The Poisson equation on Riemannian manifolds with weighted Poincaré inequality at infinity

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green’s function vanishing at infinity. On the source function, we assume a sharp pointwise decay depending on the weight appearing in the Poincaré inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold. In comparison with previous works, we can deal with a more general setting on the curvature bounds and without any spectral assumption.


Introduction
The existence of solutions to the Poisson equation on a complete Riemannian manifold (M, g), for a given function f on M, is a classical problem which has been the object of deep interest in the literature. Malgrange [11] obtained solvability of the Poisson equation for any smooth function f with compact support, as a consequence of the existence of a Green's function for −Δ on every complete Riemannian manifold. Under integrability assumptions on f, existence of solutions has been established −Δu = f 1 3 by Strichartz [17] and Ni-Shi-Tam [16,Theorem 3.2] (see also [15,Lemma 2.3]). Moreover, in the same paper, the authors proved an existence result for the Poisson problem on manifolds with nonnegative Ricci curvature under a sharp integral assumption involving suitable averages of f. This condition in particular is satisfied if for some C > 0 and > 2 , where r(x) ∶= dist(x, p) is the distance function of any x ∈ M from a fixed reference point p ∈ M . In fact, they proved a more general result where the decay rate of f is just assumed to be of order 1 + . Note that this result is sharp on the flat space ℝ n .
From now on let us consider solutions u of the Poisson equation −Δu = f which can be represented as where G(x, y) is a Green's function of −Δ on M (see Sect. 2 for further details). Muntenau-Sesum [12] addressed the case of manifolds with positive spectrum, i.e., 1 (M) > 0 , and Ricci curvature bounded from below, obtaining existence of solutions under the pointwise decay assumption for some C > 0 and > 1 . Note that this result is sharp on ℍ n . Their proof relies on very precise integral estimates on the minimal positive Green's function, which are inspired by the work of Li-Wang [10]. Note that in [12,13] the authors also study the behavior of the solution at infinity.
In [4] the authors generalized the existence result in [12], obtaining existence of solutions on manifolds with positive essential spectrum, i.e., ess 1 (M) > 0 , for source functions f satisfying for any R > 0 , where R (m) is a function related to a lower bound on the Ricci curvature, locally on geodesic balls with center p and radius 2R + m . In particular, the authors showed in [4,Corollary 1.3] existence of solutions on Cartan-Hadamard manifolds with strictly negative Ricci curvature, whenever for some C > 0 and 1 , 2 ≥ 0 with > 1 + 1 2 − 2 . Observe that the results in [4,12] cannot be used whenever the Ricci curvature tends to zero at infinity fast enough (see [19]) since, in this case, one has ess 1 (M) = 0 (and so 1 (M) = 0 ). In particular, the case of ℝ n is not covered. On the other hand, the result in [16] does not apply on manifolds with negative curvature. The purpose of our paper is to obtain a general result which includes, as special cases, both manifolds with strictly negative curvature and manifolds with Ricci curvature vanishing at infinity. Moreover, our result is sharp on spherically symmetric manifolds, and in particular on ℝ n and ℍ n . Note that the condition 1 (M) > 0 is equivalent to the validity of the Poincaré inequality for any u ∈ C ∞ c (M) . On the other hand, one has positive essential spectrum if and only if, for some compact subset K ⊂ M , one has 1 (M ⧵ K) > 0 and for any u ∈ C ∞ c (M ⧵ K) . Generalizing the previous inequalities, one says that (M, g) satisfies a weighted Poincaré inequality with a nonnegative weight function if ) and for ≡ R , we say that (M, g) satisfies a weighted Poincaré inequality at infinity. In addition, inspired by [10], we say that (M, g) satisfies the property P ∞ w , if a weighted Poincaré inequality at infinity holds for the family of weights R and the conformal R -metric defined by is complete for every R ≥ R 0 . The validity of a weighted Poincaré inequality on some classes of manifolds has been investigated in the literature. It is well known that on ℝ n inequality (1) holds with (x) = (n−2) 2 4 1 r 2 (x) . It is also called Hardy inequality. More in general, it holds on every Cartan-Hadamard manifold with (x) = C r 2 (x) , for some C > 0 (see [1,3] for some refinement of this result).
In order to state our main results, we need to introduce a (increasing) function (s) related to the value of the Ricci curvature on the annulus B 5 4 s (p) ⧵ B 3 4 s (p) (see (4) for the precise definition). In this paper, we prove the following result.
for some ≥ 0 . Then, it is direct to see that for every R > 0 and the property P ∞ w w.r.t. the family of weights R , R ≥ R 0 , holds for every R with R (x) = 1 (M ⧵ B R (p)) . Thus, therefore, our existence result is in accordance with those in [4,12].
We recall that by [

