Some functional inequalities under lower Bakry–Émery–Ricci curvature bounds with ${\varepsilon }$-range

Abstract

For n-dimensional weighted Riemannian manifolds, lower m-Bakry–Émery–Ricci curvature bounds with ${\varepsilon }$-range, introduced by Lu-Minguzzi-Ohta (Anal Geom Metr Spaces 10(1):1–30, 2022), integrate constant lower bounds and certain variable lower bounds in terms of weight functions. In this paper, we prove a Cheng type inequality and a local Sobolev inequality under lower m-Bakry–Émery–Ricci curvature bounds with ${\varepsilon }$-range. These generalize those inequalities under constant curvature bounds for $m \in (n,\infty )$ to $m\in (-\infty ,1]\cup \{\infty \}$.

Appendix: Upper bound of the \(L^p\)-spectrum for deformed measures

Although we considered the Riemannian distance d, it is also possible to study comparison theorems associated with a metric deformed by using the weight function (we refer [5, 6, 26], for example). In this “Appendix”, we start from a volume comparison theorem in [6] and prove a variant of Cheng type inequality for the $L^p$-spectrum.

Let $(M,g,\mu = \textrm{e}^{-\psi }v_g)$ be an n-dimensional weighted Riemannian manifold, $m\in (-\infty ,1]\cup [n,+\infty ]$ and ${\varepsilon }\in {\mathbb {R}}$ in the range (1). We fix a point $q\in M$. We define lower semi continuous functions $s_q:M\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \quad s_q(x):=\inf _{\gamma }\int ^{d(q,x)}_0\,\textrm{e}^{ -\frac{2(1-{\varepsilon })\psi (\gamma (\xi ))}{n-1} }\,\textrm{d}\xi , \end{aligned}$$

where the infimum is taken over all unit speed minimal geodesics $\gamma :[0,d(q,x)]\rightarrow M$ from q to x. For $r>0$, we define

$$\begin{aligned} B_{\psi ,q}(r):=\left\{ \,x\in M \mid s_q(x) < r \,\right\} , \end{aligned}$$

and also define measures

$$\begin{aligned} \mu :=\textrm{e}^{-\psi }\,v_g,\quad \nu :=\textrm{e}^{ - \frac{2(1-{\varepsilon })\psi }{n-1} }\mu . \end{aligned}$$

We set

$$\begin{aligned} {\mathcal {S}}_{-K}(r):=\int ^{r}_{0}\,{\textbf{s}}^{1/c}_{-K}(s)\,\textrm{d}s \end{aligned}$$

for $K > 0$. In [6], they obtained the following theorem.

Theorem 9

([6, Proposition 4.6], Volume comparison) Let $(M,g,\mu )$ be an n-dimensional weighted Riemannian manifold. We assume ${{\,\textrm{Ric}\,}}^{m}_{\psi } \ge -K\textrm{e}^{ \frac{4({\varepsilon }-1)\psi }{n-1} }g$ for $K > 0$. Then for all $r,R>0$ with $r\le R$ we have

$$\begin{aligned} \frac{\nu (B_{\psi ,q}(R))}{\nu (B_{\psi ,q}(r))} \le \frac{{\mathcal {S}}_{-cK}(R)}{{\mathcal {S}}_{-cK}(r)}. \end{aligned}$$

In the following argument, we start from Theorem 9 instead of Theorem 1 to prove a Cheng type inequality of the $L^p$-spectrum for the deformed measure $\nu$.

Theorem 10

Let $(M,g,\mu )$ be a complete weighted Riemannian manifold. We assume that $s_q$ is smooth and there exists a constant $k > 0$ such that

$$\begin{aligned} |\nabla s_q(x)|\le k \end{aligned}$$

holds for arbitrary $x\in M$. We also assume

$$\begin{aligned} {{\,\textrm{Ric}\,}}^{m}_{\psi } \ge -K\,\textrm{e}^{ \frac{4({\varepsilon }-1)\psi }{n-1} }g \end{aligned}$$

for $K > 0$. Then we have

$$\begin{aligned} \lambda _{\nu ,p}(M) \le \left( \frac{k}{p}\sqrt{\frac{K}{c}}\right) ^p. \end{aligned}$$

(19)

Proof

We apply the argument in Theorem 6.

