Stability of Curvature-Dimension Condition for Negative Dimensions Under Concentration Topology

Abstract

In this paper, we prove the stability of metric measure spaces satisfying the curvature-dimension condition for negative dimensions under the concentration topology. This result is an analog of the result by Funano–Shioya with respect to the dimension parameter.

1 Introduction

A metric measure space is a metric space with the structure of a measure space, defined as a generalization of a Riemannian manifold with the Riemannian distance and the volume measure determined by the Riemannian metric. In particular, it has been studied whether the properties and structure of Riemannian manifolds hold in metric measure spaces. There are many definitions of metric measure spaces, but this paper deals specifically with metric measure spaces with probability measures, which we call mm-spaces for short.

There are two main concepts in this paper. The first is the curvature-dimension condition (${{\,\textrm{CD}\,}}(K,N)$ condition, $K\in {\mathbb {R}}$, $N\in (1,\infty ]$), which is a condition that the weighted Ricci curvature ${{\,\textrm{Ric}\,}}_N$ of an n-dimensional weighted Riemannian manifold defined as

$$\begin{aligned} {{\,\textrm{Ric}\,}}_N(v, v):={{\,\textrm{Ric}\,}}_g(v,v)+{{\,\textrm{Hess}\,}}f(v,v)-\frac{\langle \nabla f(x),v\rangle ^2}{N-n}\qquad (v\in T_xM) \end{aligned}$$

satisfies

$$\begin{aligned} {{\,\textrm{Ric}\,}}_N\ge Kg. \end{aligned}$$

It is well known that this condition is equivalent to “Ricci curvature greater than or equal to K and dimension of manifold less than or equal to N" for Riemannian manifolds. The extension of this condition to the metric measure spaces is given independently by Lott–Villani [6] and Sturm [12, 13]. A metric measure space satisfying this condition is called a ${{\,\textrm{CD}\,}}(K,N)$ space. It is known that ${{\,\textrm{CD}\,}}(K,N)$ spaces inherit many properties of Riemannian manifolds with Ricci curvature bounded from below by K and dimension at most N. Then, by considering the weighted Ricci curvature as $N<0$, the ${{\,\textrm{CD}\,}}(K,N)$ condition for $N<0$ is defined, which is weaker than the ${{\,\textrm{CD}\,}}(K,\infty )$ condition, giving an even wider class of spaces. Furthermore, The ${{\,\textrm{CD}\,}}(K,N)$ condition for $N<0$ was extended to a metric measure space by Ohta [9], and this allows us to consider a metric measure space satisfying the ${{\,\textrm{CD}\,}}(K,N)$ condition for $N<0$.

The second is the convergence of metric measure spaces, in particular, the $\Box$-convergence and concentration introduced by Gromov [3], which are the convergences of metric measure spaces with probability measure (called mm-spaces). There are various types of convergence of metric measure spaces known today; some examples are

• (1) pointed measured Gromov-Hausdorff convergence ($\textrm{pmGH}$ convergence),

• (2) pointed measured Gromov convergence ($\textrm{pmG}$ convergence),

• (3) Sturm’s ${\mathbb {D}}$-convergence,

• (4) $\Box$-convergence,

• (5) Concentration.

Whether these convergences can be considered depends on the definition of the metric measure spaces, but, in mm-spaces, all convergences can be considered, and (5) is known to be the weakest convergence. Funano–Shioya [1] and Kazukawa–Ozawa–Suzuki [4] obtained the following result on the question of whether the curvature dimension condition is stable for the concentration topology which is the weakest convergence.

Theorem 1.1

([1, Theorem 1.2], [4, Theorem 1.1]) If a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ satisfying the ${{\,\textrm{CD}\,}}(K,\infty )$ condition for some $K\in {\mathbb {R}}$ concentrates to an mm-space Y, then Y is also a ${{\,\textrm{CD}\,}}(K,\infty )$ space.

This shows that the ${{\,\textrm{CD}\,}}(K,\infty )$ condition is stable in the concentration topology. A result on the question of whether the ${{\,\textrm{CD}\,}}(K,N)$ condition for $N<0$ is stable for convergence of metric measure spaces is also given by Magnabosco–Rigoni–Sosa [7].

Theorem 1.2

[7, Theorem 4.1] Suppose that a sequence $\{(X_n,d_{X_n},\mu _{X_n})\}_{n\in {\mathbb {N}}}$ of metric measure spaces satisfying the ${{\,\textrm{CD}\,}}(K,N)$ condition for some $K\in {\mathbb {R}}$ and $N<0$ converges to a metric measure space $(Y,d_Y,\mu _Y)$ in the $\textrm{pmG}$ sense. If the following assumption

$$\begin{aligned} \limsup _{n\rightarrow \infty }{{\,\textrm{diam}\,}}X_n<\frac{\pi }{\sqrt{-K}} \end{aligned}$$

holds when $K<0$, then Y is also a ${{\,\textrm{CD}\,}}(K,N)$ space.

Note that Magnabosco et al. [7] proved the above theorem for a broader definition of the metric measure spaces than that in this paper, and Theorem 1.2 is an application of their theorem to the definition in this paper. The main theorem in this paper is that the ${{\,\textrm{CD}\,}}(K,N)$ condition for $N<0$ is stable under the concentration topology weaker than the topology of $\textrm{pmG}$ convergence.

Main Theorem 1.3

Suppose that a sequence $\{X_n\}_{n\in {\mathbb {N}}}$ of mm-spaces satisfying the ${{\,\textrm{CD}\,}}(K,N)$ condition for some $K\in {\mathbb {R}}$ and $N<0$ concentrates to an mm-space Y. If the following assumption

$$\begin{aligned} \limsup _{n\rightarrow \infty }{{\,\textrm{diam}\,}}X_n<\frac{\pi }{\sqrt{-K}} \end{aligned}$$

(1)

holds when $K<0$, then Y is also a ${{\,\textrm{CD}\,}}(K,N)$ space.

Remark 1.4

If $K\ge 0$, then Main Theorem 1.3 with the ${{\,\textrm{CD}\,}}(K,N)$ condition replaced by the ${{\,\textrm{CD}\,}}^*(K,N)$ condition can be proved similarly.

It can also be shown that it is necessary to assume (1) for $K<0$.

Main Theorem 1.5

Let K, N be negative numbers, and let $D\ge \pi \sqrt{(N-1)/K}$. Then, there exist a sequence $\{f_n\}_{n\in {\mathbb {N}}}$ of smooth functions on the circle ${\mathbb {S}}^1$ and an mm-space $(Y, d_Y, \mu _Y)$ consisting of two points with ${{\,\textrm{diam}\,}}Y=D$ such that the sequence of mm-spaces $\{(X_n,d_{X_n},\mu _{X_n})\}_{n\in {\mathbb {N}}}$ defined as

$$\begin{aligned} (X_n,d_{X_n},\mu _{X_n}):=\left( {\mathbb {S}}^1, d_g, e^{-f_n}{{\,\textrm{vol}\,}}_g\right) , \end{aligned}$$

where g is the canonical metric on ${\mathbb {S}}^1$ scaled to ${{\,\textrm{diam}\,}}{\mathbb {S}}^1=D$, satisfies the ${{\,\textrm{CD}\,}}(K,N)$ condition and $\Box$-converges to $(Y, d_Y, \mu _Y)$.

In particular, since an mm-space consisting of two points is not a ${{\,\textrm{CD}\,}}(K,N)$ space, we can say that $X_n$ and Y constructed in Main Theorem 1.5 are counterexamples of Main Theorem 1.3 without assuming (1). This result is clearly different from the case $1<N\le \infty$ where the concentration limit of ${{\,\textrm{CD}\,}}(K,N)$ spaces is also a ${{\,\textrm{CD}\,}}(K,N)$ space for any $K\in {\mathbb {R}}$ without the assumption (1) as in Theorem 1.1.

Finally, we describe the structure of this paper. In Sect. 2, we state some basic propositions on measure spaces and Wasserstein spaces. In Sect. 3, we state definitions, examples, and properties of mm-spaces, curvature-dimension conditions, and concentration topology, and in Sect. 4, we give some results on the estimates of observable diameter defined in Sect. 3. In Sect. 5, we state the necessary lemmas for the proof of Main Theorem 1.3, such as the properties of Rényi entropy. In Sects. 6 and 7, we give the proofs of Main Theorems 1.3 and 1.5, respectively. In Sect. 8 we present some results for metric measure spaces that satisfy the ${{\,\textrm{CD}\,}}(K,N)$ condition for some $N<0$ and are not mm-spaces.

2 Notations

We list the notations we will use throughout this paper.

• For $x,y\in {\mathbb {R}}$, we write $x\vee y:=\max \{x,y\}$ and $x\wedge y:=\min \{x,y\}$.

• ${\mathcal {B}}_X$ denotes the set of all Borel subsets of a topological space X.

• ${\mathcal {P}}(X)$ denotes the set of all Borel probability measures on X.

• $C_b(X)$ denotes the set of all bounded continuous functions on X.

• ${\mathcal {L}}{{ ip}}_L(X)$ denotes the set of all L-Lipschitz functions on X.

• For $i=0,1$, ${{\,\textrm{proj}\,}}_i:X\times X\rightarrow X$ denotes the projection defined by ${{\,\textrm{proj}\,}}_i(x_0,x_1):=x_i$

• For $\mu ,\nu \in {\mathcal {P}}(X)$, we write $\nu \ll \mu$ to mean that $\nu$ is absolutely continuous with respect to $\mu$, and if it is not.

• ${\textbf{1}}_A$ denotes the characteristic function of a subset $A\subset X$.

• For maps $p,q:X\rightarrow Y$, $p\times q:X\times X\rightarrow Y\times Y$ denotes the product of p and q defined by $(p\times q)(x_0,x_1):=(p(x_0),q(x_1))$.

• $B_X(x,r)$ denotes the open ball in a metric space $(X,d_X)$ with center $x\in X$ and radius $r>0$.

• For Borel subsets A, B in a metric space $(X,d_X)$, we write

  $$\begin{aligned} d_X(A,B):=\inf _{x\in A, y\in B}d_X(x,y) \end{aligned}$$

  .

• $N_\varepsilon (A)$ denotes the $\varepsilon$-neighborhood of a subset A of a metric space $(X,d_X)$, namely

  $$\begin{aligned} N_\varepsilon (A):=\bigcup _{a\in A}B_X(a,\varepsilon ) \end{aligned}$$

3 Preliminaries

In this section, we give some basic definitions and propositions about metric spaces, probability measure space, and Wasserstein space, e.g. [14].

Proposition 2.1

Let $(X,d_X)$ be a metric space and let $(Y,d_Y)$ be a complete metric space. If a map $f:A\rightarrow Y$ on a subset A of X is L -Lipschitz for $L>0$, then there exists a unique L -Lipschitz extension of f on the closure ${\overline{A}}$ of A.

This unique extension of f is also denoted by f.

Definition 2.2

(Weakly convergence) Let X be a topological space. We say that $\{\mu _n\}_{n\in {\mathbb {N}}}\subset {\mathcal {P}}(X)$ converges to $\mu \in {\mathcal {P}}(X)$ weakly if for any $f\in C_b(X)$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _Xfd\mu _n=\int _Xfd\mu \end{aligned}$$

holds and write $\mu _n\rightharpoonup \mu$.

There are several conditions that are equivalent to this definition of weak convergence, and the equivalence of these conditions is known as the Portmanteau theorem. It is also known that when X is a separable metric space, the topology on ${\mathcal {P}}(X)$ which is determined by weak convergence, can be metrized by the Prokhorov distance $d_P$. Furthermore, when X is complete, Prokhorov’s theorem says that ${\mathcal {K}}\subset {\mathcal {P}}(X)$ is relatively compact if and only if ${\mathcal {K}}$ is tight, i.e., for any $\varepsilon >0$, there exists a compact subset $K\subset X$ such that

$$\begin{aligned} \sup _{\mu \in {\mathcal {K}}}\mu (X\setminus K)<\varepsilon . \end{aligned}$$

In the following, we assume that X is a complete and separable metric space.

Definition 2.3

(Coupling) Let $\mu _0, \mu _1\in {\mathcal {P}}(X)$. We say that $\pi \in {\mathcal {P}}(X\times X)$ is a coupling of $\mu _0$ and $\mu _1$ if the push-forward $({{\,\textrm{proj}\,}}_i)_*\pi$ of $\pi$ by ${{\,\textrm{proj}\,}}_i$ equals $\mu _i$ for any $i=0,1$. The set of all couplings of $\mu _0$ and $\mu _1$ is denoted by ${{\,\textrm{Cpl}\,}}(\mu _0,\mu _1)$.

Lemma 2.4

Let $\{\mu _n\}_{n\in {\mathbb {N}}}, \{\nu _n\}_{n\in {\mathbb {N}}}\subset {\mathcal {P}}(X)$, $\mu ,\nu \in {\mathcal {P}}(X)$ and $\pi _n\in {{\,\textrm{Cpl}\,}}(\mu _n,\nu _n)$. If $\mu _n$, $\nu _n$ weakly converge to $\mu$, $\nu$ respectively, then the following hold.

• $\{\pi _n\}_{n\in {\mathbb {N}}}$ is tight.

• If $\pi _n$ converges to some $\pi \in {\mathcal {P}}(X\times X)$ weakly, then $\pi$ is a coupling of $\mu$ and $\nu$.

Definition 2.5

(Wasserstein space) We define

$$\begin{aligned} {\mathcal {P}}_2(X):=\left\{ \mu \in {\mathcal {P}}(X)\left| \int _Xd_X(x,x_0)^2d\mu (x)<\infty \text { for some }x_0\in X\right\} \right. , \end{aligned}$$

and $W_2:{\mathcal {P}}_2(X)\times {\mathcal {P}}_2(X)\rightarrow [0,\infty )$ defined by

$$\begin{aligned} W_2(\mu , \nu ):=\inf _{\pi \in {{\,\textrm{Cpl}\,}}(\mu ,\nu )}\left( \int _{X\times X}d_X(x,y)^2d\pi (x,y)\right) ^{\frac{1}{2}} \end{aligned}$$

(2)

is a metric on ${\mathcal {P}}_2(X)$. We call the metric space $({\mathcal {P}}_2(X),W_2)$ the Wasserstein space. $\pi \in {{\,\textrm{Cpl}\,}}(\mu ,\nu )$ attaining the infimum of the right-hand side of (2) is called an optimal coupling of $\mu$ and $\nu$. The set of all optimal couplings of $\mu$ and $\nu$ is denoted by ${{\,\textrm{Opt}\,}}(\mu ,\nu )$.

It is known that $({\mathcal {P}}_2(X),W_2)$ is a complete and separable metric space.

Proposition 2.6

Let $\{\mu _n\}_{n\in {\mathbb {N}}}\subset {\mathcal {P}}_2(X)$ and $\mu \in {\mathcal {P}}_2(X)$. The following are equivalent.

• (1) $\mu _n$ $W_2$-converges to $\mu$ i.e. $\displaystyle \lim _{n\rightarrow \infty }W_2(\mu _n,\mu )=0$.

• (2) $\mu _n$ converges to $\mu$ weakly, and

  $$\begin{aligned} \lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\int _{X\setminus B_X(x_0, R)}d_X(x,x_0)^2d\mu _n(x)=0 \end{aligned}$$

  holds for some $x_0\in X$.

Lemma 2.7

Let $\{\mu _n\}_{n\in {\mathbb {N}}}, \{\nu _n\}_{n\in {\mathbb {N}}}\subset {\mathcal {P}}_2(X)$. If $\mu _n$, $\nu _n$ weakly converge to $\mu , \nu \in {\mathcal {P}}_2(X)$ respectively, then we have

$$\begin{aligned} W_2(\mu ,\nu )\le \liminf _{n\rightarrow \infty }W_2(\mu _n,\nu _n). \end{aligned}$$

In addition, suppose that $\mu _n$ and $\nu _n$ $W_2$-converge to $\mu$ and $\nu$ respectively. If $\pi _n\in {{\,\textrm{Opt}\,}}(\mu _n, \nu _n)$ converges to some $\pi \in {\mathcal {P}}(X\times X)$ weakly, then $\pi$ is an optimal coupling of $\mu$ and $\nu$.

Definition 2.8

Let $\mu \in {\mathcal {P}}(X)$. For $B\in {\mathcal {B}}_X$ with $\mu (B)>0$, we define $\mu _B\in {\mathcal {P}}(X)$ by

$$\begin{aligned} \mu _B(A)=(\mu )_B(A):=\frac{\mu (B\cap A)}{\mu (B)} \end{aligned}$$

for $A\in {\mathcal {B}}_X$.