Remark 1.3
We explicitly observe that in Theorem 1.2 the completeness of the conformal metric g = g is not required. As it was observed in [10], the completeness of g would hold if G(p, x) → 0 as r(x) → ∞ , a condition that we do not need to assume here.
It is well known that ℝ n is a non-parabolic manifold if n ≥ 3 , with minimal positive Green's function G(x, y) = c n |x−y| n−2 for some positive constant c n . Moreover, the weighted Poincaré -Hardy's inequality holds on ℝ n with In this case, using the definition (4) of the function (s) , it is easy to see that Hence, we can apply Theorem 1.2, with and the convergence of the series follows, whenever |f (x)| ≤ C∕(1 + r(x)) for some > 2 . This condition is optimal, as it can be easily verified by explicit computations.
In general, concerning Cartan-Hadamard manifolds, by using Theorem 1.1 we improve [4, Corollary 1.3] allowing the Ricci curvature to approach zero at infinity.

Remark 1.5
In the special case 1 = 2 = ≥ 0 the condition on in the previous corollary becomes In particular, in (M, g) is the standard hyperbolic space ℍ n , and then = 0 . Thus, we need that > 1 and this condition is sharp as observed above. We will consider also the case < 0 in Sect. 6.2 on model manifolds.
The paper is organized as follows: In Sect. 2 we collect some preliminary results and we define precisely the function ; in Sect. 3 we prove a refined local gradient estimates for positive harmonic functions; in Sect. 4 we prove key estimates on the positive minimal Green's function G(x, y) of a non-parabolic manifold, by means of the property P ∞ w w.r.t. the family of weights R , R ≥ R 0 ; in Sect. 5 we prove Theorem 1.1; finally, in Sect. 6 we prove Corollary 1.4 and show the optimality of the assumption in Theorem 1.2 for rotationally symmetric manifolds.
Finally, we note that some results concerning the Poisson equation on some manifolds satisfying a weighted Poincaré inequality have been very recently obtained in [14]. However, their assumptions and results apparently are completely different to ours.

Preliminaries
Let (M, g) be a complete non-compact n-dimensional Riemannian manifold. For any x ∈ M and R > 0 , we denote by B R (x) the geodesic ball of radius R with center x and let Vol(B R (x)) be its volume. We denote by Ric the Ricci curvature of g. For any x ∈ M , let (x) be the smallest eigenvalue of Ric at x. Thus, for any ) . For any z ∈ M , let be the minimal geodesic connecting p to z. We define the function for a given a > 0 . Note that t ↦ (t) is increasing and so invertible. Under (2), we know that Moreover, let Cut(p) be the cut locus of p ∈ M. It is known that every complete Riemannian manifold admits a Green's function (see [11]), i.e., a smooth function defined in . We say that (M, g) is non-parabolic if there exists a minimal positive Green's function G(x, y) on (M, g), and parabolic otherwise.
We say that (M, g) satisfies a weighted Poincaré inequality with a nonnegative weight function if (r( (s)) ds, ) and for ≡ R , we say that (M, g) satisfies a weighted Poincaré inequality at infinity. In addition, inspired by [10], we say that (M, g) satisfies the property P ∞ R if a weighted Poincaré inequality at infinity holds for the family of weights R and the conformal R -metric defined by is complete. With this metric we consider the -distance function where the infimum of the lengths is taken over all curves joining x and y, with respect to the metric g . For the fixed reference point p ∈ M , we denote by with Ω an open subset of M, to be the first eigenvalue of −Δ in Ω with zero Dirichlet boundary conditions. It is well known that 1 (Ω) is decreasing with respect to the inclusion of subsets. In particular, For any x ∈ M , for any s > 0 and for any 0 ≤ a < b ≤ +∞ , we define

