SOLUTIONS

. Let M be a complete Riemannian manifold of dimension n without boundary and with Ricci curvature bounded below by − K; where K (cid:21) 0 : If b is a vector (cid:12)eld such that k b k (cid:20) γ and r b (cid:20) K (cid:3) on M; for some nonnegative constants γ and K (cid:3) ; then we show that any positive C 1 ( M ) solution of the equation (cid:1) u ( x ) + ( b jr u = 0 satis(cid:12)es the estimate on M , for all w 2 (0 ; 1) : In particular, for the case when K = K (cid:3) = 0 ; this estimate is advantageous for small values of k b k and when b (cid:17) 0 it recovers the celebrated Liouville theorem of Yau ( Comm. Pure Appl. Math. 28 (1975), 201{228).


Introduction
In this paper we investigate the behaviour of positive C ∞ (M ) solutions of the equation ∆u(x) + (b(x)|∇u(x)) = 0 (1.1) on M , where M is an n-dimensional complete Riemannian manifold without boundary.
We require smoothness of the manifold, uniform bound on the norm of the vector field b as well as lower bounds on the tensor fields of the Ricci curvature and ∇b where ∇b(X, Y ) = (∇ X b|Y ), ∀X, Y ∈ X(M ), (1.2) where X(M ) denotes the Lie algebra of vectors fields on M and ∇ X b the associated (Levi-Civita) Riemannian covariant derivative of b with respect to X.
Our main result is a gradient estimate for positive C ∞ (M ) solutions of equation (1.1), namely, on M , for any w ∈ (0, 1), where the Ricci curvature is bounded below by −K, ∇b ≤ K * and b ≤ γ for some nonnegative constants K, K * and γ.
For the particular case when K = K * = 0 inequality (1.3) yields Note that this simple estimation is independent of the dimension of M and for the case when b ≡ 0 it recovers the Liouville theorem of Yau [10]. The proof of (1.3), and thus of (1.4), is essentially along the lines of Li and Yau [7] and Davies [3,Chap. 5].
In order to start, however, we need an extension of the Bochner-Lichnèrowicz-Weitzenböck formula for the operator L b = ∆ + (b|∇ ). This remarkable fact is proved as an independent lemma. It is known for drift vectors b = ∇φ of gradient form; see e.g. the monograph of Deuschel and Stroock (cf. [4], §6.2).
Let ψ be a C ∞ (R) function such that and 0 ≤ ψ(r) ≤ 1, ∀r ∈ R. Denote by > 0 and ν > 0 some constants with is the distance between p and x.
Using an argument of Calabi [1] (see also Cheng and Yau [2] and Setti [9]), we can assume without loss of generality that the function φ, with support in B p (2R), is of class C 2 . Let (a, s) be the point in B p (2R) × [0, t] at which φF takes its maximum value, and assume that this value is positive (otherwise the proof is trivial). Then at (a, s) one has ∇(φF ) = 0, ∆(φF ) ≤ 0, F t ≥ 0.
Therefore at (a, s) one has This inequality together with the estimates From (2.8) the left-hand side of (2.9) satisfies Denoting µ = ∇f 2 (a, s) F (a, s) , using (2.9) and the last inequality, we obtain Multiplying this inequality by sφ and since φ 2 ≤ 1, we obtain On the other hand, for any w ∈ (0, 1) we have (2.11) From (2.11) inequality (2.10) becomes As in [3,Lemma 5.3.3], we use the estimate , and so we obtain and using (2.12), estimate (2.5) holds.
From Theorem 2.1 one obtains the next global gradient estimate Corollary 2.1. Let M be a complete Riemannian manifold of dimension n without boundary and assume that the Ricci curvature of M is bounded from below by −K with K ≥ 0. Also we suppose that the vector field b satisfies b ≤ γ and that the tensor field ∇b, given by (1.2), is bounded from above by K * , for some nonnegative constants γ and K * . If u(x) is a positive C ∞ (M ) solution of equation (1.1), then for any w ∈ (0, 1), the following estimate holds on M : Proof. Letting R → ∞ and t → ∞ in (2.5) one has , (2.13) on M . Setting α = 2 (which minimizes the right-hand side of (2.13)), the result holds.
Remark 2.2. If u(x) is a positive C ∞ (M) solution of ∆u(x) + (b(x) | ∇u(x)) = 0, and assuming that Ric ≥ 0, ∇b ≤ 0 and b ≤ γ, for some γ ≥ 0, it follows from Corollary 2.1 above that , (2.14) for any w ∈ (0, 1). Setting w = 1/2 (which minimizes the right-hand side of (2.14)) one obtains ∇u 2 u 2 ≤ 4γ 2 . Remark 2.3. Let M = R be the one-dimensional Euclidean space with its standard Riemannian metric. It is a complete Riemannian manifold without boundary and with Ricci curvature identically zero. In this setting consider the equation