Geometric analysis on manifolds with ends

Alexander Grigor’yan Department of Mathematics University of Bielefeld 33501, Bielefeld, Germany [email protected] , Satoshi Ishiwata Department of Mathematical Sciences Yamagata University, Yamagata 990-8560, Japan [email protected] and Laurent Saloff-Coste Department of Mathematics Cornell University, Ithaca, NY, 14853-4201, USA [email protected]

Key words and phrases:

manifold with ends, heat kernel, Poincaré constant

2010 Mathematics Subject Classification:

Primary 58-02, Secondary 35K08, 58J65, 58J35

Partially supported by SFB 1283 of the German Research Council

Partially supported by JSPS, KAKENHI 17K05215

Partially supported by NSF grant DMS–1707589

1. Introduction

2. The state of the art

3. Manifold with ends with oscillating volume functions

References

1. Introduction

In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel estimates from those provided by known results.

Throughout the history of geometric analysis, manifolds with ends have appeared in several contexts. For example, Cai [2], Kasue [19] and Li-Tam [22] et. al. studied manifolds with non-negative Ricci (sectional) curvature outside a compact set where manifolds with ends play an important role. It should be pointed out that there are other recent works on manifolds with ends. See, for instance, Carron [3], Doan [5], Duong, Li and Sikora [6], Hassel, Nix and Sikora [17], Hassel and Sikora [18].

Because of the bottleneck structure inherent to most manifolds with ends, geometric and analytic properties of manifolds with ends are very different from a manifold such as $\mathbb{R}^{n}$ . For example, in 1979, Kuz’menko and Molchanov [21] proved the following:

Theorem 1.1.

On $M=\mathbb{R}^{3}\#\mathbb{R}^{3}$ , the connected sum of two copies of $\mathbb{R}^{3}$ , the weak Liouville property does not hold. Namely, there exists a non-trivial bounded harmonic function.

It is a well-known fact that the parabolic Harnack inequality ((PHI) in short) implies the weak Liouville property. See [14, Section 2.1] and [26, 5.4.5] for details. By a contraposition argument, the above theorem implies that (PHI) does not hold on $\mathbb{R}^{3}\#\mathbb{R}^{3}$ .

Denote by $p\!\left(t,x,y\right)$ the heat kernel of a non-compact weighted manifold $\!(M,d,\mu)$ , that is, the minimal positive fundamental solution of the heat equation $\partial_{t}u=\Delta u$ , where $\Delta$ is the weighed Laplacian. In 1986, Li and Yau proved in [23] that

(LY)\qquad\qquad p(t,x,y)\asymp\frac{c}{V(x,\sqrt{t})}e^{-bd^{2}(x,y)/t}

holds on non-compact manifold with non-negative Ricci curvature. Here $V(x,r):=\mu(B(x,r))$ , the measure of the open geodesic ball $B(x,r)=\{y\in M~{}:~{}d(x,y)<r\}$ and the sign $\asymp$ means that both $\leq$ and $\geq$ hold but with different values of the positive constants $C$ and $b$ . We call this estimate a Li-Yau type bound and write (LY) in short. The following theorem is a combined result of [7], [25] based on previous contributions of Moser [24], Kusuoka–Stroock [20] et al.

Theorem 1.2.

On a geodesically complete, non-compact weighted manifold $M$ , the following conditions are equivalent:

(1) The Li-Yau type heat kernel estimates (LY).

(2) The parabolic Harnack inequality (PHI).

(3) The Poincaré inequality $(PI)$ : there exists $C,\kappa>0$ such that for any $x\in M$ , $r>0$ and $f\in C^{\infty}\left(\overline{B(x,r)}\right)$ ,

(PI)~{}~{}~{}\int_{B(x,r)}|f-f_{B(x,r)}|^{2}d\mu\leq C\int_{B(x,r)}|\nabla f|^{2}d\mu,

where $f_{B(x,r)}=\frac{1}{V(x,r)}\int_{B(x,r)}fd\mu$ , and the volume doubling property $(VD)$ : there exists $C>0$ such that for any $x\in M$ , $r>0$ ,

V(x,2r)\leq CV(x,r).

Combining the above two theorems, the connected sum $M=M_{1}\#M_{2}=\mathbb{R}^{3}\#\mathbb{R}^{3}$ satisfies neither (PI) nor (LY). Indeed, the function

f(x)=\left\{\begin{array}[]{cl}1&x\in M_{1},\\ -1&x\in M_{2}\end{array}\right.

implies that $f_{B(o,r)}=0$ for a central reference point $o\in M$ and

\int_{B(o,r)}|f-f_{B(o,r)}|^{2}d\mu\simeq r^{3},\quad\int_{B(o,r)}|\nabla f|^{2}d\mu\simeq const,

which fails (PI). Here $f\simeq g$ means

cf\leq g\leq Cf

with some positive constants $0<c\leq C$ on a suitable range of functions $f$ , $g$ . Moreover, Benjamini, Chavel and Feldman [1] obtained in 1996 the following heat kernel estimate.

Theorem 1.3.

For $n\geq 3$ , let $M=M_{1}\#M_{2}=\mathbb{R}^{n}\#\mathbb{R}^{n}$ . There exists $\varepsilon>0$ such that for $x\in M_{1}$ , $y\in M_{2}$ with $|x|\simeq|y|\simeq\sqrt{t}$ ,

p(t,x,y)\leq\frac{1}{t^{\frac{n+\varepsilon}{2}}}\ll\frac{1}{t^{n/2}}.

This theorem asserts that the heat kernel between two different ends is significantly smaller than that on one end because of a bottleneck effect.

In view of the above facts, it is natural to ask on a manifold with ends the behavior of the heat kernel $p(t,x,y)$ and the estimate of

\Lambda(B(x,r)):=\sup_{\begin{subarray}{c}f\in C^{1}(\overline{B(x,r)})\\ f\neq\mathrm{const}\end{subarray}}\frac{\int_{B(x,r)}|f-f_{B(x,r)}|^{2}d\mu}{\int_{B(x,r)}|\nabla f|^{2}d\mu},

(1.1)

which is called the Poincaré constant.

