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Let (Mn , g, e−f dv) be a smooth metric measure space of dimensional n. Suppose that v is a positive weighted pharmonic functions on M, namely e f div(e −f |∇v| p−2∇v) = 0. We first give a local gradient estimate for v, as a consequence we show that if Ricm f ≥ 0 then v is constant provided that v is of sublinear growth. At the same time, we prove a Harnack inequality for such a weighted pharmonic function v. Moreover, we show a global sharp gradient estimate for weighted peigenfunctions. Then we use this estimate to study geometric structure at infinity when the first eigenvalue λ1,p obtains its maximal value
Local and global sharp gradient estimates for weighted p-harmonic functions Nguyen Thac Dung and Nguyen Duy Dat May 28, 2015 Abstract n −f Let (M , g, e dv) be a smooth metric measure space of dimensional n. Suppose that v is a positive weighted p-harmonic functions on M , namely ef div(e−f |∇v|p−2 ∇v) = 0. We first give a local gradient estimate for v, as a consequence we show that if Ricm f ≥ 0 then v is constant provided that v is of sublinear growth. At the same time, we prove a Harnack inequality for such a weighted p-harmonic function v. Moreover, we show a global sharp gradient estimate for weighted p-eigenfunctions. Then we use this estimate to study geometric structure at infinity when the first eigenvalue λ1,p obtains its maximal value. 2000 Mathematics Subject Classification: 53C23, 53C24 Keywords and Phrases: Gradient estimates, weighted p-harmonic functions, smooth metric measure spaces, Liouville property, Harnack inequality 1. Introduction The local Cheng-Yau gradient estimate is a standard result in Riemannian geometry, see [6], also see [22]. It asserts that if M be an n dimensional complete Riemannian manifold with Ric ≥ −(n − 1)κ for some κ ≥ 0, for u : B(o, R) ⊂ M → R harmonic and positive then there is a constant cn depending only on n such that √ 1 + κR |∇u| ≤ cn . (1.1) sup R B(o,R/2) u Here B(o, R) stands for the geodesic centered at a fixed point o ∈ M . Notice that when κ = 0, this implies that a harmonic function with sublinear growth on a manifold with non-negative Ricci curvature is constant. This result is clearly sharp since on Rn there exist harmonic functions which are linear. Cheng-Yau’s method is then extended and generalized by many mathematicians. For example, Li-Yau (see [9]) obtained a gradient estimate for heat equations. Cheng 1 (see [5]) and H. I. Choi (see [7]) proved gradient estimates for harmonic mappings, etc. We refer the reader to survey [8] for an overview of the subject. When (M n , g, e−f dv) is a smooth metric measure space, it is very natural to find similar results. Recall that the triple (M n , g, e−f dµ) is called a smooth metric measure space if (M, g) is a Riemannian manifold, f is a smooth function on M and dµ is the volume element induced by the metric g. On M , we consider the differential operator ∆f , which is called f −Laplacian and given by ∆f · := ∆ · − f, · . It is symmetric with repect to the measure e−f dµ. That is, (∆f ϕ)ψe−f , ϕ, ψ e−f = − M M for any ϕ, ψ ∈ C0∞ (M ). Smooth metric measure spaces are also called manifolds with ´ density. By m-dimensional Bakry-Emery Ricci tensor we mean Ricm f = Ric + Hessf − ∇f ⊗ ∇f , m−n ´ for m ≥ n. Here m = n iff f is constant. The ∞−Barky-Emery tensor is refered as Ricf = Ric + Hessf. Brighton (see [2]) gave a gradient estimate of positive weighted harmonic function, as a consequence, he proved that any bounded weighted harmonic function on a smooth metric measure space with Ricf ≥ 0 has to be constant. Later, Munteanu and Wang refined Brighton’s argument and proved that positive f -harmonic function of sub-exponential growth on smooth metric measure space with nonnegative Ricf must be a constant function. Moreover, Munteanu and Wang also applied the De GiorgiNash-Moser theory to get a sharp gradient estimate for any positive f -harmonic function provided that the weighted function f is at most linear growth (see [18, 19] for further results). On the other hand, Wu derived a Li-Yau type estimate for parabolic equations. He also made some results for heat kernel (see [28, 30] for the details.). From a variational point of view, p-harmonic function, or more general weighted pharmonic functions are natural extensions of harmonic functions, or weight harmonic functions, respectively. Compared with the theory for (weighted) harmonic functions, the study of (weighted) p-harmonic functions is generally harder, even though elliptic, is degenerate and the regularity results are far weaker. We refer the reader to [17, 11] for the connection between p-harmonic functions and the inverse mean curvature flow. For the weighted p-harmonic function, Wang (see [24]) estimated eigenvalues of this 2 operator. On the other hand, Wang, Yang and Chen (see [27]) shown gradient estimates