choice of (3.2). With this choice, we have Lemma 3.1. For the parameter function F , we have (L − ∂t )F + 2Γ (f, F ) ≥

a′ F. a

Now we are ready to prove Theorem 2.1. Proof of Theorem 2.1. The above lemma indicates that ( ) (L − ∂t )(a(t)uF ) = a(t)u (L − ∂t )F + 2Γ (f, F ) − a′ uF (u) + a(t)F (L − ∂t )u ≥ a(t)uF V. This implies (LV − ∂t )(a(t)uF ) ≥ 0. Notice that a(0) = 0, the desired result follows by the maximum principle, see [15,12].

□

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4. Harnack inequalities In this section, we shall derive the Harnack inequalities for the positive solutions to the Schr¨odinger equation (2.2), which are the consequence of the Li–Yau type inequality (2.7). Harnack inequality is one of the main techniques in the regularity theory of PDEs over the past several decades. To this end, we introduce the metric associated to Schr¨odinger operator LV , see [19] for elliptic operators on Riemannian manifolds. For δ > 1, denote } { ∫ m ∫ 1 1∑ δ 2 V (γ(s), (1 − s)t2 + st1 )ds ρδ (x, y, t) = inf a (s)ds + t γ∈Γ0 4t 0 i=1 i 0 where Γ0 is a set of admissible curves γ : [0, 1] → M satisfying γ(0) = x, γ(1) = y, and γ ′ (s) = ∑m ∑ 2 i=1 ai (s)Xi (γ(s)) a.e.s. Since i ai (s) does not depend on the choice of the Xj ’s (see section 2 in [16]), nor does the distance function ρδ . In the case of V ≡ 0, ρδ is nothing less than the Carnot–Carath´eodory distance dCC induced by the subelliptic operator L. Theorem 4.1. Let u be a positive solution to the Schr¨ odinger equation (2.2). There exist positive constants ′ Ci , i = 1, . . . , 6 and δ0 = δ0 (ρ2 , k) > 1, we have for all 0 < t1 < t2 , x, y ∈ M and δ > δ0 , ′

( ) ( ) C1 5 ′ ∑ (i ) Ci+1 t2 δ i exp ρδ (x, y, t2 − t1 ) + t2 − t1 . u(x, t1 ) ≤ u(y, t2 ) t1 i i=1 Proof . Let γ be any admissible curve given by γ : [0, 1] → M , with γ(0) = y and γ(1) = x. We define η : [0, 1] → M × [t1 , t2 ] by η(s) = (γ(s), (1 − s)t2 + st1 ), clear that η(0) = (y, t2 ) and η(1) = (x, t1 ). d Integrating ds log u along η, we get ∫

1

d log uds ds 0 ∫ 1∑ m = ai Xi log u(x, s) − (t2 − t1 )(log u)s ds.

log u(x, t1 ) − log u(u, t2 ) =

0

i=1

Applying (2.4) in Corollary 2.3, there exist some constants Ci′ , i = 1, . . . , 6 and δ0 = δ0 (ρ2 , k) > 1 such that for δ > δ0 , t = (1 − s)t2 + st1 , we have ( ) ∫ 1∑ m 6 ∑ u(x, t1 ) 1 ′ i−2 log ≤ |ai ||Xi log u| + (t2 − t1 ) − Γ (log u) + Ci t + V (γ(s), t) ds u(x, t2 ) δ 0 i=1 i=1 ) ∫ 1∑ m ( t2 − t1 2 ≤ |ai (s)||Xi log u| − |Xi log u| + (t2 − t1 )V (γ(s), t)ds δ 0 i=1 4

+ C2′ (t2 − t1 ) + C1′ log ∫ ≤ 0

1

′ ( i+1 ) t2 ∑ Ci+2 + t2 − ti+1 1 t1 i=1 i + 1

m

∑ δ a2 (s) + (t2 − t1 )V (γ(s), (1 − s)t2 + st1 )ds 4(t2 − t1 ) i=1 i

+ C1′ log

5 ′ (i ) t2 ∑ Ci+1 + t2 − ti1 , t1 i=1 i

taking exponential in the both sides of the above inequality gives the desired result. □

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Applying Corollary 2.5 in the corresponding proof, we have Theorem 4.2. Let u be a positive solution to the Schr¨ odinger equation LV u = 0. There exists positive constant C = C(θ, γ1 , γ2 , ρ1 , ρ2 , k, d), for any x, y ∈ M , we have

where d(x, y) = inf γ∈Γ0

u(x) ≤ u(y) exp(Cd(x, y)), } ∫1 2 i=1 ai (s)ds + 0 V (γ(s))ds .

