Sobolev Inequalities, Riesz Transforms, and the Ricci Flow

Abstract

A number of results about deriving further Sobolev inequalities from a given Sobolev inequality are presented. Various techniques are employed, including Bessel potentials and Riesz transforms. Combining these results with the $W^{1,2}$ Sobolev inequality along the Ricci flow established by the author in earlier papers then yields various new Sobolev inequalities along the Ricci flow.

1 The Main Results

In this section, we present the main results of this paper. Most proofs will be given in the subsequent sections. We divide the various results into several parts.

1.1 Part A

In [9–12], logarithmic Sobolev inequalities along the Ricci flow were obtained. As a consequence, $W^{1,p}$ Sobolev inequalities with $p=2$ along the Ricci flow were derived. Let $M$ be a closed manifold of dimension $n \ge 2$. Let $g=g(t)$ be a smooth solution of the Ricci flow

$$\begin{aligned} \frac{\partial g}{\partial t}=-2 \mathrm{Ric} \end{aligned}$$

(1.1)

on $M \times [0, T)$ for some (finite or infinite) $T>0$. In the case $\lambda _0(g_0) >0$, where $\lambda _0(g_0)$ denotes the first eigenvalue of the operator $-\Delta +\frac{R}{4}$ of the initial metric $g_0=g(0)$, the Sobolev inequality takes the following form (in the case $n \ge 3$)

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{dvol}\right) ^{\frac{n-2}{n}} \le A \int _M \left( |\nabla u|^2 +\frac{R}{4}u^2\right) \mathrm{dvol}, \end{aligned}$$

(1.2)

where the positive constant $A$ depends on the initial metric in terms of rudimentary geometric data. If the condition $\lambda _0(g_0)>0$ is not assumed, then the Sobolev inequality takes the form (again in the case $n\ge 3$)

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{dvol}\right) ^{\frac{n-2}{n}} \le A \int _M \left( |\nabla u|^2 +\frac{R}{4}u^2\right) \mathrm{dvol}+B\int _M u^2 \mathrm{dvol}, \end{aligned}$$

(1.3)

where the positive constants $A$ and $B$ depend on a finite upper bound for $T$ and the initial metric in terms of rudimentary geometric data. (If $\lambda _0(g_0)=0$, then the dependence on $T$ is not needed.)

As is well known, the case $p=2$ of the $W^{1,p}(M)$ Sobolev inequalities is used most often in analysis and geometry. However, it is of high interest to understand the situations $1<p<2$ and $2<p<n$, both from the point of view of a deeper understanding of the theory and the point of view of further applications. In this paper, we derive $W^{1,p}$ and $W^{2,p}$ Sobolev inequalities for general $p$ along the Ricci flow in several different ways. We will take a general point of view, and study the general problem of deriving further Sobolev inequalities from a given Sobolev inequality. In particular, we will include non-compact manifolds and manifolds with boundary, which require additional care. Our first result is the following one.

Theorem 1.1

Let $(M, g)$ be a Riemannian manifold of dimension $n \ge 2$, with or without boundary. (It is not assumed to be compact or complete.) Assume that for some $1\le p_0<n$ and non-negative constants $A$ and $B$ the inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{p_0n}{n-p_0}} \mathrm{dvol}\right) ^{\frac{n}{n-p_0}} \le A \int _M |\nabla u|^{p_0} \mathrm{dvol}+\frac{B}{\mathrm{vol}_g(M)^{\frac{p_0}{n}}} \int _M |u|^{p_0} \mathrm{dvol}\qquad \end{aligned}$$

(1.4)

holds true for all $u\in W^{1, p_0}(M)$. Then we have for all $p_0<p<n$ and $u \in W^{1,p}(M)$

$$\begin{aligned} \left( \int _M |u|^{\frac{np}{n-p}}\mathrm{dvol}\right) ^{\frac{n-p}{n}} \le C_1 \int _M |\nabla u|^p \mathrm{dvol} +\frac{C_2}{\mathrm{vol}_g(M)^{\frac{p}{n}}} \int _M |u|^p \mathrm{dvol}, \end{aligned}$$

(1.5)

where the positive constants $C_1=C_1(n,p_0, p,A,B)$ and $C_2=C_2(n,p_0, p,A,B)$ depend only on $n,p_0, p, A$ and $B$. Their dependence on $p$ is in terms of an upper bound for $\frac{1}{n-p}$. (If $\mathrm{vol}_g(M)=\infty$, then it is understood that the second term on the right hand side in (1.4) and (1.5) is zero, and $B$ is absent elsewhere.)

This theorem is proved by an induction scheme based on the Hölder inequality. The principle that Sobolev inequalities of lower $p$ lead to Sobolev inequalities of higher $p$ is known. For example, it is well-known that the Sobolev inequality

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \Vert \nabla u\Vert _p \end{aligned}$$

(1.6)

for $u \in W^{1,p}_0(\mathbf{R}^n)$, $1<p<n$, can be derived from the case $p=1$, see e.g., [4]. However, the result in Theorem 1.1 is new, and the proof is more involved. Combining this theorem with the results in [9, 10] and [11] we then obtain $W^{1,p}$ Sobolev inequalities along the Ricci flow for $p>2$ in various situations. To keep this paper streamlined, we only state the results in the situation of [9]. The results in the situations of [10] and [11] are similar and obvious. In the following two theorems, we consider as before a smooth solution $g=g(t)$ of the Ricci flow on a closed manifold $M$ of dimension $n \ge 3$ for $0\le t<T$, where $T>0$ is allowed to be $\infty$.

Theorem 1.2

Assume $T<\infty$. Let $2<p<n$. Then there holds for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$

$$\begin{aligned} \left( \int _M |u|^{\frac{np}{n-p}} \mathrm{dvol} \right) ^{\frac{n-p}{n}} \le A\left[ 1\!+\!(\max R^+\!+\!1)\mathrm{vol}(M)^{\frac{2}{n}}\right] ^{\frac{r(p) p }{2}} \int _M (|\nabla u|^p\!+\!|u|^p)\mathrm{dvol}, \nonumber \\ \end{aligned}$$

(1.7)

where all geometric quantities are associated with $g(t)$, except the constant $A$, which can be bounded from above in terms of the dimension $n$, a non-positive lower bound for $R_{g_0}$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $T$, and an upper bound for $\frac{1}{n-p}$. The number $r(p)$ is defined in Theorem 2.1 below with $p_0=2$.

Theorem 1.3

Assume that $\lambda _0(g_0) > 0$. Let $2<p<n$. Then there holds for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$

$$\begin{aligned} \left( \int _M |u|^{\frac{np}{n-p}} \mathrm{dvol} \right) ^{\frac{n-p}{n}} \le A\left[ (\max R^++1)\mathrm{vol}(M)^{\frac{2}{n}}\right] ^{\frac{r(p)p}{2}} \int _M (|\nabla u|^p+|u|^p) \mathrm{dvol}, \nonumber \\ \end{aligned}$$

(1.8)

where all geometric quantities are associated with $g(t)$, except the constant $A$, which can be bounded from above in terms of the dimension $n$, a non-positive lower bound for $R_{g_0}$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, a positive lower bound for $\lambda _0(g_0)$, and an upper bound for $\frac{1}{n-p}$. The number $r(p)$ is the same as in Theorem 1.2.

A positive lower bound of $\lambda _0(g_0)$ implies an upper bound for $T$, hence Theorem 1.3 can be viewed as a corollary of Theorem 1.2. We state it as a separate theorem because of the geometric significance of this special case. Note that the distinction between (1.2) and (1.3) regarding the form of the Sobolev inequalities is not present between Theorem 1.2 and Theorem 1.3. The sole difference lies in the dependence of the constant $A$.

We would like to mention that these results as well as the other results in the subsequent parts regarding Sobolev inequalities along the Ricci flow easily extend (with some minor modifications) to the volume-normalized Ricci flow and the Kähler-Ricci flow. (This is a straightforward consequence of the relevant results in [9].) They also extend to the Ricci flow with surgeries in dimension 3 as constructed in [6]. This will be presented elsewhere.

1.2 Part B

First we present a result on non-local Sobolev inequalities which is analogous to [9, Theorem C.5], but is formulated in terms of the canonical $(1,p)$-Bessel norm for Sobolev functions.

Definition

Let $(M, g)$ be a metrically complete Riemannian manifold, with or without boundary. Let $1<p<\infty$. Let the Bessel–Sobolev space $L_B^{1,p}(M)$ be the completion of $C^{\infty }_c(M)$ with respect to the norm $\Vert (-\Delta +1)^{\frac{1}{2}}\Vert _p$. (See Sect. 3 for the construction of the operator $(-\Delta +1)^{\frac{1}{2}}$. ) We shall say that $g$ is a $p$-Bessel metric, and $(M,g)$ is a $p$-Bessel (Riemannian) manifold, if $L^{1,p}_B(M)$ is equivalent to $W^{1,p}(M)$, i.e.,

$$\begin{aligned} c_1 \Vert u\Vert _{1,p} \le \Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p \le c_2 \Vert u\Vert _{1,p} \end{aligned}$$

(1.9)

for all $u \in C^{\infty }_{c}(M)$ and some positive constants $c_1$ and $c_2$, where $\Vert u\Vert _{1,p}=\Vert u\Vert _p+\Vert \nabla u\Vert _p$ is the $W^{1,p}$ norm of $u$.