and entropy formulae for weighted p-heat equations. Their works generalized Li’s and Kotschwar-Ni’s results (see [16, 11]). In this paper, motived by Wang-Zhang’s gradient estimate for the p-harmonic function, we give the following result. Theorem 1.1. Let (M n , g, e−f ) be a smooth metric measure space of dimension n with Ricm f ≥ −(m − 1)κ. Suppose that v is a positive smooth weighted p-harmonic function on the ball BR = B(o, R) ⊂ M . Then there exists a constant C = C(p, m, n) such that √ |∇v| C(1 + κR) ≤ on B(o, R/2). (1.2) v R We note that this theorem is very much motivated by and follows the ideas of Wang and Zhang in the paper [26]. Our argument is close to the one used in [26]. In the gerenal case, we obtain a sharp gradient estimate for weighted p-eigenfunction as follows. Theorem 1.2. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold with Ricm f ≥ −(m − 1). If v is a positive weighted p-eigenfunction with respect to the first eigenvalue λ1,p , that is, ef div(e−f |∇v|p−2 ∇v) = −λ1,p v p−1 then |∇ ln v| ≤ y. Here y is the unique positive root of the equation (p − 1)y p − (m − 1)y p−1 + λ1,p = 0. A directly consequence of the theorem 1.2 is a sharp gradient estimate for positive weighted p-harmonic function. Corollary 1.3. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold with Ricm f ≥ −(m − 1). If v is a positive weighted p-harmonic function then |∇ ln v| ≤ m−1 . p−1 The sharpness of the estimate is demonstrated by the below example. Example 1.4. Let M n = R × N n−1 with a warped product metric ds2 = dt2 + e2t ds2N , 3 where N is a complete manifold with non-negative Ricci curvature. Then it can be directly checked that RicM ≥ −(n−1) (See [14] for details of computation). Moreover, we have ∂2 ∂ ∆ = 2 + (n − 1) + e−2t ∆N . ∂t ∂t ´ Choose weighted function f = (m − n)t, then the m-dimensional Bakry-Emery curvature is bounded from below by −(m − 1). Let v(t, x) = e−at , where m−1 ≤ a ≤ m−1 , we have p p−1 |∇v| = a. v It is easy to show that |∇v|p−2 ∇v, ∇f = −(m − n)ap−1 v p−1 ∇|∇v|p−2 , ∇v = (p − 2)ap v p−1 |∇v|p−2 ∆v = (1 − n + a)ap−1 v p−1 . Hence, ef div(e−f |∇v|p−2 ∇v) = ((p − 1)a − (m − 1))ap−1 v p−1 . This implies that λ1,p = (m − 1 − (p − 1)a)ap−1 , or equivalently, (p − 1)ap − (m − 1)ap−1 + λ1,p = 0. It is also very interesting to ask what is geometric structure of manifolds with λ1,p acheiving its maximal value. When f is constant, this problem has been studied by Li-Wang, Sung-Wang in [12, 13, 23]. In this paper, we prove a generalization of their result. Theorem 1.5. Let (M n , g, e−f dv) be a smooth metric measure space of dimension n ≥ 2. Suppose that Ricm f ≥ −(m − 1) and λ1,p = m−1 p p . Then either M has no p-parabolic ends or M = R × N n−1 for some compact manifold N . Here the definition of p-parabolic ends is given in the section 3. This paper is organized as follows. In the section 2, we give a proof of the main theorem 1.1 by using the Moser’s iteration. As its applications, we show a Liouville property and a Harnack inequality for weighted p-harmonic functions. In the section 3, we prove the theorem 1.2. The proof the theorem 1.5 is given in the section 4. 4 2. Local gradient estimates for weighted p-harmonic functions on (M, g, e−f dµ) Let v be a positive weighted p-eigenfunction function with respect to the first eigenvalue λ1,p , namely, v is a smooth solution of weighted p-Laplacian equation, ∆p,f v := ef div(e−f |∇v|p−2 ∇v) = −λ1,p v p−1 . (2.1) Note that, when λ1,p = 0 then v is a weighted p-harmonic function. Let u = −(p − 1) log v, then v = e−u/(p−1) . It is easy to see that u satisfies ef div(e−f |∇u|p−2 ∇u) = |∇u|p + λ1,p (p − 1)p−1 . Put h := | u|2 , the above equation can be rewritten as follows. p − 1 hp/2−2 ∇h, ∇u + hp/2−1 ∆f u = hp/2 + λ1,p (p − 1)p−1 . 2 (2.2) Assume that h > 0. As in [11], [17], we consider the below operator Lf (ψ) := ef div e−f hp/2−1 A(∇ψ) − php/2−1 ∇u, ∇ψ , where A = id + (p − 2) ∇u ⊗ ∇u . |∇u|2 We have the following lemma and the proof is by direct computation. Lemma 2.1. Lf (h) = 2hp/2−1 (u2ij + Ricf (∇u, ∇u)) + p − 1 hp/2−2 |∇h|2 . 2 (2.3) Proof. By the definition of Lf , we have p Lf (h) =ef div e−f h 2 −1 ∇h + (p − 2) p ∇u, ∇h ∇u h p − ph 2 −1 ∇u, ∇h p =h 2 −1 ∆h + (p − 2)h 2 −1 div h−1 ∇u, ∇h ∇u p p + ef ∇ e−f h 2 −1 , ∇h + (p − 2)h−1 ∇u, ∇h ∇u − ph 2 −1 ∇u, ∇h p p p =h 2 −1 ∆h + (p − 2)h 2 −2 ∇u, ∇h ∆u − (p − 2)h 2 −3 ∇u, ∇h 2 p p p p p + (p − 2)h 2 −2 (uij hi uj + hij ui uj ) + − 1 h 2 −2 |∇h|2 + (p − 2) − 1 h 2 −3 ∇u, ∇h 2 2 p p p −1 −2 −1 2 2 2 −h ∇f, ∇h − (p − 2)h ∇u, ∇h ∇f, ∇u − ph ∇u, ∇h . 