Lemma 2.9

Let $\{\nu _0^n\}_{n\in {\mathbb {N}}}, \{\nu _1^n\}_{n\in {\mathbb {N}}}\subset {\mathcal {P}}(X)$, $\nu _0, \nu _1 \in {\mathcal {P}}(X)$, $\pi ^n\in {{\,\textrm{Cpl}\,}}(\nu _0^n,\nu _1^n)$, $\varphi \in C_b(X\times X)$, and let $f_i\in L^1(X,\nu _i)$ $(i=0,1)$. Suppose that $\nu _0^n$, $\nu _1^n$, and $\pi ^n$ weakly converge to $\nu _0$, $\nu _1$, and some $\pi \in {{\,\textrm{Cpl}\,}}(\nu _0,\nu _1)$ respectively. If either of the following three conditions

• (a) for any $i=0,1$, $f_i$ is a Borel simple function such that for any $y\in {\mathbb {R}}$, $\nu _i(\partial f_i^{-1}(y))=0$, where $\partial f_i^{-1}(y)$ is the boundary of $f_i^{-1}(y)$;

• (b) there exists $C>0$ such that $\nu _i^n\le C\nu _i$ for $i=0,1$;

• (c) there exists $\mu \in {\mathcal {P}}(X)$ such that $f_i\in {\mathcal {L}}^{\infty }(X,\mu )$, $\nu _i^n=\rho _i^n\mu$, $\nu _i=\rho _i\mu$ and $\rho _i^n$ converges to $\rho _i$ in $L^1(X,\mu )$ for $i=0,1$;

holds, then we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }\int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi ^n(x_0,x_1)\le \int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi (x_0,x_1) \end{aligned}$$

(3)

for any $i=0,1$.

Proof

If (a) holds, for any $i=0,1$, we put

$$\begin{aligned} f_i=\sum _{j\in {\mathcal {J}}}a_{i,j}{\textbf{1}}_{B_{i,j}}, \end{aligned}$$

where a finite set ${\mathcal {J}}$, $\{a_{i,j}\}_{j\in {\mathcal {J}}}\subset {\mathbb {R}}$ and mutually disjoint Borel subsets $\{B_{i,j}\}_{j\in {\mathcal {J}}}$ satisfy $\nu _i(\partial B_{i,j})=0$ for any $j\in {\mathcal {J}}$. Then, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{{\tilde{B}}_{i,j}}\varphi d\pi ^n=\int _{{\tilde{B}}_{i,j}}\varphi d\pi , \end{aligned}$$

where ${\tilde{B}}_{i,j}:={{\,\textrm{proj}\,}}_i^{-1}(B_{i,j})$. Indeed, it is obvious if $\pi ({\tilde{B}}_{i,j})=0$. If $\pi ({\tilde{B}}_{i,j})>0$, $(\pi ^n)_{{\tilde{B}}_{i,j}}$ converges to $\pi _{{\tilde{B}}_{i,j}}$ weakly by $\pi (\partial {\tilde{B}}_{i,j})=\nu _i(\partial B_{i,j})=0$. Hence,

$$\begin{aligned} \int _{{\tilde{B}}_{i,j}}\varphi d\pi ^n=\pi ^n({\tilde{B}}_{i,j})\int _{X\times X}\varphi d(\pi ^n)_{{\tilde{B}}_{i,j}}\rightarrow \pi ({\tilde{B}}_{i,j})\int _{X\times X}\varphi d\pi _{{\tilde{B}}_{i,j}}=\int _{{\tilde{B}}_{i,j}}\varphi d\pi \end{aligned}$$

holds. Thus, we get

$$\begin{aligned}&\lim _{n\rightarrow \infty }\int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi ^n(x_0,x_1)\\&\quad =\lim _{n\rightarrow \infty }\sum _{j\in {\mathcal {J}}}a_{i,j}\int _{{\tilde{B}}_{i,j}}\varphi d\pi ^n =\int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi (x_0,x_1). \end{aligned}$$

In particular, (3) holds.

Next, we take any $\varepsilon >0$. Then, there exists $f_i^{\varepsilon }\in C_b(X)$ such that

$$\begin{aligned} \int _X|f_i-f_i^{\varepsilon }|d\nu _i<\varepsilon . \end{aligned}$$

If (b) holds, we get

$$\begin{aligned}&\int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi ^n(x_0,x_1)\\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi ^n(x_0,x_1)\\&\qquad +\int _{X\times X}|\varphi (x_0,x_1)|\cdot |f_i(x_i)-f_i^\varepsilon (x_i)|d\pi ^n(x_0,x_1)\\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi ^n(x_0,x_1)+\Vert \varphi \Vert _{\infty }\int _X|f_i-f_i^\varepsilon |d\nu _i^n\\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi ^n(x_0,x_1)+C\cdot \Vert \varphi \Vert _{\infty }\cdot \varepsilon \end{aligned}$$

for any $i=0,1$. Since $\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)$ is bounded continuous on $X\times X$, we have

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi ^n(x_0,x_1)\\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi (x_0,x_1)+C\cdot \Vert \varphi \Vert _{\infty }\cdot \varepsilon \\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi (x_0,x_1)+(C+1)\Vert \varphi \Vert _{\infty }\cdot \varepsilon . \end{aligned}$$

Thus, as $\varepsilon \rightarrow 0$, we obtain (3).

If (c) holds, we get

$$\begin{aligned}&\int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi ^n(x_0,x_1)\\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi ^n(x_0,x_1)+\Vert \varphi \Vert _{\infty }\int _X|f_i-f_i^\varepsilon |d\nu _i^n\\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi ^n(x_0,x_1)\\&\qquad +\Vert \varphi \Vert _{\infty }\left( \int _X|f_i-f_i^\varepsilon |\cdot |\rho _i^n-\rho _i|d\mu +\int _X|f_i-f_i^\varepsilon |d\nu _i\right) \\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi ^n(x_0,x_1)\\&\qquad +\Vert \varphi \Vert _{\infty }\left( \left( \Vert f_i\Vert _{\infty }+\Vert f_i^\varepsilon \Vert _{\infty }\right) \int _X|\rho _i^n-\rho _i|d\mu +\varepsilon \right) \end{aligned}$$

for any $i=0,1$. Since $\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)$ is bounded continuous on $X\times X$, we have

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi ^n(x_0,x_1)\\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i^{\varepsilon }(x_i)d\pi (x_0,x_1)+\Vert \varphi \Vert _{\infty }\cdot \varepsilon \\&\quad \le \int _{X\times X}\varphi (x_0,x_1)f_i(x_i)d\pi (x_0,x_1)+2\Vert \varphi \Vert _{\infty }\cdot \varepsilon . \end{aligned}$$

Thus, as $\varepsilon \rightarrow 0$, we obtain (3). $\square$

4 Mm-Spaces and Curvature-Dimension Conditions

In this section, we give the definition and basic properties of mm-spaces and the curvature-dimension condition.

Definition 3.1

(mm-space) A triple $(X,d_X,\mu _X)$ is called an mm-space if $(X,d_X)$ is a complete separable metric space and $\mu _X$ is a Borel probability measure on X.

Definition 3.2

(mm-isomorphism) Two mm-spaces $(X,d_X,\mu _X)$, $(Y,d_Y,\mu _Y)$ are said to be mm-isomorphic if there exists an isometry $f:{{\,\textrm{supp}\,}}\mu _X\rightarrow {{\,\textrm{supp}\,}}\mu _Y$ such that $f_*\mu _X$ equals $\mu _Y$. Such an f is called an mm-isomorphism.

Let ${\mathcal {X}}$ denote the set of mm-isomorphism classes of mm-spaces.

Definition 3.3

(Parameter) Let $I:=[0,1)$ and let X be an mm-space. A Borel map $\varphi :I\rightarrow X$ is called a parameter of X if $\varphi$ satisfies $\varphi _*{\mathcal {L}}^1=\mu _X$, where ${\mathcal {L}}^1$ is the Lebesgue measure on I.

Proposition 3.4

[11, Lemma 4.2] Any mm-space has a parameter.

Definition 3.5

(Box distance between mm-spaces) The box distance $\Box (X,Y)$ between two mm-spaces X and Y is defined by the infimum of $\varepsilon >0$ satisfying that there exist a Borel set ${\tilde{I}}\subset I$ and parameters $\varphi :I\rightarrow X$ and $\psi :I\rightarrow Y$ of X and Y such that

• (1) ${\mathcal {L}}^1({\tilde{I}})\ge 1-\varepsilon$,

• (2) $|d_X(\varphi (s),\varphi (t))-d_Y(\psi (s),\psi (t))|\le \varepsilon$      for any $s,t\in {\tilde{I}}$.

It is known that $\Box$ is a metric on ${\mathcal {X}}$ and $({\mathcal {X}},\Box )$ is a complete metric space [11, Theorem 4.10, Theorem 4.14]. If a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ converges to an mm-space Y with respect to $\Box$, we say that $\{X_n\}_{n\in {\mathbb {N}}}$ $\Box$-converges to Y and write $X_n\xrightarrow {\Box }Y$.

Proposition 3.6

[11, Proposition 4.12] Let X be a complete separable metric space. For any two Borel probability measures $\mu$ and $\nu$ on X, we have

$$\begin{aligned} \frac{1}{2}\Box ((X,\mu ),(X,\nu ))\le \Box ((2^{-1}X,\mu ),(2^{-1}X,\nu ))\le d_P(\mu ,\nu ). \end{aligned}$$

In particular, if a sequence $\{\mu _n\}_{n\in {\mathbb {N}}}\subset {\mathcal {P}}(X)$ converges to some $\mu \in {\mathcal {P}}(X)$ weakly, then we have $(X,\mu _n)\xrightarrow {\Box }(X,\mu )$.

Definition 3.7

(Observable diameter) Let X be an mm-space and let $\alpha \ge 0$. We define the partial diameter ${{\,\textrm{diam}\,}}(X;\alpha )$ of X by

$$\begin{aligned} {{\,\textrm{diam}\,}}(X;\alpha )={{\,\textrm{diam}\,}}(\mu _X;\alpha ):=\inf \{{{\,\textrm{diam}\,}}A\mid A\in {\mathcal {B}}_X \text { with } \mu _X(A)\ge \alpha \}. \end{aligned}$$

For $\kappa \ge 0$, we define

$$\begin{aligned} {{\,\textrm{ObsDiam}\,}}(X;-\kappa )&:=\sup \{{{\,\textrm{diam}\,}}(f_*\mu _X;1-\kappa )\mid f\in {\mathcal {L}}{{ ip}}_1(X)\},\\ {{\,\textrm{ObsDiam}\,}}(X)&:=\inf _{\kappa >0}(\kappa \vee {{\,\textrm{ObsDiam}\,}}(X;-\kappa )). \end{aligned}$$

We call ${{\,\textrm{ObsDiam}\,}}(X;-\kappa )$ (resp. ${{\,\textrm{ObsDiam}\,}}(X)$) the $\kappa$-observable diameter of X (resp. observable diameter of X).

Remark 3.8

The $\kappa$-observable diameter is nonincreasing for $\kappa$, and we usually consider only when $\kappa <1$ because ${{\,\textrm{ObsDiam}\,}}(X;-\kappa )=0$ if $\kappa \ge 1$.

Definition 3.9

(Lévy family) A sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ is called a Lévy family if

$$\begin{aligned} \lim _{n\rightarrow \infty }{{\,\textrm{ObsDiam}\,}}(X_n)=0 \end{aligned}$$

holds, or equivalently

$$\begin{aligned} \lim _{n\rightarrow \infty }{{\,\textrm{ObsDiam}\,}}(X_n;-\kappa )=0 \end{aligned}$$

holds for any $\kappa >0$.

Definition 3.10

(Separation distance) Let X be an mm-space and let $N\in {\mathbb {N}}$, $\kappa _0, \kappa _1,\ldots ,\kappa _N>0$. We define the separation distance ${{\,\textrm{Sep}\,}}(X;\kappa _0, \kappa _1,\ldots ,\kappa _N)$ of X as the supremum of $\min _{i\ne j}d_X(A_i,A_j)$ , where $A_0, A_1,\ldots ,A_N$ run over all Borel subsets satisfying $\mu _X(A_i)\ge \kappa _i$ for $i=0,1,\ldots ,N$.

Proposition 3.11

[11, Proposition 2.26] Let X be an mm-spaces. For any $\kappa>\kappa '>0$, we have

• ${{\,\textrm{ObsDiam}\,}}(X;-2\kappa )\le {{\,\textrm{Sep}\,}}(X;\kappa , \kappa )$,

• ${{\,\textrm{Sep}\,}}(X;\kappa , \kappa )\le {{\,\textrm{ObsDiam}\,}}(X;-\kappa ')$.

Definition 3.12

(Ky Fan metric) The Ky Fan distance ${ d}_{\textrm{KF}}(f,g)$ between two measurable functions $f,g:\Omega \rightarrow {\mathbb {R}}$ on a probability measure space $(\Omega , \mu )$ is defined by

$$\begin{aligned} { d}_{\textrm{KF}}(f,g):=\inf \left\{ \varepsilon \ge 0\mid \mu (\{x\in \Omega \mid |f(x)-g(x)|>\varepsilon \})\le \varepsilon \right\} . \end{aligned}$$

This distance function ${ d}_{\textrm{KF}}$ is called the Ky Fan metric.

Definition 3.13

(Observable distance) The observable distance ${ d}_{\textrm{conc}}(X,Y)$ between two mm-spaces X and Y is defined as

$$\begin{aligned} { d}_{\textrm{conc}}(X,Y):=\inf _{\varphi , \psi }{ d}_{ H}^{\textrm{KF}}\left( \varphi ^*{\mathcal {L}}{{ ip}}_1(X),\psi ^*{\mathcal {L}}{{ ip}}_1(Y)\right) , \end{aligned}$$

where $\varphi$ and $\psi$ run over all parameters of X and Y, $\varphi ^*{\mathcal {L}}{{ ip}}_1(X):=\{f\circ \varphi \mid f\in {\mathcal {L}}{{ ip}}_1(X)\}$ which is a subset of the set of measurable functions on $(I, {\mathcal {L}}^1)$, and ${ d}_{ H}^{\textrm{KF}}$ is the Hausdorff distance with respect to the Ky Fan metric.

It is also known that ${ d}_{\textrm{conc}}$ is a metric on ${\mathcal {X}}$ ( [11, Theorem 5.16]). If a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ converges to an mm-space Y with respect to ${ d}_{\textrm{conc}}$, we say that $\{X_n\}_{n\in {\mathbb {N}}}$ concentrates to Y and write $X_n\xrightarrow {{\textrm{conc}}}Y$.

Proposition 3.14

[11, Proposition 5.5] Any $X,Y\in {\mathcal {X}}$ satisfy ${ d}_{\textrm{conc}}(X,Y)\le \Box (X,Y)$. In particular, if a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ $\Box$-converges to an mm-space Y, then $\{X_n\}_{n\in {\mathbb {N}}}$ concentrates to Y.

A topology on ${\mathcal {X}}$ induced by ${ d}_{\textrm{conc}}$ is called the concentration topology.

Proposition 3.15

[11, Proposition 5.7, Corollary 5.8] Let $*$ be an mm-space consisting of one point, i.e. $*:=(\{*\}, d_*, \delta _*)$. We have

$$\begin{aligned} { d}_{\textrm{conc}}(X,*)\le {{\,\textrm{ObsDiam}\,}}(X)\le 2{ d}_{\textrm{conc}}(X,*) \end{aligned}$$

for any $X\in {\mathcal {X}}$. In particular, for a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$, $X_n \xrightarrow {{\textrm{conc}}}*$ is equivalent to $\{X_n\}_{n\in {\mathbb {N}}}$ being a Lévy family.

Proposition 3.16

[11, Corollary 5.35, Proposition 9.31] Suppose that a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ concentrates to an mm-space Y. Then, for any $n\in {\mathbb {N}}$, there exist a Borel map $p_n:X_n\rightarrow Y$, a compact set ${\tilde{X}}_n\subset X_n$ and $\varepsilon _n>0$ such that

• (1) ${ d}_{ H}^{\textrm{KF}}({\mathcal {L}}{{ ip}}_1(X_n),p_n^*{\mathcal {L}}{{ ip}}_1(Y))\le \varepsilon _n$ and $\varepsilon _n\rightarrow 0$,

• (2) $(p_n)_*\mu _{X_n}\rightharpoonup \mu _Y$,

• (3) $d_Y(p_n(x),p_n(x'))\le d_{X_n}(x,x')+\varepsilon _n$      for any $x,x'\in {\tilde{X}}_n$,

• (4) $\mu _{X_n}({\tilde{X}}_n)\ge 1-\varepsilon _n$,

• (5) $\displaystyle \limsup _{n\rightarrow \infty }\sup _{x\in X_n\setminus {\tilde{X}}_n}d_Y(p_n(x),y) <\infty$      for any $y\in Y$.

