A quantitative second order estimate for (weighted) $p$-harmonic functions in manifolds under curvature-dimension condition

We build up a quantitative second order Sobolev estimate of $ \ln w$ for positive $p$-harmonic functions $w$ in Riemannian manifolds under Ricci curvature bounded from blow and also for positive weighted $p$-harmonic functions $w$ in weighted manifolds under the Bakry-\'{E}mery curvature-dimension condition.


Introduction
Let (M n , g) be a complete non-compact Riemannian manifold with dimension n ≥ 2. Suppose that the Ricci curvature is bounded from below, that is, Ric g ≥ −κ for some κ ≥ 0. For any positive harmonic function w in a domain Ω ⊂ M n , Cheng-Yau [2] established the following famous gradient estimate: in B(z, r) ⊂ B(z, 2r) ⊂ Ω. (1.1) Recall that a harmonic function w in Ω is a weak solution to the Laplace equation ∆w := div(∇w) = 0 in Ω.
Precisely, if (M n , g) is flat (that is, the Euclidean space R n ) or its sectional curvature is bounded from below by −κ, via Cheng-Yau's approach Moser [16] and Kotschwar-Ni [11] showed that any positive p-harmonic function w in Ω satisfies where the constant C(n) > 0 is independent of p ∈ (1, ∞).Under the Ricci curvature lower bound Ric g ≥ −κ, it was asked in [11] whether (1.2) holds or not.Some progress was made as below.Based on Cheng-Yau's argument, Wang-Zhang [21] proved that |∇ ln w| p−γ 2 ∈ W 1,2 loc with γ < 0 (1.3) and the following weaker revision of (1.2): |∇ ln w| ≤ C(n, p) where the constant C(n, p) > 0 blows up as p → 1.Recently, with the aid of the fake distance coming from capacity, C(n, p) was proved by Mari-Rigoli-Setti [15] to be bounded by n−1 p−1 as p → 1.Moreover, (1.3) and (1.4) were generalized to weighted manifolds (M n , g, e −h dvol g ).A weighted p-harmonic function w in a domain Ω ⊂ M n is a weak solution to the weighted p-harmonic equation ∆ p,h w := e h div(e −h |∇w| p−2 ∇w) = 0 in Ω.
Under the Bakry-Émery curvature-dimension condition Ric N h ≥ −κ for some N ∈ [n, ∞) and κ ≥ 0 (see Section 2 for details), Dung-Dat [5] showed that if w > 0, then |∇ ln w| The main aim of this paper is to build up a quantitative second-order Sobolev estimate of ln w for positive p-harmonic functions w in Riemannian manifolds under Ricci curvature bounded from below and also for positive weighted p-harmonic functions w in weighted manifolds under the Bakry-Émery curvature-dimension condition.See Theorem 1.1 and Theorem 1.2 separately.These improve the corresponding second-order Sobolev regularity in [21,5] mentioned above.
To be precise, under the Ricci curvature lower bound, we have the following result.For convenience, below we write -E f dm as the average of f in the set E with respect to the measure m, that is, (1.7) whenever B(z, 4r) ⋐ Ω.
Here and throughout the paper for domains A and B, the notation A ⋐ B stands for that A is a bounded subdomain of B and its closure A ⊂ B.