5 2 Hence, p p − 1 h 2 −2 |∇h|2 2 p p p −2 2 − 2 h 2 −3 ∇h, ∇u 2 + (p − 2)h ∇h, ∇u ∆f u + (p − 2) 2 p p −2 2 2 + (p − 2)h (uij hi uj + hij ui uj ) − ph −1 ∇u, ∇h p p p − 1 h 2 −2 |∇h|2 =2h 2 −1 u2ij + Ricf (∇u, ∇u) + 2 p p p p + (p − 2)h 2 −2 ∇h, ∇u ∆f u + 2h 2 −1 ∇∆f u, ∇u + (p − 2) − 2 h 2 −3 ∇h, ∇u 2 p p −2 −1 2 2 + (p − 2)h (uij hi uj + hij ui uj ) − ph ∇u, ∇h . p Lf (h) =h 2 −1 ∆f h + Here we used the Bochner identity ∆f h = ∆f |∇u|2 = 2 u2ij + Ricf (∇u, ∇u) + 2 ∇∆f u, ∇u in the last equation. On the other hand, differentiating both side of (2.2) then multiplying the obtained results by ∇u, we have p p p p p −1 h2 ∇h, ∇u = − 1 h 2 −2 ∇h, ∇u ∆f u + h 2 −1 ∇∆f u, ∇u 2 2 p p p p p + −2 − 1 h 2 −3 ∇h, ∇u 2 + − 1 h 2 −2 (uij hi uj + hijui uj ) 2 2 2 Combining this equation and the above equation, we are done. Now, suppose that v is a weighted p-harmonic function, so we can assume λ1,p = 0. We choose a local orthonormal frame {ei } with e1 = ∇u/|∇u| then n u21i = 2hu11 = ∇u, ∇h , i=1 1 |∇h|2 . 4 h Hence, (2.2) takes the following form n (p − 1)u11 + uii = h + ∇f, ∇u . i=2 6 2 Therefore n u2ij ≥ u211 + 2 n u2ii u21i + i=2 i=2 n ≥ u211 + 2 i=2 n = u211 + 2 1 u21i + n−1 i=2 uii i=2 u21i + 1 (h − (p − 1)u11 + ∇f, ∇u )2 n−1 u21i + 1 n−1 i=2 n ≥ u211 + 2 2 n ∇u, ∇f (h − (p − 1)u11 )2 − m−n m−n 1 + n−1 n−1 1 2(p − 1) (p − 1)2 ≥ h2 − hu11 + 1 + m−1 m−1 m−1 ≥ 2(p − 1) 1 h2 − hu11 + a0 m−1 m−1 n 2 n u211 u21i − +2 i=2 ∇f, ∇u m−n 2 2 u21i − i=1 ∇f, ∇u , m−n (p−1)2 where a0 = max 1 + m−1 , 2 > 1. Note that we used (a − b)2 ≥ in the fourth inequality. Again, by using the identities n u21i = 2hu11 = ∇u, ∇h , i=1 a2 1+δ − b2 δ for δ > 0 1 |∇h|2 4 h we conclude that 1 p−1 a0 |∇h|2 ∇f, ∇u 2 2 ≥ h − ∇u, ∇h + − . m−1 m−1 4 h m−n Assume that Ricm f ≥ −(m − 1)κ, we infer p − 1 hp/2−2 |∇h|2 Lf (h) = 2hp/2−1 (u2ij + Ricf (∇u, ∇u)) + 2 1 p−1 a0 |∇h|2 df ⊗ df ≥ 2hp/2−1 h2 − ∇u, ∇h + + Ricf − m−1 m−1 4 h m−n p + − 1 hp/2−2 |∇h|2 2 p + a0 ≥ −2(m − 1)κhp/2 + − 1 |∇h|2 hp/2−2 2 2 2(p − 1) p/2−1 + hp/2+1 − h ∇u, ∇h m−1 m−1 2 2(p − 1) p/2−1 ≥ −2(m − 1)κhp/2 + hp/2+1 − h ∇u, ∇h m−1 m−1 u2ij 7 (∇u, ∇u) The above equation holds wherever h is strictly positive. Let K = {x ∈ M, h(x) = 0}. Then for any non-negative function ψ with compact support in Ω \ K, we have hp/2−1 ∇h + (p − 2)hp/2−2 ∇u, ∇h ∇u, ∇ψ e−f Ω hp/2−1 ∇u, ∇h ψe−f + +p Ω hp/2 ψe−f ≤ 2(m − 1)κ Ω 2(p − 1) + m−1 2 m−1 hp/2+1 ψe−f Ω hp/2−1 ∇u, ∇h ψe−f (2.4) Ω In order to consider the cases h = 0, for ε > 0, b > 2 we choose ψ = hbε η 2 where hε = (h − ε)+ , η ∈ C0∞ (BR ) is non-negative, 0 ≤ η ≤ 1, b is to be determined later. Then direct computation shows that 2 b ∇ψ = bhb−1 ε ∇hη + 2hε η∇η. Plugging this identity into (2.4), we have p 2 h 2 −1 hb−1 ε |∇h| + (p − 2)h b p−2 −2 2 hb−1 ∇u, ∇h ε 2 η 2 e−f Ω p p h 2 −1 hbε ∇η, ∇h ηe−f + 2(p − 2) +2 Ω +p h 2 −2 hbε ∇u, ∇h ∇u, ∇η ηf −f Ω h p −1 2 p 2 h 2 +1 hbε η 2 e−f m−1 Ω p 2(p − 1) h 2 −1 hbε η 2 e−f . + m−1 Ω hbε ∇u, ∇h η 2 e−f + Ω p h 2 hbε η 2 e−f ≤ 2(m − 1)κ Ω Let if p ≥ 2 , if 1 < p < 2 1 p−1 a1 = it is easy to see that p h 2 −1 hεb−1 |∇h|2 + (p − 2)h p−2 −2 2 hb−1 ∇u, ∇h ε 2 p 2 ≥ a1 h 2 −1 hb−1 ε |∇h| . Hence, by passing ε to 0, we obtain p h 2 +b−2 |∇h|2 η 2 e−f ba1 Ω p Ω +p h p h 2 +b−2 ∇u, ∇h ∇u, ∇η ηe−f + 2 + 2(p − 2) p +b−1 2 Ω p h 2 +b η 2 e−f Ω 2 h η 2 e−f m−1 Ω p 2(p − 1) + h 2 +b−1 ∇u, ∇h η 2 e−f . m−1 Ω p +b+1 2 ∇u, ∇h η 2 e−f + Ω ≤ 2(m − 1)κ h 2 +b−1 ∇h, ∇η ηe−f 8 (2.5) From now on, we use a1 , a2 , . . . to denote constants depending only on m, p, n. Combining terms in (2.5) and using the definition of h, we infer p h 2 +b−2 |∇h|2 η 2 e−f + a1 b Ω ≤ 2(m − 1)κ h p +b 2 2 m−1 + a3 h Ω η 2 e−f + a2 Ω p +b−1 2 p h 2 +b+1 η 2 e−f h p−1 +b 2 |∇h|η 2 e−f Ω |∇h||∇η|ηe−f . (2.6) Ω For i, j = 1, 2, 3, . . ., again, from now on we use Ri , Lj to denote the i-th term on the right hand side, the j-th term in the left hand side. For the third term on the 2 right hand side, using the Cauchy’s inequality 2αβ ≤ α4c + cβ 2 for c > 0, we have the following estimate. ba1 4 |R3 | ≤ p h 2 +b−2 |∇h|2 η 2 e−f + Ω a4 b p |∇η|2 h 2 +b e−f . Ω By Cauchy’s inequality again, R2 can be estimated by ba1 4 |R2 | ≤ p h 2 +b−2 |∇h|2 η 2 e−f + Ω a5 b p h 2 +b+1 η 2 e−f . Ω Combining these two inequalities togather with (2.6), we obtain ba1 2 p 2 h 2 +b+1 η 2 e−f m−1 Ω Ω p p a4 a5 |∇η|2 h 2 +b e−f + ≤ 2(m − 1)κ h 2 +b η 2 e−f + b Ω b Ω p h 2 +b−2 |∇h|2 η 2 e−f + p h 2 +b+1 η 2 e−f . (2.7) Ω By requiring a5 1 ≤ , b m−1 (2.8) we have that (2.7) implies ba1 2 p 1 2 h 2 +b+1 η e−f m−1 Ω Ω p p a4 ≤ 2(m − 1)κ h 2 +b η 2 e−f + |∇η|2 h 2 +b e−f . b Ω Ω p h 2 +b−2 |∇h|2 η 2 e−f + By Schwarz inequality, it is easy to see that p b ∇(h 4 + 2 η) 2 ≤ 1 p +b 2 2 2 p p h 2 +b−2 |∇h|2 η 2 + 2h 2 +b |∇η|2 . From (2.9), we have following lemma. 