Notation. Throughout this article, the letters $c,c^{\prime},C,C^{\prime},C^{\prime\prime}$ denote positive constants whose values may be different at different instances. When the value of a constant is significant, it will be explicitly stated.

2. The state of the art

2.1. Setting

First of all, we begin with the definition of what we call a manifold with finitely many ends. For a fixed integer $k\geq 2$ , let $M_{1},...,M_{k}$ be a sequence of geodesically complete, non-compact weighted manifolds of the same dimension.

Definition 2.1.

We say that a weighted manifold $M$ is a manifold with $k$ ends $M_{1},M_{2},\ldots M_{k}$ and write

M=M_{1}\#...\#M_{k}

(2.1)

if there is a compact set $K\subset M$ so that $M\setminus K$ consists of $k$ connected components $E_{1},E_{2},\ldots,E_{k}$ such that each $E_{i}$ is isometric (as a weighted manifold) to $M_{i}\setminus K_{i}$ for some compact set $K_{i}\subset M_{i}$ (see Fig. 1). Each $E_{i}$ (or $M_{i}$ ) will be referred to as an end of $M$ .

Figure 1. Manifold with ends

Here we remark that the definition of end given above is different from the usual notion defined as a connected component of the ideal boundary.

We say that a manifold $M$ is parabolic if any positive superharmonic function on $M$ is constant, and non-parabolic otherwise. See [8] for details.

Throughout this article, we always assume that each end $M_{i}$ satisfies (VD) and (PI). Moreover, if the end $M_{i}$ is parabolic, then we also assume that $M_{i}$ satisfies the relatively connected annuli condition defined as follows.

Definition 2.2 ((RCA)).

A weighted manifold $M$ satisfies relatively connected annuli condition ((RCA) in short) with respect to a reference point $o\in M$ if there exists a positive constant $A>1$ such that for any $r>A^{2}$ and all $x,y\in M$ with $d(o,x)=d(o,y)=r$ , there exists a continuous path from $x$ to $y$ staying in $B(o,Ar)\backslash B(o,A^{-1}r)$ . See Fig. 2 and 3 for typical positive and negative examples.

Figure 2. Manifold with (RCA)

Figure 3. Manifold without (RCA)

The assumption (RCA) seems technical but it makes it possible to obtain an optimal estimates of the first exit (hitting) probability and the Dirichlet heat kernel in the exterior of a compact set of a parabolic manifold satisfying (LY). See [12] and [13] for details.

2.2. Heat kernel estimates

2.2.1. Off-diagonal estimates

Let $M=M_{1}\#\cdots\#M_{k}$ be a manifold with $k$ ends. For $t>0$ , $x\in E_{i}$ and $y\in M$ , let $p^{D}_{E_{i}}(t,x,y)$ be the extended Dirichlet heat kernel on an end $E_{i}$ , that is, the Dirichlet heat kernel in $y\in E_{i}$ and extension to $0$ if $y\not\in E_{i}$ . Let $\tau_{E_{i}}$ be the first exit time of the Brownian motion from $E_{i}$ and then $\mathbb{P}_{x}(\tau_{E_{i}}<t)$ is the first exit probability starting from $x$ by time $t$ from $E_{i}$ . We will use the following theorem to estimate the off-diagonal heat kernel estimates.

Theorem 2.3 (Grigor’yan and Saloff-Coste [15, Theorem 3.5]).

Let
$M=M_{1}\#\cdots\#M_{k}$ be a manifold with $k$ ends and fix a central reference point $o\in K$ . For $x\in E_{i}$ , $y\in E_{j}$ and $t>1$ ,

	$\displaystyle p(t,x,y)\simeq$	$\displaystyle p^{D}_{E_{i}}(t,x,y)+p(t,o,o)\mathbb{P}_{x}(\tau_{E_{i}}<t)\mathbb{P}_{y}(\tau_{E_{j}}<t)$
		$\displaystyle\!\!\!+\int_{0}^{t}p(s,o,o)ds\left(\partial_{t}\mathbb{P}_{x}(\tau_{E_{i}}<t)\mathbb{P}_{y}(\tau_{E_{j}}<t)+\mathbb{P}_{x}(\tau_{E_{i}}<t)\partial_{t}\mathbb{P}_{y}(\tau_{E_{j}}<t)\right).$		(2.2)

Under the assumption of (PI), (VD) on each end, and, in addition, (RCA) on each parabolic end, applying results in [12] and [13], the quantities $p^{D}_{E_{i}}(t,x,y)$ , $\mathbb{P}_{x}(\tau_{E_{i}}<t)$ , $\mathbb{P}_{y}(\tau_{E_{j}}<t)$ , $\partial_{t}\mathbb{P}_{x}(\tau_{E_{i}})$ and $\partial_{t}\mathbb{P}_{y}(\tau_{E_{j}}<t)$ can be estimated. Hence, estimating $p(t,o,o)$ becomes the key missing estimate to obtain off-diagonal bounds on manifolds with ends.

2.2.2. Non-parabolic case

First we consider heat kernel estimates on $M=M_{1}\#\cdots\#M_{k}$ , where $M$ is non-parabolic, namely, at least one end $M_{i}$ is non-parabolic.

For a fixed reference point $o_{i}\in K_{i}\subset M_{i}$ , let

V_{i}(r):=\mu(B(o_{i},r))

and

h_{i}(r):=1+\left(\int_{1}^{r}\frac{sds}{V_{i}(s)}\right)_{+}.

Here we remark that, under the assumption (LY), $M_{i}$ is parabolic if and only if

\lim_{r\rightarrow\infty}h_{i}(r)=\infty.

In 2009, Grigor’yan and Saloff-Coste [15] obtained the following (see also [16]).

Theorem 2.4.