Conversely, if there exist $p_n$ and $\varepsilon _n$ satisfying (1) and (2), then $X_n$ concentrates to Y.

Proposition 3.17

[4, Proposition 2.6] Suppose that a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ concentrates to an mm-space Y. Then we have

$$\begin{aligned} {{\,\textrm{diam}\,}}Y\le \liminf _{n\rightarrow \infty }{{\,\textrm{diam}\,}}X_n. \end{aligned}$$

Proposition 3.18

[11, Proposition 2.38] Let M be a compact Riemannian manifold and let $k\in {\mathbb {N}}$. For any $\kappa _0, \kappa _1,\ldots ,\kappa _k>0$, we have

$$\begin{aligned} {{\,\textrm{Sep}\,}}(M;\kappa _0, \kappa _1,\ldots ,\kappa _k)\le \frac{2}{\sqrt{\lambda _k(M)\min _{i=0,1,\ldots ,k}\kappa _i}}, \end{aligned}$$

where $\lambda _k(M)$ is the kth eigenvalue of the Laplacian of M.

Definition 3.19

(k-Lévy family) Let $k\in {\mathbb {N}}$. A sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ is a k-Lévy family if

$$\begin{aligned} \lim _{n\rightarrow \infty }{{\,\textrm{Sep}\,}}(X_n;\kappa _0, \kappa _1,\ldots ,\kappa _k)=0 \end{aligned}$$

holds for any $\kappa _0, \kappa _1,\ldots ,\kappa _k>0$.

Remark 3.20

By Proposition 3.11, a 1-Lévy family is the same as a Lévy family. A sequence of compact Riemannian manifolds $\{M_n\}_{n\in {\mathbb {N}}}$ satisfying $\lim _{n\rightarrow \infty }\lambda _k(M_n)=\infty$ for some $k\in {\mathbb {N}}$ is a k-Lévy family by Proposition 3.18.

For an mm-space $X=(X,d_X,\mu _X)$ and $t>0$, we define tX by

$$\begin{aligned} tX:=(X,td_X,\mu _X). \end{aligned}$$

Proposition 3.21

([1, Theorem 4.4], [11, Theorem 9.40]) Let $k\in {\mathbb {N}}$ and let $\{X_n\}_{n\in {\mathbb {N}}}$ be a k-Lévy family. We have only one of the following.

• $\{X_n\}_{n\in {\mathbb {N}}}$ is a Lévy family.

• There exist a subsequence $\{n_i\}_{i\in {\mathbb {N}}}$, a sequence $\{t_i\}_{i\in {\mathbb {N}}}\subset (0,1]$, and an mm-space Y with $|Y|\in \{2,\ldots ,k\}$ such that $t_iX_{n_i}\xrightarrow {{\textrm{conc}}}Y$ holds.

Next, we define some functions necessary to define the curvature-dimension condition.

Definition 3.22

Let $\kappa \in {\mathbb {R}}$. We define

$$\begin{aligned} \omega _{\kappa }:= {\left\{ \begin{array}{ll} \dfrac{\pi }{\sqrt{\kappa }}&{}\quad (\kappa >0)\\ \infty &{}\quad (\kappa \le 0) \end{array}\right. } \end{aligned}$$

and $s_{\kappa }:[0,\infty )\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} s_{\kappa }(\theta ):= {\left\{ \begin{array}{ll} \dfrac{\sin \sqrt{\kappa }\theta }{\sqrt{\kappa }\theta }&{}\quad (\kappa >0)\\ 1 &{}\quad (\kappa = 0)\\ \dfrac{\sinh \sqrt{-\kappa }\theta }{\sqrt{-\kappa }\theta }&{}\quad (\kappa <0). \end{array}\right. } \end{aligned}$$

For $t\in [0,1]$, we define $\sigma _{\kappa }^{(t)}:[0,\omega _{\kappa })\rightarrow [0,\infty ]$ as

$$\begin{aligned} \sigma _{\kappa }^{(t)}(\theta ):= {\left\{ \begin{array}{ll} t\dfrac{s_{\kappa }(t\theta )}{s_{\kappa }(\theta )} &{}\quad (\theta \in [0,\omega _{\kappa }))\\ \infty &{}\quad (\theta \in [\omega _{\kappa },\infty )) \end{array}\right. } \end{aligned}$$

and $\tau _{K, N}^{(t)}:[0,\infty )\rightarrow [0,\infty ]$ by

$$\begin{aligned} \tau _{K, N}^{(t)}(\theta ):=t^{\frac{1}{N}}\left( \sigma _{K/(N-1)}^{(t)}(\theta )\right) ^{1-\frac{1}{N}}= {\left\{ \begin{array}{ll} t\left( \dfrac{s_{K/(N-1)}(t\theta )}{s_{K/(N-1)}(\theta )} \right) ^{1-\frac{1}{N}}&{}\quad (\theta \in [0,\omega _{K/(N-1)}))\\ \infty \ {} &{}\quad (\theta \in [\omega _{K/(N-1)},\infty )) \end{array}\right. } \end{aligned}$$

for $K\in {\mathbb {R}}$ and $N<0$. For simplicity of notation in the following, we put $\sigma _{K/N}^{(t),0}:=\sigma _{K/N}^{(1-t)}$, $\sigma _{K/N}^{(t),1}:=\sigma _{K/N}^{(t)}$, $\tau _{K, N}^{(t),0}:=\tau _{K, N}^{(1-t)}$, $\tau _{K, N}^{(t),1}:=\tau _{K, N}^{(t)}$.

We put $C_{K, N}^{(t), i}:=\sup _{\theta }\tau _{K, N}^{(t),i}\left( \theta \right) >0$ for $K\in {\mathbb {R}}$. This value varies depending on the range over which $\theta$ moves. For instance, when K is nonnegative, then we have $C_{K, N}^{(t), i}\le t<\infty$, and when K is negative and $\sup \theta <\pi \sqrt{(N-1)/K}$, then we have $C_{K, N}^{(t), i}<\infty$. In particular, we have $C_{K, N}^{(t), i}<\infty$ for any $N<0$ if $\sup \theta <\pi /\sqrt{-K}$.

Definition 3.23

(Rényi entropy) Let X be a topological space and let $\mu$ be an element in ${\mathcal {P}}(X)$. For $N<0$, we define $S_{N,\,\mu }:{\mathcal {P}}(X)\rightarrow [1,\infty ]$ by

We call it the Rényi entropy of $\nu$ with respect to $\mu$. Also, we define

$$\begin{aligned} {\mathcal {D}}(S_{N,\,\mu }):=\{\nu \in {\mathcal {P}}(X)\mid S_{N,\,\mu }(\nu )<\infty \}. \end{aligned}$$

Definition 3.24

(cf. [9, Definition 4.4] (Curvature-dimension condition)) Let $(X,d_X, \mu _X)$ be an mm-space and let $K\in {\mathbb {R}}$, $N<0$. We say that X is a ${{\,\textrm{CD}\,}}(K,N)$ space (or satisfies ${{\,\textrm{CD}\,}}(K,N)$ condition) if for any $\nu _0=\rho _0\mu _X, \nu _1=\rho _1\mu _X\in {\mathcal {P}}_2(X)\cap {\mathcal {D}}(S_{N,\,\mu _X})$, there exist a geodesic $\{\nu _t\}_{t\in [0,1]}$ in $({\mathcal {P}}_2(X),W_2)$ from $\nu _0$ to $\nu _1$ and $\pi \in {{\,\textrm{Opt}\,}}(\nu _0,\nu _1)$ such that we have

$$\begin{aligned} S_{N',\mu _X}(\nu _t)\le \sum _{i=0}^1\int _{X\times X}\tau _{K,N'}^{(t),i}\left( d_X(x_0,x_1)\right) \rho _i(x_i)^{-\frac{1}{N'}}d\pi (x_0, x_1) \end{aligned}$$

(4)

for any $t\in [0,1]$ and for any $N'\in [N,0)$ with $\nu _0, \nu _1\in {\mathcal {D}}(S_{N',\,\mu _X})$.

Definition 3.25

(cf. [9, Definition 4.5]) Reduced curvature-dimension condition) Let $(X,d_X, \mu _X)$ be an mm-space and let $K\in {\mathbb {R}}$, $N<0$. We say that X is a ${{\,\textrm{CD}\,}}^*(K,N)$ space (or satisfies ${{\,\textrm{CD}\,}}^*(K,N)$ condition) if for any $\nu _0=\rho _0\mu _X, \nu _1=\rho _1\mu _X\in {\mathcal {P}}_2(X)\cap {\mathcal {D}}(S_{N,\,\mu _X})$, there exist a geodesic $\{\nu _t\}_{t\in [0,1]}$ in $({\mathcal {P}}_2(X),W_2)$ from $\nu _0$ to $\nu _1$ and $\pi \in {{\,\textrm{Opt}\,}}(\nu _0,\nu _1)$ such that we have

$$\begin{aligned} S_{N',\mu _X}(\nu _t)\le \sum _{i=0}^1\int _{X\times X}\sigma _{K/N'}^{(t),i}\left( d_X(x_0,x_1)\right) \rho _i(x_i)^{-\frac{1}{N'}}d\pi (x_0, x_1) \end{aligned}$$

(5)

for any $t\in [0,1]$ and for any $N'\in [N,0)$ with $\nu _0, \nu _1\in {\mathcal {D}}(S_{N',\,\mu _X})$.

It is clear that ${{\,\textrm{CD}\,}}(0,N)$ condition is equivalent to ${{\,\textrm{CD}\,}}^*(0,N)$ condition for any $N<0$, and Proposition 4.7 in [9] implies that a ${{\,\textrm{CD}\,}}(K,N)$ space is a ${{\,\textrm{CD}\,}}^*(K,N)$ space for any $K\in {\mathbb {R}}$ and $N<0$.

Example 3.26

[8, Theorem 1.1] Let $n\ge 2$, $\alpha >0$, and let g be the canonical Riemannian metric on ${\mathbb {S}}^n$ induced by ${\mathbb {R}}^{n+1}$. If we define a function $\varphi :{\mathbb {S}}^n\rightarrow [0,\infty )$ for $x\in {\mathbb {R}}^{n+1}$ with $|x|<1$ by

$$\begin{aligned} \varphi (y):=\frac{c_x^{n,\alpha }}{|y-x|^{n+\alpha }}, \end{aligned}$$

where $c_x^{n,\alpha }>0$ is the normalizing constant, then, the mm-space $({\mathbb {S}}^n,d_g,\varphi {{\,\textrm{vol}\,}}_g)$, where $d_g$ is the Riemannian distance induced by g, is a ${{\,\textrm{CD}\,}}(n-1-\frac{n+\alpha }{4}, -\alpha )$ space.

Proposition 3.27

Let $t>0$. If an mm-space X is a ${{\,\textrm{CD}\,}}(K,N)$ space, tX is a ${{\,\textrm{CD}\,}}(t^{-2}K,N)$ space.

Proposition 3.28

[9, Theorem 4.10] Let $(M^n, g, e^{-f}{{\,\textrm{vol}\,}}_g)$ be a complete weighted Riemannian manifold. For any $K\in {\mathbb {R}}$ and $N<0$, the following are equivalent.

• ${{\,\textrm{Ric}\,}}_N\ge Kg$.

• $(M^n, d_g, e^{-f}{{\,\textrm{vol}\,}}_g)$ is a ${{\,\textrm{CD}\,}}(K,N)$ space.

• $(M^n, d_g, e^{-f}{{\,\textrm{vol}\,}}_g)$ is a ${{\,\textrm{CD}\,}}^*(K,N)$ space.

Definition 3.29

[9, Sect. 2.1] Let $(M^n,g)$ be an n-dimensional Riemannian manifold and let $f\in C^{\infty }(M^n)$. We say that f is (K, N)-convex for some $K\in {\mathbb {R}}$ and $N<0$ if the condition

$$\begin{aligned} {{\,\textrm{Hess}\,}}f_N\ge -\frac{K}{N}f_N\cdot g \end{aligned}$$

holds, where $f_N:=\exp (-f/N)\in C^{\infty }(M^n)$.

Proposition 3.30

[9, Corollary 4.12] Let $n\in {\mathbb {N}}$, $K_1,K_2\in {\mathbb {R}}$ and $N_1,N_2\in {\mathbb {R}}$ with $N_1\ge n$ and $N_2<-N_1$. If $(M^n,g,\mu _M)$ is an n-dimensional complete weighted Riemannian manifold with ${{\,\textrm{Ric}\,}}_{N_1}\ge K_1g$ and f is a smooth $(K_2,N_2)$-convex function on $M^n$, then $(M^n,g,e^{-f}\mu _M)$ is a ${{\,\textrm{CD}\,}}(K_1+K_2,N_1+N_2)$ space.

5 Some Results on the Estimates of the Observable Diameter

The next theorems and corollary are extensions of Proposition 9.26 and Corollary 9.27 in [11] to ${{\,\textrm{CD}\,}}(K,N)$ (and ${{\,\textrm{CD}\,}}^*(K,N)$) spaces for $N<0$.

Theorem 4.1

Let X be a ${{\,\textrm{CD}\,}}(K,N)$ space for some $K>0$ and $N<0$. For any $\kappa _0, \kappa _1>0$ with $\kappa _0+\kappa _1<1$ and for any $\kappa \in (0,1)$,

• (1) $\displaystyle {{\,\textrm{Sep}\,}}(X;\kappa _0,\kappa _1)\le 2\sqrt{\frac{1-N}{K}}\cosh ^{-1}\left( \left( \frac{\kappa _0^{1/N}+\kappa _1^{1/N}}{2}\right) ^{\frac{-N}{1-N}}\right)$,

• (2) $\displaystyle {{\,\textrm{ObsDiam}\,}}(X;-\kappa )\le 2\sqrt{\frac{1-N}{K}}\cosh ^{-1}\left( \left( 2\kappa ^{-1}\right) ^{\frac{1}{1-N}}\right)$

hold, where $\cosh ^{-1}$ is the inverse function of $\cosh$ given by $\cosh ^{-1}(x)=\log (x+\sqrt{x^2-1})$ for $x\ge 1$.

We obtain similar results for ${{\,\textrm{CD}\,}}^*(K,N)$ spaces.

Theorem 4.2

Let X be a ${{\,\textrm{CD}\,}}^*(K,N)$ space for some $K>0$ and $N<0$. For any $\kappa _0, \kappa _1>0$ with $\kappa _0+\kappa _1<1$ and for any $\kappa \in (0,1)$,

• (3) $\displaystyle {{\,\textrm{Sep}\,}}(X;\kappa _0,\kappa _1)\le 2\sqrt{\frac{-N}{K}}\cosh ^{-1}\left( \frac{\kappa _0^{1/N}+\kappa _1^{1/N}}{2}\right)$,

• (4) $\displaystyle {{\,\textrm{ObsDiam}\,}}(X;-\kappa )\le 2\sqrt{\frac{-N}{K}}\cosh ^{-1}\left( (2\kappa ^{-1})^{-\frac{1}{N}}\right)$

hold.

Since Theorem 4.2 can be proved in the same way as Theorem 4.1 by replacing $\tau _{K,N}^{(t)}$ by $\sigma _{K/N}^{(t)}$, we will not write the proof.