9 (2.9) Lemma 2.2. Let M be a smooth metric measure space with Ricm f ≥ −(m − 1)κ, for some κ ≥ 0, and Ω ⊂ M is an open set and v is a smooth weighted p-harmonic function on M . Let u = −(p − 1) log v and h = |∇u|2 . Then for any b > 2, there exist c1 , c2 , c3 depending on b, m, n such that p b 2 p ∇(h 4 + 2 η) e−f +c1 h 2 +b+1 η 2 e−f Ω Ω p p h 2 +b η 2 e−f + c3 ≤ κc2 Ω h 2 +b |∇η|2 e−f , (2.10) Ω for any η ∈ C0∞ (BR ), where BR is a geodesic ball centered at a fixed point o ∈ M . Moreover, we have c1 ∼ b, c2 ∼ b (Here c1 ∼ b means c1 is comparable to b, c2 ∼ b is understood the same way). In [1], Bakry and Qian proved the following generalized Laplacian comparison theorem (also see Remark 3.2 in [16]) √ √ (2.11) ∆f ρ := ∆ − ∇f, ∇ρ ≤ (m − 1) κ coth( κρ) provided that Ricm f ≥ −(m − 1)κ. Here ρ(x) := dist(o, x) stands for the distance between x ∈ M and a fixed point o ∈ M . This implies the volume comparison VHm (r2 ) Vf (Bx (r2 )) ≤ Vf (Bx (r1 )) VHm (r1 ) (2.12) (see [29] for details). It turns out that we have the local f -volume doubling property. Then we follow the Buser’s proof [4] or the Saloff-Coste’s alternate proof (Theorem 5.6.5 in [20]), we can easily get a local Neumann Poincar´e inequality in the setting of smooth metric measure spaces. Using the volume comparison theorem, the local Neumann Poincar´e inequality and following the argument in [21], we obtain a local Sobolev inequality as belows. Theorem 2.3. Let (M, g, e−f dµ) be an n-dimensional complete noncompact smooth metric measure space. If Ricm f ≥ −(m − 1)κ for some nonnegative constants κ, then for any p > 2, there exists a constant c = c(n, p, m) > 0 depending only on p, n, m such that |ϕ| 2p p−2 −f e BR √ p−2 p ≤ R2 .ec(1+ κR) Vf (BR ) 2 p |∇ϕ|2 + R−2 ϕ2 e−f BR for any ϕ ∈ C0∞ (Bp (R)). Proof. We refer the reader to [28] for the details of the argument. 10 From now on, we suppose Ω = BR . Theorem 2.3 implies h n(p/2+b) n−2 2n n−2 η e n−2 n −f BR ≤ ec(1+ √ κR) 2 Vf (BR )− n p b 2 p ∇(h 4 + 2 ) e−f + R2 h 2 +b η 2 e−f BR (2.13) BR √ where c(n, p, m) > 0 depends only on n, p. Let b0 = C(n, p, m)(1 + κR) with C(n, p, m) ≥ c(n, p, m) large enough to make b0 satisfy (2.8) and b0 > 2, then from (2.10) and (2.13) we infer h n(p/2+b) n−2 η 2n n−2 n−2 n −f e p + a6 beb0 R2 Vf (BR )−2/n h 2 +b+1 η 2 e−f BR BR p p h 2 +b η 2 e−f + a8 eb0 Vf (BR )−2/n R2 ≤ a7 b20 beb0 Vf (BR )−2/n h 2 +b |∇η|2 e−f . BR BR (2.14) Lemma 2.4. Let b1 = b0 + p 2 n . n−2 Then there exists d = d(n, p, m) > 0 such that ||h||Lb1 (B3R/4 ) ≤ d b20 Vf (BR )1/b1 . R2 Proof. First let us choose b = b0 . We observe that if h > (2.15) 2a7 b2 a6 R2 then p p 1 a7 b3 h 2 +b < a6 bR2 h 2 +b+1 . 2 Hence, in (2.14), R1 can be estimated by p R1 = a7 b3 eb Vf (BR )−2/n BR ∩ h≤ ≤ p+2b b R ab9 b3 eb 2a7 b2 a6 R2 2 Vf (BR )1− n + p h 2 +b η 2 e−f + a7 b3 eb Vf (BR )−2/n BR ∩ h> 2a7 b2 a6 R 2 h 2 +b η 2 e−f L2 . 2 Now, (2.14) can be read as follows. h n(p/2+b) n−2 η 2n n−2 e −f n−2 n + BR ≤ ab9 b3 eb b R p+2b a6 b 2 be R Vf (BR )−2/n 2 p h 2 +b+1 η 2 e−f BR 2 p Vf (BR )1− n + a8 eb Vf (BR )−2/n R2 h 2 +b |∇η|2 e−f . BR 11 (2.16) Now, we want to control R2 in terms of the left hand side. We choose η1 ∈ C0∞ (BR ) such that C(n) 0 ≤ η1 ≤ 1, η1 ≡ 1, in B3R/4 , |∇η1 | ≤ . R p Let η = η12 +b+1 . Direct computation shows 2b+p R2 |∇η|2 ≤ a10 b2 η b+p/2+1 . By the above inequality and Young’s inequality, the second term of the right hand side of (2.16) can be estimated by p e−b Vf (BR )2/n R2 = a8 R2 h 2 +b |∇η|e−f BR 2b+p p h 2 +b η b+p/2+1 e−f ≤ a8 a10 b2 BR ≤ a8 a10 b 2 h p +b+1 2 b+p/2 b+p/2+1 2 −f η e 1 Vf (BR ) b+p/2+1 BR ba6 2 R 2 ≤ b+p/2 b p h 2 +b+1 η 2 e−f + a11 BR b+p/2+2 R2b+p Vf (BR ). Plugging this inequality into (2.16), and note that b > 2, we obtain n−2 n h n(p/2+b) n−2 ≤ ab12 b3 eb B3R/4 n Recall that b = b0 , b1 = b0 + p2 n−2 = b + p2 1 root on both sides of (2.17), we have the b+p/2 ||h||Lb1 (B3R/4 ) ≤ d b R p+2b n n−2 and b0 = C(1 + Vf (BR )1−2/n . (2.17) √ κR). Taking 1 b20 b1 V (B ) f R R2 here we used the boundedness of b1/b . The proof is complete. Now, we give a proof of the main theorem. Proof of Theorem 1.1. By (2.14), we have h n(p/2+b) n−2 η 2n n−2 e −f n−2 n ≤ a13 BR eb0 p 2/n Vf (BR ) b20 bη 2 + R2 |∇η|2 h 2 +b e−f . (2.18) BR In order to apply the Moser iteration, let us put bk+1 = bk n , n−2 Bk = B o, 12 R R + k 2 4 , k = 1, 2, . . . and choose ηk ∈ C0∞ (BR ) such that ηk ≡ 1 in Bk+1 , η ≡ 0 in BR \ Bk , |∇ηk | ≤ C1 4k , R 0 ≤ ηk ≤ 1, where C1 is a certain constant. Hence in (2.18), by letting b + obtain 1 bk+1 bk+1 −f h ≤ e a13 Bk+1 p 2 = bk , η = ηk , we 1 bk eb0 b20 bk Vf (BR )2/n 2 2 + R |∇ηk | bk −f h e 1 bk . Bk By assumption of |∇ηk |, this implies 1 bk b0 ||h||Lbk+1 (Bk+1 ) ≤ ∞ It is easy to see that k=1 ||h||L∞ (BR/2 ) ≤ a13 1 bk = e n . 2b1 e Vf (BR ) e 2b1 1 (b20 bk + 16k ) bk ||h||Lbk (Bk ) . Vf (BR )2/n The above inequality leads to ∞ 1 b k=1 k b0 nb0 ≤ a14 a13 ∞ b30 2/n k=1 n n−2 ∞ k=1 k bk + 16k ||h||Lb1 (B3R/4 ) 3n b 2b1 ||h||Lb1 (B3R/4 ) 1/b1 0 Vf (BR ) here we used that 1 bk k (2.19) converges. Now, by Lemma 2.4 and (2.19), we conclude ||h||L∞ (BR/2 ) ≤ a15 b20 . R2 The proof is complete. As a consequence, we obtain an important folowing theorem ralating to Liouvilleproperty for weighted p-harmonic functions. Theorem 2.5. Assume that (M, g) is a smooth metric measure space with Ricm f ≥ 0. If u is a weighted p-harmonic function bounded from below on M and u is of sublinear growth then u is constant. Moreover, let x, y ∈ M be arbitrary points. There is a minimal geodesic γ(s) joining x and y, γ : [0, 1] → M, γ(0) = x, γ(1) = y. By integrating (1.2) over this geodesic, we obtain the below Harnack inequality. 