Local gradient estimate for harmonic functions
In this section, we improve [4, Lemma 3.1]. We set ).
for z ∈ M and R > 0; Proof Following the classical argument of Yau, let v ∶= log u . Then, Then, from classical Bochner-Weitzenböch formula and Newton inequality, one has Moreover, by Laplacian comparison, since (z)) and weakly on B R (z) . Thus, . At q, we obtain So Thus, for any ∈ B R∕2 (z), We get for some positive constant C > 0 . By standard Calabi trick (see [2,5]), the same estimate can be obtained when q ∈ Cut(z) . This concludes the proof of the lemma. ◻ As a corollary, we have the following

Remark 4.2 One has
for any y ∈ B a (p) . This follows from Lemma 4.1 with y ∈ B a (p) and the maximum principle, since y ↦ G(p, y) is (weakly) superharmonic in B a (p) . In particular,

Remark 4.3 We also note that
In fact, let y ∈ M ⧵ B a (p) and take j > r(y) . Since G j (p, y) ≤ G(p, y) and G j (p, ⋅) ≡ 0 on B j (p) , by Lemma 4.1, we have L p (A exp (B (a)), ∞) ⊂ B a (p).
note that the right-hand side is independent of y. Since y ↦ G j (x, y) is harmonic in B j (p) ⧵ B a (p) , by maximum principle, Sending j → ∞ , by (7), we obtain and the claim follows. For the proof see [12]. Moreover, we get the following weighted integrability property for the Green's function. Proof In order to simplify the notation, let ≡ m . Fix R 1 > 0 such that B m (p) ⊂ B R 1 (p) and let be defined as Let R > 2R 1 and G R (p, y) be the Green's function of −Δ in B R (p) satisfying zero Dirichlet boundary conditions on B R (p) . Following the proof in [10], since G R is harmonic in B R (p) , one has

Auxiliary estimates
where the last equality follows by integration by parts and the fact that G R (p, y) vanishes on B R (p) . Hence, the weighted Poincaré inequality yields Letting R → ∞ , by Fatou's lemma and uniform convergence of G R → G on compact subsets, we get and the thesis follows. ◻ We expect a decay estimate similar to the one in [10, Theorem 2.1]. However, we leave out this refinement since it is not necessary in our arguments.

Integral estimates on level sets
We begin by noting that using Remark 4.3 and the fact that G(p, ⋅) ∈ L 1 loc (M) one has the following integral estimate on large level sets.   exp (B (a)),∞) G(p, y) dy < ∞.

Proof
We follow the general argument in [10,12]; however, some relevant differences are in order, due to the use of the property P ∞ w . Let ∶= with and for any fixed R > 0

