New characterizations of Ricci curvature on RCD metric measure spaces

We prove that on a large family of metric measure spaces, if the $L^p$-gradient estimate for heat flows holds for some $p>2$, then the $L^1$-gradient estimate also holds. This result extends Savar\'e's result on metric measure spaces, and provides a new proof to von Renesse-Sturm theorem on smooth metric measure spaces. As a consequence, we propose a new analysis object based on Gigli's measure-valued Ricci tensor, to characterize the Ricci curvature of ${\rm RCD}$ space in a local way. The argument is a new iteration technique based on non-smooth Bakry-\'Emery theory, which is a new method to study the curvature dimension condition of metric measure spaces.

where H t f is the solution to the heat equation with initial datum f .
In non-smooth setting, the notions of synthetic Ricci curvature bounds, or nonsmooth curvature-dimension conditions, were proposed by Lott-Villani and Sturm (see [13] and [16]) using optimal transport theory. Later on, by assuming the infinitesimally Hilbertianity (i.e. the Sobolev space W 1,2 is a Hilbert space), RCD condition (or RCD(K, ∞) condtion to emphasize the curvature) which is a refinement of Lott-Sturm-Villani's curvature-dimension condition, was proposed by Ambrosio-Gigli-Savaré (see [4] and [1]). It is known that RCD(K, ∞) spaces are generalizations of Riemannian manifolds with lower Ricci curvature bound and their limit spaces, as well as Alexandrov spaces with lower curvature bound.
Is is known that Lott-Sturm-Villani's synthetic Ricci bound and 2-gradient estimate (for heat flows) are equivalent in non-smooth setting. Let (X, d, m) be a RCD(K, ∞) space, it is proved in [4] that (1. 3) for any f ∈ W 1,2 and t > 0, where H t f is the heat flow from f and |Df | is the minimal weak upper gradient (or weak gradient for simplicity) of f . In particular, by Hölder inequality we know |DH t f | p ≤ e −pKt H t |Df | p , m − a.e. (1.4) for any p ≥ 2. Furthermore, it is proved in [15] that inequality (1.3) can be improved as |DH t f | ≤ e −Kt H t |Df |, m − a.e.. (1.5) In conclusion, inequality (1.4) holds for any p ∈ [1, ∞].
Conversely, it is shown in [5] that a space satisfying inequality (1.3) is RCD(K, ∞). Let (X, d, m) be an infinitesimally Hilbertian space, we have a well-defined Dirichlet energy: for any f ∈ W 1,2 (X, d, m). We denote the L 2 -gradient flow of E(·) starting from f by (H t f ) t . Assume further that the space (X, d, m) has Sobolev-to-Lipschitz property: for any function f ∈ W 1,2 with|Df | ∈ L ∞ , we can find a Lipschitz continuous functionf such that f =f m-a.e. and Lip(f ) = ess sup |Df |. If |DH t f | 2 ≤ e −2Kt H t |Df | 2 , m − a.e. (1.6) for any f ∈ W 1,2 and t > 0, then (X, d, m) is RCD(K, ∞).
The main goal of this paper is to prove that for any p > 2, p-gradient estimate (1.4) can also characterize the curvature-dimension condition of metric measure spaces. We prove a non-smooth version of 2) ⇒ 3) in von Renesse-Sturm's result, thus we complete the circle 1) ⇔ 2) ⇔ 3) in non-smooth setting. Now, we introduce our main result in this paper. When p = 2, it is proved in [15] that there exists a space of test functions TestF(X, d, m) which is a dense subspace of W 1,2 (X) defined as such that ∆|Df | 2 is a well-defined measure (see Definition 3.1) for any f ∈ TestF. So it is reasonable to the following assumption (Assumption 3.5, see a similar assumption in [17]): there exists a dense subspace A in TestF with respect to the graph norm We remark that we do not need to assume the density of A in W 1,2 .
Theorem 1.1 (Theorem 3.6, Improved Bakry-Émery theory). Let M := (X, d, m) be a metric measure space such that there exists an algebra A as described above. If for some p ∈ (1, ∞). Then (1.7) holds for p = 1. In particular, M is a RCD(K, ∞) space.
Since we do not have second order differentiation formula for relative entropy along Wasserstein geodesics, or Taylor's expansion in non-smooth setting, we can not simply use the argument in smooth metric measure space (see the proofs in [14]). The argument we adopt here is the so-called 'self-improvement' method in Bakry-Emery's Γ-calculus, which was used in [15] to deal with the non-smooth problems. We remark that we not only use 'self-improvement' technique, but an improved iteration method based on this technique. We believe that this method also has potential application in the future.