Assume that $(M,g)$ is $p$-Bessel. We define the $(1,p)$-Bessel norm for $f \in W^{1,p}(M)$ to be $\Vert f\Vert _{B, 1, p} =\Vert (-\Delta +1)^{\frac{1}{2}} f\Vert _p$.

The operators $(-\Delta +1)^{\alpha }$ are called Bessel potentials, which is the reason for the above terminologies involving “Bessel”. Note that every metrically complete $(M, g)$ is $2$-Bessel because of the identity

$$\begin{aligned} \langle (-\Delta +1)^{\frac{1}{2}}u, (-\Delta +1)^{\frac{1}{2}}u\rangle _2 =\int _M (|\nabla u|^2+u^2)\mathrm{dvol} \end{aligned}$$

(1.10)

for $u\in C^{\infty }_c(M)$ (and then also for $u \in W^{1,2}(M)$), where $\langle \cdot , \cdot \rangle _2$ is the $L^2$ inner product. By the arguments in [9, Appendix C] (see also [8]), $(M,g)$ is $p$-Bessel for each $1<p<\infty$ if $M$ is compact.

Theorem 1.4

Let $(M,g)$ be a metrically complete manifold with or without boundary of dimension $n \ge 2$. Let $1<\mu <\infty$. Assume the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{2\mu }{\mu -2}}\mathrm{dvol}\right) ^{\frac{\mu -2}{\mu }} \le A\int _M |\nabla u|^{2}\mathrm{dvol} +B \int _M |u|^{2}\mathrm{dvol} \end{aligned}$$

(1.11)

for all $u \in W^{1,2}(M)$. Let $1<p<\mu$. Then there holds

$$\begin{aligned} \Vert (-\Delta +1)^{-\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu , A, B, p) \Vert u\Vert _p \end{aligned}$$

(1.12)

for all $u \in L^p(M)$, where the constant $C(\mu , A, B, p)$ can be bounded from above in terms of upper bounds for $A$, $B$, $\mu$, $\frac{1}{\mu -p}$ and $\frac{1}{p-1}$. Consequently, there holds for a given $1<p<\mu$

$$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu , A, B, p) \Vert u\Vert _{B,1,p} \end{aligned}$$

(1.13)

for all $u \in W^{1,p}(M)$, provided that $(M, g)$ is $p$-Bessel.

Combining this theorem with the results in [9, 10] and [11] we obtain non-local Sobolev inequalities along the Ricci flow which are analogous to Theorem C.6 and C.7 in [9], but are formulated in terms of the canonical $(1,p)$-Bessel norm. Again we only state the results in the situation of [9]. As before, let $g=g(t)$ be a smooth solution of the Ricci flow on $M \times [0, T)$ for a closed manifold $M$ of dimension $n \ge 3$ and some $T>0$. We also state the special case $\lambda _0(g_0)>0$ as a separate theorem.

Theorem 1.5

Assume $T<\infty$ and $1<p<n$. There is a positive constant $C$ depending only on the dimension $n$, a non-positive lower bound for $R_{g_0}$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $T$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-p}$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \left( 1+ R_{\mathrm{max}}^+\right) ^{\frac{1}{2}} \Vert u\Vert _{B,1,p}. \end{aligned}$$

(1.14)

Theorem 1.6

Assume that $\lambda _0(g_0)>0$. Let $1<p<n$. There is a positive constant $C$ depending only on the dimension $n$, a positive lower bound for $\lambda _0(g_0)$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-p}$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \left( 1+R_{\mathrm{max}}^+\right) ^{\frac{1}{2}}\Vert u\Vert _{B,1,p}. \end{aligned}$$

(1.15)

The inequality (1.12) in Theorem 1.4 is a special case of the following more general result.

Theorem 1.7

Let $(M,g)$ be a metrically complete manifold, possibly with boundary. Let $\Psi \in L^{\infty }(M)$ and $\mu >1$. Assume that $\Psi \ge 0$ and the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{2\mu }{\mu -2}} \mathrm{dvol} \right) ^{\frac{\mu -2}{\mu }} \le A\int _M \left( |\nabla u|^2+\Psi u^2\right) \mathrm{dvol} \end{aligned}$$

(1.16)

for some $A>0$ and all $u \in W^{1,2}(M)$. Set $H=-\Delta +\Psi$. Let $1<p<\mu$. Then there holds

$$\begin{aligned} \Vert H^{-\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \le C(A, \mu , p) \Vert u\Vert _p \end{aligned}$$

(1.17)

for all $u \in L^p(M)$, where the positive constant $C(\mu , c, p)$ can be bounded from above in terms of upper bounds for $A$, $\mu$, $\frac{1}{\mu -p}$ and $\frac{1}{p-1}$. Consequently, there holds for each $1<p<\infty$

$$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C(A, \mu , p) \Vert H^{\frac{1}{2}}u\Vert _p \end{aligned}$$

(1.18)

for all $u \in W^{1,p}(M)$, provided that $(M, g)$ is compact.

1.3 Part C

Next we have the following consequence of Theorem 1.7.

Theorem 1.8

Let $(M, g)$ be a compact Riemannian manifold of dimension $n\ge 2$, with or without boundary. Assume $\Psi \in L^{\infty }(M)$ and $\mu >2$. Assume $\Psi \ge 0$ and the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{2\mu }{\mu -2}} \mathrm{dvol}\right) ^{\frac{\mu -2}{\mu }} \le A \int _M \left( |\nabla u|^2+\Psi u^2\right) \mathrm{dvol} \end{aligned}$$

(1.19)

for all $u\in W^{1,2}(M)$. Let $1<p<\frac{\mu }{2}$. Then there holds

$$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -2p}} \le C(\mu ,A, p) \Vert \Delta u+\Psi u\Vert _p \end{aligned}$$

(1.20)

for all $u \in W^{2,p}(M)$, where the constant $C(\mu ,A, p)$ can be bounded from above in terms of upper bounds for $\mu$, $A$, $\frac{1}{\mu -2p}$ and $\frac{1}{p-1}$.

Combining this theorem with the results in [9, 10] and [11] we then obtain $W^{2,p}$ Sobolev inequalities along the Ricci flow. Again, we only state the results in the situation of [9]. Let $g=g(t)$ be a smooth solution of the Ricci flow on $M \times [0, T)$ for a closed manifold of dimension $n \ge 3$ and some $0<T \le \infty$, with a given initial metric $g_0$.

Theorem 1.9

Assume $T<\infty$ and $1<p<\frac{n}{2}$. There is a positive constant $C$ depending only on the dimension $n$, a non-positive lower bound for $R_{g_0}$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $T$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-2p}$, such that for each $t \in [0, T)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-2p}} \le C \Vert \Delta u-\left( \frac{R}{4}-\frac{\min R^-_{g_0}}{4}+1\right) u\Vert _{p} \end{aligned}$$

(1.21)

for all $u \in W^{2, p}(M)$.

Theorem 1.10

Assume that $R_{g_0} \ge 0$ and $\lambda _0(g_0)>0$ (thus $R_{g_0}$ is somewhere positive). Let $1<p<\frac{n}{2}$. There is a positive constant $C$ depending only on the dimension $n$, a positive lower bound for $\lambda _0(g_0)$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-2p}$, such that for each $t \in [0, T)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-2p}} \le C \Vert \Delta u-\frac{R}{4}u\Vert _{p} \end{aligned}$$

(1.22)

for all $u \in W^{2, p}(M)$.

1.4 Part D

Now we address the issue of converting the non-local $W^{1,p}$ Sobolev inequalities in Part B into conventional $W^{1,p}$ Sobolev inequalities. Let $H$ denote the operator $-\Delta +\Psi$. Obviously, the desired conversion requires an estimate of the following kind

$$\begin{aligned} \Vert H^{\frac{1}{2}}u\Vert _p \le C \Vert u\Vert _{1,p} \end{aligned}$$

(1.23)

for all $u \in W^{1,p}(M)$. Assume that $M$ is compact. Since $H$ is a pseudo-differential operator of order 1 [7], the inequality (1.23) holds true for some $C$, as mentioned before for the special case $\Psi =1$. But the constant $C$ obtained this way depends on $M$ and the metric $g$ in rather complicated ways. Our purpose is to obtain a constant $C$ which has clear and rudimentary geometric dependence. For this purpose, the general theory of pseudo-differential operators does not seem to give any information.