{∫ ∑ 1 m 0

5. Perelman type entropy and its monotonicity Throughout this section, we assume that the operator L satisfies the generalized curvature-dimension inequality CD(ρ1 , ρ2 , k, d) for some ρ1 ≥ 0, ρ2 > 0, k ≥ 0 and d < ∞. Consequently, the invariant measure is finite, see [4] Theorem 1.3. Without loss of any generality, we suppose µ is a probability measure. Let ∫ u(x, t) be the positive solution of the heat equation ∂t u = Lu on M × [0, ∞), we can assume M udµ = 1. Denote g(x, t) the function u(x, t) =

e−g(x,t) (4πt)

τD 2

(5.1)

)2 ( 3k with D = d 1 + 2ρ , where τ is some positive constant determined later. We define the so-called Nash-type 2 entropy, see [22,23], ∫ N (u, t) = − u log udµ M

and ˜ (u, t) = N (u, t) − τ D (log(4πt) + 1). N (5.2) 2 Thanks to the Li–Yau type inequality established in Section 2, we have the monotonicity of the Nash entropy formulae: Theorem 5.1. For all t > 0 and τ ≥ 1, we have ( ( ) ) ∫ d ˜ 3k 3kτ D N (u, t) = u Γ (g) + 1 + gt + dµ ≤ 0. dt 2ρ2 4ρ2 t M Proof . Denote f = log u as above, we have f = −g −

τD 2

log(4πt), it yields

τD , Γ (f ) = Γ (g). 2t ∫ Thanks to the fact M udµ = 1, the integration by parts formula gives ∫ ∫ d ˜ τD τD N (u, t) = uΓ (log u)dµ − = uΓ (g)dµ − , dt 2t 2t M M ∫ ∫ ∫ τD τD τD ugt dµ = − − uft dµ = − − ut dµ = − . 2t 2t 2t M M M ft = −gt −

It follows ( ) (∫ ) 3k τD τD u (Γ (g)) dµ + 1 + ugt dµ + − 2ρ t 2t 2 M ) ) ( ∫M ( 3kτ D 3k = gt + dµ. u Γ (g) + 1 + 2ρ2 4ρ2 t M

d ˜ N (u, t) = dt

∫

(5.3)

B. Qian / Nonlinear Analysis 155 (2017) 163–175

172

By (2.10) and (5.3), we have (

3k Γ (g) + 1 + 2ρ2

)

(

3k gt + 1 + 2ρ2

)

τD D − ≤ 0. 2t 2t

For τ ≥ 1, it is clear that d ˜ N (u, t) ≤ 0. dt The desired conclusion follows.

□

Following Perelman [24] (see also [23,20,9,14,3] etc.), we define ∫ ) d ( ˜ W(u, t) = tN (u, t) , u (tΓ (g) + g − τ D) dµ = dt M

(5.4)

and Wς (u, t) = W(u, t) + ςt2

∫

uΓ Z (log u)dµ,

M

where ς is some positive constant to be determined. To study the monotonicity of the functional Wς , let us give the following useful lemma. Lemma 5.2. For any positive solution to the heat equation ∂t u = Lu, we have ∫ d2 N (u, t) = −2 uΓ2 (log u)dµ, dt2 M this gives ∫ ∫ d τD W(u, t) = 2 uΓ (log u)dµ − 2t uΓ2 (log u)dµ − . dt 2t M M ∫ Moreover, denote B(u, t) = M uΓ Z (log u)dµ, we have ∫ B ′ (u, t) = −2 uΓ2Z (log u)dµ. M

Proof . Through computation, we get d2 d2 N (u, t) = N (u, t) dt2 dt2 ∫ d =− uL log udµ dt M ∫ ∫ =− LuL log udµ − u∂t (L log u)dµ M ∫M ∫ = −2 LuL log udµ − u(∂t − L)(L log u)dµ. M

M

Notice that (∂t − L)L log u = L(∂t − L) log u = LΓ (log u), this yields d2 N (u, t) = − dt2

∫

∫ uLΓ (log u)dµ + 2

M ∫

= −2

uΓ2 (log u)dµ. M

uΓ (log u, L log u)dµ M

(5.5)

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173

Hence ) d2 ( ˜ d (u, t) W(u, t) = 2 tN dt dt = tN ′′ (u, t) + 2N ′ (u, t) − (tτ D(1 + log(4πt)))′′ ∫ ∫ τD = −2t uΓ2 (log u)dµ + 2 . uΓ (log u)dµ − 2t M M Meanwhile ∫ ∂t uΓ Z (log u)dµ + 2 uΓ (log u, ∂t log u)dµ ∫M ∫M ( ) = LuΓ Z (log u)dµ + 2 uΓ Z log u, L log u + Γ (log u) dµ ∫ ∫M ∫M ( ) ( ) Z uΓ Z log u, Γ (log u) dµ uΓ Z log u, L log u dµ + 2 uLΓ (log u)dµ + 2 = ∫M ∫M ∫M ( ) ( ) Z Z = uLΓ (log u)dµ + 2 uΓ log u, L log u dµ + 2 uΓ log u, Γ Z (log u) dµ,

B ′ (u, t) =

∫

M

M

M

where in the last equality we apply the Assumption. Notice that ∫ ∫ ∫ uLΓ Z (log u)dµ = − Γ (u, Γ Z (log u))dµ = − uΓ (log u, Γ Z (log u))dµ, M

M

M

it follows ∫

′

∫

Z

B (u, t) = −

uLΓ (log u)dµ + 2 ∫M

= −2

( ) uΓ Z log u, L log u dµ

M

uΓ2Z (log u)dµ.