13 Theorem 2.6. Let (M, g, e−f dv) be a complete smooth metric measure space of dimension n ≥ 2 with Ricm f ≥ −(m − 1)κ. Suppose that v is a positive weighted pharmonic function on the geodesic ball B(o, R) ⊂ M . There exists a constant Cp,n,m depending only on p, n, m such that v(x) ≤ eCp,n,m (1+ √ κR) v(y), ∀x, y ∈ B(o, R/2). If κ = 0, we have a uniform constant cp,n,m (independent of R) such that sup v ≤ cp,n B(o,R/2) 3. inf v. B(o,R/2) Global sharp gradient estimate for weighted p-Laplacian Recall that a function v is an eigenfunction of p-Laplacian with corresponding eigenvalue λ1,p ≥ 0 if ef div(e−f |∇v|p−2 ∇v) = −λ1,p |v|p−2 v. (3.1) In this section, we only consider positive solution v. Set u = −(p−1) ln v, the equation (3.1) can be rewritten as follows ef div(e−f |∇u|p−2 ∇u) = |∇u|p + λ1,p (p − 1)p−1 . (3.2) Put h := | u|2 , assume that h > 0. As in [23], we consider L(ψ) := ef div e−f hp/2−1 A(∇ψ) which is a slight modification of L(ψ). By Lemma 2.1, we have that L(h) = 2hp/2−1 (u2ij + Ricf (∇u, ∇u)) + p − 1 hp/2−2 |∇h|2 2 p + ph 2 −1 ∇u, ∇h Let {e1 , e2 , . . . , en } be an orthonormal frame on M with |∇u|e1 = ∇u. Then (3.2) can be read as n uii = h + ∇f, ∇u + |∇u|2−p λ1,p (p − 1)p−1 . (p − 1)u11 + i=2 14 Therefore, n u2ij ≥ i=1 n ≥ 1 u21i + n−1 (h + λ1,p (p − 1)p−1 |∇u|2−p + ∇f, ∇u − (p − 1)u11 )2 n−1 u21i + 1 n−1 u21i + ∇f, ∇u (h + λ1,p (p − 1)p−1 |∇u|2−p )2 − m−1 m−n i=1 n ≥ uii i=2 u21i + i=1 n ≥ 2 n i=1 − (h + λ1,p (p − 1)p−1 |∇u|2−p − (p − 1)u11 )2 ∇u, ∇f − m−n m−n 1 + n−1 n−1 2 2(p − 1) 2λ1,p (p − 1)p hu11 − |∇u|2−p u11 . m−1 m−1 Note that we used (a − b)2 ≥ using the identities a2 1+δ − b2 δ for δ > 0 in the fourth inequality. Again, by n u21i = 2hu11 = ∇u, ∇h , i=1 1 |∇h|2 4 h we conclude that u2ij 2 1 |∇h|2 (h + λ1,p (p − 1)p−1 |∇u|2−p )2 ∇f, ∇u 2 ≥ + − 4 h m−1 m−n 2λ1,p (p − 1)p ∇h, ∇u 2(p − 1) ∇h, ∇u − |∇u|2−p . − m−1 m−1 h 15 Assume that Ricm f ≥ −(m − 1)κ, we infer L(h) =2hp/2−1 (u2ij + Ricf (∇u, ∇u)) + p − 1 hp/2−2 |∇h|2 2 p + ph 2 −1 ∇u, ∇h 1 |∇h|2 (h + λ1,p (p − 1)p−1 |∇u|2−p )2 df ⊗ df ≥2hp/2−1 + + Ricf − 4 h m−1 m−n p p + − 1 hp/2−2 |∇h|2 + ph 2 −1 ∇u, ∇h 2 4(p − 1) p −1 2λ1,p (p − 1)p ∇h, ∇u h2 − ∇h, ∇u − m−1 m−1 h p−1 2−p 2 (h + λ1,p (p − 1) |∇u| ) ≥2hp/2−1 − (m − 1)h m−1 p p − 1 p/2−2 h |∇h|2 + ph 2 −1 ∇u, ∇h + 2 4(p − 1) p −1 2λ1,p (p − 1)p ∇h, ∇u − h2 . ∇h, ∇u − m−1 m−1 h (∇u, ∇u) (3.3) To show the sharp estimate, let x be the unique positive root of the equation p x 2 − (m − 1)x p−1 2 + λ1,p (p − 1)p−1 = 0. For any δ > 0, we consider ω= h − (x + δ), h > x + δ . 0, otherwise To show global sharp estimate of weighted p-eigenfunction, we need to have a upper bound of λ1,p as follows Lemma 3.1. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold with Ricm f ≥ −(m − 1). Then λ1,p ≤ m−1 p p . In order to prove lemma 3.1, let us recall a definition. Definition 3.2. (see [3]) Let (M n , g, e−f dv) be a smooth metric measure space. For a fixed point o ∈ M , let B o (r) = {q ∈ M : dist(o, q) ≤ r}. An end E is an unbounded component E of M \ B o (r0 ) for some r0 ≥ 0. For any 1 ≤ p < ∞. The end E is said to be p-parabolic if for each K M and ε > 0 there exists a Lipschitz function φ with 16 compact support, φ ≥ 1 on K, such that Here gφ (x) is defined as E gφp < ε. Otherwise, E is p-nonparabolic. dist(φ(y), φ(x)) . dist(y, x) y∈Bx (r) gφ (x) = lim inf sup + r→0 Proof of lemma 3.1. Without loss of generality, we may assume that λ1,p is positive. By the variational characterization of λ1,p , we know that M has infinite f -volume. The theorem 0.1 in [3] implies M is p-nonparabolic, moreover 1/p Vf (B(r)) ≥ Cepλ1,p r , for all sufficiently large r and C is a constant dependent on r. On the other hand, the volume comparison theorem (2.12) infers Vf (B(r)) ≤ C1 e(m−1)r . Here C1 is a constant dependent on r. Therefore, we obtain 1/p Cepλ1,p r ≤ C1 e(m−1)r , or equivalently, 1/p λ1,p ≤ 1 ln pr C1 C + m−1 p for all sufficiently large r. Lettting r → ∞, we have λ1,p ≤ m−1 p p . The proof is complete. We have following key lemma. Lemma 3.3. Let (M n , g, e−f dv) be an n-dimensional complete noncompact manifold with Ricm f ≥ −(m − 1). Then there are some positive constants a and b depending on p, n, m and δ such that L(ω) ≥ aω − b|∇ω|, (3.4) in the weak sense, namely L(φ)ωe−f ≥ M φ(aω − b|∇ω|)e−f M for any non-negative function φ with compact support on M . 