The issue at hand can be understood in terms of the Riesz transform of $H$, which is defined to be ${R}_H=\nabla H^{-\frac{1}{2}}$. An $L^p$ inequality for the Riesz transform

$$\begin{aligned} \Vert {R}_Hu\Vert _p \le c \Vert u\Vert _p \end{aligned}$$

(1.24)

for all $u \in L^p(M)$ means the same as

$$\begin{aligned} \Vert \nabla u\Vert _p \le c \Vert H^{\frac{1}{2}}u\Vert _p \end{aligned}$$

(1.25)

for all $u \in W^{1,p}(M)$. On the other hand, by duality, the inequality (1.25) implies (1.23) for the dual exponent under suitable conditions on $\Psi$. (In the special case $\Psi =0$, it leads to $\Vert H^{\frac{1}{2}}u\Vert _q \le c\Vert \nabla u\Vert _q$ for the dual exponent $q$.) In general, the Riesz transform ${R}_L$ of a non-negative symmetric elliptic operator $L$ (of second order) is defined in the same way as ${R}_H$. A fundamental problem in harmonic analysis and potential theory is to obtain $L^p$ boundedness for Riesz transforms ${R}_L$, or inequalities $\Vert \nabla u\Vert _p \le C \Vert L^{\frac{1}{2}}u\Vert _p$ and $\Vert L^{\frac{1}{2}}u\Vert _p \le C\Vert u\Vert _{1,p}$. From a geometric point of view, $L^p$-boundedness alone is not enough. It is crucial to obtain geometric estimates for the constants.

Based on the $L^p$ estimates for Riesz transforms due to  Bakry [1], we can convert the non-local $W^{1,p}$ Sobolev inequalities in Theorem 1.4 to conventional $W^{1,p}$ Sobolev inequalities which depend on a lower bound for the Ricci curvature. The situation of Theorem 1.7 with a general $\Psi$ is more complicated, our corresponding result will be presented elsewhere.

Theorem 1.11

Let $(M,g)$ be a complete manifold (without boundary) of dimension $n \ge 2$. Let $1<\mu <\infty$. Assume the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{2\mu }{\mu -2}}\mathrm{dvol}\right) ^{\frac{\mu -2}{\mu }} \le A\int _M |\nabla u|^{2}\mathrm{dvol} +B \int _M |u|^{2}\mathrm{dvol} \end{aligned}$$

(1.26)

for all $u \in W^{1,2}(M)$. Assume $\mathrm{Ric} \ge -a^2 g$ with $a \ge 0$. Let $1<p<\mu$. Then there holds

$$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu , A, B, p) (\Vert \nabla u\Vert _{p}+(1+a) \Vert u\Vert _p) \end{aligned}$$

(1.27)

for all $u \in W^{1,p}(M)$, where the constant $C(\mu , A, B, p)$ can be bounded from above in terms of upper bounds for $A$, $B$, $\mu$, $\frac{1}{\mu -p}$ and $\frac{1}{p-1}$.

We formulate the corresponding results for the Ricci flow in the situation of [9]. The results in the situations of [10] and [11] can be formulated in a similar way.

Consider a smooth solution $g=g(t)$ of the Ricci flow on $M \times [0, T)$, with initial metric $g_0$, where $M$ is a closed manifold of dimension $n \ge 3$. Let $\kappa =\kappa (t)$ denote $(-\min \{0, \min \mathrm{Ric}\})^{1/2}$ at time $t$.

Theorem 1.12

Assume $T<\infty$ and $1<p<n$. There is a positive constant $C$ depending only on the dimension $n$, a non-positive lower bound for $R_{g_0}$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $T$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-p}$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \left( 1+R_{\mathrm{max}}^+\right) ^{\frac{1}{2}} \left( \Vert \nabla u\Vert _{p}+(1+\kappa )\Vert u\Vert _p\right) . \end{aligned}$$

(1.28)

Theorem 1.13

Assume $\lambda _0(g_0)>0$. Let $1<p<n$. There is a positive constant $C$ depending only on the dimension $n$, a positive lower bound for $\lambda _0(g_0)$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{n-p}$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \left( 1+R_{\mathrm{max}}^+\right) ^{\frac{1}{2}} \left( \Vert \nabla u\Vert _{p}+(1+\kappa )\Vert u\Vert _p\right) . \end{aligned}$$

(1.29)

We would like to point out the obvious similarities, as well as delicate differences between these results and those in Part A. One should also compare these results and the results in Part E with Gallot’s estimates of the isoperimetric constant [4, 5] which imply estimates for the Sobolev inequalities. In contrast to Gallot’s estimates, no upper bound for the diameter nor positive lower bound for the volume of $g(t)$ is assumed. Moreover, in (1.28) the lower bound for the Ricci curvature does not appear in front of $\Vert \nabla u\Vert _p$. (Gallot’s estimates lead to Sobolev inequalities in which the lower bound for the Ricci curvature gets involved with $\Vert \nabla u\Vert _p$.)

1.5 Part E

Based on the $L^p$ estimates of Riesz transforms due to  Li [5], we obtain a variant of the results in Part D in the case $1<p<2$. The non-positive lower bound for the Ricci curvature is replaced by the $L^{\frac{n}{2}+\epsilon }$ bound for the (adjusted) negative part of the Ricci curvature, where $\epsilon >0$. For a Riemannian manifold $(M, g)$ we set $\mathrm{Ric}_{\mathrm{min}}(x)=\min \{\mathrm{Ric}(v,v): v \in T_xM, |v|=1\}$.

Theorem 1.14

Let $(M,g)$ be a complete manifold (without boundaryrm ) of dimension $n \ge 3$. Assume the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}}\mathrm{dvol}\right) ^{\frac{n-2}{n}} \le A\int _M |\nabla u|^{2}\mathrm{dvol} +B \int _M |u|^{2}\mathrm{dvol} \end{aligned}$$

(1.30)

for all $u\in W^{1,2}(M)$. Let $c\ge 0$ and $\epsilon >0$ and assume $(\mathrm{Ric}_{\mathrm{min}}+c)^- \in L^{\frac{n}{2}+\epsilon }(M)$. Let $1<p<2$. Then there holds

$$\begin{aligned} \Vert u\Vert _{\frac{n p}{n-p}} \le C \left( \Vert \nabla u\Vert _{p}+(1+\gamma )\Vert u\Vert _p\right) \end{aligned}$$

(1.31)

for all $u \in W^{1,p}(M)$, where

$$\begin{aligned} \gamma =\left( \int _M |(\mathrm{Ric}_{\mathrm{min}}+c)^-|^{\frac{n}{2}+\epsilon }\mathrm{dvol} \right) ^{\frac{1}{2\epsilon }}, \end{aligned}$$

(1.32)

and the constant $C$ can be bounded from above in terms of upper bounds for $n$, $A$, $B$, $c$, $\frac{1}{\epsilon }$ and $\frac{1}{p-1}$.

Next we consider a smooth solution $g=g(t)$ of the Ricci flow on $M \times [0, T)$ for a closed manifold $M$ of dimension $n\ge 3$ and $T>0$.

Theorem 1.15

Assume $T<\infty$. Let $\epsilon >0$ and $1<p<2$. There is a positive constant $C$ depending only on the dimension $n$, a non-positive lower bound for $R_{g_0}$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $T$, an upper bound for $\frac{1}{p-1}$, and an upper bound for $\frac{1}{\epsilon }$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \left( 1+R_{\mathrm{max}}^+\right) ^{\frac{1}{2}} \left( \Vert \nabla u\Vert _{p}+(1+\gamma )\Vert u\Vert _p\right) , \end{aligned}$$

(1.33)

where $\gamma =\gamma (t)$ is the same quantity as in Theorem 1.14.

Theorem 1.16

Assume $\lambda _0(g_0)>0$. Let $\epsilon >0$ and $1<p<2$. There is a positive constant $C$ depending only on the dimension $n$, a positive lower bound for $\lambda _0(g_0)$, a positive lower bound for $\mathrm{vol}_{g_0}(M)$, an upper bound for $C_S(M,g_0)$, an upper bound for $\frac{1}{p-1}$ and an upper bound for $\frac{1}{\epsilon }$, such that for each $t \in [0, T)$ and all $u \in W^{1,p}(M)$ there holds

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le C \left( 1+R_{\mathrm{max}}^+\right) ^{\frac{1}{2}} \left( \Vert \nabla u\Vert _{p}+(1+\gamma )\Vert u\Vert _p\right) , \end{aligned}$$

(1.34)

where

$$\begin{aligned} \gamma =\gamma (t)=\left( \int _M \left| \left( \mathrm{Ric}_{\mathrm{min}}-\frac{1}{n}\min R_{g_0}^-\right) ^-\right| ^{\frac{n}{2}+\epsilon }\mathrm{dvol} \right) ^{\frac{1}{2\epsilon }}. \end{aligned}$$

(1.35)

2 From $W^{1,p}$ for Lower $p$ to $W^{1,p}$ for Higher $p$

Theorem 2.1

Consider a Riemannian manifold $(M, g)$ of dimension $n \ge 2$, with or without boundary. Let $1 \le p_0<n$. Assume that the Sobolev inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{np_0}{n-p_0}}\mathrm{dvol}\right) ^{\frac{n-p_0}{n}} \le A \int _M |\nabla u|^{p_0}\mathrm{dvol} +\frac{B}{\mathrm{vol}_g(M)^{\frac{2}{n}}}\int _M |u|^{p_0}\mathrm{dvol}\qquad \end{aligned}$$