Proof of Theorem 4.1

(2) follows from (1) and Proposition 3.11. We prove only (1). We take any $A_0,A_1\in {\mathcal {B}}_X$ with $\mu _X(A_0)\ge \kappa _0$ and $\mu _X(A_1)\ge \kappa _1$. For $R>0$, $i=0,1$ and a fixed $a\in X$, we put

$$\begin{aligned} A_i(R)&:=A_i\cap B_X(a,R),\\ \kappa _i(R)&:=\mu _X(A_i(R)). \end{aligned}$$

Since $\kappa _i(R)$ converges to $\mu _X(A_i)>0$ as $R\rightarrow \infty$, we have $\kappa _i(R)>0$ for sufficiently large $R>0$. For such $R>0$ and $i=0,1$, we put

$$\begin{aligned} \nu _{i,R}:=\frac{{\textbf{1}}_{A_i(R)}}{\kappa _i(R)}\mu _X. \end{aligned}$$

Then, the finiteness of $S_{N,\,\mu _X}(\nu _{i,R})$ and the boundedness of $A_i(R)$ give $\nu _{i,R}\in {\mathcal {P}}_2(X)\cap {\mathcal {D}}(S_{N,\,\mu _X})$. Since X is a ${{\,\textrm{CD}\,}}(K,N)$ space, there exist a geodesic $\{\nu _{t,R}\}_{t\in [0,1]}$ on $({\mathcal {P}}_2(X),W_2)$ from $\nu _{0,R}$ to $\nu _{1,R}$ and $\pi \in {{\,\textrm{Opt}\,}}(\nu _{0,R},\nu _{1,R})$ such that for any $t\in [0,1]$,

$$\begin{aligned} S_{N,\,\mu _X}(\nu _{t,R})\le \sum _{i=0}^1\int _{X\times X}\tau _{K,N}^{(t),i}\left( d_X(x_0,x_1)\right) \left( \frac{{\textbf{1}}_{A_i(R)}(x_i)}{\kappa _i(R)}\right) ^{-\frac{1}{N}}d\pi (x_0, x_1) \end{aligned}$$

holds. If $t=1/2$, we have

$$\begin{aligned} \tau _{K,N}^{(t),0}(\theta )=\tau _{K,N}^{(t),1}(\theta )&=\frac{1}{2}\left( \dfrac{2\sinh \left( \frac{\theta }{2}\sqrt{\frac{K}{1-N}}\right) }{\sinh \sqrt{\frac{K}{1-N}}\theta }\right) ^{1-\frac{1}{N}} =\frac{1}{2}\left( \cosh \left( \frac{\theta }{2}\sqrt{\frac{K}{1-N}}\right) \right) ^{\frac{1}{N}-1} \end{aligned}$$

holds, and for any $x_0\in A_0(R), x_1\in A_1(R)$, $A_i(R)\subset A_i$ implies

$$\begin{aligned} d_X(x_0,x_1)\ge d_X(A_0(R),A_1(R))\ge d_X(A_0,A_1). \end{aligned}$$

Hence, $\pi (X\times X\setminus (A_0(R)\times A_1(R)))=0$ implies

$$\begin{aligned} 1&\le S_{N,\,\mu _X}(\nu _{1/2,R})\\&\le \sum _{i=0}^1\int _{X\times X}\tau _{K,N}^{(1/2),i} \left( d_X(x_0,x_1)\right) \left( \frac{{\textbf{1}}_{A_i(R)}(x_i)}{\kappa _i(R)} \right) ^{-\frac{1}{N}}d\pi (x_0, x_1)\\&\le \sum _{i=0}^1\int _{A_0(R)\times A_1(R)}\frac{1}{2} \left( \cosh \left( \frac{d_X(A_0,A_1)}{2}\sqrt{\frac{K}{1-N}} \right) \right) ^{\frac{1}{N}-1}\kappa _i(R)^{1/N}d\pi (x_0, x_1)\\&= \frac{\kappa _0(R)^{1/N}+\kappa _1(R)^{1/N}}{2}\left( \cosh \left( \frac{d_X(A_0,A_1)}{2}\sqrt{\frac{K}{1-N}}\right) \right) ^{\frac{1}{N}-1}. \end{aligned}$$

Using $\mu _X(A_i)^{1/N}\le \kappa _i^{1/N}$, we take $R\rightarrow \infty$ to obtain

$$\begin{aligned} 1\le \frac{\kappa _0^{1/N}+\kappa _1^{1/N}}{2}\left( \cosh \left( \frac{d_X(A_0,A_1)}{2}\sqrt{\frac{K}{1-N}} \right) \right) ^{\frac{1}{N}-1}. \end{aligned}$$

Hence, we have

$$\begin{aligned} d_X(A_0,A_1)\le 2\sqrt{\frac{1-N}{K}}\cosh ^{-1} \left( \left( \frac{\kappa _0^{1/N}+\kappa _1^{1/N}}{2}\right) ^{\frac{-N}{1-N}}\right) . \end{aligned}$$

Thus, we have (1) by the arbitrariness of $A_0$ and $A_1$. $\square$

Corollary 4.3

Let $\{K_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {R}}$ and $\{N_n\}_{n\in {\mathbb {N}}}\subset (-\infty ,0)$. Suppose that $K_n$ diverges to infinity. If a sequence of mm-spaces $\{X_n\}_{n\in {\mathbb {N}}}$ satisfies either of the following conditions, then, $\{X_n\}_{n\in {\mathbb {N}}}$ is a Lévy family.

• $X_n$ is a ${{\,\textrm{CD}\,}}(K_n,N_n)$ space.

• $X_n$ is a ${{\,\textrm{CD}\,}}^*(K_n,N_n)$ space and $K_n\cdot N_n\rightarrow -\infty$ as $n\rightarrow \infty$.

Proof

We take any $\kappa \in (0,1)$. For any sufficiently large n, we have $K_n>0$. If $X_n$ is a ${{\,\textrm{CD}\,}}(K_n,N_n)$ space, then

$$\begin{aligned} {{\,\textrm{ObsDiam}\,}}(X_n;-\kappa )&\le 2\sqrt{\frac{1-N_n}{K_n}}\cosh ^{-1} \left( \left( 2\kappa ^{-1}\right) ^{\frac{1}{1-N_n}}\right) \\&\le 2\sqrt{\frac{1-N_n}{K_n}}\cdot \sqrt{2\left( \left( 2\kappa ^{-1} \right) ^{\frac{1}{1-N_n}}-1\right) } \end{aligned}$$

holds by Theorem 4.1 (2) and $\cosh ^{-1}(x)\le \sqrt{2(x-1)}$ as $x\ge 1$. If we put $a_n=(1-N_n)^{-1}\in (0,1)$, then, we have $((2\kappa ^{-1})^{a_n}-1)/a_n<2\kappa ^{-1}-1$ since the function $x\mapsto ((2\kappa ^{-1})^x-1)/x$ is increasing by $2\kappa ^{-1}>2$. Thus, we get

$$\begin{aligned} {{\,\textrm{ObsDiam}\,}}(X_n;-\kappa )&\le \frac{2\sqrt{2}}{\sqrt{K_n}}\cdot \sqrt{\frac{\left( \left( 2\kappa ^{-1}\right) ^{a_n}-1\right) }{a_n}} \le \frac{2\sqrt{2}}{\sqrt{K_n}}\cdot \sqrt{2\kappa ^{-1}-1}. \end{aligned}$$

By $K_n\rightarrow \infty$, we have ${{\,\textrm{ObsDiam}\,}}(X_n;-\kappa )\rightarrow 0$.

On the other hand, if $X_n$ is a ${{\,\textrm{CD}\,}}^*(K_n,N_n)$ space and $K_n\cdot N_n\rightarrow -\infty$ as $n\rightarrow \infty$, then, we can assume $N_n\ge -1$ for any $n\in {\mathbb {N}}$ since the ${{\,\textrm{CD}\,}}^*(K_n,N_n)$ condition implies the ${{\,\textrm{CD}\,}}^*(K_n,(-1)\vee N_n)$ condition. Theorem 4.2 (4) and $\cosh ^{-1}(a^x)\le x\log a+\log 2$ as $a>1$ and $x>0$ imply

$$\begin{aligned} {{\,\textrm{ObsDiam}\,}}(X_n;-\kappa )&\le 2\sqrt{\frac{-N_n}{K_n}} \left( -\frac{1}{N_n}\log (2\kappa ^{-1})+\log 2\right) \\&\le \frac{2\log (2\kappa ^{-1})}{\sqrt{-K_n\cdot N_n}} +\frac{2\log 2}{\sqrt{K_n}}. \end{aligned}$$

By $K_n\rightarrow \infty$ and $K_n\cdot N_n\rightarrow -\infty$, we have ${{\,\textrm{ObsDiam}\,}}(X_n;-\kappa )\rightarrow 0$. Thus, $\{X_n\}_{n\in {\mathbb {N}}}$ is a Lévy family in both cases. $\square$

Remark 4.4

The mm-space $({\mathbb {S}}^n,d_g,\varphi {{\,\textrm{vol}\,}}_g)$ given in Example 3.26 is a ${{\,\textrm{CD}\,}}(n-1-\frac{n+\alpha }{4}, -\alpha )$ space, and by

$$\begin{aligned} \lim _{n\rightarrow \infty }K_n=\lim _{n\rightarrow \infty }\left( n-1-\frac{n+\alpha }{4}\right) =\infty , \end{aligned}$$

Corollary 4.3 implies that $\{({\mathbb {S}}^n,d_g,\varphi {{\,\textrm{vol}\,}}_g)\}_{n\in {\mathbb {N}}}$ is a Lévy family. This result can also be proved by Theorem 8.1 in [5]. In other words, this is not a new result, but it gives another proof.

6 Estimates of Entropy

In this section, we give several lemmas which are necessary for the proof of the main theorem. Since these lemmas are rewritings of the propositions in the references for the case that the entropy is replaced by the Rényi entropy for $N<0$, most of the proofs can be done in the same way. Thus, we shall give no proof or only brief outlines of the proofs.

Lemma 5.1

(cf. [7, Proposition 4.8]) Let $N'$ be a negative number. The Rényi entropy $S_{N',\cdot }(\cdot ):{\mathcal {P}}(X)\times {\mathcal {P}}(X)\rightarrow [1,\infty ]$ is lower-semicontinuous with respect to the weak convergence topology.

Lemma 5.2

(cf. [2, Proposition 4.1]) For any $N'<0, \varepsilon >0$ and $E\in (0,\infty )$, there exists $\delta >0$ such that $\nu (X\setminus {\tilde{X}})<\varepsilon$ if $\mu ,\nu \in {\mathcal {P}}(X), {\tilde{X}}\in {\mathcal {B}}_X$ satisfy $\mu (X\setminus {\tilde{X}})<\delta$ and $S_{N',\mu }(\nu )\le E$. In particular, for any $\{\mu _n\}_{n=1}^{\infty }, \{\nu _n\}_{n=1}^{\infty }\subset {\mathcal {P}}(X)$, if $\{\mu _n\}_{n=1}^{\infty }$ is tight and $\sup _{n\in {\mathbb {N}}}S_{N',\mu _n}(\nu _n)$ $<\infty$, then $\{\nu _n\}_{n=1}^{\infty }$ is also tight.

Lemma 5.3

(cf. [11, Lemma 9.15]) Let $p:X\rightarrow Y$ be a Borel map between two complete separable metric spaces X and Y. If $\mu , \nu \in {\mathcal {P}}(X)$ and $\nu \ll \mu$, then $p_*\nu \ll p_*\mu$ and

$$\begin{aligned} S_{N',\, p_*\mu }(p_*\nu )\le S_{N',\, \mu }(\nu ) \end{aligned}$$

for any $N'<0$.

Lemma 5.4

(cf. [10, Lemma 28]) Let $N'$ be a negative number. For any $\mu , \nu \in {\mathcal {P}}(X)$ and $B\in {\mathcal {B}}_X$ with $\nu (B)>0$, we have

$$\begin{aligned} \nu (B)^{1-\frac{1}{N'}}S_{N',\mu }(\nu _B)\le S_{N', \mu }(\nu ). \end{aligned}$$

Lemma 5.5

[10, Lemma 29] Let $Z_n$, Y be complete separable metric spaces and $q_n:Z_n\rightarrow Y$ be a Borel map. Suppose that $B_n\subset Z_n$, $\mu _n\in {\mathcal {P}}(Z_n)$ and $\nu \in {\mathcal {P}}(Y)$ satisfy $\mu _n(B_n)\rightarrow 1$ and $(q_n)_*\mu _n\rightharpoonup \nu$ as $n\rightarrow \infty$. Then $(q_n)_*\left( (\mu _n)_{B_n}\right)$ converges to $\nu$ weakly. If $\nu \in {\mathcal {P}}_2(Y)$ and $W_2((q_n)_*\mu _n, \nu )\rightarrow 0$, then $W_2((q_n)_*\left( (\mu _n)_{B_n}\right) , \nu )\rightarrow 0$ as $n\rightarrow \infty$.

Lemma 5.6

(cf. [10, Corollary 30]) Let $N'$ be a negative number and let $\mu , \nu$ be elements in ${\mathcal {P}}(X)$. If $\{B_n\}_{n=1}^{\infty }\subset {\mathcal {B}}_X$ satisfies $\nu (B_n)\rightarrow 1$, then, $\nu _{B_n}$ converges to $\nu$ weakly and $S_{N',\mu }(\nu _{B_n})$ converges to $S_{N',\mu }(\nu )$ as $n\rightarrow \infty$.

Lemma 5.7

(cf. [10, Lemma 31]) Let $\mu \in {\mathcal {P}}(X)$, $\nu \in {\mathcal {P}}_2(X)$ and $N'<0$. If a countable family of mutually disjoint subsets $\{B_j\}_{j\in {\mathcal {J}}}\subset {\mathcal {B}}_X$ satisfies $\nu (X\setminus \bigcup _{j\in {\mathcal {J}}}B_j)=0$ and $\mu (B_j)>0$ for any $j\in {\mathcal {J}}$, then we have

$$\begin{aligned} W_2(\nu , {\underline{\nu }})\le 2D\quad \text {and}\quad S_{N', \mu }(\nu )\ge S_{N', \mu }({\underline{\nu }}), \end{aligned}$$

where $D:=\sup _{j\in {\mathcal {J}}}{{\,\textrm{diam}\,}}B_j$ and ${\underline{\nu }}:=\sum _{j\in {\mathcal {J}}}\nu (B_j)\mu _{B_j}\in {\mathcal {P}}(X)$.

Lemma 5.8

[10, Lemma 32] Let $(X,d_X)$ be a metric space. If $\{\nu _n\}_{n=1}^{\infty }, \{\nu _n'\}_{n=1}^{\infty }\subset {\mathcal {P}}(X)$ and $\nu \in {\mathcal {P}}(X)$ satisfy $W_2(\nu _n,\nu _n')\rightarrow 0$ and $\nu _n\rightharpoonup \nu$ as $n\rightarrow \infty$, then, we have $\nu _n'\rightharpoonup \nu$.

Lemma 5.9

(cf. [10, Lemma 38]) Let X be a topological space and $N'<0$. Suppose that $\mu$, $\nu _n=\rho _n\mu$, $\nu =\rho \mu \in {\mathcal {P}}(X)$ satisfy $\nu _n\rightharpoonup \nu$ and $\limsup _{n\rightarrow \infty }S_{N', \mu }(\nu _n)\le S_{N', \mu }(\nu )<\infty$, then, we have $\int _{X}|\rho _n-\rho |d\mu \rightarrow 0$.

Lemma 5.10

[10, Lemma 42] Let $(Y,d_Y,\mu _Y)$ be an mm-space. For any $\delta >0$ and $S\subset Y$ with $\mu _Y(S)>0$, there exists a countable family of mutually disjoint subsets $\{B_j\}_{j\in {\mathcal {J}}}\subset {\mathcal {B}}_Y$ such that ${{\,\textrm{diam}\,}}B_j\le \delta$, $\mu _Y(S\cap B_j)>0$, $\mu _Y(\partial B_j)=0$ for any $j\in {\mathcal {J}}$ and $\mu _Y(S\setminus \bigcup _{j\in {\mathcal {J}}}B_j)=0$.

Definition 5.11

Let $(X,d_X,\mu _X)$ be an mm-space. We define

$$\begin{aligned} {\mathcal {P}}^{ac}(X)&:=\{\nu \in {\mathcal {P}}(X)\mid \nu \ll \mu _X\},\quad {\mathcal {P}}_2^{ac}(X):={\mathcal {P}}^{ac}(X)\cap {\mathcal {P}}_2(X),\\ {\mathcal {P}}_{cb}(X)&:=\left\{ \nu \in {\mathcal {P}}^{ac}(X)\left| {{\,\textrm{supp}\,}}\nu \text { is compact}, \frac{d\nu }{d\mu _X}\text { is bounded}\right. \right\} . \end{aligned}$$

Remark 5.12

For an mm-space $(X,d_X,\mu _X)$ and $B\in {\mathcal {B}}_X$ with $\mu _X(B)>0$, $\left( \mu _X\right) _B$ is simply denoted by $\mu _B$.

Lemma 5.13

(cf. [10, Lemma 44]) Let $\nu _0$ and $\nu _1$ be elements in ${\mathcal {P}}^{ac}(Y)$. For any $m\in {\mathbb {N}}$, let $\{B_{j,m}\}_{j\in {\mathcal {J}}'_m}$ be a countable family given by Lemma 5.10 as $\delta =m^{-1}$ and $S={{\,\textrm{supp}\,}}\nu _0 \cup {{\,\textrm{supp}\,}}\nu _1$, and let ${\mathcal {J}}_m\subset {\mathcal {J}}'_m$ be a finite set such that $\mu _Y(S\setminus \bigcup _{j\in {\mathcal {J}}_m}B_{j,m})<m^{-1}$. If we put

$$\begin{aligned} U_m:=\bigcup _{j\in {\mathcal {J}}_m}B_{j,m}\subset Y, \quad {\underline{\nu }}_i^m:=\sum _{j\in {\mathcal {J}}_m}\frac{\nu _i(B_{j, m})}{\nu _i(U_m)}\mu _{B_{j,m}}\in {\mathcal {P}}_2(Y), \end{aligned}$$

then, $\nu _i(U_m)$ converges to 1, ${\underline{\nu }}_i^m$ converges to $\nu _i$ weakly and $S_{N',\mu _Y}({\underline{\nu }}_i^m)$ converges to $S_{N',\mu _Y}(\nu _i)$ as $m\rightarrow \infty$. In addition, if $\nu _i$ is an element in ${\mathcal {P}}_2(Y)$, then, ${\underline{\nu }}_i^m$ $W_2$-converges to $\nu _i$.