17 (3.5) Proof. Since h = |∇u|2 , by the theorem 1.1, (see also the proof of Lemma 2.3 in [24]) we have x + δ ≤ h ≤ c(n, p, m). Denote by Ω = {h ≥ x + δ}, then (3.3) implies that there are positive constants c1 , c2 depending only on n, p, m such that on Ω p L(ω) ≥ c1 h 2 − (m − 1)h p−1 2 + λ1,p (p − 1)p−1 − c2 |∇ω|. (3.6) Now, we can follow an argument in [23] to prove that on Ω p h 2 − (m − 1)h p−1 2 + λ1,p (p − 1)p−1 ≥ c3 ω (3.7) for some positive constant c3 depending only on n, p, m, δ. Indeed, we consider both sides of (3.7) as a function of h. It is easy to see that (3.7) is valid when h = x for any choice of c3 . Now, we view the left hand side of (2.10) as a function of h, its derivative is 1 p p −1 (m − 1)(p − 1) p−3 1 p−3 h2 − h 2 = h 2 ph 2 − (p − 1)(n − 1) . 2 2 2 m−1 p By lemma 3.1, we have λ1,p ≤ h ≥ x + δ, we conclude that p , this implies x ≥ (p − 1)2 m−1 p 2 . Since 1 ph 2 − (p − 1)(m − 1) ≥ c(n, p, m, δ) > 0. Hence, (3.7) holds true on Ω for some 0 < c3 < c(n, p, m, δ). From (3.6), (3.7), we obtain L(ω) ≥ aω − b|∇ω| (3.8) on Ω. Here a, b are positive constants depending only on p, n, m, δ. Using the integration by parts, we have L(φ)ωe−f = M L(φ)ωe−f Ω p φLωe−f + = M p h 2 −1 A(∇φ), ν ωe−f − ∂Ω h 2 −1 A(∇ω), ν φe−f . ∂Ω ∇h ∇ω Here ν is the outward unit normal vector of ∂Ω. Since ν = − |∇h| = − |∇ω| and ω = 0 18 on ∂Ω, this implies L(φ)ωe−f = p φL(ω)e−f + M h 2 −1 Ω ∂Ω φ A(∇ω), ∇ω −f e |∇ω| φL(ω)e−f . ≥ Ω φ(aω − b|∇ω|)e−f ≥ Ω φ(aω − b|∇ω|)e−f . = M where we used (3.8) in the third inequality. The proof is complete. Now, we give a proof of the theorem 1.2. Proof of Theorem 1.2. We follow the proof in [23]. First, we will prove that ω ≡ 0. Indeed, for any cut-off function φ on M , and for any q > 0, by using (3.5), we have (aφ2 ω q+1 − bφ2 ω q |∇ω|)e−f . ωL(φ2 ω q )e−f ≥ M M Integration by parts implies p ωL(φ2 ω q )e−f = − A(∇(φ2 ω q )), ∇ω h 2 −1 e−f . Ω M Therefore, φ2 ω q |∇ω|e−f + C(n, p, mδ) φ2 ω q+1 e−f ≤ b a Ω M φ|∇φ||∇ω|ω q e−f Ω qφ2 ω q−1 A(∇ω), ∇ω e−f . − Ω It is easy to see that A(∇ω), ∇ω = |∇ω|2 + (p − 2) ∇u, ∇ω |∇u|2 ≥ (p − 1)|∇ω|2 . Hence, for any ε > 0, we have φ2 ω q+1 e−f ≤ bε a M φ2 ω q+1 e−f + Ω c + 4ε b 4ε φ2 ω q−1 |∇ω|2 e−f + ε Ω Ω φ2 ω q−1 |∇ω|2 e−f − c Ω φ2 ω q−1 |∇ω|2 e−f , Ω 19 |∇φ|2 ω q+1 e−f where c and c are constants depending only on n, p, m, δ. Choose q such that b + c = 4εc then (a − bε) φ2 ω q+1 e−f ≤ ε |∇φ|2 ω q+1 e−f . M M Now a standard argument implies either ω ≡ 0 or for all R ≥ 1, ω q+1 e−f ≥ c1 eR ln c2 ε (3.9) B(R) for some positive constants c1 and c2 independent of ε. Since ω is bounded and the f -volume of the ball B(R) satisfies Vf (B(R)) ≤ (m−1)R ce (by (2.12)), if ε > 0 is chosen sufficiently small, (3.9) can not hold. Hence, ω ≡ 0. This implies h ≤ x since δ is arbitrary. Thus, |∇ ln v| ≤ y. The proof is complete. Let λ1,p = m−1 p p , it is easy to see that the equation (p − 1)y p − (m − 1)y p−1 + λ1,p = 0 has the unique positive solution y = m−1 . p Hence, we have the following corollary. Corollary 3.4. Let (M n , g, e−f ) be an n-dimensional complete noncompact manifold with Ricm f ≥ −(m − 1). Suppose that u is a positive solution of f e div e Then 4. −f p−2 |∇u| ∇u = − m−1 p p up−1 . |∇u| m−1 ≤ . u p Rigidity of manifolds with maximal λ1,p In this section, we study structure at infinity of manifolds with maximal λ1,p . Our main purpose is to prove the theorem 1.5 stated in the introduction part. Theorem 4.1. Let (M n , g, e−f dv) be a smooth metric measure space of dimension n ≥ 2. Suppose that Ricm f ≥ −(m − 1) and λ1,p = m−1 p p . Then either M has no p-parabolic ends or M = R × N n−1 for some compact manifold N . Proof. Our argument is close to the argument in [23]. Suppose that M has a pparabolic end E. Let β be the Busemann function associated with a geodesic ray γ containd in E, namely, β(x) = lim (t − dist(x, γ(t))). t→∞ 20 Using the Laplacian comparison theorem (2.11), we have ∆f β ≥ −(m − 1). Hence, ∆p,f e m−1 β p m−1 p −f f = e div e = p−1 m−1 p e p−2 m−1 (p−1)β p e m−1 (p−2)β p p−1 m−1 β p m−1 β p m−1 p ∆f β + p−1 m−1 p (p − 1)e m−1 (p−1)β p p−1 −1 p m−1 m−1 e p (p−1)β ≥ (m − 1) p p m−1 m−1 =− e p (p−1)β . p Therefore, let ω := e ∇e , we obtain ∆p,f (ω) ≥ −λ1,p ω p−1 . Suppose that φ is a nonnegative compactly supported smooth function on M . Then by the variational principle, |∇(φω)|p e−f . (φω)p e−f ≤ λ1,p M M Noting that, integration by parts implies M M M φp−1 ω ∇φ, ∇ω |∇ω|p−2 e−f φp |∇ω|e−f − p φp ω∆p,f (ω)e−f = − and |∇(φω)|p = |∇φ|2 ω 2 + 2φω ∇φ, ∇ω + φ2 |∇ω|p−2 p 2 ≤ φp |∇ω|p + pφω ∇φ, ∇ω φp−2 |∇ω|p−2 + c|∇φ|2 ω p for some constant c depending only on p, we infer φp ω(∆p,f (ω) + λ1,p ω p−1 )e−f M (φω)p e−f − = λ1,p M φp |∇ω|p e−f − p M |∇(φω)|p e−f − ≤ M M φp |∇ω|p e−f − p M φp−1 ω ∇φ, ∇ω |∇ω|p−2 e−f M |∇φ|2 ω p e−f . ≤c φp−1 ω ∇φ, ∇ω |∇ω|p−2 e−f (4.1) M 21 |∇β|2 Now, we choose φ= such that |∇φ| ≤ 2 . R in B(R) on M \ B(2R) Then we conclude |∇φ|2 ω p e−f = M 1 0, |∇φ|2 e(m−1)β e−f M 4 ≤ 2 R 4 = 2 R e(m−1)β e−f B(2R)\B(R) e(m−1)β e−f + E∩(B(2R)\B(R)) 4 R2 e(m−1)β e−f . (M \E)∩(B(2R)\B(R)) (4.2) Since λ1,p = m−1 p p , by theorem 0.1 in [3], it turns out that Vf (E \ B(R)) ≤ ce−(m−1)R . Hence, the first term of (4.2) tends to 0 as R goes to ∞. On the other hand, by [15] we have β(x) ≤ −r(x) + c on M \ E. The volume comparison (2.12) implies Vf (B(R)) ≤ c e(m−1)R . It turns out that the second term of (4.2) also goes to 0 as R → ∞. Therefore, (4.1) infers ∆p,f (ω) + λ1,p ω p−1 ≡ 0. This implies ∆f β = −(m − 1). Moreover, all the inequalities used to prove ∆f β ≥ −(m − 1) become equalities (see Theorem 1.1 in [16]). By the proof of the Theorem 1.1 in [16] and the argument in [15], we conclude that M = R × N n−1 for some compact manifold N of dimension n. The proof is complete. For p-nonparabolic end, we have the following theorem. Theorem 4.2. Let (M n , g, e−f dv) be a smooth metric measure space of dimension n ≥ 3. Suppose that Ricm f ≥ −(m − 1) and λ1,p = m−1 p p for some 2 ≤ p ≤ (m−1)2 . 