(2.1)

holds true for all $u \in W^{1, p_0}(M)$ with some $A>0$ and $B>0$. Then we have

$$\begin{aligned} \left( \int _M |u|^{\frac{pn}{n-p}} \right) ^{\frac{n-p}{n}}&\le 2^{\frac{p-p_0}{p_0}} A^{\frac{p}{p_0}}(r(p)^{p_0}+B)^{\frac{p}{p_0}} \int _M |\nabla u|^p +\frac{2^{\frac{p-p_0}{p_0}}B^{\frac{2p}{p_0}}}{\mathrm{vol}_g(M)^{\frac{p}{n}}} \int _M u^p\nonumber \\ \end{aligned}$$

(2.2)

for each $p_0<p \le \frac{n^2p_0}{(n-p_0)^2+np_0}$ and all $u \in W^{1, p}(M)$, where $r(p)=\frac{p(n-p_0)}{p_0(n-p)}$ and the notation of the volume form is omitted. If $\mathrm{vol}_g(M)=\infty$, it is understood that the second term on the right hand side in (2.1) and (2.2) is zero, and $B$ is absent elsewhere.

Proof

We present the case $\mathrm{vol}_g(M)<\infty$, while the case $\mathrm{vol}_g(M)=\infty$ is similar and much easier. By scaling invariance we can assume $\mathrm{vol}_g(M)=1$. Consider $u \in C_c^{\infty }(M)$ and set $v=|u|^r$ for $r>1$. Then we have

$$\begin{aligned} \left( \int _M |u|^{\frac{np_0r}{n-p_0}}\right) ^{\frac{n-p_0}{n}} \le A r^{p_0} \int _M |u|^{p_0(r-1)} |\nabla u|^{p_0} +B\int _M |u|^{p_0r}. \end{aligned}$$

(2.3)

For a given $p_0<p<n$ we choose $r=r_p$ and hence $r_p-1=\frac{n(p-p_0)}{p_0(n-p)}$. Then we infer by Hölder’s inequality

$$\begin{aligned} \left( \int _M |u|^{\frac{np}{n-p}}\right) ^{\frac{n-p_0}{n}}&\le Ar_p^{p_0} \int _M |u|^{\frac{n(p-p_0)}{n-p}} |\nabla u|^{p_0} + B\int _M |u|^{\frac{p(n-p_0)}{n-p}} \nonumber \\&\le Ar_p^{p_0} \left( \int _M |u|^{\frac{np}{n-p}} \right) ^{\frac{p-p_0}{p}} \cdot \left( \int _M |\nabla u|^p \right) ^{\frac{p_0}{p}}+ B\int _M |u|^{\frac{p(n-p_0)}{n-p}}.\nonumber \\ \end{aligned}$$

(2.4)

Now we assume $p_0<p\le \frac{n^2p_0}{(n-p_0)^2+np_0}$. Then $\frac{p(n-p_0)}{n-p}\le \frac{np_0}{n-p_0)}$. Since $\mathrm{vol}_g(M)=1$ we have by Hölder’s inequality

$$\begin{aligned} \int _M |u|^{\frac{p(n-p_0)}{n-p}} \le \left( \int _M |u|^{\frac{np_0}{n-p_0}} \right) ^{\frac{p(n-p_0)^2}{np_0(n-p)}} \end{aligned}$$

(2.5)

and

$$\begin{aligned} \int _M |u|^{\frac{np_0}{n-p_0}} \le \left( \int _M |u|^{\frac{np}{n-p}}\right) ^{\frac{p_0(n-p)}{(n-p_0)p}}. \end{aligned}$$

(2.6)

We deduce

$$\begin{aligned} \left( \int _M |u|^{\frac{pn}{n-p}} \right) ^{\frac{(n-p_0)}{n}-\frac{p-p_0}{p}}&\le Ar_p^{p_0} \left( \int _M |\nabla u|^p \right) ^{\frac{p_0}{p}} \nonumber \\&+\, B \left( \int _M |u|^{\frac{np_0}{n-p_0}} \right) ^{\frac{p(n-p_0)^2}{np_0(n-p)}-\frac{(n-p_0)(p-p_0)}{p_0(n-p)}}, \end{aligned}$$

(2.7)

which leads to

$$\begin{aligned} \left( \int _M |u|^{\frac{pn}{n-p}} \right) ^{\frac{p_0(n-p)}{np}}&\le Ar_p^{p_0} \left( \int _M |\nabla u|^p \right) ^{\frac{p_0}{p}}+B \left( \int _M |u|^{\frac{np_0}{n-p_0}} \right) ^{\frac{n-p_0}{n}} \\&\le Ar_p^{p_0} \left( \int _M |\nabla u|^p \right) ^{\frac{p_0}{p}}+AB \int _M |\nabla u|^{p_0} +B^2 \int _M u^{p_0} \\&\le A(r_p^{p_0}+B) \left( \int _M |\nabla u|^p \right) ^{\frac{p_0}{p}} +B^2 \left( \int _M u^p\right) ^{\frac{p_0}{p}}. \end{aligned}$$

It follows that

$$\begin{aligned} \left( \int _M |u|^{\frac{pn}{n-p}} \right) ^{\frac{n-p}{n}}&\le 2^{\frac{p-p_0}{p_0}} A^{\frac{p}{p_0}}\left( r_p^{p_0}+B\right) ^{\frac{p}{p_0}} \int _M |\nabla u|^p +2^{\frac{p-p_0}{p_0}}B^{\frac{2p}{p_0}} \int _M u^p.\qquad \qquad \end{aligned}$$

(2.8)

By approximation, this holds for all $u \in W^{1,p}(M)$. $\square$

Remark

We can also consider the following assumption

$$\begin{aligned} \left( \int _M |u|^{\frac{np_0}{n-p_0}}\mathrm{dvol}\right) ^{\frac{n-p_0}{n}} \le A \int _M |\nabla u|^{p_0}\mathrm{dvol} +\int _M f |u|^{p_0}\mathrm{dvol} \end{aligned}$$

(2.9)

with a given function $f$. It is easy to adapt the above proof to obtain Sobolev inequalities for higher $p$ in terms of an $L^q$ bound of $f$ for a suitable $q$. This can be applied to the Ricci flow to yield Sobolev inequalities in terms of an $L^q$ bound of the scalar curvature.

Lemma 2.2

Let $1\le p_0<n$. We set $p_{k+1}=\frac{n^2p_k}{(n-p_k)^2+np_k}$ for $k\ge 0$. Then $1\le p_k <n$ for all $k$. Moreover, the sequence $p_k$ is increasing and converges to $n$.

Proof

The inequality $p_{k+1} <n$ is equivalent to $(n-p_k)^2>0$, while the inequality $p_k\ge 1$ is equivalent to $(n^2+n)p_k+p_k^2 \ge n^2$. Hence $1\le p_k<n$ follows from the induction. Since $p_k<n$, we have $(n-p_k)^2+np_k<n^2$, and hence $p_{k+1}>p_k$. Let $p_*$ denote the limit of $p_k$. Then $p_*=\frac{n^2 p_*}{(n-p_*)^2+np_*}$. It follows that $p_*=n$. $\square$

Proof of Theorem 1.1

Applying Theorem 2.1 repeatedly, starting with $p_0=2$. By induction and Lemma 2.2, we then arrive at the desired Sobolev inequalities. $\square$

Proof of Theorem 1.3

We first observe the following property of the inequality (1.5): if $A=\alpha A_1$ and $B=\alpha B_1$ for some $\alpha \ge 1$, then we have

$$\begin{aligned} C_1(n, p_0, p, A, B) \le \alpha ^{\frac{m(p)p}{p_0}} C_1(n, p_0, p, A_1, B_1) \end{aligned}$$

(2.10)

and

$$\begin{aligned} C_2(n, p_0, p, A, B) \le \alpha ^{\frac{m(p)p}{p_0}} C_2(n, p_0, p, A_1, B_1), \end{aligned}$$

(2.11)

where $m(p)=2^{k+1}$ for $p \in (p_k, p_{k+1}]$ (see Lemma 2.2 for $p_k$). This follows from the formula (2.2). By [Theorem $\text{ D }^*$, 12], the Sobolev inequality (1.2) holds true, where $A$ has the dependence as stated in Theorem 1.3, without reference to $p$. We then have

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}} \mathrm{dvol}\right) ^{\frac{n-2}{n}} \le A\left( 1+\max R^+ \mathrm{vol}(M)^{\frac{2}{n}}\right) \int _M \left( |\nabla u|^2 +\frac{u^2}{\mathrm{vol}(M)^{\frac{2}{n}}}\right) \mathrm{dvol}.\nonumber \\ \end{aligned}$$

(2.12)

Applying Theorem 1.1 and the above observation we then arrive at the desired Sobolev inequality. $\square$

Theorem 1.2 can be proved in the same way.