Recall that if (M n , g) is flat, that is, the Euclidean space R n , p-harmonic functions w in a domain Ω ⊂ R n are proved to satisfy |∇w| p−γ 2 ∇w ∈ W 1,2 loc (Ω) with some quantative bound whenever γ < 3 + p−1 n−1 see [13,9,4,14] and also the references therein for some earlier partial results.In particular, if 1 < p < 3 + 2 n−2 , noting p < 3 + p−1 n−1 and taking γ = p, one has w ∈ W 2,2 loc (Ω).When n ≥ 3 and p ≥ 3 + 2 n−2 , it is not clear whether w ∈ W 2,2 loc (Ω) or not.When n = 2, the range γ < 3 + p−1 n−1 = p + 2 is optimal as witnessed by some construction in [9].Moreover, we extend Theorem 1.1 to weighted manifolds satisfying Bakry-Émery curvaturedimension condition, Theorem 1.2.Let (M n , g, e −h vol g ) be a weighted manifold with Ric N h ≥ −κ for some n ≤ N < ∞ and κ ≥ 0. Let 1 < p < ∞ and γ < 3 + p−1 N −1 .For any positive weighted p-harmonic function w in a domain Ω ⊂ M , we have |∇ ln w| ∈ W 1,2 loc for all γ < 2 (see (1.3) and the line above (1.5)).Thus our range for γ obviously improves the one obtained in [21,5] respectively.Now we sketch the ideas to prove Theorem 1.1 and Theorem 1.2.Note that when N = n and h ≡ 1, we have Ric N h = Ric g , and hence Theorem 1.1 corresponds to the special case N = n and h ≡ 1 in Theorem 1.2.We only need to prove Theorem 1.2.As usual, we approximate u = −(p − 1) ln w by smooth solution u ǫ to the standard approximation/regularized equation (i) Using Bochner formula and the approximation equation (3.3), for 0 < η < 1/2 we bound the integral of ]φ by the integral of some first order terms, where all integrals are against e −h dvol g .A standard argument then leads to the proof of Theorem 1.2.
Finally, we also notice that the Cheng-Yau gradient estimate (1.1) was generalized to positive harmonic functions w in Alexandrov spaces with curvature bounded from below by Zhang-Zhu in [22], where the authors showed |∇ ln w| 2 ∈ W 1,2 loc (Ω) as a key step.Furthermore, one could study the regularity of p-harmonic functions in more general metric measure spaces.In these spaces, a natural generalization of the (weighted) Ricci curvature bound is the curvature-dimension condition RCD(κ, N ) in the sense of Bakry-Émery or Ambrosio-Gigli-Savaré.The two senses turned out to be equivalent by the work of Erbar-Kuwada-Sturm [6] (in the finite dimensional case) and Ambrosio-Gigli-Savaré [1] and the spaces satisfying one of the two equivalent conditions are known as RCD(κ, N ) spaces.Some progress was made in RCD(κ, N ) spaces.The Cheng-Yau gradient estimate was established by Jiang in [10] for positive harmonic functions w in RCD(κ, N ) spaces; recently, Gigli-Violo in [7] However, when p = 2, it remains open to prove the Cheng-Yau type gradient estimates for positive p-harmonic functions in Alexandrov spaces and also RCD(κ, N ) spaces.

Preliminaries
Let n ≥ 2 and M n be a Riemannian manifold, and g be the Riemannian metric.By abuse of notation we also write |ξ| 2 = g(ξ, ξ) and ξ, η = g(ξ, η) for all ξ, η ∈ T x M n .The corresponding Riemannian volume measure is written as dvol g , and the volume of a set E is written as vol g (E).Denote by Ric g the Ricci curvature 2-tensor and write For 1 < p < ∞, the p-Laplace operator ∆ p in M n is given by Obviously, ∆ 2 is exactly the Laplace-Beltrami operator ∆ in (M n , g).
Note that 2-harmonic functions are the well-known harmonic functions.
Next we recall some basic facts of weighted Riemannian manifolds (M n , g, e −h dvol g ), where the weight h is a positive smooth function in M n .The weighted measure dvol h = e −h dvol g can be viewed as the volume form of a suitable conformal change of the metric g.Denote by vol h (E) the weighted volume of a set E. For n ≤ N < ∞, the corresponding N -Bakry-Émery curvature tensor is where when N = n, by convention, h is a constant function and hence Ric N h = Ric g .We say that (M n , g, e −h dvol g ) satisfies the Bakry-Émery curvature-dimension condition By [18], under Ric N h ≥ −κ, one has the following volume comparison result For 1 < p < ∞, the weighted p-Laplacian ∆ h,p is defined as In the case p = 2, one writes ∆ 2,h as ∆ h , and hence 2).We also recall the following Bochner formula in (M n , g, e −h dvol g ): which will be used in Section 3. Finally, we recall the following fundamental inequality; see for example [21,5,14].For the reader's convenience we include it here.Recall that ∆ ∞ f = ∇ 2 f ∇f, ∇f .Lemma 2.1.Let n ≥ 2 and Ω be a domain of M n .For any f ∈ C 2 (Ω), we have where when n = 2, "≥" becomes "=".