2(m−2) Then either M has only one p-nonparabolic end or M = R × N n−1 for some compact manifold N . To prove theorem 4.2, let us recall a fact on weighted Poincar´e inequality in [14]. 22 Proposition 4.3 ([14], Proposition 1.1). Let M be a complete Riemannian manifold. If there exists a nonnegative function h defined on M , that is not identically 0, satisfying ∆h(x) + ∇g, ∇h (x) ≤ −ρ(x)h(x), for some nonnegative function ρ, then the weighted Poincar´e inequality ρ(x)φ2 (x)eg(x) ≤ |∇φ|2 (x)eg(x) M M must be hold true for all compactly supported smooth function φ ∈ C0∞ (M ). Now we give a proof of theorem 4.2. Proof of theorem 4.2. We follow Sung-Wang’s argument in [23]. For the completeness, 2 we give a detail of the proof here. If p = 2 then we have λ1 (M ) = λ1,2 = (m−1) . 4 Hence, by Theorem 1.5 in [29], we are done. Therefore, we may assume p > 2. Now let u be a positive weighted p-eigenfunction of the weighted p-Laplacian satisfying ∆p,f u = ef div(e−f |∇u|p−2 ∇u) = −λ1,p up−1 . This implies ∆u + −∇f + (p − 2) = −λ1,p up−2 u. |∇u|p−2 ρ = λ1,p up−2 . |∇u|p−2 ∇|∇u| , ∇u |∇u| Let g = −f + (p − 2) ln |∇u|, and By proposition 4.3, we obtain λ1,p M up−2 2 φ |∇u|p−2 e−f ≤ p−2 |∇u| |∇φ|2 |∇u|p−2 e−f or equivalently, up−2 φ2 e−f ≤ λ1,p M |∇u|p−2 |∇φ|2 e−f , M for any compactly supported smooth function φ on M . Therefore φ2 e−f = λ1,p λ1,p M φu− p−2 2 2 up−2 e−f M ∇ φu− ≤ M 23 p−2 2 2 |∇u|p−2 e−f By a direct computation, we have ∇ φu− 2 p−2 2 |∇u|p−2 e−f M |∇φ|2 = M p−2 |∇u|p−2 −f e + φ2 ∇ u− 2 p−2 u φu− +2 p−2 2 ∇φ, ∇ u− p−2 2 2 |∇u|p−2 e−f |∇u|p−2 e−f M |∇φ|2 = M |∇u|p−2 −f e + up−2 p−2 2 2 M |∇u|p 2 −f 1 φe + up 2 ∇φ2 , ∇u−(p−2) |∇u|p−2 e−f . M (4.3) We claim that 1 2 ∇φ2 , ∇v −2 |∇u|p−2 e−f = − p−2 2 λ1,p + (p − 1) M |∇u|p up φ2 e−f . (4.4) Suppose that the claim (4.4) is verified. By (4.3), (4.4), we obtain 2 −f λ1,p φe ≤ M M 2 p−2 |∇u|p 2 −f + φe 2 up M |∇u|p λ1,p + (p − 1) p φ2 e−f . u |∇u|p−2 |∇φ|2 p−2 e−f u − p−2 2 M This means p λ1,p 2 Since λ1,p = φ2 e−f + M m−1 p p(p − 2) 4 M |∇u|p −f e ≤ up p , the corollary 3.4 implies (m − 1)2 2p Hence, λ1 (M ) ≥ φ2 (m−1)2 2p ∇u u ≤ M m−1 . p |∇u|p−2 |∇φ|2 e−f . up−2 Therefore, |∇φ|2 e−f . φ2 e−f ≤ M M . Here λ1 (M ) is the first eigenvalue of the weighted Laplacian. (m−1)2 , 2(m−2) Since p ≤ we have λ1 (M ) ≥ (m − 2). By a Wang’s theorem (see [25]), we are done. The rest of the proof is to verify the claim (4.4). Indeed, we have 1 2 ∇φ2 , ∇u−(p−2) |∇u|p−2 e−f M =− 1 2 ∆f (u−(p−2) )|∇u|p−2 φ2 e−f − M 1 2 24 ∇u−(p−2) , ∇|∇u|p−2 φ2 e−f . M (4.5) Note that ef div e−f |∇u|p−2 ∇u = −λ1,p up−1 we have |∇u|p−2 ∆f u + ∇|∇u|p−2 , ∇u = −λ1,p up−1 . Therefore, |∇u|p−2 ∆f (u2−p ) + ∇(u2−p , ∇(|∇p|p−2 )) =(2 − p)|∇u|p−2 u1−p ∆f u + (2 − p)(1 − p)|∇u|p−2 u−p |∇u|2 + (2 − p)u1−p ∇u, ∇(|∇u|p−2 ) = (2 − p)u1−p (−λ1,p up−1 ) + (2 − p)(1 − p) |∇u|p |∇u|p = (p − 2) λ + (p − 1) 1,p up up (4.6) By (4.5) and (4.6), the claim 4.4 is verified. Since the above-mentioned result of Wang plays a critical role in the proof of theorem 4.2, we reformulate it here for reader’s convenience. Theorem 4.4 ([25]). Let (M n , g, e−f dv) be a smooth metric measure space of dimension n ≥ 3. Suppose that the lower bound of the spectrum λ1 (M ) of the weighted Laplacian is positive and m−1 Ricm λ1 (M ). f ≥ − m−2 Then either M has only one p-nonparabolic end or M = R × N n−1 for some compact manifold N of dimension n − 1 with the product metric ds2M = dt2 + cosh λ1 (M ) 2 tdsN . m−2 Acknowledgment A part of this paper was done during a visit of the first author to Vietnam Institute for Advanced Study in Mathematics (VIASM). 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Chen, Gradient estimates and entropy formulae for weighted p-heat equations on smooth metric measure spaces, Acta Math. Scientia 33B (2013), 963 - 974 [28] J. Y. Wu, Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds, Jour. Math. Anal. Appl., 369 (2010), no.1, 400-407 [29] J. Y. Wu, A note on the splitting theorem for the weighted measure, Ann. Glob. Anal. Geom., 43 (2013) no. 3, 287-298. 