3 Non-local Sobolev Inequalities in Terms of the $(1,p)$-Bessel Norm

First we extend the general results in [9] on the heat semi-group and the non-local Sobolev inequalities to general metrically complete manifolds with or without boundary. Consider a Riemannian manifold $(M,g)$ of dimension $n \ge 2$, and a function $\Psi \in L^{\infty }(M)$. We set as in [9] $H=-\Delta +\Psi$ and $Q(u)=\int _M (|\nabla u|^2+\Psi u^2)\mathrm{dvol}.$

Theorem 3.1

Let $(M, g)$ be metrically complete manifold possibly with boundary. Let $0<\sigma ^*\le \infty$. Assume that for each $0<\sigma <\sigma ^*$ the logarithmic Sobolev inequality

$$\begin{aligned} \int _M u^2 \ln u^2 \mathrm{dvol} \le \sigma Q(u)+ \beta (\sigma ) \end{aligned}$$

(3.1)

holds true for all $u \in W^{1,2}(M)$ with $\Vert u\Vert _2=1$, where $\beta$ is a non-increasing continuous function. Assume that

$$\begin{aligned} \tau (t)=\frac{1}{2t}\int ^t_0 \beta (\sigma )d\sigma \end{aligned}$$

(3.2)

is finite for all $0<t < \sigma ^*$. Then there holds

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{\tau (t)-\frac{3t}{4}\inf \Psi ^-} \Vert u\Vert _2 \end{aligned}$$

(3.3)

for each $0<t< \frac{1}{4}\sigma ^*$ and all $u \in L^2(M)$. There also holds

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le e^{2\tau (\frac{t}{2})-\frac{3t}{4} \inf \Psi ^-} \Vert u\Vert _1 \end{aligned}$$

(3.4)

for each $0<t< \frac{1}{4}\sigma ^*$ and all $u \in L^1(M)$.

Theorem 3.2

Let $(M,g)$ be a metrically complete manifold, possibly with boundary.

• 1) Let $\mu >1$. Assume that $\Psi \ge 0$ and for some $c>0$ the inequality

  $$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le c t^{-\frac{\mu }{4}} \Vert u\Vert _2 \end{aligned}$$

  (3.5)

  holds true for each $t>0$ and all $u\in L^2(M)$. Let $1<p<\mu$. Then there holds

  $$\begin{aligned} \Vert H^{-\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \le C(c, \mu , p) \Vert u\Vert _p \end{aligned}$$

  (3.6)

  for all $u \in L^p(M)$, where the positive constant $C(\mu , c, p)$ can be bounded from above in terms of upper bounds for $c$, $\mu$, $\frac{1}{\mu -p}$ and $\frac{1}{p-1}$. Consequently, there holds

  $$\begin{aligned} \Vert u\Vert _{\frac{2\mu }{\mu -2}} \le C(c, 2, p) \left( \int _M (|\nabla u|^2+\Psi u^2)\mathrm{dvol}\right) ^{\frac{1}{2}} \end{aligned}$$

  (3.7)

  for all $u \in W^{1,2}(M)$. Moreover, there holds for a given $1<p<\infty$

  $$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C(c, \mu , p) \Vert H^{\frac{1}{2}}u\Vert _p \end{aligned}$$

  (3.8)

  for all $u \in W^{1,p}(M)$, provided that $(M, g)$ is $p$-Bessel.

• 2) Let $\mu >1$. Assume that for some $c>0$ the inequality

  $$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le c t^{-\frac{\mu }{4}}\Vert u\Vert _2 \end{aligned}$$

  (3.9)

  holds true for each $0<t<1$ and all $u\in L^2(M)$. Set $H_0=H-\inf \Psi ^-+1$. Let $1<p<\mu$. Then there holds

  $$\begin{aligned} \Vert H_0^{-\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu ,c, p) \Vert u\Vert _p \end{aligned}$$

  (3.10)

  for all $u \in L^p(M)$, where the positive constant $C(\mu ,c, p)$ has the same property as the $C(\mu , c, p)$ above. Consequently, there holds

  $$\begin{aligned} \Vert u\Vert _{\frac{2\mu }{\mu -2}} \le C(\mu , 2, p) \left( \int _M (|\nabla u|^2 +(\Psi -\inf \Psi ^-+1) u^2)\mathrm{dvol}\right) ^{\frac{1}{2}} \end{aligned}$$

  (3.11)

  for all $u \in W^{1,2}(M)$. Moreover, there holds for a given $1<p<\infty$

  $$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu , c, p) \Vert H_0^{\frac{1}{2}}u\Vert _p \end{aligned}$$

  (3.12)

  for all $u \in W^{1,p}(M)$, provided that $(M, g)$ is $p$-Bessel.

To establish these two results, we need the following two ingredients: the construction of the heat semi-group $\mathrm{e}^{-tH}$ and the $L^p$ contraction properties of $\mathrm{e}^{-tH}$ for all $1<p<\infty$. In [9], since the manifold is assumed to be closed, the heat semi-group is constructed by the spectral representation in terms of the eigenfunctions. This works equally well on a compact manifold with boundary, where the eigenfunctions satisfy the Neumann boundary condition. For a general metrically complete manifold, we follow the construction in [8] based on the general theory of spectral representation of self-adjoint operators. The case of $H=-\Delta$ on a complete manifold without boundary is treated in [8], but the arguments extend to general metrically complete manifolds and $H=-\Delta +\Psi$ with $\Psi \in L^{\infty }(M)$ and $\Psi \ge 0$, as explained below.

Consider $\Psi \in L^{\infty }(M)$ with $\Psi \ge 0$. The initial domain for $H=-\Delta +\Psi$ is the space $\Omega _H=C^{\infty }_{c,N}(M)=\{ u \in C^{\infty }_c(M): \frac{\partial u}{\partial \nu }=0\}$, where $\nu$ denotes the inward unit normal of $\partial M$. Note that $\Omega _H$ is dense in $L^2(M)$. Let $H_{\mathrm{min}}$ denote the $L^2$ closure of $H$, whose domain $D(H_{\mathrm{min}})$ consists of all $u \in L^2(M)$ such that there is a sequence $u_i \in \Omega _H$ such that $u_i \rightarrow u$ in $L^2(M)$ and $Hu_i$ converges in $L^2(M)$ to some function, which we can write $Hu$. Let $H_\mathrm{max}$ be the adjoint of $H_{\mathrm{min}}$ in $L^2(M)$, and $D(H_\mathrm{max}) \subset L^2(M)$ its domain. We have the following extension of [8, Lemma 2.3].

Lemma 3.3

Let $(M, g)$ be metrically complete. Assume that $u \in D(H_{\mathrm{max}})$ satisfies $Hu=\lambda u$ for some $\lambda <0$. Then $u \equiv 0$.

Proof

By basic elliptic regularity, we have $u\in W^{2,p}_{loc}(M)$ for all $p>0$ and $\frac{\partial u}{\partial \nu }=0$. ( By the Sobolev embedding we have $u \in C^1(M)$. ) Fix $x_0 \in M$ and let $\varphi (x)=\varphi _{r_1, r_2}(x) =\psi ((r_2-r_1)^{-1}(d(x_0,x)+r_2-2r_1))$ for a smooth function $\psi (t)$ which is $1$ for $t\le 1$ and $0$ for $t\ge 2$. Then $\varphi (x)=1$ if $d(x_0, x) \le r_1$, $\varphi (x)=0$ if $d(x_0, x) \ge r_2$, and $|\nabla \varphi | \le c(r_2-r_1)^{-1}$ for a constant $c$.

Now we have

$$\begin{aligned} \lambda \langle \varphi ^2u, u\rangle _2&= \langle \varphi ^2 u, Hu\rangle _2 = -\langle \varphi ^2 u, \Delta u\rangle _2 +\langle \varphi ^2 u, \Psi u\rangle \nonumber \\&\ge -\langle \varphi ^2 u, \Delta u\rangle _2 =\Vert \varphi ^2\nabla u\Vert _2^2+2\langle u \nabla \phi , \phi \nabla u\rangle _2. \end{aligned}$$

(3.13)

It follows that

$$\begin{aligned} \Vert \varphi ^2\nabla u\Vert _2^2 \le \lambda \langle \varphi ^2 u, u\rangle _2 +2\langle u \nabla \phi , \phi \nabla u\rangle _2. \end{aligned}$$

(3.14)

By Schwarz inequality we then deduce

$$\begin{aligned} \Vert \varphi ^2\nabla u\Vert _2^2 \le 2\lambda \langle \varphi ^2 u, u\rangle _2 +\frac{4c^2}{(r_2-r_1)^2}\Vert u\Vert ^2_2. \end{aligned}$$

(3.15)

Letting first $r_2 \rightarrow \infty$ and then $r_1 \rightarrow \infty$ we arrive at

$$\begin{aligned} \Vert \nabla u\Vert ^2_2 \le 2\lambda \Vert u\Vert _2^2. \end{aligned}$$

(3.16)

Since $\lambda <0$, we conclude $u \equiv 0$. $\square$

By this lemma and [8, Lemma 2.1] we infer that $H_{\mathrm{max}}=H_{\mathrm{min}}$, which is the self-adjoint extension of $H$. Now we can apply the spectral theorem for self-adjoint operators to obtain the heat semi-group $\mathrm{e}^{-tH}$ and other potentials of $H$ such as $H^{-\frac{1}{2}}$ and $H^{\frac{1}{2}}$.