By the weighted Poincaré inequality at infinity, we get
We estimate where we used Lemma 4.4 in the last equality. On the other hand, Now we let R → ∞ and use Lemma 4.5. The thesis now follows. ◻ In the special case when M is non-parabolic with positive minimal Green's function G and with weight (x) = |∇G(p,x)| 2 4G 2 (p,x) , we have the following refinement of Proposition 4.8.
(y) ∶= In order to prove Theorem 1.1, we will show that with v ∈ C 0 (M) . We divide the proof in two parts, we first consider the case when (M, g) is non-parabolic, and then, the case when it is parabolic.
where C 2 (x) can be chosen as the constant in the Harnack's inequality for the ball B r(x)+1 (p) . For any a > 0 , we estimate By Proposition 4.7, Remark 4.3 we get for some positive constant C 3 (a) . To estimate the first integral, we observe that for any m 0 = m 0 (x) ≥ a one has We need the following lemma.  A exp (B (a))) G(p, y) |f (y)| dy (A exp (B (a)),∞) G(p, y) |f (y)| dy. A exp (B (a))) G(p, y) |f (y)| dy + C 3 (a) A exp (B (a))) G(x, y) |f (y)| dy.    Let G(x, y) be a Green's function on M (which is positive inside a certain ball, and negative outside). Fix any R > 0 and let ≡ R 0 . Note that arguing as in the proof of (8), it is sufficient to estimate since G(p, ⋅) ∈ L 1 loc (M) and f is locally bounded. We have that where each E i is an end with respect to B R (p) . Note that every end E i is parabolic. In fact, if at least one end E i is non-parabolic, then (M, g) is non-parabolic (see [8] for a nice A exp (B (a))) G(x, y) |f (y)| dy ≤ C 4 (a, m 0 ).
overview), but we are in the case that (M, g) is parabolic. Since every E i is parabolic, every E i has finite weighted volume (see [9]), i.e., Now choose R large enough so that we can apply Lemma 4.5 obtaining This concludes the proof of Theorem 1.1. ◻ Proof of Theorem 1. 2 We start as in the proof of Theorem 1.1 using (8), (9), (10) and (13).
Then, similar to (15), using Proposition 4.9, we obtain Then and the proof of Theorem 1.2 is complete. ◻

Cartan-Hadamard and model manifolds
We consider Cartan-Hadamard manifolds, i.e., complete, non-compact, simply connected Riemannian manifolds with non-positive sectional curvatures everywhere. Observe that on Cartan-Hadamard manifolds the cut locus of any point p is empty. Hence, for any x ∈ M ⧵ {p} one can define its polar coordinates with pole at p, namely r(x) = dist(x, p) and ∈ n−1 . We have for a specific positive function A which is related to the metric tensor [6,Sect. 3]. Moreover, it is direct to see that the Laplace-Beltrami operator in polar coordinates has the form where m(r, ) ∶= r (log A) and Δ is the Laplace-Beltrami operator on B r (p) . We have

Let
We say that (M, g) is a rotationally symmetric manifold or a model manifold if the Riemannian metric is given by where d 2 is the standard metric on n−1 and ∈ A . In this case, Note that (r) = r corresponds to M = ℝ n , while (r) = sinh r corresponds to M = ℍ n , namely the n-dimensional hyperbolic space. The Ricci curvature in the radial direction is given by

Cartan-Hadamard manifolds
Concerning the validity of the property P ∞ w w.r.t. the family of weights R , R ≥ R 0 on a Cartan-Hadamard manifold we have the following result. .
Ric(∇r, ∇r)(x) ≤ −C 1 + r(x) Note that |∇ k | ≤ 1 and for all x ∈ M ⧵ B 1 (p) , ∑ k k = 1 and x ∈ supp k at most for two integers k. If suppu ⊂ M ⧵ B 1 (p) , we have where in the last passage we used (17) where in the last passage we used (17) with R = 1 . Hence, the weighted Poincaré inequality holds for functions with support in M ⧵ B 1 (p).
Finally, the completeness of the metric g R ∶= R g follows. In fact, for any curve (s) parametrized by arclength with 0 ≤ s ≤ T , the length of with respect tp g R is given by ◻ Let us write some estimates which will be useful both in the proof of Corollary 1.4 and in Sect. 6.2. Choose ∈ A as in (16) with = 1 obtaining and for r(x) > R > 1 . A simple computation shows that for R = r(x)∕4 , one has

Optimality on rotationally symmetric manifolds
We show that the assumptions in Theorem 1.2 are sharp on model manifolds. Let (M, g) be a rotationally symmetric manifold with ∈ A defined as in (16)  Case 1: > −2 . With our choice of , by the change of variable s = t 1+ 2 , it is easily seen that for any r > 0 sufficiently large