27 [30] J. Y. Wu, Lp -Liouville theorems on complete smooth metric measure spaces, Bull. Sci. Math., 318 (2014), 510-539. Nguyen Thac Dung, Department of Mathematics - Mechanics - Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Vietnam. E-mail address: dungmath@yahoo.co.uk Nguyen Duy Dat, Department of Mathematics - Mechanics - Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Vietnam. E-mail address: duynguyendat@gmail.com 28 [...]... L F Wang, A splitting theorem for the weighted measure, Ann Glob Anal Geom 42 (2012) 79 - 89 [26] X Wang, and L Zhang, Local gradient estimate for p−harmonic functions on Riemannian manifolds, preprint, Comm Anal Geom., 19 (2011), no 4, 759-771, see also arXiv 1010.2889 v1 [math.DG] 14 Oct 2010 [27] Y Wang, J Yang and, W Chen, Gradient estimates and entropy formulae for weighted p-heat equations on... Saloff-Coste, A note on Poincare, Sobolev, and Harnack inequalities, Internat Math Res Notices (1992), 27-38 [22] R Schoen and S T Yau, Lectures on Differential Geometry, International Press, Boston, 1994 [23] C J Anna Sung and J Wang, Sharp gradient estimate and spectral rigidity for p-Laplacian, to appear in MRL [24] L F Wang, Eigenvalue estimate for the weighted p-Laplacian, Annali di Matematica... Perelman’s entropy formula for the Witten Laplacian on Riemannian ´ manifolds via Bakry-Emery Ricci curvature, Math Ann., 353 (2012), 403-437 [11] B Kotschwar and L Ni, Gradient estimate for p-harmonic function, 1/H flow ´ Norm Sup´er., 42 (2009), 1-36 and an entropy formula, Ann Sci Ec [12] P Li and J Wang, Complete manifolds with positive spectrum, Jor Diff Geom., 58 (2001) 501-534 [13] P Li and J Wang,... 147-151 [6] S Y Cheng and S T Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm Pure Appl Math., 28 (1975), 333354 [7] H I Choi, On the Liouville theorem for harmonic maps, Proc Amer Math Soc., 85 (1982), no.1, 91-94 [8] P Li, Harmonic functions and applications to complete manifolds, Preprint (available on the author’s homepage) [9] P Li and S T Yau, On the parabolic... (2.19) converges Now, by Lemma 2.4 and (2.19), we conclude ||h||L∞ (BR/2 ) ≤ a15 b20 R2 The proof is complete As a consequence, we obtain an important folowing theorem ralating to Liouvilleproperty for weighted p-harmonic functions Theorem 2.5 Assume that (M, g) is a smooth metric measure space with Ricm f ≥ 0 If u is a weighted p-harmonic function bounded from below on M and u is of sublinear growth then... 1295-1361 [17] R Moser, The inverse mean curvature flow and p−harmonic functions, Jour Eur Math Soc., 9 (2007), 77-83 [18] O Munteanu, and J Wang, Smooth metric measure spaces with nonnegative curvature, Comm Anal Geom., 19 (2011), no 3, 451-486, see arXiv: 1103.0746v1 [math.DG] 3 Mar 2011 [19] O Munteanu, and J Wang, Analysis of the weighted Laplacian and applications to Ricci solitons, Comm Anal Geom.,... There exists a constant Cp,n,m depending only on p, n, m such that v(x) ≤ eCp,n,m (1+ √ κR) v(y), ∀x, y ∈ B(o, R/2) If κ = 0, we have a uniform constant cp,n,m (independent of R) such that sup v ≤ cp,n B(o,R/2) 3 inf v B(o,R/2) Global sharp gradient estimate for weighted p-Laplacian Recall that a function v is an eigenfunction of p-Laplacian with corresponding eigenvalue λ1,p ≥ 0 if ef div(e−f |∇v|p−2... would like to express his thanks to staffs there for the excellent working conditions, and financial support He also would like to express his deep gratitude to his advisor Prof Chiung Jue Sung for her constant support The authors also thank Munteanu and J Y Wu for their usefull comments on the earlier manuscript of this paper 25 References [1] D Bakry and Z M Qian, Volume comparison theorems withour... M is p-nonparabolic, moreover 1/p Vf (B(r)) ≥ Cepλ1,p r , for all sufficiently large r and C is a constant dependent on r On the other hand, the volume comparison theorem (2.12) infers Vf (B(r)) ≤ C1 e(m−1)r Here C1 is a constant dependent on r Therefore, we obtain 1/p Cepλ1,p r ≤ C1 e(m−1)r , or equivalently, 1/p λ1,p ≤ 1 ln pr C1 C + m−1 p for all sufficiently large r Lettting r → ∞, we have λ1,p... positive spectrum II, Jour Diff Geom., 62 (2002) 143-162 [14] P Li and J Wang, Weighted Poincar´e inequality and rigidity of complete ´ Norm Sup., 39 (2006) 921 - 982 manifolds, Ann Scient Ec 26 [15] P Li and J Wang, Connectedness at infinity of complete K¨ahler manifolds, Amer Jour Math., 131 (2009), 771-817 [16] X D Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, ... property and a Harnack inequality for weighted p-harmonic functions In the section 3, we prove the theorem 1.2 The proof the theorem 1.5 is given in the section 4 Local gradient estimates for weighted. .. shown gradient estimates and entropy formulae for weighted p-heat equations Their works generalized Li’s and Kotschwar-Ni’s results (see [16, 11]) In this paper, motived by Wang-Zhang’s gradient. .. more general weighted pharmonic functions are natural extensions of harmonic functions, or weight harmonic functions, respectively Compared with the theory for (weighted) harmonic functions, the