In [9], the $L^p$ contraction property of $\mathrm{e}^{-tH}$ is derived in terms of the $L^2$ contraction property and the $L^{\infty }$ contraction property, with the latter implied by the maximum principle. This argument can be extended to compact manifolds with boundary. But the maximum principle may not hold on general complete, non-compact manifolds. Instead, we follow the arguments in [8] for obtaining the $L^p$ contraction property. By the arguments in Sect. 3 of [8], in order to show that $\mathrm{e}^{-tH}$ is a contraction on $L^p(M) \cap L^2(M)$ for each $1 \le p \le \infty$, it suffices to establish the following two lemmas.

Lemma 3.4

For each $1<p<\infty$ the operator $H$ with domain $\Omega _H$ is dissipative, i.e., for each nonzero $u \in \Omega _H$, there is a function $v \in L^q$ with $q=\frac{p}{p-1}$ such that $\Vert v\Vert _q=\Vert u\Vert _p, \langle u,v\rangle _2=\Vert u\Vert _p$ and $\langle Hu, v\rangle _2 \le 0$.

Lemma 3.5

Let $1<p\le q<\infty$. Assume that $u \in L^p(M) \cap L^q(M)$ satisfies $Hu=\lambda u$ for some $\lambda <0$ (this contains the assumption that $u$ lies in the domain of the closure of $H$ in $L^p(M)$ and that in $L^q(M)$.) Then $u\equiv 0$.

Since $\Psi \ge 0$, the proofs of these two lemmas in [8] (for the special case that $M$ has no boundary and $\Psi =0$) can easily be adapted to our situation. This is similar to the above proof of Lemma 3.3.

Having established the desired construction of $\mathrm{e}^{-tH}$ and the $L^p$ contraction properties we make two more remarks. First, the construction of the heat semi-group $\mathrm{e}^{-tH}$ for a general $\Psi \in L^{\infty }(M)$ follows via the formula $\mathrm{e}^{-tH}=e^{-t \inf \Psi ^-} e^{-tH_1}$, where $H_1=-\Delta +\Psi -\inf \Psi ^-$. Second, in [9], the space $L^{\infty }(M)$ is used in the formulations of the Marcinkiewicz interpolation theorem and the Riesz-Thorin interpolation theorem. In the case of a general Riemannian manifold, we replace $L^{\infty }(M)$ by $L^{\infty }(M) \cap L^1(M)$.

Proof of Theorem 3.1

Consider $u_0 \in L^2(M)$. We claim that $\mathrm{e}^{-tH}u_0 \in W^{1,2}(M)$ for $t>0$. Indeed we have for $u=e^{-tH}u_0$

$$\begin{aligned} \frac{\partial u}{\partial t}=Hu, \, \, \, \frac{\partial u}{\partial \nu }=0. \end{aligned}$$

(3.17)

Then we have

$$\begin{aligned} \frac{d}{dt} \int _M \varphi ^2 u^2 \!=\! 2\int _M \varphi ^2 u Hu = -2\int _M \varphi ^2 |\nabla u|^2 -2\int _M \varphi u \nabla u \cdot \nabla \varphi -2\int _M \Psi \varphi ^2 u^2,\nonumber \\ \end{aligned}$$

(3.18)

where $\varphi =\varphi _{r_2,r_1}$ is the function in the proof of Lemma 3.3. It follows that

$$\begin{aligned} \frac{d}{dt} \int _M \varphi ^2 u^2 \le -\int _M \varphi ^2 |\nabla u|^2 +\int _M u^2 |\nabla \varphi |^2 -\int _M \Psi \varphi ^2 u^2, \end{aligned}$$

(3.19)

and then

$$\begin{aligned} \int _M \varphi ^2 u^2 +\int _0^t \int _M \varphi ^2 |\nabla u|^2 \le \int _M \varphi ^2 u^2 |_{t=0} + \int _0^t \int _M (|\nabla \varphi |^2-\Psi )u^2.\qquad \end{aligned}$$

(3.20)

Letting $r_2 \rightarrow \infty$ and then $r_1 \rightarrow \infty$ we arrive at

$$\begin{aligned} \int _M u^2 +\int _0^t \int _M |\nabla u|^2 \le \int _M u^2 |_{t=0} -\int _0^t \int _M \Psi u^2. \end{aligned}$$

(3.21)

It follows that $\int _M |\nabla u(\cdot , t)|^2 <\infty$ for a.e. $t>0$. By continuity, $\int _M |\nabla u(\cdot , t)|^2<\infty$ for all $t>0$. Hence $\mathrm{e}^{-tH}u_0 \in W^{1,2}(M)$ for all $t>0$.

Now we can carry over the proof of Theorem 5.3 in [9]. Some modification is necessary because $M$ is possibly non-compact. Let $\varphi =\varphi _{r_1, r_2}$ be the function in the proof of Lemma 3.3. In place of [9, (B.13)] we have now for $u_s=e^{-sH}u_0$ for a given $u_0 \in W^{1,2}(M) \cap L^{\infty }(M)$

$$\begin{aligned}&\frac{d}{ds}\ln \left( e^{-N(s)}\Vert \varphi u_s \Vert _{p(s)}\right) =\frac{d}{ds}\left( -N(s)+\frac{1}{p(s)} \ln \Vert \varphi u_s\Vert _{p(s)}^{p(s)}\right) \nonumber \\&\quad =\frac{\Gamma }{\sigma }-\frac{1}{p^2}\frac{p}{\sigma } \ln \Vert \varphi u_s\Vert _p^p +\frac{1}{p} \Vert \varphi u_s\Vert _p^{-p} \left( -pQ\left( \varphi u_s, u_s^{p-1}\right) +\frac{p}{\sigma } \int _M \varphi u_s^p \ln u_s\right) \nonumber \\&\quad =\frac{1}{\sigma }\Vert \varphi u_s\Vert ^{-p}_p \left( \int _M \varphi u_s^p \ln u_s - \sigma Q\left( \varphi u_s, u_s^{p-1}\right) \right. \nonumber \\&\left. \qquad -\Gamma \Vert \varphi u_s\Vert _p^p-\Vert \varphi u_s\Vert _p^p \ln \Vert \varphi u_s\Vert _p \right) . \end{aligned}$$

(3.22)

It follows that

$$\begin{aligned}&\ln \left( e^{-N(t_2)}\Vert \varphi u_{t_2} \Vert _{p(t_2)}\right) \le \ln \left( e^{-N(t_1)}\Vert \varphi u_{t_1} \Vert _{p(t_1)}\right) \nonumber \\&+ \int _{t_1}^{t_2} \frac{1}{\sigma }\Vert \varphi u_s\Vert ^{-p}_p \left( \int _M \varphi u_s^p \ln u_s\right. \nonumber \\&\left. - \sigma Q\left( \varphi u_s, u_s^{p-1}\right) -\Gamma \Vert \varphi u_s\Vert _p^p-\Vert \varphi u_s\Vert _p^p \ln \Vert \varphi u_s\Vert _p \right) ds \end{aligned}$$

(3.23)

for $t_2>t_1>0$. Letting first $r_2 \rightarrow \infty$ and then $r_1 \rightarrow \infty$ we then arrive at (3.23) without the presence of $\phi$. $\square$

Proof of Theorem 3.2

Given the $L^p$ contraction property established above, the proof of Theorem C.5 in [9] carries over straightforwardly. The Sobolev inequality (3.7) follows because of the identity $\langle H^{\frac{1}{2}}u, H^{\frac{1}{2}}u\rangle _2 =\int _M (|\nabla u|^2+\Psi u^2)\mathrm{dvol}$ for all $u \in C^{\infty }_c(M)$ (then also for $u \in W^{1,2}(M)$). The Sobolev inequality (3.11) follows in the same fashion. $\square$

Proof of Theorem 1.7

Assume $\int _M u^2 =1$. As in the proof of Theorem 3.1 in [9] we have

$$\begin{aligned} \ln \int _M |u|^{\frac{2\mu }{\mu -2}} =\ln \int _M u^2 |u|^{\frac{4}{\mu -2}} \ge \int _M u^2 \ln |u|^{\frac{4}{\mu -2}}. \end{aligned}$$

(3.24)

It follows that

$$\begin{aligned} \int _M u^2 \ln u^2&\le \frac{\mu }{2} \ln \left( \int _M |u|^{\frac{2\mu }{\mu -2}}\right) ^{\frac{\mu -2}{\mu }} \le \frac{\mu }{2} \ln \left( A\int _M (|\nabla u|^2 +\Psi \int _M u^2 \right) \nonumber \\&\le \frac{\mu }{2} \ln A +\frac{\mu }{2} \ln \int _M \left( |\nabla u|^2+\Psi u^2\right) . \end{aligned}$$