Hereafter, unless otherwise noted, let $X_n$ and Y be mm-spaces, and let $p_n$, $\varepsilon _n$, ${\tilde{X}}_n$ be given by Proposition 3.16 under the assumption $X_n\xrightarrow {{\textrm{conc}}}Y$.

Lemma 5.14

(cf. [10, Lemma 35]) Suppose that $X_n$ concentrates to Y. If $\nu _n, \nu _n'\in {\mathcal {P}}_2(X_n)$ and $\nu , \nu '\in {\mathcal {P}}(Y)$ satisfy $(p_n)_*\nu _n\rightharpoonup \nu$, $(p_n)_*\nu _n'\rightharpoonup \nu '$ and $\limsup _{n\rightarrow \infty }\left( S_{N',\mu _{X_n}}(\nu _n)+S_{N',\mu _{X_n}}(\nu _n')\right) <\infty$ for some $N'<0$, then, we have

$$\begin{aligned} W_2(\nu ,\nu ')\le \liminf _{n\rightarrow \infty }W_2(\nu _n, \nu _n'). \end{aligned}$$

Lemma 5.15

([1, Lemma 3.13], [11, Lemma 9.33]) Suppose that $X_n$ concentrates to Y. For any sufficiently small $\delta >0$ and any $B_0, B_1\in {\mathcal {B}}_Y$ with

$$\begin{aligned} {{\,\textrm{diam}\,}}B_i\le \delta ,\quad \mu _Y(B_i)>0, \quad \text {and} \quad \mu _Y(\partial B_i)=0 \end{aligned}$$

for $i=0,1$, there exist $\xi _0^n, \xi _1^n\in {\mathcal {P}}(X_n)$ and $\pi _{01}^n\in {{\,\textrm{Cpl}\,}}(\xi _0^n,\xi _1^n)$ such that for all $n\in {\mathbb {N}}$ large enough, we have

• (1) $\xi _i^n\le (1+\theta _1(\delta ^{1/2}))\mu _{{\tilde{B}}_i}$   $(i=0,1)$;

• (2) $d_{X_n}({\tilde{B}}_0, {\tilde{B}}_1)\ge d_Y(B_0,B_1)-\varepsilon _n$;

• (3) ${{\,\textrm{supp}\,}}\pi _{01}^n\subset \{(x,x')\in X_n\times X_n\mid d_{X_n}(x,x')\le d_Y(B_0,B_1)+\delta ^{1/2}\}$;

• (4) $-\varepsilon _n\le W_2(\xi _0^n, \xi _1^n)-d_Y(B_0,B_1)\le \delta ^{1/2}$;

where ${\tilde{B}}_i:=p_n^{-1}(B_i)\cap {\tilde{X}}_n$ and $\theta _1$ is a function $\theta _1:[0,\infty )\rightarrow [0,\infty )$ with $\theta _1(x)\rightarrow 0$ as $x\rightarrow 0$.

Lemma 5.16

(cf. [10, Lemma 43]) Suppose that $X_n$ concentrates to Y, and let $\nu _0, \nu _1 \in {\mathcal {P}}_2^{ac}(Y)$. If the same condition in Lemma 5.13, i.e. for any $m\in {\mathbb {N}}$, let $\{B_{j,m}\}_{j\in {\mathcal {J}}'_m}$ be a countable family given by Lemma 5.10 as $\delta =m^{-1}$ and $S={{\,\textrm{supp}\,}}\nu _0 \cup {{\,\textrm{supp}\,}}\nu _1$, and let ${\mathcal {J}}_m\subset {\mathcal {J}}'_m$ be a finite set such that $\mu _Y(S\setminus \bigcup _{j\in {\mathcal {J}}_m}B_{j,m})<m^{-1}$, and we put

$$\begin{aligned} U_m:=\bigcup _{j\in {\mathcal {J}}_m}B_{j,m}\subset Y, \quad {\underline{\nu }}_i^m:=\sum _{j\in {\mathcal {J}}_m}\frac{\nu _i(B_{j, m})}{\nu _i(U_m)}\mu _{B_{j,m}}\in {\mathcal {P}}_2(Y). \end{aligned}$$

Then, for sufficiently large $n\in {\mathbb {N}}$, there exist $\nu _0^{mn}, \nu _1^{mn} \in {\mathcal {P}}(X_n)$ such that

$$\begin{aligned}{} & {} \limsup _{n\rightarrow \infty }W_2((p_n)_*\nu _i^{mn}, {\underline{\nu }}_i^m)\le \theta _2(m^{-1}),\\{} & {} \limsup _{n\rightarrow \infty }\left| W_2(\nu _0^{mn},\nu _1^{mn})-W_2({\underline{\nu }}_0^m,{\underline{\nu }}_1^m)\right| \le \theta _2(m^{-1}) \end{aligned}$$

hold for $i=0,1$. In addition, for any $N'<0$ with $\nu _0, \nu _1\in {\mathcal {D}}(S_{N',\mu _Y})$, we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }\left| S_{N',\mu _{X_n}}(\nu _i^{mn})-S_{N',\mu _Y}({\underline{\nu }}_i^m)\right| \le \theta _2(m^{-1}), \end{aligned}$$

where $\theta _2$ is a function $\theta _2:[0,\infty )\rightarrow [0,\infty )$ with $\theta _2(x)\rightarrow 0$ as $x\rightarrow 0$.

Sketch of the proof

We take any $(j,k)\in {\mathcal {J}}_m\times {\mathcal {J}}_m$. By applying Lemma 5.15 for $B_0=B_{j,m}$, $B_1=B_{k,m}$ and $\delta =m^{-1}$, we get $\xi _{jk}^{mn}\in {\mathcal {P}}(X_n)$ satisfying

$$\begin{aligned} \xi _{jk}^{mn}\le (1+\theta _1(m^{-1}))\mu _{{\tilde{B}}_{j,m}} \quad \text { and }\quad W_2(\xi _{jk}^{mn},\xi _{kj}^{mn})\le d_Y(B_{j,m},B_{k,m})+m^{-1/2}. \end{aligned}$$

Then, the tightness of $\{(p_n)_*\xi _{jk}^{mn}\}_{n\in {\mathbb {N}}}$ implies that there exists $\xi _{jk}^m\in {\mathcal {P}}(Y)$ such that $(p_n)_*\xi _{jk}^{mn}$ converges to $\xi _{jk}^m$ weakly as $n\rightarrow \infty$ for any $(j,k)\in {\mathcal {J}}_m\times {\mathcal {J}}_m$. We fix a $\pi _m\in {{\,\textrm{Opt}\,}}({\underline{\nu }}_0^m,{\underline{\nu }}_1^n)$ and put

$$\begin{aligned} w_{jk}^0&:=\pi _m(B_{j,m}\times B_{k,m}), \quad w_{jk}^1:=w_{kj}^0,\\ \nu _i^{mn}&:=\sum _{j,k\in {\mathcal {J}}_m}w_{jk}^i\xi _{jk}^{mn}\in {\mathcal {P}}_2(X_n), \quad \nu _i^m:=\sum _{j,k\in {\mathcal {J}}_m}w_{jk}^i\xi _{jk}^m\in {\mathcal {P}}_2(Y) \end{aligned}$$

for $i=0,1$. Considering $(p_n)_*\nu _i^{mn}\rightharpoonup \nu _i^m$, $S_{N',\mu _Y}({\underline{\nu }}_i^m)\le S_{N',\mu _Y}(\nu _i^m)$, etc., we can prove this lemma in the same way as in [10]. $\square$

7 Proof of Main Theorem 1.3

In this section, we give the proof of Main Theorem 1.3. For this purpose, we first give an equivalent condition of the ${{\,\textrm{CD}\,}}(K,N)$ condition.

Lemma 6.1

Let $K\in {\mathbb {R}}$, $N<0$, and let Y be an mm-space with ${{\,\textrm{diam}\,}}Y<\pi /\sqrt{-K}$ $(K<0)$. Then, the following are equivalent.

• (A) Y is a ${{\,\textrm{CD}\,}}(K, N)$ space.

• (B) For any $\nu _0=\rho _0\mu _Y$, $\nu _1=\rho _1\mu _Y\in {\mathcal {P}}_{cb}(Y)$, there exist $\{\nu _t\}_{t\in [0,1]\cap {\mathbb {Q}}}\subset {\mathcal {P}}_2(Y)$ and $\pi \in {{\,\textrm{Opt}\,}}(\nu _0, \nu _1)$ such that

  $$\begin{aligned}{} & {} W_2(\nu _s, \nu _t)\le |t-s|W_2(\nu _0, \nu _1),\\{} & {} S_{N',\mu _Y}(\nu _t)\le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1) \end{aligned}$$

  for any $s, t\in [0,1]\cap {\mathbb {Q}}$ and $N'\in [N,0)$.

Proof

(A) $\Rightarrow$ (B) is obvious by ${\mathcal {P}}_{cb}(Y)\subset {\mathcal {P}}_2(Y)\cap {\mathcal {D}}(S_{N, \mu _Y})$. Hence, it suffices to prove (B) $\Rightarrow$ (A). We take any $\nu _0=\rho _0\mu _Y$, $\nu _1=\rho _1\mu _Y\in {\mathcal {P}}_2(Y)\cap {\mathcal {D}}(S_{N, \mu _Y})$. Then, for any $n\in {\mathbb {N}}$ and $i\in \{0,1\}$, we put

$$\begin{aligned} \nu _i^n=\rho _i^n\mu _Y,\quad \rho _i^n:=(c_{i,n})^{-1}(\rho _i\wedge n){\textbf{1}}_{K_n},\quad c_{i,n}:=\int _{K_n}(\rho _i\wedge n)\,d\mu _Y, \end{aligned}$$

where $K_n$ is a compact set satisfying $K_n\subset K_{n+1}$ and $\mu _Y(Y\setminus K_n)\rightarrow 0$. Then, $\nu _i^n$ is an element in ${\mathcal {P}}_{cb}(Y)$ and $W_2$-converges to $\nu _i$ for any $i=0,1$. Now, by the assumption of (B), there exist $\{\nu _t^n\}_{t\in [0,1]\cap {\mathbb {Q}}}\subset {\mathcal {P}}_2(Y)$ and $\pi ^n\in {{\,\textrm{Opt}\,}}(\nu _0^n, \nu _1^n)$ such that

$$\begin{aligned}{} & {} W_2(\nu _s^n, \nu _t^n)\le |t-s|W_2(\nu _0^n, \nu _1^n),\\{} & {} S_{N',\mu _Y}(\nu _t^n)\le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i^n(y_i)^{-\frac{1}{N'}}d\pi ^n(y_0, y_1) \end{aligned}$$

for any $s, t\in [0,1]\cap {\mathbb {Q}}$ and for any $N'\in [N,0)$. First, from Proposition 2.4, we have $\{\pi ^n\}_{n=1}^{\infty }$ is tight, there exist $\pi \in {\mathcal {P}}(Y\times Y)$ and a subsequence $\{n_k^0\}_{k\in {\mathbb {N}}}$ such that $\pi ^{n_k^0}\rightharpoonup \pi$, and we also know that $\pi$ is an element in ${{\,\textrm{Opt}\,}}(\nu _0, \nu _1)$ by Proposition 2.7. For any $N'\in [N, 0)$ with $\nu _0, \nu _1\in {\mathcal {D}}(S_{N', \mu _Y})$,

$$\begin{aligned} S_{N', \mu _Y}(\nu _t^{n_k^0})&\le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i^{n_k^0}(y_i)^{-\frac{1}{N'}}d\pi ^{n_k^0}(y_0, y_1)\\&\le \sum _{i=0}^1\left( c_{i,n_k^0}\right) ^{\frac{1}{N'}}\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi ^{n_k^0}(y_0, y_1) \end{aligned}$$

for any $t\in [0,1]\cap {\mathbb {Q}}$. Then, the condition (b) of Lemma 2.9 holds by $\nu _i^{n_k^0}\le (c_{i,n_k^0})^{-1}\nu _i$, $c_{i,n_k^0}\rightarrow 1$ as $k\rightarrow \infty$ and $C_{K, N'}^{(t),i}<\infty$. Thus, Lemma 2.9 implies

$$\begin{aligned}&\limsup _{k\rightarrow \infty }S_{N', \mu _Y}(\nu _t^{n_k^0}) \le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0,y_1). \end{aligned}$$

Now, since this right-hand side is finite, $\sup _{k\in {\mathbb {N}}}S_{N', \mu _Y}(\nu _t^{n_k^0})$ is finite. Thus, Lemma 5.2 gives that $\{\nu _t^{n_k^0}\}_{k\in {\mathbb {N}}}$ is tight. Now, let $(0,1)\cap {\mathbb {Q}}=\{t_l\mid l\in {\mathbb {N}}\}$, then, by the above argument, $\{\nu _{t_1}^{n_k^0}\}_{k\in {\mathbb {N}}}$ will be tight. Thus, by Prokhorov’s theorem, there exist a subsequence $\{n_k^1\}_{k\in {\mathbb {N}}}\subset \{n_k^0\}_{k\in {\mathbb {N}}}$ and $\nu _{t_1}\in {\mathcal {P}}(Y)$ such that $\nu _{t_1}^{n_k^1}\rightharpoonup \nu _{t_1}$. Similarly, since $\{\nu _{t_2}^{n_k^1}\}_{k\in {\mathbb {N}}}$ is tight, there exist a subsequence $\{n_k^2\}_{k\in {\mathbb {N}}}\subset \{n_k^1\}_{k\in {\mathbb {N}}}$ and $\nu _{t_2}\in {\mathcal {P}}(Y)$ such that $\nu _{t_2}^{n_k^2}\rightharpoonup \nu _{t_2}$. By repeating this operation, we obtain $\{n_k^l\}_{k,l\in {\mathbb {N}}}$ and $\{\nu _{t_l}\}_{l\in {\mathbb {N}}}\cup \{\nu _0,\nu _1\}=\{\nu _t\}_{t\in [0,1]\cap {\mathbb {Q}}}$, while $n_k:=n_k^k$, we obtain that for any $t\in [0,1]\cap {\mathbb {Q}}$, $\nu _t^{n_k}$ converges to $\nu _t$ weakly. Then, for any $s,t\in [0,1]\cap {\mathbb {Q}}$ and $N'\in [N, 0)$ with $\nu _0, \nu _1\in {\mathcal {D}}(S_{N', \mu _Y})$

$$\begin{aligned} W_2(\nu _s, \nu _t)&\le \liminf _{k\rightarrow \infty }W_2(\nu _s^{n_k}, \nu _t^{n_k})\qquad (\text {Lemma}~2.7)\\&\le \liminf _{k\rightarrow \infty }|t-s|W_2(\nu _0^{n_k}, \nu _1^{n_k})\\&= |t-s|W_2(\nu _0, \nu _1) \end{aligned}$$

and

$$\begin{aligned} S_{N', \mu _Y}(\nu _t)&\le \liminf _{k\rightarrow \infty }S_{N', \mu _Y}(\nu _t^{n_k})\quad (\text {Lemma}~5.1)\\&\le \limsup _{k\rightarrow \infty }S_{N', \mu _Y}(\nu _t^{n_k^0})\\&\le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0,y_1) \end{aligned}$$

hold. If we consider $\nu$ as a map $\nu :[0,1]\cap {\mathbb {Q}}\rightarrow {\mathcal {P}}_2(Y); t\mapsto \nu _t$, then $\nu$ is $W_2(\nu _0, \nu _1)$-Lipschitz. Since $({\mathcal {P}}_2(Y), W_2)$ is a complete metric space, there is a unique extension of $\nu$ to $\overline{[0,1]\cap {\mathbb {Q}}}=[0,1]$ by Proposition 2.1. This extension $\nu$ is a geodesic from $\nu _0$ to $\nu _1$ on $({\mathcal {P}}_2(Y), W_2)$. For any $t\in [0,1]$, by the definition of $\nu$, there exists $\{t_k\}_{k\in {\mathbb {N}}}\subset {\mathbb {Q}}$ such that $t_k\rightarrow t$ and $W_2(\nu _{t_k},\nu _t)\rightarrow 0$ as $k\rightarrow \infty$. Hence, the continuity of $\tau _{K, N'}^{(t)}$ for t and Lebesgue’s convergence theorem imply

$$\begin{aligned} S_{N', \mu _Y}(\nu _t)&\le \liminf _{k\rightarrow \infty }S_{N', \mu _Y}(\nu _{t_k})\\&\le \limsup _{k\rightarrow \infty }\sum _{i=0}^1\int _{Y\times Y}\tau _{K, N' }^{(t_k),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1)\\&= \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1) \right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1). \end{aligned}$$

Thus Y is a ${{\,\textrm{CD}\,}}(K, N)$ space i.e. we have the condition (A). $\square$

Lemma 6.1 shows that it is sufficient to prove the condition (B) instead of proving the ${{\,\textrm{CD}\,}}(K,N)$ condition.