Let $M=M_{1}\#\cdots\#M_{k}$ be a manifold with $k$ ends. Assume that each end $M_{i}$ satisfies (PI) and (VD) and that each parabolic end satisfies (RCA). Assume also that $M$ is non-parabolic. Then for all $t>0$ ,

p(t,o,o)\simeq\frac{1}{\min_{i}V_{i}(\sqrt{t})h_{i}^{2}(\sqrt{t})}.

If all ends $M_{1},\ldots,M_{k}$ are non-parabolic, then all functions $h_{1},\ldots,h_{k}$ are bounded. Hence, the above theorem implies that

p(t,o,o)\simeq\frac{1}{\min_{i}V_{i}(\sqrt{t})},

namely, the behavior of the heat kernel at the central reference point is determined by the smallest end!

As a typical example, let $M$ be $M_{1}\#M_{2}=\mathbb{R}^{n}\#\mathbb{R}^{n}$ , the connected sum of two copies of $\mathbb{R}^{n}$ with $n\geq 3$ . Then the above theorem implies that

p(t,o,o)\simeq\frac{1}{t^{n/2}}.

Substituting this estimates into Theorem 2.3, we obtain that for $x\in M_{1}$ , $y\in M_{2}$ ,

p(t,x,y)\simeq\frac{1}{t^{n/2}}\left(\frac{1}{|x|^{n-2}}+\frac{1}{|y|^{n-2}}\right)e^{-bd^{2}(x,y)/t},

where $|x|=1+d(x,K)$ .

2.2.3. Parabolic case

Next, we consider the case of manifolds with ends, $M=M_{1}\#\cdots\#M_{k}$ , which are parabolic, that is, for which all ends $M_{1},\ldots,M_{k}$ are parabolic. To prove an optimal heat kernel estimates, we need the following assumptions on each end.

Definition 2.5 (c.f. [10]).

An end $M_{i}$ is called subcritical if

h_{i}(r)\leq C\frac{r^{2}}{V_{i}(r)}\quad~{}(\forall r>1)

and regular if there exist $\gamma_{1},\gamma_{2}>0$ satisfying $2\gamma_{1}+\gamma_{2}<2$ such that

c\left(\frac{R}{r}\right)^{2-\gamma_{2}}\leq\frac{V_{i}(R)}{V_{i}(r)}\leq C\left(\frac{R}{r}\right)^{2+\gamma_{1}}\quad(\forall 1<r\leq R).

(2.3)

For example, a manifold $M$ with volume function $V(r)=r^{\alpha}\left(\log r\right)^{\beta}$ is parabolic if and only if either $\alpha<2$ or $\alpha=2$ and $\beta\leq 1$ . Moreover, $M$ is subcritical if $\alpha<2$ and regular if $\alpha=2$ and $\beta\leq 1$ . We remark that if $M_{i}$ satisfies (VD), then the reverse doubling property holds and that implies that for any subcritical end, there exists $\delta>0$ such that

V_{i}(r)\leq Cr^{2-\delta}~{}(\forall r>0).

(2.4)

For $r>0$ , let $m=m(r)$ be a number so that

V_{m}(r)=\max_{i}V_{i}(r).

(2.5)

We can now state the following result.

Theorem 2.6 (Grigor’yan, Ishiwata, Saloff-Coste [10]).

Let $M=M_{1}\#\!\cdots\!\#M_{k}$ be a manifold with $k$ parabolic ends. Assume that each end $M_{i}$ satisfies (PI), (VD), (RCA) and is either subcritical or regular. If there exist both of subcritical and regular ends, assume also that the constant $\delta$ in (2.4) satisfies $\delta>\gamma_{2}$ , namely, for any subcritical volume function $V_{i}(r)$ and any regular volume function $V_{j}(r)$ ,

V_{i}(r)\leq Cr^{2-\delta}\leq C^{\prime}r^{2-\gamma_{2}}\leq C^{\prime\prime}V_{j}(r)~{}(\forall r>0).

Moreover, assume that there exists an end $M_{m}$ such that for all $i=1,\ldots,k$ and for all $r>0$

V_{m}(r)\geq cV_{i}(r)~{}\mbox{and}~{}V_{m}(r)h_{m}^{2}(r)\leq CV_{i}(r)h_{i}^{2}(r).

(2.6)

Then for $t>0$

p(t,o,o)\simeq\frac{1}{V_{m}(\sqrt{t})}.

(2.7)

This means that the on-diagonal heat kernel estimates at the central reference point is determined by the largest end!

Remark 2.7.

In our approach, we require the existence of a fixed dominating end given by (2.6) for the optimal estimates in (2.7) to hold. Indeed, more generally, on a manifold with either regular or subcritical ends, we obtain for $t>1$ , (see [10] for the detail)

p(t,o,o)\leq C\frac{\min_{i}h_{i}^{2}(\sqrt{t})}{\min_{i}V_{i}(\sqrt{t})h_{i}^{2}(\sqrt{t})}.

(2.8)

The assumption in (2.6) implies that for all $r>1$

\frac{\min_{i}h_{i}^{2}(r)}{\min_{i}V_{i}(r)h_{i}^{2}(r)}\leq\frac{C}{V_{m}(r)},

(2.9)

which allows to apply [4, Theorem 7.2] for the matching lower bound. In Section 3, we construct manifolds with ends without a fixed dominating end and, in such cases, the estimates in (2.9) does not hold.

As illustrative examples, let $M=M_{1}\#\cdots\#M_{k}$ be a manifold with parabolic ends, where each end $M_{i}$ satisfies (PI), (VD), (RCA). Let $\alpha_{i}$ and $\beta_{i}$ be sequences satisfying

(\alpha_{1},\beta_{1})\geq(\alpha_{2},\beta_{2})\geq\cdots\geq(\alpha_{k},\beta_{k})>(0,+\infty)

in the sense of lexicographical order, namely $(\alpha_{i},\beta_{i})>(\alpha_{j},\beta_{j})$ means that

\alpha_{i}>\alpha_{j}~{}\mbox{or}~{}\alpha_{i}=\alpha_{j}~{}\mbox{and}~{}\beta_{i}>\beta_{j}

and we assume that

V_{i}(r)\simeq r^{\alpha_{i}}\left(\log r\right)^{\beta_{i}},\quad r>2.