It can be seen that Assumption 3.5 is satisfied in the following cases, where we can apply our main result. Example 1. Smooth metric measure space: obviously, C ∞ c (M), the space of smooth functions with compact support is a good algebra in Assumption 3.5. Hence we obtain a new quick proof to von Renesse-Sturm's theorem, without using Taylor's expansion method.
Example 2. RCD(K, ∞) metric measure space: it is proved in Lemma 3.2 [15] that |Df | 2 ∈ M ∞ for any f ∈ TestF. By Theorem 1.1 we obtain the following proposition which deals with the optimal comstant K in the curvature-dimension condition. It is also a complement to Savaré's result in [15]. Proposition 1.2 (Self-improvement of gradient estimate). Let (X, d, m) be a RCD(K, ∞) metric measure space. If for any f ∈ W 1,2 ∩ Lip(X) ∩ L ∞ (X) we have the gradient estimate In [10], Gigli defines measure valued Ricci tensor on RCD metric measure space (see also [12]) as  [10] for details). He shows that Ricci(∇f, ∇f ) ≥ K|Df | 2 m if and only if the space is RCD(K, ∞). However, we do not know if Ricci has locality in the sense that Ricci(∇f, ∇f ) | {|Df |=0} = 0. Then the following characterizations are equivalent.
We remark that this naive extension is non-trivial, because 2) is not a direct consequence of 3) due to lack of the locality of Ricci(·, ·). From this proposition, we know that Ricci(∇f, ∇f ) := |Df | 2 Ricci ac (∇f, ∇f ) m characterizes the Ricci curvature of (X, d, m) and Ricci has locality in the sense that

Preliminaries
First of all, we summarize the basic hypothesis on the metric measure space (X, d, m) below in Assumption 2.1 below, the notions and concepts in in this assumption will be explained later.
Assumption 2.1. We assume that: there exits a unique heat kernel p t (x, y).
where lip(f n ) is the local Lipschitz constant of f n . It is known that there exists a minimal function G in m-a.e. sense. We call the minimal G the minimal weak upper gradient (or weak gradient for simplicity) of the function f , and denote it by |Df |. It is known that the locality holds for |Df |, i.e. |Df | = |Dg| a.e. on the set {f = g}. Furthermore, we have the lower semi-continuity: if {f n } n ⊂ W 1,2 (X, d, m) is a sequence converging to some f in m-a.e. sense and (|Df n |) n is bounded in L 2 (X, m), then f ∈ W 1,2 (X, d, m) and We equip W 1,2 (X, d, m) with the norm We say that (X, d, m) is an infinitesimally Hilbertian space if W 1,2 is a Hilbert space (see [4], [11] for more discussions).
On an infinitesimally Hilbertian space, we have a natural 'carré du champ' op- It can be seen that Γ(·, ·) is symmetric, bilinear and continuous. We denote Γ(f, f ) by Γ(f ). We have the following chain rule and Leibnitz rule (Lemma 4.7 and Proposition 4.17 in [1], see also Corollary 7.1.2 in [8]) We say that a metric measure space M = (X, d, m) has Sobolev-to-Lipschitz property if: for any function f ∈ W 1,2 with |Df | ∈ L ∞ , we can find a Lipschitz continuous functionf such that f =f m-a.e. and Lip(f ) = ess sup |Df |.
We define the Dirichlet (energy) form E : It is proved (see [2,3]) that Lipschitz functions are dense in energy: for It can be proved that E is a strongly local, symmetric, quasi-regular Dirichlet form (see [2,4,5]). The Markov semigroup (H t ) t≥0 generated by E is called the heat flow. There exists heat kernel which is a family of functions p t (x, y) : Here the Laplacian is defined in the following way (see [11] for the compatibility of different definitions of Laplacian): Definition 2.2 (Measure valued Laplacian, [10,11,15]). The domain of the Laplacian D(∆) ⊂ W 1,2 consists of f ∈ W 1,2 such that there is a measure µ ∈ Meas(M) satisfying In this case the measure µ is unique and we denote it by ∆f . If ∆f ≪ m, we denote its density with respect to m by ∆f .