Proof of Main Theorem 1.3

We can assume $\sup _{n\in {\mathbb {N}}}{{\,\textrm{diam}\,}}X_n<\pi /\sqrt{-K}$ as $K<0$ by the assumption (1) and taking a subsequence first. Hence, this assumption gives ${{\,\textrm{diam}\,}}Y<\pi /\sqrt{-K}$ and $C_{K,N'}^{(t)}<\infty$ for any $N'<0$ by Proposition 3.17. We take any $\nu _0, \nu _1\in {\mathcal {P}}_{cb}(Y)$. For this $\nu _i$ and sufficiently large $m,n\in {\mathbb {N}}$, let $\nu _i^{mn}, {\underline{\nu }}_i^m$ be families of measures constructed in Lemma 5.16. Since $\nu _i^{mn}$ is an element in ${\mathcal {P}}_2(X_n)\cap {\mathcal {D}}(S_{N,\,\mu _Y})$ and $X_n$ satisfies the ${{\,\textrm{CD}\,}}(K,N)$ condition, there are a geodesic $\{\nu _t^{mn}\}_{t\in [0,1]}\subset {\mathcal {P}}_2(X_n)$ from $\nu _0^{mn}$ to $\nu _1^{mn}$ on $({\mathcal {P}}_2(X_n), W_2)$ and $\pi ^{mn}\in {{\,\textrm{Opt}\,}}(\nu _0^{mn}, \nu _1^{mn})$ such that

$$\begin{aligned} S_{N',\mu _{X_n}}(\nu _t^{mn})\le \sum _{i=0}^1\int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1)\right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1) \end{aligned}$$

for any $t\in [0,1]$ and $N'\in [N,0)$. Then, Lemmas 5.16 and 5.13 imply

$$\begin{aligned}{} & {} \limsup _{n\rightarrow \infty }\left| W_2(\nu _0^{mn},\nu _1^{mn})-W_2({\underline{\nu }}_0^m,{\underline{\nu }}_1^m)\right| \le \theta _2(m^{-1}),\nonumber \\{} & {} \limsup _{n\rightarrow \infty }\left| S_{N',\mu _{X_n}}(\nu _i^{mn})-S_{N',\mu _Y}({\underline{\nu }}_i^m)\right| \le \theta _2(m^{-1})\qquad (i=0,1),\end{aligned}$$

(6)

$$\begin{aligned}{} & {} S_{N',\mu _Y}({\underline{\nu }}_i^m)\rightarrow S_{N',\mu _Y}(\nu _i)<\infty \qquad (m\rightarrow \infty ),\\{} & {} W_2({\underline{\nu }}_i^m, \nu _i)\rightarrow 0\qquad (m\rightarrow \infty )\nonumber \end{aligned}$$

(7)

and we have

$$\begin{aligned}&\limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }W_2(\nu _0^{mn},\nu _1^{mn})\\&\quad \le \limsup _{m\rightarrow \infty }\left( W_2({\underline{\nu }}_0^m,{\underline{\nu }}_1^m) +\theta _2(m^{-1})\right) \\&\quad =W_2(\nu _0,\nu _1). \end{aligned}$$

Then, since $(p_n\times p_n)_*\pi ^{mn}$ is a coupling of $(p_n)_*\nu _0^{mn}$ and $(p_n)_*\nu _1^{mn}$ and $(p_n)_*\nu _i^{mn}$ converges to $\nu _i^m$ weakly, $\{(p_n\times p_n)_*\pi ^{mn}\}_{n\in {\mathbb {N}}}$ is tight for each $m\in {\mathbb {N}}$. $(p_n\times p_n)_*\pi ^{mn}$ converges to some $\pi ^m\in {{\,\textrm{Cpl}\,}}(\nu _0^m, \nu _1^m)$ weakly by taking a subsequence. Furthermore, since $\{\pi ^m\}_{m\in {\mathbb {N}}}$ is tight by $\nu _i^m \rightharpoonup \nu _i$, $\pi ^m$ converges to some $\pi \in {{\,\textrm{Cpl}\,}}(\nu _0, \nu _1)$ by taking a subsequence. Then,

$$\begin{aligned} \int _{Y\times Y}d_Y(y,y')^2d\pi (y,y')&\le \liminf _{m\rightarrow \infty }\liminf _{n\rightarrow \infty }\int _{Y\times Y}d_Y(y,y')^2d((p_n\times p_n)_*\pi ^{mn})(y,y')\\&=\liminf _{m\rightarrow \infty }\liminf _{n\rightarrow \infty }\int _{{\tilde{X}}_n\times {\tilde{X}}_n}d_Y(p_n(x),p_n(x'))^2d\pi ^{mn}(x,x')\\&\le \liminf _{m\rightarrow \infty }\liminf _{n\rightarrow \infty }\int _{{\tilde{X}}_n\times {\tilde{X}}_n}(d_{X_n}(x,x')+\varepsilon _n)^2d\pi ^{mn}(x,x')\\&\le \liminf _{m\rightarrow \infty }\liminf _{n\rightarrow \infty }\left( W_2(\nu _0^{mn},\nu _1^{mn})+\varepsilon _n\right) ^2\\&\le W_2(\nu _0,\nu _1)^2 \end{aligned}$$

implies that $\pi$ is an optimal coupling. Here, the following claim holds.

Claim 6.2

Let m, n be the above subsequences. Then

$$\begin{aligned}&\limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\sum _{i=0}^1\int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1)\right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\\&\quad \le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1) \end{aligned}$$

holds.

The proof of this claim is given after the proof of the theorem. This claim and Lemma 5.3 imply

$$\begin{aligned}&\limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }S_{N', (p_n)_*\mu _{X_n}}((p_n)_*\nu _t^{mn})\nonumber \\&\quad \le \limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }S_{N',\mu _{X_n}}(\nu _t^{mn})\nonumber \\&\quad \le \limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\sum _{i=0}^1\int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1)\right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\nonumber \\&\quad \le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1)<\infty \end{aligned}$$

(8)

for any $t\in [0,1]$. Hence Lemma 5.2 implies that $\{(p_n)_*\nu _t^{mn}\}_{m,n\in {\mathbb {N}}}$ is tight for any $t\in [0,1]$ because $\{(p_n)_*\mu _{X_n}\}_{n\in {\mathbb {N}}}$ is tight. Here, we label rational numbers in (0, 1) as $\{t_l \mid l\in {\mathbb {N}}\}$ and fix any $m\in {\mathbb {N}}$. Since $\{(p_n)_*\nu _{t_1}^{mn}\}_{n\in {\mathbb {N}}}$ is tight by Inequality (8), Prokhorov’s theorem implies that there are a subsequence $\{n_k^1\}_{k\in {\mathbb {N}}}$ and $\nu _{t_1}^m\in {\mathcal {P}}(Y)$ such that $(p_{n_k^1})_*\nu _{t_1}^{mn_k^1}\rightharpoonup \nu _{t_1}^m$ as $k\rightarrow \infty$. By a diagonal argument, we can get a subsequence $\{n_k\}_{k\in {\mathbb {N}}}$ and $\{\nu _{t_l}^m\}_{l\in {\mathbb {N}}}\cup \{\nu _0^m,\nu _1^m\}=\{\nu _t^m\}_{t\in [0,1]\cap {\mathbb {Q}}}$ such that $(p_{n_k})_*\nu _t^{mn_k}\rightharpoonup \nu _t^m$ for any $t\in [0,1]\cap {\mathbb {Q}}$.

Similarly, by Lemma 5.1, Lemma 5.2 and formula (8), $\{\nu _t^m\}_{m\in {\mathbb {N}}}$ is tight for any $t\in [0,1]\cap {\mathbb {Q}}$. Hence, there are a subsequence $\{m_k^1\}_{k\in {\mathbb {N}}}$ and $\nu _{t_1}\in {\mathcal {P}}(Y)$ such that $\nu _{t_1}^{m_k^1}\rightharpoonup \nu _{t_1}$ as $k\rightarrow \infty$. By a diagonal argument, we can get a subsequence $\{m_k\}_{k\in {\mathbb {N}}}$ and $\{\nu _{t_l}\}_{l\in {\mathbb {N}}}\cup \{\nu _0,\nu _1\}=\{\nu _t\}_{t\in [0,1]\cap {\mathbb {Q}}}$ such that $\nu _t^{m_k}\rightharpoonup \nu _t$ for any $\nu _t^{m_k}\rightharpoonup \nu _t$. By Lemma 5.1 and Lemma 5.14, for any $s,t\in [0,1]\cap {\mathbb {Q}}$ and $N'\in [N, 0)$,

$$\begin{aligned} S_{N', \mu _Y}(\nu _t)&\le \liminf _{k'\rightarrow \infty }\liminf _{k\rightarrow \infty }S_{N', (p_{n_k})_*\mu _{X_{n_k}}}((p_{n_k})_*\nu _t^{m_{k'}n_k})\\&\le \limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }S_{N', (p_n)_*\mu _{X_n}}((p_n)_*\nu _t^{mn})\\&\le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1)<\infty \end{aligned}$$

and

$$\begin{aligned} W_2(\nu _s, \nu _t)&\le \liminf _{k'\rightarrow \infty }\liminf _{k\rightarrow \infty }W_2(\nu _s^{m_{k'}n_k}, \nu _t^{m_{k'}n_k})\\&\le |t-s|\limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }W_2(\nu _0^{mn},\nu _1^{mn})\\&\le |t-s|W_2(\nu _0,\nu _1) \end{aligned}$$

hold. Thus Y satisfies ${{\,\textrm{CD}\,}}(K,N)$ condition. $\square$

Finally, we give the proof of Claim 6.2. Before that, we first prove a simple lemma.

Lemma 6.3

Let q be a positive number. For any $\varepsilon >0$ and $M>0$, there exists $C(M,\varepsilon )>0$ such that

$$\begin{aligned} |x^q-y^q|\le C(M,\varepsilon )|x-y|+\varepsilon \end{aligned}$$

for any $x,y\in [0,M]$.

Proof

We take any $\varepsilon >0$ and $M>0$. Since the function $x\mapsto x^q$ is uniformly continuous on [0, M], there exists $\delta >0$ such that any $x,y\in [0,M]$ with $|x-y|<\delta$ satisfy $|x^q-y^q|<\varepsilon$. We put

$$\begin{aligned} C(M,\varepsilon ):=\left( \frac{M^q-\varepsilon }{\delta }\right) \vee 1 \end{aligned}$$

and take any $x,y\in [0,M]$. Then,

$$\begin{aligned} |x^q-y^q|<\varepsilon \le C(M,\varepsilon )|x-y|+\varepsilon \end{aligned}$$

if $|x-y|<\delta$, and otherwise

$$\begin{aligned} |x^q-y^q|\le M^q\le C(M,\varepsilon )\cdot \delta +\varepsilon \le C(M,\varepsilon )|x-y|+\varepsilon \end{aligned}$$

holds. $\square$

Proof of Claim 6.2

First, we consider the case where $K=0$. Then, we have

$$\begin{aligned}&\sum _{i=0}^1\int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1)\right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\\&\quad =(1-t)S_{N',\mu _{X_n}}(\nu _0^{mn})+ tS_{N',\mu _{X_n}}(\nu _1^{mn}) \end{aligned}$$

and

$$\begin{aligned}&\sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1)\\&\quad =(1-t)S_{N',\mu _Y}(\nu _0)+ tS_{N',\mu _Y}(\nu _1). \end{aligned}$$

Therefore, by (6) and (7), we obtain

$$\begin{aligned}&\limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\left( (1-t)S_{N',\mu _{X_n}}(\nu _0^{mn})+ tS_{N',\mu _{X_n}}(\nu _1^{mn})\right) \\&\quad \le \limsup _{m\rightarrow \infty }\left( (1-t)S_{N',\mu _Y}({\underline{\nu }}_i^m)+ tS_{N',\mu _Y}({\underline{\nu }}_i^m)+\theta _2(m^{-1})\right) \\&\quad =(1-t)S_{N',\mu _Y}(\nu _0)+ tS_{N',\mu _Y}(\nu _1). \end{aligned}$$

Thus, Claim 6.2 is proved as $K=0$.

Next, we consider the case where $K\ne 0$. For any sufficiently small $\eta >0$, we put the set $A_n^{\eta }$ as

$$\begin{aligned} A_n^{\eta }:= \{(x_0, x_1)\in X_n\times X_n\mid ({{\,\textrm{sgn}\,}}K)\cdot (d_Y(p_n(x_0), p_n(x_1))-d_{X_n}(x_0, x_1))> \eta \}, \end{aligned}$$

where ${{\,\textrm{sgn}\,}}K$ is the sign of K. Since $A_n^{\eta }$ is contained in $(X_n\times X_n)\setminus ({\tilde{X}}_n\times {\tilde{X}}_n)$ as $K>0$ for any n such that $\varepsilon _n\le \eta$,

$$\begin{aligned} \limsup _{n\rightarrow \infty }\pi ^{mn}(A_n^{\eta })=0 \end{aligned}$$

holds. On the other hand, using

$$\begin{aligned}&\int _{X_n\times X_n}d_{X_n}(x_0, x_1)\,d\pi ^{mn}(x_0,x_1) \le W_2(\nu _0^{mn}, \nu _1^{mn}) \end{aligned}$$

and

$$\begin{aligned} W_2(\nu _0^m,\nu _1^m)^2&\le \int _{Y\times Y}d_Y(y_0, y_1)^2d\pi ^m(y_0,y_1)\\&\le \liminf _{n\rightarrow \infty }\int _{Y\times Y}d_Y(y_0, y_1)^2d((p_n\times p_n)_*\pi ^{mn})(y_0,y_1), \end{aligned}$$

we get

$$\begin{aligned} \pi ^{mn}(A_n^{\eta })&=\int _{A_n^{\eta }}d\pi ^{mn}\\&<\frac{1}{\eta ^2}\int _{A_n^{\eta }}(d_{X_n}(x_0, x_1)-d_Y(p_n(x_0), p_n(x_1)))^2d\pi ^{mn}(x_0,x_1)\\&\le \frac{1}{\eta ^2}\int _{A_n^{\eta }}\left( d_{X_n}(x_0, x_1)^2-d_Y(p_n(x_0), p_n(x_1))^2\right) d\pi ^{mn}(x_0,x_1)\\&=\frac{1}{\eta ^2}\int _{X_n\times X_n}\left( d_{X_n}(x_0, x_1)^2-d_Y(p_n(x_0), p_n(x_1))^2\right) d\pi ^{mn}(x_0,x_1)\\&\quad +\frac{1}{\eta ^2}\int _{(X_n\times X_n)\setminus A_n^{\eta }}\left( d_Y(p_n(x_0), p_n(x_1))^2-d_{X_n}(x_0, x_1)^2\right) d\pi ^{mn}(x_0,x_1)\\&\le \frac{1}{\eta ^2}\left( W_2(\nu _0^{mn}, \nu _1^{mn})^2-\int _{Y\times Y}d_Y(y_0, y_1)^2d((p_n\times p_n)_*\pi ^{mn})(y_0,y_1)\right) \\&\quad +\frac{1}{\eta ^2}\int _{X_n\times X_n}\varepsilon _n\left( 2\cdot d_{X_n}(x_0, x_1)+\varepsilon _n\right) d\pi ^{mn}(x_0,x_1) \end{aligned}$$

as $K<0$. Hence,

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\pi ^{mn}(A_n^{\eta })\\&\quad \le \limsup _{n\rightarrow \infty }\frac{1}{\eta ^2}\left( W_2(\nu _0^{mn}, \nu _1^{mn})^2-\int _{Y\times Y}d_Y(y_0, y_1)^2d((p_n\times p_n)_*\pi ^{mn})(y_0,y_1)\right) \\&\qquad +\limsup _{n\rightarrow \infty }\frac{\varepsilon _n}{\eta ^2}\left( 2\cdot W_2(\nu _0^{mn}, \nu _1^{mn})+\varepsilon _n\right) \\&\quad \le \frac{1}{\eta ^2}\left( (W_2({\underline{\nu }}_0^m, {\underline{\nu }}_1^m)+\theta _2(m^{-1}))^2-\liminf _{n\rightarrow \infty }\int _{Y\times Y}d_Y(\cdot , \cdot )^2d((p_n\times p_n)_*\pi ^{mn})\right) \\&\quad \le \frac{1}{\eta ^2}\left( (W_2({\underline{\nu }}_0^m, {\underline{\nu }}_1^m)+\theta _2(m^{-1}))^2-W_2(\nu _0^m,\nu _1^m)^2\right) \end{aligned}$$

holds and if we define $\theta _3$ as $\theta _3(m^{-1}):=(W_2({\underline{\nu }}_0^m, {\underline{\nu }}_1^m)+\theta _2(m^{-1}))^2-W_2(\nu _0^m,\nu _1^m)^2$, $\theta _3(m^{-1})$ converges to 0 as $m\rightarrow \infty$ by $W_2({\underline{\nu }}_0^m, {\underline{\nu }}_1^m), W_2(\nu _0^m, \nu _1^m)\rightarrow W_2(\nu _0, \nu _1)$. Thus we have

$$\begin{aligned} \lim _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\pi ^{mn}(A_n^{\eta })=0. \end{aligned}$$