Here we need $(\alpha_{1},\beta_{1})\leq(2,1)$ so that all ends $M_{1},\ldots,M_{k}$ are parabolic. Then the above theorem implies that

p(t,o,o)\simeq\frac{1}{t^{\alpha_{1}/2}\left(\log t\right)^{\beta_{1}}},\quad t>2.

As an explicit example, suppose that $k=2$ and $(\alpha_{1},\beta_{1})=(2,0)$ and $(\alpha_{2},\beta_{2})=(1,0)$ . Substituting the above estimates into (2.2), we obtain for $x\in E_{1}$ , $y\in E_{2}$ and $t>1$

p(t,x,y)\simeq\left\{\begin{array}[]{ll}\frac{1}{t}e^{-bd^{2}(x,y)/t}&\mbox{if }|x|>\sqrt{t}\\ \frac{1}{t}\left(1+\frac{|y|}{\sqrt{t}}\log\frac{e\sqrt{t}}{|x|}\right)&\mbox{if }|x|,|y|\leq\sqrt{t}\\ \frac{1}{t}\left(\log\frac{e\sqrt{t}}{|x|}\right)e^{-bd^{2}(x,y)/t}&\mbox{if }|x|\leq\sqrt{t}<|y|.\end{array}\right.

Remark 2.8.

Assume that all ends of a manifold $M=M_{1}\#\cdots\#M_{k}$ are subcritical. Then for $x\in E_{i}$ and $y\in E_{j}$ with $i\neq j$ and $t>1$ ,

p(t,x,y)\asymp\frac{C}{V_{m}(\sqrt{t})}e^{-bd^{2}(x,y)/t}

(see [9, Theorem 2.3]).

2.3. Poincaré constant estimates

In this section, we consider the estimates of the Poincaré constant defined in (1.1). Recall that $M=M_{1}\#\cdots\#M_{k}$ is a manifold with ends $M_{1},\ldots,M_{k}$ , where each end satisfies (VD) and (PI). Let $o\in K$ be a central reference point. Our main interest is to obtain the Poincaré constant $\Lambda(B(o,r))$ at the central point $o$ . In fact, by the monotonicity of $\Lambda$ together with a Whitney covering argument (see [10]), for $r>2|x|$

\Lambda(B(x,r))\simeq\Lambda(B(o,r)).

For $r>0$ , let $n=n(r)$ be the number so that

\displaystyle V_{n}(r)

\displaystyle=\max_{i\neq m}V_{i}(r),

where $m=m(r)$ is the number of the largest end (see (2.5)). Then we obtain the following.

Theorem 2.9 (Grigor’yan, Ishiwata, Saloff-Coste [10]).

Let $M=M_{1}\#\!\cdots\!\#M_{k}$ be a manifold with $k$ non-parabolic ends. Assume that each end $M_{i}$ satisfies (VD) and (PI). Then for sufficiently large $r>1$

\Lambda(B(o,r))\leq CV_{n}(r).

Moreover, if for all $r>0$

rV^{\prime}(r)\leq CV(r),

then for sufficiently large $r>1$

\Lambda(B(o,r))\simeq V_{n}(r).

When $M$ has at least one parabolic end, we assume the following additional condition (see [10] for details).

Definition 2.10 ((COE)).

We say that a manifold $M=\#_{i\in I}M_{i}$ has critically ordered ends and write (COE) in short if there exist $\varepsilon,\delta,\gamma_{1},\gamma_{2}>0$ such that

\gamma_{1}<\varepsilon,~{}~{}\gamma_{1}+\gamma_{2}<\delta<2,~{}~{}2\gamma_{1}+\gamma_{2}<2,

and a decomposition

I=I_{super}\sqcup I_{middle}\sqcup I_{sub}

such that the following conditions are satisfied:

$\left(a\right)$

For each $i\in I_{super}$ and all $r\geq 1$ ,

$V_{i}(r)\geq cr^{2+\epsilon}\ .$
$\left(b\right)$

For each $i\in I_{sub}$ , $V_{i}$ is subcritical (see Definition 2.5) and

$V_{i}(r)\leq Cr^{2-\delta}\ .$
$\left(c\right)$

For each $i\in I_{middle}$ , $V_{i}$ is regular (see (2.3)). Moreover, for any pair $i,j\in I_{middle}$ we have either $V_{i}\geq cV_{j}$ or $V_{j}\geq cV_{i}$ (i.e., the ends in $I_{middle}$ can be ordered according to their volume growth uniformly over $r\in[1,\infty)$ ) and $V_{i}\geq cV_{j}$ implies that $V_{i}h_{i}\geq c^{\prime}V_{j}h_{j}$ . Besides, if $M$ is parabolic (i.e., all ends are parabolic) then $V_{i}\geq CV_{j}$ also implies $V_{i}h_{i}^{2}\leq C^{\prime}V_{j}h_{j}^{2}$ .

Theorem 2.11 ([10]).

Let $M=M_{1}\#\cdots\#M_{k}$ be a manifold with $k$ ends, where each end satisfies (VD) and (PI). Suppose that there exists at least one parabolic end and each parabolic end satisfies (RCA). If $M$ admits (COE), then for sufficiently large $r>1$

\Lambda(B(o,r))\leq CV_{n}(r)h_{n}(r).

If, in addition, each $V_{i}$ satisfies that

rV_{i}^{\prime}(r)\leq CV_{i}(r)~{}~{}(\forall r>1),

then, for sufficiently large $r>1$

\Lambda(B(o,r))\simeq V_{n}(r)h_{n}(r).

These results say that the Poincaré constant $\Lambda(B(o,r))$ is determined by the second largest end!

As an explicit example, let $M=\mathbb{R}^{n}\#\mathbb{R}^{n}$ with $n\geq 2$ . Then Theorems 2.9 and 2.11 imply that

\Lambda(B(o,r))\simeq\left\{\begin{array}[]{ll}r^{n}&n\geq 3,\\ r^{2}\log r&n=2.\end{array}\right.