We define TestF(X, d, m) ⊂ W 1,2 (X, d, m), the space of test functions as It is known from [15] and [4] that TestF(M) is an algebra and it is dense in W 1,2 (X, d, m) when (X, d, m) is a RCD metric measure space. We will see in Lemma 3.4 that TestF is dense in W 1,2 even when L p -gradient estimate for heat flow holds for some p > 2.  Γ(g, h)).
We have the following lemma.

Main Results
Firstly, we discuss more about the measure-valued Laplacian. Since E is quasiregular, we know (see Remark 1.3.9 (ii), [9]) that every function f ∈ W 1,2 has an quasi-continuous representative f . And f is unique up to quasi-everywhere equality, i.e. iff is another quasi-continuous representative, thenf = f holds in a complement of an E-polar set. For more details, see Definition 2.1 in [15] and the references therein.
In particular, every E-polar set is (∆f )-negligible and the measure ϕ∆f is welldefined.
In the next lemma we study the measure ∆Γ(f ) p 2 . Since Γ(f ) is not necessarily continuous, and Φ(x) = x p 2 is not C 2 (R), we can not use Lemma 2.4 directly.
if and only if and Proof. Since p > 2, it can be seen that (3.2) is equivalent to for any Lipschitz function ϕ with bounded support.
The following lemma will be used in the proof of Theorem 3.6. , and a 0 ≥ 0 be an arbitrary initial datum, we define (a n ) n∈N recursively by the formula a n+1 = P (a n ).
Then there exists an integer N 0 such that 0 ≤ a N 0 < 1 and − 1 4 ≤ a N 0 +1 < 0. Conversely, for any a ∈ [0, 1) and b > a, there exists a sequence a 0 , ..., a N 0 defined by the recursive function P such that a 0 > b and a N 0 = a.
As we mentioned in the Introduction, the space of test functions is dense in W 1,2 (X, d, m) when L p -gradient estimate for heat flow holds. Lemma 3.4 (Density of test functions in W 1,2 (X, d, m), Remark 2.5 [5]). Let (X, d, m) be a metric measure space satisfying Assumption 2.1. Assume that for any f ∈ W 1,2 ∩ Lip ∩L ∞ (X, d, m) we have the L p -gradient estimate (3.7) for some p ∈ [1, ∞). Then the space of test functions TestF(X, d, m) is dense in W 1,2 .
Proof. As we discussed in the preliminary section, the space is dense in W 1,2 . We also know that the in dense in L 2 , and V 1 ∞ is invariant under the action (H t ) t by (3.7) and Sobolevto-Lipschitz property. Hence by an approximation argument (see e.g. Lemma 4.9 in [4]), we know V 1 ∞ is dense in W 1,2 . Similarly, by a semigroup mollification (see e.g. page 351, [5]) we can prove that We now introduce the following technical assumption, which is important in our proof. It can be proved that Riemannian manifolds and RCD(K, ∞) spaces satisfy this assumption.
Assumption 3.5 (Existence of good algebra). We assume the existence of a dense subspace A in TestF(X, d, m) with respect to the graph norm such that Γ(f ) ∈ M ∞ for any f ∈ A.
It can be seen that A is an algebra (i.e. A is closed w.r.t. pointwise multiplication), if it is non-trivial. In particular, by Lemma 3.4 we know that A is dense in W 1,2 if L p gradient estimate holds. Theorem 3.6 (Improved Bakry-Émery theory). Let (X, d, m) be a metric measure space satisfying Assumption 2.1 and Assumption 3.5. If for any f ∈ W 1,2 ∩ Lip ∩L ∞ (X, d, m) we have the gradient estimate (3.8) Proof. If p ≤ 2, by the result in [5] we know (X, d, m) is a RCD(K, ∞). Now we assume p > 2.
As a corollary, we have the following proposition. We recall (see [10]) that the measure-valued Ricci tensor on RCD metric measure space is defined as where Γ 2 (f ) := 1 2 ∆Γ(f ) − Γ(f, ∆f ) m and |Hess[f ]| HS is the minimal L 2 function G such that | i,j Hess[f ](g i , h j )| ≤ G i,j Γ 2 (g i , h j ) for any (g i ), (h j ) ⊂ TestF (see [10] and [15] for details). It is proved that Ricci is well defined for any f ∈ TestF(X, d, m) when (X, d, m) is RCD. 2) for any test function f ∈ TestF we have Ricci(∇f, ∇f ) ≥ K|Df | 2 m in the sense that Ricci ac (∇f, ∇f ) ≥ K|Df | 2 m − a.e.