Here, we put the function $d_{\eta }:Y\times Y\rightarrow [0,\infty )$ with

$$\begin{aligned} d_{\eta }(y_0,y_1):= {\left\{ \begin{array}{ll} (d_Y(y_0,y_1)-\eta )\vee 0&{}\quad (K>0)\\ d_Y(y_0,y_1)+\eta &{}\quad (K<0). \end{array}\right. } \end{aligned}$$

Then, for any $(x_0, x_1)\in (X_n\times X_n)\setminus A_n^{\eta }$,

$$\begin{aligned} \tau _{K,N'}^{(t),i}\left( d_{X_n}(x_0, x_1)\right) \le \tau _{K,N'}^{(t),i}\left( d_{\eta }(p_n(x_0), p_n(x_1))\right) \le C_{K,N'}^{(t),i}<\infty \end{aligned}$$

holds. We put

$$\begin{aligned} C_{m,n}':=\max _{j\in {\mathcal {J}}_m}\frac{\mu _Y(B_{j,m})}{\mu _{X_n}({\tilde{B}}_{j,m})} \end{aligned}$$

and, $C_{m,n}'$ converges to 1 as $n\rightarrow \infty$ by $\mu _{X_n}({\tilde{B}}_{j,m})\rightarrow \mu _Y(B_{j,m})$. Also, we put

$$\begin{aligned} {\tilde{U}}_m:=\bigcup _{j\in {\mathcal {J}}_m}{\tilde{B}}_{j,m} \end{aligned}$$

and it satisfies $\pi ^{mn}({\tilde{U}}_m)=1$ and

$$\begin{aligned} {\underline{\rho }}_i^m\circ p_n=\sum _{j\in {\mathcal {J}}_m}\frac{{\underline{\nu }}_i^m(B_{j,m})}{\mu _Y(B_{j,m})}({\textbf{1}}_{B_{j,m}}\circ p_n)=\sum _{j\in {\mathcal {J}}_m}\frac{{\underline{\nu }}_i^m(B_{j,m})}{\mu _Y(B_{j,m})}{\textbf{1}}_{{\tilde{B}}_{j,m}} \end{aligned}$$

on ${\tilde{U}}_m$. Hence,

$$\begin{aligned} \rho _i^{mn}\le \sum _{j,k\in {\mathcal {J}}_m}w_{jk}^i\frac{1+\theta _1(m^{-1})}{\mu _{X_n}({\tilde{B}}_{j,m})}{\textbf{1}}_{{\tilde{B}}_{j,m}}\le (1+\theta _1(m^{-1}))\cdot C_{m,n}'\cdot {\underline{\rho }}_i^m\circ p_n \end{aligned}$$

holds. By $\nu _i(B_{j,m})\le \Vert \rho _i\Vert _{\infty }\cdot \mu _Y(B_{j,m})$,

$$\begin{aligned} {\underline{\rho }}_i^m\le \frac{\Vert \rho _i\Vert _{\infty }}{\nu _i(U_m)} \end{aligned}$$

holds. As we put $C_{m,n}:=\left( (1+\theta _1(m^{-1}))C_{m,n}'\right) ^{-\frac{1}{N'}}$ and $B_{j,m,i}:={{\,\textrm{proj}\,}}_i^{-1}(B_{j,m})$

$$\begin{aligned}&\sum _{i=0}^1\int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1)\right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\\&\quad \le C_{m,n}\sum _{i=0}^1\int _{(X_n\times X_n)\setminus A_n^{\eta }}\tau _{K, N'}^{(t),i}\left( d_{\eta }(p_n(x_0), p_n(x_1))\right) {\underline{\rho }}_i^m(p_n(x_i))^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\\&\qquad +C_{m,n}\sum _{i=0}^1\int _{A_n^{\eta }}C_{K, N'}^{(t),i}\cdot \left( \frac{\Vert \rho _i\Vert _{\infty }}{\nu _i(U_m)}\right) ^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\\&\quad \le C_{m,n}\sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_{\eta }(y_0, y_1)\right) {\underline{\rho }}_i^m(y_i)^{-\frac{1}{N'}}d((p_n\times p_n)_*\pi ^{mn})(y_0, y_1)\\&\qquad +C_{m,n}\sum _{i=0}^1C_{K, N'}^{(t),i}\cdot \left( \frac{\Vert \rho _i\Vert _{\infty }}{\nu _i(U_m)}\right) ^{-\frac{1}{N'}}\cdot \pi ^{mn}(A_n^{\eta }) \end{aligned}$$

holds. Then, $\tau _{K, N'}^{(t),i}\circ d_{\eta }$ is bounded continuous on $Y\times Y$, ${\underline{\rho }}_i^m$ is a simple function, and $\nu _i^m(\partial B_{j,m})=0$ holds by $\nu _i^m\ll \mu _Y$ and $\mu _Y(\partial B_{j,m})=0$. Hence, since the condition (a) of Lemma 2.9 holds, as we put the constant $C_{m,N'}:=\left( 1+\theta _1(m^{-1})\right) ^{-\frac{1}{N'}}$, Lemma 2.9 gives

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\sum _{i=0}^1\int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1)\right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\nonumber \\&\quad \le C_{m,N'}\sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_{\eta }(y_0, y_1)\right) {\underline{\rho }}_i^m(y_i)^{-\frac{1}{N'}}d\pi ^m(y_0, y_1)\nonumber \\&\qquad +C_{m,N'}\cdot \frac{\theta _3(m^{-1})}{\eta ^2}\sum _{i=0}^1C_{K, N'}^{(t),i}\cdot \left( \frac{\Vert \rho _i\Vert _{\infty }}{\nu _i(U_m)}\right) ^{-\frac{1}{N'}}\nonumber \\&\quad \le C_{m,N'}\sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i}\left( d_{\eta }(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi ^m(y_0, y_1) \end{aligned}$$

(9)

$$\begin{aligned}&\qquad +C_{m,N'}\sum _{i=0}^1C_{K,N'}^{(t),i}\int _Y\left| ({\underline{\rho }}_i^m)^{-\frac{1}{N'}}-(\rho _i)^{-\frac{1}{N'}}\right| d\nu _i^m \end{aligned}$$

(10)

$$\begin{aligned}&\qquad +C_{m,N'}\cdot \frac{\theta _3(m^{-1})}{\eta ^2}\sum _{i=0}^1C_{K, N'}^{(t),i}\cdot \left( \frac{\Vert \rho _i\Vert _{\infty }}{\nu _i(U_m)}\right) ^{-\frac{1}{N'}}. \end{aligned}$$

(11)

Finally, we estimate these terms (9), (10) and (11). We take any $\varepsilon >0$ and there exists $M>0$ such that $\rho _i, {\underline{\rho }}_i^m\le M$ for any sufficiently large m and $i=0,1$ by $\nu _i(U_m)\rightarrow 1$. Let $C(M,\varepsilon )>0$ be the constant given by Lemma 6.3 as $q=-\frac{1}{N'}$. Then, Lemma 5.9 gives

$$\begin{aligned} \int _Y|{\underline{\rho }}_i^m-\rho _i|d\mu _Y,\quad \int _Y|\rho _i^m-\rho _i|d\mu _Y\rightarrow 0\qquad (m\rightarrow \infty ) \end{aligned}$$

because ${\underline{\nu }}_i^m$ and $\nu _i^m$ converge to $\nu _i$ weakly and the entropies of these also converge to $S_{N',\mu _Y}(\nu _i)$. Hence

$$\begin{aligned}&\int _Y\left| ({\underline{\rho }}_i^m)^{-\frac{1}{N'}}-(\rho _i)^{-\frac{1}{N'}}\right| d\nu _i^m\\&\quad \le \int _Y\left| ({\underline{\rho }}_i^m)^{-\frac{1}{N'}}-(\rho _i)^{-\frac{1}{N'}}\right| \cdot \rho _id\mu _Y +\int _Y\left| ({\underline{\rho }}_i^m)^{-\frac{1}{N'}}-(\rho _i)^{-\frac{1}{N'}}\right| \cdot |\rho _i^m-\rho _i|d\mu _Y\\&\quad \le \Vert \rho _i\Vert _{\infty }\left( C(M,\varepsilon )\int _Y\left| {\underline{\rho }}_i^m-\rho _i\right| d\mu _Y+\varepsilon \right) +M^{-\frac{1}{N'}}\int _Y|\rho _i^m-\rho _i|d\mu _Y \end{aligned}$$

holds about (10). Thus, taking the limit of this inequality as $m\rightarrow \infty$ and $\varepsilon \rightarrow 0$, we have

$$\begin{aligned}&\limsup _{m\rightarrow \infty }C_{m,N'}\sum _{i=0}^1C_{K,N'}^{(t),i}\int _Y \left| ({\underline{\rho }}_i^m)^{-\frac{1}{N'}}-(\rho _i)^{-\frac{1}{N'}}\right| d\nu _i^m\\&\quad \le \lim _{\varepsilon \rightarrow 0}\sum _{i=0}^1C_{K,N'}^{(t),i}\cdot \Vert \rho _i\Vert _{\infty }\cdot \varepsilon =0. \end{aligned}$$

Next, about (9), since the condition (c) of Lemma 2.9 holds, Lemma 2.9 implies

$$\begin{aligned}&\limsup _{m\rightarrow \infty }C_{m,N'}\sum _{i=0}^1\int _{Y\times Y} \tau _{K, N'}^{(t),i}\left( d_{\eta }(y_0,y_1)\right) \rho _i (y_i)^{-\frac{1}{N'}}d\pi ^m(y_0, y_1)\\&\quad \le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i} \left( d_{\eta }(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1). \end{aligned}$$

Finally, (11) satisfies

$$\begin{aligned} \limsup _{m\rightarrow \infty }C_{m,N'}\cdot \frac{\theta _3(m^{-1})}{\eta ^2} \sum _{i=0}^1C_{K, N'}^{(t),i}\cdot \left( \frac{\Vert \rho _i\Vert _{\infty }}{\nu _i(U_m)}\right) ^{-\frac{1}{N'}}=0. \end{aligned}$$

Thus, we obtain

$$\begin{aligned}&\limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\sum _{i=0}^1 \int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1) \right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\\&\quad \le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i} \left( d_{\eta }(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1). \end{aligned}$$

Since $d_{\eta }(y_0,y_1)$ converges to $d_Y(y_0,y_1)$ as $\eta \rightarrow 0$, by Lebesgue’s convergence theorem, we obtain

$$\begin{aligned}&\limsup _{m\rightarrow \infty }\limsup _{n\rightarrow \infty }\sum _{i=0}^1 \int _{X_n\times X_n}\tau _{K, N'}^{(t),i}\left( d_{X_n}(x_0,x_1) \right) \rho _i^{mn}(x_i)^{-\frac{1}{N'}}d\pi ^{mn}(x_0, x_1)\\&\quad \le \sum _{i=0}^1\int _{Y\times Y}\tau _{K, N'}^{(t),i} \left( d_Y(y_0,y_1)\right) \rho _i(y_i)^{-\frac{1}{N'}}d\pi (y_0, y_1) \end{aligned}$$

i.e. we get Claim 6.2. $\square$

Remark 6.4

As noted in Remark 1.4, if $K\ge 0$, Main Theorem 1.3 for ${{\,\textrm{CD}\,}}^*(K,N)$ spaces can also be proved by replacing $C_{K,N'}^{(t),i}$ by $\sup _{\theta \in [0,\infty )}\sigma _{K/N'}^{(t),i}(\theta )<\infty$.

Theorem 1.3 can imply the following theorem.

Theorem 6.5

(cf. [1, Corollary 1.4]) Let N be a negative number. Suppose that a sequence of compact Riemannian manifolds $\{M_n\}_{n\in {\mathbb {N}}}$ satisfies following two conditions:

• (d) $M_n$ is a ${{\,\textrm{CD}\,}}(0,N)$ space.

• (e) $\displaystyle \lim _{n\rightarrow \infty }\lambda _k(M_n)=\infty$ for some $k\in {\mathbb {N}}$.

Then, $\{M_n\}_{n\in {\mathbb {N}}}$ is a Lévy family.

Proof

By the condition (e) and Remark 3.20, $\{M_n\}_{n\in {\mathbb {N}}}$ is a k-Lévy family. Hence, we have only one of (1) or (2) of Proposition 3.21. If (2) holds, there exist a subsequence $\{n_i\}_{i\in {\mathbb {N}}}$, a sequence $\{t_i\}_{i\in {\mathbb {N}}}\subset (0,1]$ and an mm-space Y with $|Y|\in \{2,\ldots ,k\}$ such that $t_iM_{n_i}\xrightarrow {{\textrm{conc}}}Y$ holds. Then, $t_iM_{n_i}$ is a ${{\,\textrm{CD}\,}}(0,N)$ space by the condition (d) and Proposition 3.27. Hence, Main Theorem 1.3 implies that Y is ${{\,\textrm{CD}\,}}(0,N)$. However, since any finite mm-space except $*$ is not a ${{\,\textrm{CD}\,}}(0,N)$ space, it is a contradiction. Thus, we obtain (1) of Proposition 3.21, i.e., $\{M_n\}_{n\in {\mathbb {N}}}$ is a Lévy family. $\square$

8 Proof of Main Theorem 1.5

Next, we give the proof of Main Theorem 1.5. Before we do so, we define a function and prove its properties.

Definition 7.1

Let $a>0$. We define a function $F_a:{\mathbb {R}}\rightarrow (0,\infty )$ by

$$\begin{aligned} F_a(x):=a^{-1}\log \left( e^{ax}+e^{-ax}\right) . \end{aligned}$$

This function $F_a$ satisfies

$$\begin{aligned} |x|<F_a(x)\le |x|+a^{-1}\log 2 \end{aligned}$$

for any $a>0$ and $x\in {\mathbb {R}}$ and this implies $F_a(x)\rightarrow |x|$ as $a\rightarrow \infty$.

Lemma 7.2

Let $a>0$, $\beta \ge 0$, and let U be a nonempty open set in ${\mathbb {R}}$. If $f\in C^{\infty }(U)$ and $x\in U$ satisfy $f''(x)+\beta f(x)\ge 0$ and $f(x)\ge 0$, then $F_a(f(x))''+\beta F_a(f(x))\ge 0$ holds.