Let $M=M_{1}\#\cdots\#M_{k}$ be a manifold with ends. Assume that each end $M_{i}$ satisfies (VD), (PI) and that each parabolic end satisfies (RCA). Suppose that for $i=1,\ldots k$

V_{i}(r)\simeq r^{\alpha_{i}}\left(\log r\right)^{\beta_{i}},

where $(\alpha_{1},\beta_{1})\geq(\alpha_{2},\beta_{2})\geq\cdots\geq(\alpha_{k},\beta_{k})$ as the lexicographical order. Then

	$\displaystyle\Lambda(B(o,r))$	$\displaystyle\simeq V_{2}(r)h_{2}(r)$
		$\displaystyle\simeq\left\{\begin{array}[]{ll}r^{\alpha_{2}}(\log r)^{\beta_{2}}&\mbox{if }(\alpha_{2},\beta_{2})>(2,1),\\ r^{2}\log r(\log\log r)^{2}&\mbox{if }(\alpha_{2},\beta_{2})=(2,1),\\ r^{2}\log r&\mbox{if }(2,-\infty)<(\alpha_{2},\beta_{2})<(2,1),\\ r^{2}&\mbox{if }(\alpha_{2},\beta_{2})<(2,-\infty).\end{array}\right.$

3. Manifold with ends with oscillating volume functions

3.1. Preliminaries

The purpose of this section is to construct manifolds with ends for which the estimate in (2.8) might not give an optimal bound. To obtain such a manifold, we need a manifold with (VD) and (PI) together with oscillating volume function. First, let us recall the following theorem.

Theorem 3.1 (Grigor’yan and Saloff-Coste [14, Theorem 5.7]).

Let $(M,\mu)$ be a complete non-compact wighted manifold with (PHI) and (RCA) at a reference point $o\in M$ . If a positive valued smooth function $W:[0,\infty)\rightarrow\mathbb{R}$ satisfies for all $r>0$

	$\displaystyle\sup_{[r,2r]}W$	$\displaystyle\leq C\inf_{[r,2r]}W,$		(3.1)
	$\displaystyle\int_{0}^{r}W^{2}(s)sds$	$\displaystyle\leq CW^{2}(r)r^{2},$		(3.2)

then the weighted manifold $(M,W^{2}(d(o,\cdot))\mu)$ also satisfies (PHI).

Let $(M_{1},\mu_{1})$ be the $2$ -dimensional Euclidean space $\mathbb{R}^{2}$ with the Euclidean measure. We denote by $(M_{2},\mu_{2})$ a weighted manifold $(\mathbb{R}^{2},W^{2}(d(o,\cdot))\mu_{1})$ , where the positive valued smooth function $W:[0,\infty)\rightarrow\mathbb{R}$ is defined as follows. For $\alpha>2$ , $0<\beta<2$ define a function $W$ so that for all $k\in\mathbb{N}$

\int_{0}^{r}W^{2}(s)sds=\left\{\begin{array}[]{ll}r^{2}&0<r<a_{1},a_{k}\leq r<b_{k},\\ \left(\frac{r}{b_{k}}\right)^{\alpha}b_{k}^{2}&b_{k}\leq r<c_{k},\\ r^{2}\log r&c_{k}\leq r<d_{k},\\ \left(\frac{r}{d_{k}}\right)^{\beta}d_{k}^{2}\log d_{k}&d_{k}\leq r<a_{k+1},\end{array}\right.

(3.3)

where the sequences $a_{k}\leq b_{k}<c_{k}\leq d_{k}<a_{k+1}$ satisfy $a_{1}>e$ and

	$\displaystyle b_{k}$	$\displaystyle=\frac{c_{k}}{(\log c_{k})^{\frac{1}{\alpha-2}}},$		(3.4)
	$\displaystyle a_{k+1}$	$\displaystyle=d_{k}(\log d_{k})^{\frac{1}{2-\beta}}$		(3.5)

(see fig. 4). These sequences will be fixed later.

Figure 4. Oscillating volume function (thick line)

Then the function $W$ satisfies that

\displaystyle W(r)\simeq\left\{\begin{array}[]{ll}1&0<r<a_{1},a_{k}\leq r<b_{k},\\ \left(\frac{r}{b_{k}}\right)^{\frac{\alpha-2}{2}}&b_{k}\leq r<c_{k},\\ \sqrt{\log r}&c_{k}\leq r<d_{k},\\ \sqrt{\log d_{k}}\left(\frac{r}{d_{k}}\right)^{-\frac{2-\beta}{2}}&d_{k}\leq r<a_{k+1},\end{array}\right.

\displaystyle W^{\prime}(r)\simeq\left\{\begin{array}[]{ll}0&0<r<a_{1},a_{k}<r<b_{k},\\ r^{\frac{\alpha-4}{2}}b_{k}^{-\frac{\alpha-2}{2}}&b_{k}<r<c_{k},\\ \frac{1}{\sqrt{\log r}}\frac{1}{r}&c_{k}<r<d_{k},\\ -r^{-\frac{4-\beta}{2}}\sqrt{\log d_{k}}d_{k}^{\frac{2-\beta}{2}}&d_{k}<r<a_{k+1}.\end{array}\right.

Since

\displaystyle\frac{W^{\prime}(r)}{W(r)}\simeq\left\{\begin{array}[]{ll}0&0<r<a_{1},a_{k}<r<b_{k},\\ \frac{1}{r}&b_{k}<r<c_{k},\\ \frac{1}{r\log r}&c_{k}<r<d_{k},\\ -\frac{1}{r}&d_{k}<r<a_{k+1},\end{array}\right.

(3.10)

the condition in (3.1) holds. Indeed, the estimates in (3.10) imply that for any $0<r_{1}<r_{2}$

-C\log\frac{r_{2}}{r_{1}}=-\int_{r_{1}}^{r_{2}}\frac{Cds}{s}\leq\int_{r_{1}}^{r_{2}}\frac{W^{\prime}(s)}{W(s)}ds=\log\frac{W(r_{2})}{W(r_{1})}\leq\int_{r_{1}}^{r_{2}}\frac{Cds}{s}=C\log\frac{r_{2}}{r_{1}}.