Proof.It suffices to prove that for any symmetric n × n matrix A one has Note that if ξ = 0, (2.5) holds obviously.Below assume that ξ = 0. Up to a scaling we may assume |ξ| = 1.By a change of coordinates, we may further assume ξ = e n = (0, • • • , 0, 1); in this case, (2.5) reads as Denoting by A n−1 the (n − 1) order principal submatrix of A, one has Noting that where when n = 2, one has |A n−1 | 2 = (trA n−1 ) 2 , one concludes (2.4).
3 Proof of Theorem 1.2 Let w be a positive weighted p-harmonic function in a domain Ω.Set u = −(p − 1) ln w.Then u is a weak solution to the equation that is, Given any smooth domain U ⋐ Ω and ǫ ∈ (0, 1], consider the approximation/regularized equation defined by It is well known that if u is the solution to (3.1), then u ∈ C 1,α (Ω) for some α ∈ (0, 1); see [3,12,19,20].Moreover, in the following lemma, we summarize some properties of the solution u to (3.1) and u ǫ to (3.3), which result from [3] as a special case.See also [19].Lemma 3.1.For any ǫ ∈ (0, 1], there exists a unique solution u ǫ ∈ C ∞ (U ) ∩ C 0 (U ) to (3.3), and moreover, u ǫ → u in C 0 (U ) and u ǫ → u in C 1,α (V ) uniformly in ǫ > 0 as ǫ → 0 for all V ⋐ U where u is the solution to (3.1).
To show Lemma 3.1, we just need to check that equations (3.1) and (3.3) are special cases of those considered in [3].We put this verification in the appendix.
By Lemma 3.1, the solution u ǫ to (3.2) is C ∞ , which implies that u ǫ satisfies (3.2) pointwise.Hence by a direct computation, (3.2) is equivalent to To prove Theorem 1.2 we first build up the following upper bound.
Lemma 3.2.Let u ǫ be the solution to (3.3).For any γ ∈ R, η > 0 and φ ∈ C ∞ c (U ), we have To prove this, we need the following identity.
Proof.Applying the Bochner formula to v, one has and hence By this, to get (3.5), it suffices to show the following identity Via integration by parts, one has Similarly, via integration by parts one also has Combining together we obtain (3.6) and hence, (3.5) as desired.
We are ready prove Lemma 3.2 as below.
Proof of Lemma 3.2.
To bound the second term in the right-hand side in (3.7), recalling (3.3), that is, by Cauchy-Schwarz's inequality one has where 0 < η < 1 is any constant.Thus The first term in the right-hand side in (3.7) is further written as For the third term in the right-hand side of (3.10), by Cauchy-Schwarz's inequality, one has For the fourth term in the right-hand side of (3.10), in a similar way, using (3.8), one has From (3.13), (3.12), (3.11) and (3.10) we attain Obviously from (3.14), (3.9) and (3.7) we conclude (3.4).
We now prove Lemma 3.4 by using Lemma 3.5.
We prove (3.26) as below.Since we rewrite we further write Thus Combining (3.15) and (3.4) we have the following.Recall that Corollary 3.6.Let u ǫ be the solution to (3.3).If γ < 3 + p−1 N −1 for some N ≥ n, then for sufficiently small η > 0 one has Under the Bakry-Émery curvature-dimension assumption, we have the following uniform upper bound.Lemma 3.7.Let u ǫ be the solution to (3.3) On the other hand, a direct calculation leads to Thus, up to a constant multiplier, the left-hand side of (3.We therefore conclude (3.28) from (3.27).
Now we are able to prove Theorem 1.2.