Proof

By differentiating $F_a(f)=a^{-1}\log \left( e^{af}+e^{-af}\right)$ twice, we get

$$\begin{aligned} F_a(f)'&=\frac{f'e^{af}-f'e^{-af}}{e^{af}+e^{-af}},\\ F_a(f)''&=\frac{f''e^{af}+a\left( f'\right) ^2e^{af}-f''e^{-af} +a\left( -f'\right) ^2e^{-af}}{e^{af}+e^{-af}}-\frac{a\left( f'e^{af} -f'e^{-af}\right) ^2}{\left( e^{af}+e^{-af}\right) ^2}\\&=\frac{f''e^{af}-f''e^{-af}}{e^{af}+e^{-af}}+\frac{4a\left( f' \right) ^2}{\left( e^{af}+e^{-af}\right) ^2}. \end{aligned}$$

Thus, $f''(x)+\beta f(x)\ge 0$ and $f(x)\ge 0$ imply

$$\begin{aligned} F_a(f(x))''\ge f''(x)\frac{e^{af(x)}-e^{-af(x)}}{e^{af(x)}+e^{-af(x)}} \ge -\beta f(x)\cdot 1\ge -\beta F_a(f(x)), \end{aligned}$$

i.e., we have $F_a(f(x))''+\beta F_a(f(x))\ge 0$. $\square$

Proof of Main Theorem 1.5

We put $r:=D/\pi$, $N':=N-1$, $\kappa :=K/N'$ and $Y:=\{(-1,0),(1,0)\}\subset {\mathbb {S}}^1$. We define a sequence of smooth functions $f_n:{\mathbb {S}}^1\rightarrow {\mathbb {R}}$ by $f_n(x,y):=-N'\log (a_n \cdot F_n(y))$, where $a_n>0$ is the normalizing constant, i.e.,

$$\begin{aligned} a_n:=\left( \int _{{\mathbb {S}}^1}F_n(y)^{N'}d{{\,\textrm{vol}\,}}_g(x,y)\right) ^{-\frac{1}{N'}}. \end{aligned}$$

Then, we have $e^{-f_n}{{\,\textrm{vol}\,}}_g\in {\mathcal {P}}({\mathbb {S}}^1)$. Indeed,

$$\begin{aligned} \left( e^{-f_n}{{\,\textrm{vol}\,}}_g\right) ({\mathbb {S}}^1)&=\int _{{\mathbb {S}}^1}e^{-f_n}d{{\,\textrm{vol}\,}}_g\\&=\int _{{\mathbb {S}}^1}\left( a_n\cdot F_n(y)\right) ^{N'}d{{\,\textrm{vol}\,}}_g(x,y)\\&=a_n^{N'}\cdot a_n^{-N'}=1. \end{aligned}$$

Hence, $({\mathbb {S}}^1, g, e^{-f_n}{{\,\textrm{vol}\,}}_g)$ is a 1-dimensional weighted Riemannian manifold, and $\{({\mathbb {S}}^1, d_g, e^{-f_n}{{\,\textrm{vol}\,}}_g)\}_{n\in {\mathbb {N}}}$ is a sequence of mm-spaces whose diameter is D. We will prove the following two conditions

• (C) $({\mathbb {S}}^1, d_g, e^{-f_n}{{\,\textrm{vol}\,}}_g)$ satisfies the ${{\,\textrm{CD}\,}}(K,N)$ condition.

• (D) $({\mathbb {S}}^1, d_g, e^{-f_n}{{\,\textrm{vol}\,}}_g)$ $\Box$-converges to $(Y, d_g, 2^{-1}(\delta _{(-1,0)}+\delta _{(1,0)}))$.

First, to show the condition (C), it suffices to prove that $f_n$ is a $(K,N')$-convex function by Proposition 3.30 since $({\mathbb {S}}^1, d_g, {{\,\textrm{vol}\,}}_g)$ satisfies ${{\,\textrm{Ric}\,}}_1\ge 0$. Since the function $\varphi :{\mathbb {R}}\rightarrow {\mathbb {S}}^1; \varphi (t):=(\cos (r^{-1}t),\sin (r^{-1}t))$ is an isometric embedding when restricted on any open interval of length less than or equal to 2D. Thus, in order to prove ${{\,\textrm{Hess}\,}}\left( e^{-f_n/N'}\right) \ge -\kappa g$, we prove $(a_n\cdot F_n(f))''+\kappa (a_n\cdot F_n(f))\ge 0$ on (0, 2D), where $f(t):=\sin (r^{-1}t)$. If $f(t)\ge 0$ i.e. $t\in (0, D]$, we have $f''(t)+\kappa f(t)=(\kappa -r^{-2})f(t)\ge 0$ by $r=D/\pi \ge \kappa ^{-1/2}$. Lemma 7.2 implies

$$\begin{aligned} (a_n\cdot F_n(f(t)))''+\kappa (a_n\cdot F_n(f(t)))=a_n(F_n(f(t))''+F_n(f(t)))\ge 0. \end{aligned}$$

If $f(t)<0$ i.e. $t\in (D, 2D)$, $h=-f$ satisfies $h(t)\ge 0$ and $h''(t)+\kappa h(t)\ge 0$. Hence, $F_n(h)=F_n(f)$ and Lemma 7.2 imply $(a_n\cdot F_n(f(t)))''+\kappa a_n\cdot F_n(f(t))\ge 0$. Thus, the condition (C) holds.

Next, we will show the condition (D). Since $(Y, d_g, 2^{-1}(\delta _{(-1,0)}+\delta _{(1,0)}))$ is mm-isomorphic to $({\mathbb {S}}^1, d_g, 2^{-1}(\delta _{(-1,0)}+\delta _{(1,0)}))$, it suffices to prove $\mu _{X_n}:=e^{-f_n}{{\,\textrm{vol}\,}}_g\rightharpoonup 2^{-1}(\delta _{(-1,0)}+\delta _{(1,0)})=:\mu _Y$ by Proposition 3.6. For that purpose, it is sufficient to show that for any $\varepsilon \in (0,D/2)$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\mu _{X_n}({\mathbb {S}}^1\setminus N_{\varepsilon }(Y))=0 \end{aligned}$$

because $f_n(x,y)$ is independent of x and an even function for y. First, we prove $a_n\rightarrow \infty$ as $n\rightarrow \infty$ beforehand. For any $\varepsilon \in (0, D/2)$,

$$\begin{aligned} \liminf _{n\rightarrow \infty }a_n&\ge \liminf _{n\rightarrow \infty }\left( \int _{\varphi ([D/2,D-\varepsilon ])}F_n\left( y\right) ^{N'}d{{\,\textrm{vol}\,}}_g(x,y)\right) ^{-\frac{1}{N'}}\\&\ge \left( \int _{D/2}^{D-\varepsilon }\left( \lim _{n\rightarrow \infty }F_n(f(t))\right) ^{N'}dt\right) ^{-\frac{1}{N'}}\\&=\left( \int _{D/2}^{D-\varepsilon }f(t)^{N'}dt\right) ^{-\frac{1}{N'}} \end{aligned}$$

and $0<f(t)\le r^{-1}(D-t)$ as $t\in [D/2,D-\varepsilon ]$ imply

$$\begin{aligned} \int _{D/2}^{D-\varepsilon }f(t)^{N'}dt&\ge \int _{D/2}^{D-\varepsilon }\left( r^{-1}(D-t)\right) ^{N'}dt\\&=\left[ -\frac{r^{-N'}}{N}(D-t)^N\right] _{D/2}^{D-\varepsilon }\\&=-\frac{r^{-N'}}{N}\left( \varepsilon ^N-(D/2)^N\right) . \end{aligned}$$

Since this value diverges to infinity as $\varepsilon \rightarrow 0$, $a_n\rightarrow \infty$ holds. We take any $\varepsilon \in (0, D/2)$. For any $t\in [\varepsilon ,D-\varepsilon ]\cup [D+\varepsilon ,2D-\varepsilon ]$, $F_n(f(t))\ge |f(t)|\ge f(\varepsilon )>0$ and ${\mathbb {S}}^1\setminus N_{\varepsilon }(Y)=\varphi ([\varepsilon ,D-\varepsilon ]\cup [D+\varepsilon ,2D-\varepsilon ])$ imply

$$\begin{aligned} \mu _{X_n}({\mathbb {S}}^1\setminus N_{\varepsilon }(Y))&=\int _{{\mathbb {S}}^1\setminus N_{\varepsilon }(Y)}\left( a_n\cdot F_n(y)\right) ^{N'}d{{\,\textrm{vol}\,}}_g(x,y)\\&=a_n^{N'}\left( \int _{\varepsilon }^{D-\varepsilon }F_n(f(t))^{N'}dt+\int _{D+\varepsilon }^{2D-\varepsilon }F_n(f(t))^{N'}dt\right) \\&\le a_n^{N'}\cdot f(\varepsilon )^{N'}\cdot (2D-4\varepsilon )\\&\rightarrow 0\qquad (n\rightarrow \infty ). \end{aligned}$$

Thus, $X_n$ is a ${{\,\textrm{CD}\,}}(K,N)$ space for any $n\in {\mathbb {N}}$ and $\Box$-converges to Y. $\square$

Appendix

In this appendix, we give our results for metric measure spaces that are not mm-spaces. A triple $(X,d_X,\mu _X)$ is a metric measure space if $(X,d_X)$ is a complete separable metric space and $\mu _X$ is a locally finite non-zero Borel measure, i.e., for any $x\in X$, there exists $r>0$ such that

$$\begin{aligned} \mu _X\left( B_X(x,r)\right) <\infty . \end{aligned}$$

The ${{\,\textrm{CD}\,}}(K,N)$ (resp. ${{\,\textrm{CD}\,}}^*(K,N)$) condition can be defined for metric measure spaces in the same way as in Definition 3.24 (resp. Definition 3.25).

Definition 8.1

(Volume growth condition) A metric measure space $(X,d_X,\mu _X)$ satisfies the volume growth condition if there exist $C>0$ and $x_0\in X$ such that

$$\begin{aligned} \int _Xe^{-Cd_X(x,x_0)^2}d\mu _X(x)<\infty . \end{aligned}$$

According to Theorems 4.24 and 4.26 in [12], every ${{\,\textrm{CD}\,}}(K,\infty )$ space with $K\in {\mathbb {R}}$ satisfies the volume growth condition, and its total measure is finite if $K>0$. However, for $N<0$, there are ${{\,\textrm{CD}\,}}(K,N)$ spaces that do not satisfy the volume growth condition even if $K>0$.

Example 8.2

For $K>0$ and $N<0$, we consider a function $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ with

$$\begin{aligned} f(x):=-(N-1)\sqrt{\frac{1}{4}-\frac{K}{N-1}}\sinh x. \end{aligned}$$

Then, $({\mathbb {R}},|\cdot |,e^{-f}{\mathcal {L}}^1)$ is a ${{\,\textrm{CD}\,}}(K,N)$ space and does not satisfy the volume growth condition.

Proof

First, we prove that $({\mathbb {R}},|\cdot |,e^{-f}{\mathcal {L}}^1)$ does not satisfy the volume growth condition. For any $C>0$ and $x_0\in {\mathbb {R}}$, the function $C|\cdot -x_0|^2+f$ is not bounded from below. Thus,

$$\begin{aligned} \int _{{\mathbb {R}}}e^{-C|x-x_0|^2}d(e^{-f}{\mathcal {L}}^1)=\int _{-\infty }^{\infty }e^{-(C|x-x_0|^2+f(x))}dx=\infty . \end{aligned}$$

Next, we prove that $({\mathbb {R}},|\cdot |,e^{-f}{\mathcal {L}}^1)$ is a ${{\,\textrm{CD}\,}}(K,N)$ space. By Proposition 3.30, it is sufficient to prove that f is a $(K,N-1)$-convex function. We put a constant $a:=\sqrt{\frac{1}{4}-\frac{K}{N-1}}$ and a function $g:=e^{-f/(N-1)}$. Then,

$$\begin{aligned} g(x)&=e^{a\sinh x},\\ g'(x)&=ae^{a\sinh x}\cosh x,\\ g''(x)&=ae^{a\sinh x}\sinh x+a^2e^{a\sinh x}(\cosh x)^2\\&=\left( a^2\left( \sinh x+(2a)^{-1}\right) ^2-4^{-1}+a^2\right) e^{a\sinh x}\\&\ge \left( a^2-4^{-1}\right) e^{a\sinh x}=-\frac{K}{N-1}g(x) \end{aligned}$$

imply that f is $(K,N-1)$-convex. $\square$

In general, it is known that the ${{\,\textrm{CD}\,}}(K,N)$ space (and ${{\,\textrm{CD}\,}}^*(K,N)$ space) becomes smaller when K is large. For instance, if X is a ${{\,\textrm{CD}\,}}(K,N)$ space for some $K>0$ and $N\in (1,\infty )$, then

$$\begin{aligned} {{\,\textrm{diam}\,}}{{\,\textrm{supp}\,}}\mu _X\le \pi \sqrt{\frac{N-1}{K}} \end{aligned}$$

holds ( [13, Corollary 2.6]). But this is not always the case when $N<0$, as in the example given above. However, when “$K=\infty$", the result is the same as when $N>1$.

Theorem 8.3

Let $(X,d_X, \mu _X)$ be a metric measure space. If there exists $N<0$ such that $(X,d_X, \mu _X)$ is a ${{\,\textrm{CD}\,}}^*(K,N)$ space for any $K>0$, then ${{\,\textrm{supp}\,}}\mu _X$ consists of one point.

We state a proposition necessary to prove this theorem.

Proposition 8.4

[9, Theorem 4.8] Let $(X,d_X,\mu _X)$ be a ${{\,\textrm{CD}\,}}^*(K,N)$ space for some $K\in {\mathbb {R}}$ and $N<0$. For any $A_0, A_1\in {\mathcal {B}}_X$ with ${{\,\textrm{diam}\,}}(A_0\cup A_1)<\pi \sqrt{N/K}$ if $K<0$, we have

$$\begin{aligned} \mu _X(A_t)^{\frac{1}{N}}\le \sum _{i=0}^1\sup _{x\in A_0, y\in A_1}\sigma _{K/N}^{(t),i}\left( d_X(x,y)\right) \mu _X(A_i)^{\frac{1}{N}} \end{aligned}$$

(12)

for any $t\in [0,1]$, where $A_t$ is defined as

$$\begin{aligned} A_t:=\{\gamma _t\in X\mid \gamma \text { is a geodesic in }X\text { with } \gamma _0\in A_0\text { and }\gamma _1\in A_1\}. \end{aligned}$$

We call this inequality (12) the Brunn-Minkowski inequality.

Proof of Theorem 8.3

We assume that there exist two points $x_0,x_1\in {{\,\textrm{supp}\,}}\mu _X$ with $x_0\ne x_1$. We take $r\in (0,d_X(x_0, x_1)/2)$ with $\mu _X(B_X(x_0, 3r))<\infty$ and $\mu _X(B_X(x_1, r))<\infty$ (The local finiteness of $\mu _X$ implies the existence of such r). We put $\varepsilon :=d_X(x_0, x_1)$ and $A_i:=B_X(x_i,r)$ for $i=0,1$. Then, since $(X,d_X, \mu _X)$ is a ${{\,\textrm{CD}\,}}^*(K,N)$ space, Proposition 8.4 implies

$$\begin{aligned} \mu _X(A_t)^{\frac{1}{N}}\le \sum _{i=0}^1\sup _{x\in A_0, y\in A_1}\sigma _{K/N}^{(t),i}\left( d_X(x,y)\right) \mu _X(A_i)^{\frac{1}{N}} \end{aligned}$$

for any $t\in [0,1]$ and $K>0$. If $t\in (0,1)$, we have that $\sigma _{K/N}^{(t),i}(\theta )$ is decreasing for $\theta$ and converges to 0 as $\theta \rightarrow \infty$ by $K>0$. Furthermore, for any $x\in A_0$, $y\in A_1$,

$$\begin{aligned} d_X(x,y)\ge d_X(x_0,x_1)-d_X(x_0,x)-d_X(y,x_1)>\varepsilon -2r>0 \end{aligned}$$

holds. Thus $\sigma _{K/N}^{(t),i}(\theta )=\sigma _{-1}^{(t),i}(\sqrt{K/(-N)}\theta )$ implies

$$\begin{aligned} \mu _X(A_t)^{\frac{1}{N}}&\le \sum _{i=0}^1\sigma _{-1}^{(t),i}\left( \sqrt{K/(-N)} \cdot (\varepsilon -2r)\right) \mu _X(A_i)^{1/N}. \end{aligned}$$

Since $A_t$ is independent of K and this inequality holds for any $K>0$, we obtain $\mu _X(A_t)^{1/N}\le 0$ i.e. $\mu _X(A_t)=\infty$ by taking $K\rightarrow \infty$. But, if we put $n\ge 2$ with $\varepsilon /(n+1)\le r< \varepsilon /n$ and $t:=(n+1)^{-1}\in (0,1/2)$, and take any $y\in A_t$, there exists a geodesic $\gamma$ in X such that $\gamma _0\in A_0$, $\gamma _1\in A_1$ and $y=\gamma _t$. Then,

$$\begin{aligned} d_X(x_0,y)&\le d_X(x_0,\gamma _0)+d_X(\gamma _0,\gamma _t)\\&< r+td_X(\gamma _0,\gamma _1)\\&\le r+t(d_X(\gamma _0,x_0)+d_X(x_0,x_1)+d_X(x_1,\gamma _1))\\&< r+t(\varepsilon +2r)< 3r \end{aligned}$$

implies $A_t\subset B_X(x_0, 3r)$, and it contradicts $\mu _X(B_X(x_0, 3r))<\infty$. Thus ${{\,\textrm{supp}\,}}\mu _X$ consists of one point. $\square$