Then we obtain for any $0<r_{1}\leq r_{2}\leq 2r_{1}$

W(r_{1})\simeq W(r_{2}).

Taking $r_{1}\leq r_{2},r_{3}\leq 2r_{1}$ so that

W(r_{2})=\sup_{[r_{1},2r_{1}]}W\mbox{ and }W(r_{3})=\inf_{[r_{1},2r_{1}]}W,

we conclude the condition in (3.1).

Moreover, we see that

\displaystyle W^{2}(r)r^{2}\simeq\left\{\begin{array}[]{ll}r^{2}&0<r<a_{1},a_{k}\leq r<b_{k},\\ r^{\alpha}b_{k}^{2-\alpha}&b_{k}\leq r<c_{k},\\ r^{2}\log r&c_{k}\leq r<d_{k},\\ r^{\beta}d_{k}^{2-\beta}\log d_{k}&d_{k}\leq r<a_{k+1},\end{array}\right.

which satisfies the condition in (3.2). Applying Theorem 3.1, the weighted manifold $(M_{2},\mu_{2})=(\mathbb{R}^{2},W^{2}(d(o,\cdot))\mu_{1})$ satisfies (PHI) ¹¹1 We need a smooth modification of the function $W$ satisfying (3.3) to apply Theorem 3.1. However, we omit the smoothing argument for simplicity. .

Let $Z:[0,\infty)\rightarrow\mathbb{R}$ be a positive valued smooth function satisfying

Z(r)=\sqrt{1+2\log r}\quad(r\geq 1).

Let $(M_{3},\mu_{3})$ be a weighted manifold $(\mathbb{R}^{2},Z^{2}(d(o,\cdot))\mu_{1})$ . Then Theorem 3.1 implies also that $(M_{3},\mu_{3})$ satisfies (PHI).

For $i=1,2,3$ , we denote by $V_{i}(r)$ the volume function on $(M_{i},\mu_{i})$ at $o\in M_{i}=\mathbb{R}^{2}$ . Then we obtain

	$\displaystyle V_{1}(r)$	$\displaystyle=\pi r^{2},$
	$\displaystyle V_{2}(r)$	$\displaystyle=2\pi\int_{0}^{r}W^{2}(s)sds,$
	$\displaystyle V_{3}(r)$	$\displaystyle=2\pi\int_{0}^{1}Z^{2}(s)sds+2\pi r^{2}\log r~{}~{}(r\geq 1).$

It is easy to obtain for sufficiently large $r>1$

h_{1}(r)\simeq\log r~{}\mbox{ and }~{}h_{3}(r)\simeq\log\log r.

Now we estimate the function $h_{2}(r)$ . Observe that

	$\displaystyle\int_{a_{k}}^{b_{k}}\frac{sds}{V_{2}(s)}$	$\displaystyle=\log\frac{b_{k}}{a_{k}},$
	$\displaystyle\int_{b_{k}}^{c_{k}}\frac{sds}{V_{2}(s)}$	$\displaystyle=\int_{b_{k}}^{c_{k}}b_{k}^{\alpha-2}s^{1-\alpha}ds=\frac{1}{\alpha-2}\left(1-\frac{1}{\log c_{k}}\right)\simeq 1,$
	$\displaystyle\int_{c_{k}}^{d_{k}}\frac{sds}{V_{2}(s)}$	$\displaystyle=\int_{c_{k}}^{d_{k}}\frac{ds}{s\log s}=\log\left(\frac{\log d_{k}}{\log c_{k}}\right),$
	$\displaystyle\int_{d_{k}}^{a_{k+1}}\frac{sds}{V_{2}(s)}$	$\displaystyle=\int_{d_{k}}^{a_{k+1}}\frac{s^{1-\beta}ds}{d_{k}^{2-\beta}\log d_{k}}=\frac{1}{(2-\beta)}\left(1-\frac{1}{\log d_{k}}\right)\simeq 1.$

Then we obtain

h_{2}(a_{n})=1+\int_{1}^{a_{1}}\frac{sds}{V_{2}(s)}+\sum_{k=1}^{n-1}\int_{a_{k}}^{a_{k+1}}\frac{sds}{V_{2}(s)}\simeq\sum_{k=1}^{n}\left(\log\frac{b_{k}}{a_{k}}+\log\frac{\log d_{k}}{\log c_{k}}+1\right),

(3.11)

which shows that the behavior of $h_{2}(r)$ depends on the choice of sequences $a_{k}\leq b_{k}<c_{k}\leq d_{k}<a_{k+1}$ satisfying (3.4) and (3.5).

3.2. Example 1

For the first case, let us choose sequences $a_{k}\leq b_{k}<c_{k}\leq d_{k}<a_{k+1}$ so that for any $k\in\mathbb{N}$

a_{k}\simeq b_{k},\quad c_{k}\simeq d_{k}.

(3.12)

In this case, we obtain the following.

Lemma 3.2.

If $a_{1}$ is large enough and the sequences $a_{k}\leq b_{k}<c_{k}\leq d_{k}<a_{k+1}$ satisfy (3.4), (3.5) and (3.12), then for sufficiently large $r>1$ ,

h_{2}(r)\simeq\frac{\log r}{\log\log r}.

Proof.

By the estimate in (3.11), we obtain

h_{2}(a_{n})\simeq n.

(3.13)

Let us consider the behavior $a_{n}$ . By the assumption in (3.4), we always have

c_{k}\simeq b_{k}\left(\log b_{k}\right)^{\frac{1}{\alpha-2}}.