(3.25)

By [9, Lemma 3.2] we then deduce each $\sigma >0$

$$\begin{aligned} \int _M u^2 \ln u^2 \le \frac{\mu }{2} \sigma \int _M \left( |\nabla u|^2+\Psi u^2\right) -\frac{\mu }{2} \ln \sigma +\frac{\mu }{2} \ln A-1, \end{aligned}$$

(3.26)

which leads to

$$\begin{aligned} \int _M u^2 \ln u^2 \le \sigma \int _M \left( |\nabla u|^2+\Psi u^2\right) -\frac{\mu }{2} \ln \sigma +\frac{\mu }{2} \ln \frac{\mu }{2}+\frac{\mu }{2} \ln A-1.\qquad \end{aligned}$$

(3.27)

By Theorem 3.1 we deduce for $H=-\Delta +1$

$$\begin{aligned} \Vert e^{-tH}u\Vert _{\infty } \le t^{-\frac{\mu }{4}} e^{\frac{\mu }{4}+\frac{A_0}{2}} \Vert u\Vert _2 \end{aligned}$$

(3.28)

for all $t>0$, where $A_0=\frac{\mu }{2} \ln \frac{\mu }{2}+\frac{\mu }{2} \ln A-1.$ Applying Theorem 3.2, we then arrive at the desired inequality (1.17). $\square$

Proof of Theorem 1.4

The Sobolev inequality (1.11) leads to

$$\begin{aligned} \left( \int _M |u|^{\frac{2\mu }{\mu -2}}\mathrm{dvol}\right) ^{\frac{\mu -2}{\mu }} \le \max \{A, B\} \int _M \left( |\nabla u|^2+u^2\right) \mathrm{dvol}. \end{aligned}$$

(3.29)

Hence we can apply Theorem 1.7. $\square$

Lemma 3.6

Let $\bar{g}=\lambda ^2 g$ for some $\lambda \ge 1$. Let $\mu >1$ and $1 \le p<\mu$. Assume the inequality

$$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le C \Vert u\Vert _{B, 1, p} \end{aligned}$$

(3.30)

for all $u \in W^{1, p}(M)$ with respect to $\bar{g}$. Then there holds

$$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -p}} \le \lambda C \Vert u\Vert _{B, 1, p} \end{aligned}$$

(3.31)

for all $u \in W^{1,p}(M)$ with respect to $g$.

Proof

We compute the scaling change of $(-\Delta +1)^{\frac{1}{2}}$. We have $\Delta _{\bar{g}}=\lambda ^{-2} \Delta$. Hence

$$\begin{aligned} -\Delta _{\bar{g}}+1=-\lambda ^{-2} \Delta +1=\lambda ^{-2}(-\Delta +\lambda ^2). \end{aligned}$$

(3.32)

By [1, Lemma 4.2], we have for $u \in L^p(M)$ and $a\ge 0$

$$\begin{aligned} c_1\left( a \Vert u\Vert _p+\Vert (-\Delta )^{\frac{1}{2}}u\Vert _p\right) \le \Vert \left( -\Delta +a^2\right) ^{\frac{1}{2}}u\Vert _p \le c_2\left( a \Vert u\Vert _p + \Vert (-\Delta )^{\frac{1}{2}}u\Vert _p\right) ,\nonumber \\ \end{aligned}$$

(3.33)

where $c_1$ and $c_2$ are universal constants. It follows that

$$\begin{aligned} \Vert \left( -\Delta +\lambda ^2\right) ^{\frac{1}{2}} u\Vert _p&\le c_2\left( \lambda \Vert u\Vert _p+\Vert (-\Delta )^{\frac{1}{2}}u\Vert _p\right) \nonumber \\&\le c_2\lambda \left( \Vert u\Vert _p+\Vert (-\Delta )^{\frac{1}{2}}u\Vert _p\right) \nonumber \\&\le c_1c_2 \lambda \Vert \left( -\Delta +1\right) ^{\frac{1}{2}}u\Vert _p. \end{aligned}$$

(3.34)

Hence

$$\begin{aligned} \Vert \left( -\Delta _{\bar{g}}+1\right) ^{\frac{1}{2}}u\Vert _p \le c_1c_2 \Vert \left( -\Delta +1\right) ^{\frac{1}{2}}u\Vert _p. \end{aligned}$$

(3.35)

Now we have

$$\begin{aligned} \Vert u\Vert _{\frac{pn}{n-p}, \bar{g}}=\Vert u\Vert _{\frac{np}{n-p}} \lambda ^{\frac{n-p}{p}} \end{aligned}$$

(3.36)

and

$$\begin{aligned} \Vert \left( -\Delta _{\bar{g}}+1\right) ^{\frac{1}{2}}u\Vert _{p, \bar{g}} = \Vert \left( -\Delta _{\bar{g}}+1\right) ^{\frac{1}{2}}u\Vert _p \lambda ^{\frac{n}{p}}. \end{aligned}$$

(3.37)

We arrive at

$$\begin{aligned} \Vert u\Vert _{\frac{np}{n-p}} \le \lambda C\Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p. \end{aligned}$$

(3.38)

$\square$

Proof of Theorem 1.5

By [9, Theorem $D$], the Sobolev inequality (1.3) holds true, where $A$ and $B$ have the same property as the $C$ in the theorem, without the reference to $p$. Let $\lambda$ and $\bar{g}$ be the same as above. Then we have for $\bar{g}$

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}}\mathrm{dvol}\right) ^{\frac{n-2}{n}}&\le A\int _M \left( |\nabla u|^2+\frac{R}{4}u^2\right) \mathrm{dvol}+\frac{B}{\lambda ^2} \int _M u^2\mathrm{dvol} \nonumber \\&\le A \int _M |\nabla u|^2 \mathrm{dvol}+ \left( \frac{A}{4}+B\right) \int _M u^2 \mathrm{dvol} \nonumber \\&\le (A+B) \int _M \left( |\nabla u|^2+u^2\right) \mathrm{dvol}. \end{aligned}$$

(3.39)

Applying Theorem 1.4 and Lemma 3.6, we arrive at the desired inequality (1.21). $\square$

Proof of Theorem 1.6

By [9, Theorem $\text{ D }^*$], the Sobolev inequality (1.2) holds true, where $A$ has the same property as the $C$ in the theorem without the reference to $p$. Set $\lambda =\lambda (t)=(1+R_{\mathrm{max}}^+)^{1/2}$ at time $t$. Then the Sobolev inequality (1.2) still holds true for $\bar{g}=\lambda ^2 g$. Since $R^+_{\mathrm{max}} \le 1$ for $\bar{g}$, we deduce

$$\begin{aligned} \left( \int _M |u|^{\frac{2n}{n-2}}\mathrm{dvol}\right) ^{\frac{n-2}{n}} \le A\int _M \left( |\nabla u|^2+ u^2\right) \mathrm{dvol}. \end{aligned}$$

(3.40)

Applying Theorem 1.4 and Lemma 3.6 we then arrive at the desired inequality (1.22). $\square$

4 $W^{2,p}$ Sobolev Inequalities

Proof of Theorem 1.8

Let $u \in W^{2,p}(M)$ for $1<p<\frac{\mu }{2}$. Since $(-\Delta +\Psi )^{\frac{1}{2}}$ is a pseudo-differential operator of order $1$ [7] on a compact manifold, it is a bounded map from $W^{2,p}(M)$ into $W^{1,p}(M)$. Hence $v=(-\Delta +\Psi )^{\frac{1}{2}}u \in W^{1,p}(M)$. Applying Theorem 1.7 to $v$ we infer

$$\begin{aligned} \Vert v\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu , A, p) \Vert (-\Delta +\Psi )^{\frac{1}{2}}v\Vert _p =C(\mu , A, p) \Vert (-\Delta +\Psi )u\Vert _p, \end{aligned}$$

(4.1)

i.e.,

$$\begin{aligned} \Vert (-\Delta +\Psi )^{\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \le C(\mu , A, p) \Vert (-\Delta +\Psi )u\Vert _p. \end{aligned}$$

(4.2)

For each $1<q<\infty$, $(-\Delta +\Psi )^{\frac{1}{2}}$ is a bounded operator from $W^{1, q}(M)$ into $L^q(M)$ with the bounded inverse $(-\Delta +\Psi )^{-\frac{1}{2}}$. Hence we deduce $u \in W^{1, \frac{\mu p}{\mu -p}}(M)$. Applying Theorem 1.7 to $u$ with the exponent $\frac{\mu p}{\mu -p}$ instead of $p$ we then infer

$$\begin{aligned} \Vert u\Vert _{\frac{\mu p}{\mu -2p}}&\le C\left( \mu , A, \frac{\mu p}{\mu -p}\right) \Vert (-\Delta +\Psi )^{\frac{1}{2}}u\Vert _{\frac{\mu p}{\mu -p}} \nonumber \\&\le C\left( \mu , A, \frac{\mu p}{\mu -p}\right) C(\mu , A, p) \Vert (-\Delta +\Psi )u\Vert _p. \end{aligned}$$

(4.3)

(Note that $1<\frac{\mu p}{\mu -p}<\mu$ because $1<p<\frac{\mu }{2}$.) $\square$

Theorems 1.9 and 1.10 follow from Theorem 1.8, and [9, Theorem D] and [9, Theorem $\text{ D }^*$], respectively.