For $R\ge 2$, let $\eta :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a nonnegative smooth function such that $\eta = 1$ on $(-(R-1),R-1)$, $\eta = 0$ on ${\mathbb {R}}\backslash (-R,R)$ and $|\eta '| \le C_3$, where $C_3$ is a constant independent of R. We set, for an arbitrary $\delta > 0$,

$$\begin{aligned} \alpha = -\frac{\sqrt{K/c} + \delta }{p} \end{aligned}$$

and

$$\begin{aligned} \phi (y):= \exp (\alpha s_q(y))\varphi (y), \end{aligned}$$

where $\varphi (y):= \eta (s_q(y))$. By the assumption of $s_{q}$, we have

$$\begin{aligned} |\nabla \varphi | = |\eta '(s_q)||\nabla s_q| \le k C_3. \end{aligned}$$

As in the proof of Theorem 6, we find for an arbitrary $\zeta > 0$,

$$\begin{aligned} |\nabla \phi |^p= & {} \left| \alpha \textrm{e}^{\alpha s_q} \varphi \nabla s_q+\textrm{e}^{\alpha s_q} \nabla \varphi \right| ^p \\\le & {} \textrm{e}^{p \alpha s_q}(-k\alpha \varphi +|\nabla \varphi |)^p \\\le & {} \textrm{e}^{p \alpha s_q}\left[ (1+\zeta )^{p-1}(-k\alpha \varphi )^p+\left( \frac{1+\zeta }{\zeta }\right) ^{p-1}|\nabla \varphi |^p\right] . \end{aligned}$$

By the definition of $\lambda _{\nu ,p}(M)$, we obtain

$$\begin{aligned} \lambda _{\nu ,p}(M)\le & {} (1+\zeta )^{p-1}(-k\alpha )^p+\left( \frac{1+\zeta }{\zeta }\right) ^{p-1} \frac{\int _M \exp (p \alpha s_q)|\nabla \varphi |^p \textrm{d}\nu }{\int _M \exp (p \alpha s_q) \varphi ^p \textrm{d}\nu }\nonumber \\= & {} (1+\zeta )^{p-1}(-k\alpha )^p+\left( \frac{1+\zeta }{\zeta }\right) ^{p-1} \frac{\int _{B_{\psi ,q}(R) \backslash B_{\psi ,q}(R-1)} \exp (p \alpha s_q)|\nabla \varphi |^p \textrm{d}\nu }{\int _{B_{\psi ,q}(R)} \exp (p \alpha s_q) \varphi ^p \textrm{d}\nu }\nonumber \\\le & {} (1+\zeta )^{p-1}(-k\alpha )^p+(kC_3)^p\left( \frac{1+\zeta }{\zeta }\right) ^{p-1} \frac{\exp (p \alpha (R-1)) \nu \left( B_{\psi ,q}(R)\right) }{\int _{B_{\psi ,q}(1)} \exp (p \alpha s_q) \textrm{d}\nu } \nonumber \\\le & {} (1+\zeta )^{p-1}(-k\alpha )^p+(kC_3)^p\left( \frac{1+\zeta }{\zeta }\right) ^{p-1} \frac{\exp (p \alpha (R-1)) \nu \left( B_{\psi ,q}(R)\right) }{\exp (p \alpha ) \nu \left( B_{\psi ,q}(1)\right) }.\nonumber \\ \end{aligned}$$

(20)

From Theorem 9 and

$$\begin{aligned} (\sqrt{cK})^{1/c}{\mathcal {S}}_{-cK}(R)\le & {} \int _0^R \left[ \frac{1}{2}\left\{ \exp (\sqrt{cK}s) - \exp (-\sqrt{cK}s)\right\} \right] ^{1/c}\textrm{d}s\\\le & {} \sqrt{\frac{c}{K}}\exp \left( \sqrt{\frac{K}{c}}R\right) , \end{aligned}$$

we deduce

$$\begin{aligned} \frac{\textrm{e}^{p\alpha (R-1)}\nu (B_{\psi ,q}(R))}{\textrm{e}^{p\alpha }\nu (B_{\psi ,q}(1))}\le & {} \frac{\textrm{e}^{p\alpha R}}{\textrm{e}^{2p\alpha }}\frac{1}{(\sqrt{cK})^{1/c}{\mathcal {S}}_{-cK}(1)}\sqrt{\frac{c}{K}}\exp \left( \sqrt{\frac{K}{c}}R\right) \\= & {} \frac{1}{\textrm{e}^{2p\alpha }(\sqrt{cK})^{1/c} {\mathcal {S}}_{-cK}(1)}\sqrt{\frac{c}{K}}\exp \left( p\alpha R + \sqrt{\frac{K}{c}}R\right) \rightarrow 0 \end{aligned}$$

as $R\rightarrow \infty$. Letting $R\rightarrow \infty$ in (20), we obtain

$$\begin{aligned} \lambda _{\nu ,p}(M) \le (1 + \zeta )^{(p-1)}(-k\alpha )^p. \end{aligned}$$

(21)

Since $\zeta > 0$ and $\delta > 0$ are arbitrary, the theorem follows. $\square$