(3.14)

Assumptions in (3.4) and (3.5) imply that for any $k\in\mathbb{N}$

	$\displaystyle a_{k+1}$	$\displaystyle=d_{k}\left(\log d_{k}\right)^{\frac{1}{2-\beta}}\simeq c_{k}\left(\log c_{k}\right)^{\frac{1}{2-\beta}}$
		$\displaystyle\simeq b_{k}\left(\log b_{k}\right)^{\frac{1}{\alpha-2}}\left(\log\left(b_{k}\left(\log b_{k}\right)^{\frac{1}{\alpha-2}}\right)\right)^{\frac{1}{2-\beta}}$
		$\displaystyle=b_{k}\left(\log b_{k}\right)^{\frac{1}{\alpha-2}}\left(\log b_{k}+\frac{1}{\alpha-2}\log\log b_{k}\right)^{\frac{1}{2-\beta}}$
		$\displaystyle\simeq b_{k}\left(\log b_{k}\right)^{\frac{1}{\alpha-2}+\frac{1}{2-\beta}}=b_{k}\left(\log b_{k}\right)^{\gamma}\simeq a_{k}\left(\log a_{k}\right)^{\gamma},$

where $\gamma=\frac{1}{\alpha-2}+\frac{1}{2-\beta}$ . Taking $a_{1}$ large enough, there exist positive constants $c,C$ and $\gamma^{\prime}>\gamma$ such that for any $n\in\mathbb{N}$

cn^{\gamma n}\leq a_{n}\leq Cn^{\gamma^{\prime}n}.

This implies that for sufficiently large $n\in\mathbb{N}$

\displaystyle\log a_{n}\simeq n\log n,\quad\log\log a_{n}\simeq\log n.

Then we obtain for sufficiently large $n\in\mathbb{N}$ ,

\frac{\log a_{n}}{\log\log a_{n}}\simeq n\simeq h_{2}(a_{n}),

which concludes the lemma. ∎

Now we estimate the heat kernel on $M=M_{1}\#M_{2}$ . By the definition of $V_{1}(r)$ , $V_{2}(r)$ and by Lemma 3.2, we obtain for sufficiently large $r>1$

$\displaystyle V_{1}(r)$	$\displaystyle=r^{2},$	$\displaystyle V_{2}(r)\simeq\left\{\begin{array}[]{ll}r^{2}&a_{k}\leq r<b_{k}\\ r^{2}\log r&c_{k}\leq r<d_{k},\end{array}\right.$
$\displaystyle h_{1}(r)$	$\displaystyle\simeq\log r,$	$\displaystyle h_{2}(r)\simeq\frac{\log r}{\log\log r},$
$\displaystyle V_{1}(r)h_{1}^{2}(r)$	$\displaystyle\simeq r^{2}(\log r)^{2},$	$\displaystyle V_{2}(r)h_{2}^{2}(r)\simeq\left\{\begin{array}[]{ll}r^{2}\frac{(\log r)^{2}}{(\log\log r)^{2}}&a_{k}\leq r<b_{k}\\ r^{2}\frac{(\log r)^{3}}{(\log\log r)^{2}}&c_{k}\leq r<d_{k}.\end{array}\right.$

According to the heat kernel upper estimate in (2.8), we obtain for $c_{k}\leq\sqrt{t}<d_{k}$

p(t,o,o)\leq\frac{\min_{i}h_{i}^{2}(\sqrt{t})}{\min_{i}V_{i}(\sqrt{t})h_{i}^{2}(\sqrt{t})}\simeq\frac{\left(\frac{\log\sqrt{t}}{\log\log\sqrt{t}}\right)^{2}}{t\left(\log\sqrt{t}\right)^{2}}\simeq\frac{1}{t(\log\log t)^{2}}.

Since $V(r)\simeq\max_{i}V_{i}(r)=r^{2}\log r$ in this interval, the above upper estimate is much larger than

\frac{1}{V(\sqrt{t})}\simeq\frac{1}{t\log t},

which makes difficult to obtain matching lower bound by using [4, Theorem 7.2].

3.3. Example 2

Next, let us choose sequences $a_{k}\leq b_{k}<c_{k}\leq d_{k}<a_{k+1}$ so that for some $\delta>1$ and for any $k\in\mathbb{N}$

a_{k}\simeq b_{k},\quad c_{k}^{\delta}\simeq d_{k}.

(3.15)

In this case, we obtain the following.

Lemma 3.3.

If $a_{1}$ is large enough and the sequences $a_{k}\leq b_{k}<c_{k}\leq d_{k}<a_{k+1}$ satisfy (3.4), (3.5) and (3.15), then for sufficiently large $r>1$ ,

h_{2}(r)\simeq\log\log r.

Proof.

The estimate in (3.11) yields that

h_{2}(a_{n})\simeq n.

By the estimates in (3.14) and (3.15), we obtain

\displaystyle d_{k}

\displaystyle\simeq c_{k}^{\delta}\simeq\left(b_{k}\left(\log b_{k}\right)^{\frac{1}{\alpha-2}}\right)^{\delta}\simeq a_{k}^{\delta}\left(\log a_{k}\right)^{\frac{\delta}{\alpha-2}}.

Assumptions in (3.4) and (3.5) imply that

	$\displaystyle a_{k+1}$	$\displaystyle=d_{k}\left(\log d_{k}\right)^{\frac{1}{2-\beta}}$
		$\displaystyle\simeq a_{k}^{\delta}\left(\log a_{k}\right)^{\frac{\delta}{\alpha-2}}\left(\delta\log a_{k}+\frac{\delta}{\alpha-2}\log\log a_{k}\right)^{\frac{1}{2-\beta}}$
		$\displaystyle\simeq a_{k}^{\delta}\left(\log a_{k}\right)^{\theta},$

where $\theta=\frac{\delta}{\alpha-2}+\frac{1}{2-\beta}$ . Hence, there exist positive constants $c,C$ and $\eta>\delta$ such that for any $k\in\mathbb{N}$

ca_{k}^{\delta}\leq a_{k+1}\leq Ca_{k}^{\eta}.

This implies that for any $n\in\mathbb{N}$

c^{\frac{\delta^{(n-1)}-1}{\delta-1}}a_{1}^{\delta^{(n-1)}}\leq a_{n}\leq C^{\frac{\eta^{(n-1)}-1}{\eta-1}}a_{1}^{\eta^{(n-1)}}.