5 Estimates of the Riesz Transform and $W^{1,p}$ Sobolev Inequalities

The following theorem is a consequence of Bakry’s result on $L^p$ estimates for the Riesz transform [1].

Theorem 5.1

Let $(M, g)$ be a complete Riemannian manifold (without boundary) of dimension $n \ge 2$ such that the Ricci curvature is bounded from below by $-a^2$ for some $0\le a <\infty$. Then there holds for each $1<p<\infty$

$$\begin{aligned} \Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p \le C(p)(\Vert \nabla u\Vert _p+(1+a)\Vert u\Vert _p) \end{aligned}$$

(5.1)

for all $u \in W^{1,p}(M)$, where the constant $C(p)$ depends only on $p$.

Proof

In [1] the operator $-\Delta +\nabla \phi \cdot \nabla$ for a given function $\phi$ is handled. It is easy to see that all the arguments in [1] go through for the operator $-\Delta +1$. Hence [1, Theorem 4.1] extends to yield for $1<q<\infty$

$$\begin{aligned} \Vert \nabla v\Vert _q \le C_q \left( \Vert (-\Delta +1)^{\frac{1}{2}}v\Vert _q+a\Vert v\Vert _q\right) \end{aligned}$$

(5.2)

for all $u \in C^{\infty }_c(M)$, where $C_q$ depends only on $q$. On the other hand, we have $\Vert \mathrm{e}^{t(\Delta -1)}v\Vert _q \le \mathrm{e}^{-t}\Vert v\Vert _q$ for $1<q<\infty$ and all $v\in L^q(M) \cap L^2(M)$. Applying the formula $(-\Delta +1)^{-\frac{1}{2}}=\Gamma (\frac{1}{2})^{-1}\int _0^{\infty } t^{-\frac{1}{2}}\mathrm{e}^{t(\Delta -1)}\mathrm{d}t$ we infer $\Vert (-\Delta +1)^{-\frac{1}{2}}v\Vert _q \le \Vert v\Vert _q$ and hence

$$\begin{aligned} \Vert v\Vert _q \le \Vert (-\Delta +1)^{\frac{1}{2}}v\Vert _q \end{aligned}$$

(5.3)

for all $v \in L^q(M)\cap L^2(M)$.

Since $(-\Delta +1)^{\frac{1}{2}}(C^{\infty }_c(M))$ is dense in $L^q(M)$ (see [7, 8]), we have for $u \in C^{\infty }_c(M)$, $1<p<\infty$ and $q=\frac{p}{p-1}$

$$\begin{aligned} \Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p&= \sup \left\{ \langle (-\Delta +1)^{\frac{1}{2}}u, (-\Delta +1)^{\frac{1}{2}}v\rangle _2: \right. \nonumber \\&\left. v \in C^{\infty }_c(M), \Vert (-\Delta +1)^{\frac{1}{2}}v\Vert _q \le 1\right\} . \end{aligned}$$

(5.4)

But

$$\begin{aligned} \langle (-\Delta +1)^{\frac{1}{2}}u, (-\Delta +1)^{\frac{1}{2}}v\rangle _2&= \langle (-\Delta +1)u, v\rangle _2= \int _M \nabla u \cdot \nabla v + \int _M uv \nonumber \\&\le \Vert \nabla u\Vert _p \Vert \nabla v\Vert _q+ \Vert u\Vert _p \Vert v\Vert _q \nonumber \\&\le (\Vert \nabla u\Vert _p+\Vert u\Vert _p) (\Vert \nabla v\Vert _q+\Vert v\Vert _q). \end{aligned}$$

(5.5)

By (5.2) and (5.3) we then deduce $\Vert \nabla v\Vert _q+\Vert v\Vert _q \le (1+(a+1)C(q))\Vert (-\Delta +1)^{\frac{1}{2}}v\Vert _q$. By (5.4) and (5.5) we then arrive at

$$\begin{aligned} \Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p \le C(p,a)(\Vert \nabla u\Vert _p+\Vert u\Vert _p), \end{aligned}$$

(5.6)

where $C(p,a)=1+(a+1)C_{\frac{p}{p-1}}$. By (5.2), (5.3) (applied to $p$) and (5.6) we conclude that $(M, g)$ is $(1,p)$-Bessel and that (5.6) holds true for all $u \in W^{1,p}(M)$.

To derive the inequality (5.13), we consider the metric $\bar{g}=\lambda ^2 g$, where $\lambda =1+a$. Since the Ricci curvature of $\bar{g}$ is bounded from below by $-\frac{a^2}{(1+a)^2} \ge -1$, we have by (5.6)

$$\begin{aligned} \Vert (-\Delta _{\bar{g}}+1)^{\frac{1}{2}}u\Vert _{p, \bar{g}} \le C(p, 1)(\Vert \nabla _{\bar{g}} u\Vert _{p, \bar{g}}+\Vert u\Vert _{p, \bar{g}}) \end{aligned}$$

(5.7)

for $1<p<\infty$ and all $u \in L^p(M)$. But $\Delta _{\bar{g}}= \lambda ^{-2} \Delta$. Hence we obtain

$$\begin{aligned} \Vert (-\Delta +\lambda ^2)^{\frac{1}{2}}u\Vert _{p, \bar{g}} \le \lambda C(p, 1) (\Vert \nabla _{\bar{g}} u\Vert _{p, \bar{g}}+\Vert u\Vert _{p, \bar{g}}). \end{aligned}$$

(5.8)

Transforming to $g$ we obtain

$$\begin{aligned} \Vert (-\Delta +\lambda ^2)^{\frac{1}{2}}u\Vert _{p} \le C(p, 1) (\Vert \nabla u\Vert _{p}+\lambda \Vert u\Vert _{p}). \end{aligned}$$

(5.9)

By (3.33) there holds

$$\begin{aligned} \Vert (-\Delta +\lambda ^2)^{\frac{1}{2}}u\Vert _{p}&\ge c_1 (\lambda \Vert u\Vert _p +\Vert (-\Delta )^{\frac{1}{2}}u\Vert _p) \ge c_1(\Vert u\Vert _p+\Vert (-\Delta )^{\frac{1}{2}}u\Vert _p) \nonumber \\&\ge c_1c_2^{-1} \Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p. \end{aligned}$$

(5.10)

It follows that

$$\begin{aligned} \Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p\le c_1^{-1}c_2 C(p, 1) (\Vert \nabla u\Vert _{p}+\lambda \Vert u\Vert _{p}) \end{aligned}$$

(5.11)

which gives rise to (5.13). $\square$

The next result is a consequence of the $L^p$ estimates for the Riesz transform due to Li [5].

Theorem 5.2

Let $(M,g)$ be a complete Riemannian manifold (without boundary) of dimension $n\ge 2$. Assume that there is a constant $c_1$ such that

$$\begin{aligned} \Vert e^{t(\Delta -1)}u\Vert _{\infty } \le c_1t^{-\frac{n}{2}}\Vert u\Vert _1 \end{aligned}$$

(5.12)

for all $u \in L^1(M)$ and $0<t\le 1$. Assume that $(\mathrm{Ric}_{\mathrm{min}}+c_2)^- \in L^{\frac{n}{2}+\epsilon }(M)$ for some $c_2\ge 0$ and $\epsilon >0$. Then there holds for each $1<p<2$

$$\begin{aligned} \Vert (-\Delta +1)^{\frac{1}{2}}u\Vert _p \le C(\Vert \nabla u\Vert _p+(1+\gamma ) \Vert u\Vert _p) \end{aligned}$$

(5.13)

for all $u \in W^{1,p}(M)$, where

$$\begin{aligned} \gamma =\left( \int _M |(\mathrm{Ric}_\mathrm{min}+c_2)^-|^{\frac{n}{2}+\epsilon }\mathrm{dvol}\right) ^{\frac{1}{2\epsilon }} \end{aligned}$$

(5.14)

and the constant $C$ can be bounded above in terms of upper bounds for $n, c_1, c_2, \frac{1}{\epsilon }$ and $\frac{1}{p-1}$.

Proof

This follows from the proof of Theorem 2.2 in [5] and the arguments in the above proof of Theorem 5.1. $\square$

Proof of Theorems 1.11–1.13 Combine Theorem 5.1 with Theorems 1.4–1.6. $\square$

Proof of Theorems 1.14–1.16 Combine Theorem 5.2 with Theorems 1.4–1.6, and also apply Theorem 3.1 and the arguments in the proof of Theorem 1.7. $\square$