M

Hence we complete the proof. □ Now we are ready to state the main result in this section. Theorem 5.3. For ρ2 ≤ ς ≤ 53 ρ2 and τ ≥ 2 + 2k ρ2 , we have ( ) ∫ d 2t D 2(k + ς) 2 Wς (u, t) ≤ − u(L log u) dµ + −τ ≤0 dt d M 2t ς holds for all t > 0. Proof . By Lemma 5.2, we have d d d Wς (u, t) = W(u, t) + 2ςtB(t) + ςt2 B(t) dt dt ∫ dt∫

uΓ2Z (log u)dµ + 2

uΓ2 (log u)dµ − 2ςt2

= −2t M ∫

M

∫ uΓ (log u)dµ M

τD + 2ςt uΓ (log u)dµ − 2t ∫ M ∫ ∫ 2t 2(k + ς) τD ≤− u(L log u)2 dµ + uΓ (log u)dµ + 2t(ς − ρ2 ) uΓ Z (log u)dµ − . d M ς 2t M M The last inequality follows from the generalized curvature-dimension inequality CD(ρ1 , ρ2 , k, d) with ρ1 ≥ 0. Applying the Li–Yau type estimate (2.10), we have for ρ2 ≤ ς ≤ 35 ρ2 , ∫ ∫ ∫ 2(k + ς) (k + ς)(2ρ2 + 3k) (k + ς)D Z uΓ (log u)dµ + 2t(ς − ρ2 ) uΓ (log u)dµ ≤ ut dµ + ς ςρ2 ςt M M M (k + ς)D = . ςt

B. Qian / Nonlinear Analysis 155 (2017) 163–175

174

This gives ∫

d 2t Wς (u, t) ≤ − dt d hence for τ ≥ 2 +

2k ρ2

≥2+

2k ς ,

D u(L log u) dµ + 2t M 2

(

2(k + ς) −τ ς

) ,

we have d Wς (u, t) ≤ 0. dt

We complete the proof. □ Remark 5.4. (i) Theorems 5.1 and 5.3 generalize Theorem 1.12 and Theorem 1.13 in [14] respectively, where they consider the case of sub-Laplacians on a closed pseudohermitian 3-manifold with pseudo-hermitian torsion vanishes and Tanaka–Webster curvature bounded below, i.e. CD(k, 21 , 1, 2) case. (ii) For the diffusion operators satisfying the curvature-dimension inequality CD(k, m) on a complete noncompact Riemannian manifold, the monotonicity of the Perelman entropy has been obtained in [20], see also [9,21]. It would be very interesting to derive monotonicity of the Perelman type entropies for subelliptic operator on complete but noncompact Riemannian manifolds, which will be studied in near (future. ) 2

3k by (iii) In the definitions of (5.1), (5.2) and (5.4), if we replace the parameter D = d 1 + 2ρ 2 ( ) 2 2 (1+γ)k ˜ = dγ D (γ > 1), using the Li–Yau type inequality (2.9) instead of (2.10) in the 4(γ−1) 1 + γρ2 corresponding proofs, we have for all t > 0, γ > 1 and τ ≥ 1, ( ) ( ) ∫ ˜ (1 + γ)k (γ + 1)kτ D d ˜ N (u, t) = u Γ (g) + 1 + dµ ≤ 0. gt + dt γρ2 2γρ2 t M

Moreover, for γ > 1, ρ2 ≤ ς ≤

3+γ 1+γ ρ2

2t d Wς (u, t) ≤ − dt d

and τ ≥ 2 + ∫ M

2k ρ2 ,

we have

u(L log u)2 dµ +

) ˜ ( 2(k + ς) D − τ ≤ 0. 2t ς

If we choose γ = 2, the above two results become Theorems 5.1 and 5.3 respectively. Acknowledgement The author would like to express his sincere thanks to Prof. F. Baudoin for the helpful discussion.The author is greatly indebted to Prof. L. M. Wu (Clermont-Ferrand University II, France), Prof. D. Bakry (Toulouse University III, France), Prof. H. Q. Li (Fudan University, China) and Prof X. D. Li (Institute of Applied Mathematics, Academia Sinica, China) for their constant encouragement and support. He is also acknowledges the financial supported by “the Fundamental Research Funds for the Central Universities”, “DHU Distinguished Young Professor Program” and by National Science Funds of China No. 11671076 and No. 11371351. References

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