Then we obtain

\displaystyle\frac{\delta^{n-1}-1}{\delta-1}\log c+\delta^{(n-1)}\log a_{1}\leq\log a_{n}\leq\frac{\eta^{(n-1)}-1}{\eta-1}\log C+\eta^{(n-1)}\log a_{1}.

Taking $a_{1}$ large enough, we obtain for sufficiently large $n\in\mathbb{N}$

\log\log a_{n}\simeq n\simeq h_{2}(a_{n}),

which concludes the lemma. ∎

Now let us consider the estimate of the heat kernel on $M=M_{2}\#M_{3}$ . By the definition of $V_{2}(r)$ , $V_{3}(r)$ and by Lemma 3.3, we obtain for sufficiently large $r>1$

$\displaystyle V_{2}(r)$	$\displaystyle\simeq\left\{\begin{array}[]{ll}r^{2}&a_{k}\leq r<b_{k},\\ r^{2}\log r&c_{k}\leq r<d_{k},\end{array}\right.$	$\displaystyle\!\!\!\!V_{3}(r)\simeq r^{2}\log r,$
$\displaystyle h_{2}(r)$	$\displaystyle\simeq\log\log r,$	$\displaystyle\!\!\!\!h_{3}(r)\simeq\log\log r,$
$\displaystyle V_{2}(r)h_{2}^{2}(r)$	$\displaystyle\simeq\left\{\begin{array}[]{ll}r^{2}(\log\log r)^{2}&a_{k}\leq r<b_{k},\\ r^{2}\log r(\log\log r)^{2}&c_{k}\leq r<d_{k}.\end{array}\right.$	$\displaystyle\!\!\!\!V_{3}(r)h_{3}^{2}(r)\simeq r^{2}\log r(\log\log r)^{2}.$

Substituting above into (2.8), we obtain for sufficiently large $k\in\mathbb{N}$ and $a_{k}\leq\sqrt{t}<b_{k}$

p(t,o,o)\leq C\frac{(\log\log\sqrt{t})^{2}}{t(\log\log\sqrt{t})^{2}}\simeq\frac{1}{t}.

Since $V(r)\simeq\max_{i}V_{i}(r)\simeq r^{2}\log r$ for all $r>1$ , the above upper estimate is much larger than

\frac{1}{V(\sqrt{t})}\simeq\frac{1}{t\log t},

which makes difficult to obtain matching lower bound by using [4, Theorem 7.2].

We hope to prove matching heat kernel lower bounds in forthcoming work.

Acknowledgments

The second author would like to thank Professor Gilles Carron for telling him how to construct manifolds with oscillating volume function with (PHI).

References

[1] I. Benjamini, I. Chavel, E. A. Feldman, Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash. Proc. London Math. Soc., 72 (1996) 215–240.
[2] M. Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set. Bull. A.M.S., 24 (1991) no. 2, 371-377.
[3] G. Carron, Riesz transforms on connected sums. Festival Yves Colin de Verdière. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2329-–2343.
[4] T. Coulhon, A. Grigor’yan, On-diagonal lower bounds for heat kernels on no-compact manifolds and Markov chains. Duke Math. J., 89 (1997) no. 1, 133-199.
[5] H. Doan, Boundedness of maximal functions on nondoubling parabolic manifolds with ends. To appear in J. Aust. Math. Soc.
[6] X. T. Duong, J. Li and A. Sikora, Boundedness of maximal functions on non-doubling manifolds with ends. Proc. Centre Math. Appl. Austral. 45 (2012), 37–-47.
[7] A. Grigor’yan, The heat equation on noncompact Riemannian manifolds (in Russian). Mat. Sb., 182 (1991) no. 1, 55-87; English translation in Math. USSR-Sb., 72 (1992) no. 1, 47–77.
[8] A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. AMS, 36 (1999) 135–249.
[9] A. Grigor’yan, S. Ishiwata and L. Saloff-Coste, Heat kernel estimates on connected sums of parabolic manifolds, J. Math. Pures Appl. 113(2018), 155-194.
[10] A. Grigor’yan, S. Ishiwata and L. Saloff-Coste, Poincaré constants on manifolds with ends, in preparation.
[11] A. Grigor’yan and L. Saloff-Coste, Heat kernel on connected sums of Riemannian manifolds. Math. Research Letters, 6 (1999) no. 3-4, 307–321.
[12] A.Grigor’yan and L. Saloff-Coste, Dirichlet heat kernel in the exteior of a compact set. Comm. Pure Appl. Math, 55 (2002), 93–133.
[13] A. Grigor’yan and L. Saloff-Coste, Hitting probabilities for Brownian motion on Riemannian manifolds. J. Math. Pures Appl., 81 (2002) no. 2, 115–142.
[14] A. Grigor’yan and L. Saloff-Coste, Stability results for Harnack inequalities. Ann. Inst. Fourier, Grenoble, 55 (2005) no. 3, 825–890.
[15] A. Grigor’yan and L. Saloff-Coste, Heat kernel on manifolds with ends. Ann. Inst. Fourier, Grenoble, 59 (2009) no. 5, 1917–1997.
[16] A. Grigor’yan and L. Saloff-Coste, Surgery of the Faber-Krahn inequality and applications to heat kernel bounds, J. Nonlinear Analysis, 131 (2016) 243–272.
[17] A. Hassel, D. Nix and A. Sikora, Riesz transforms on a class of non-doubling manifolds II. arXiv:1912.06405.
[18] A. Hassel and A. Sikora, Riesz transforms on a class of non-doubling manifolds. Comm. Partial Differential Equations 44 (2019) no. 11, 1072–-1099.
[19] A. Kasue, Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature I, Geometry and Analysis on Manifolds (Ed. by T. Sunada), Lecture Notes in Math., 1339, Springer-Verlag (1988), 158-181.
[20] S. Kusuoka, D. Stroock, Application of Malliavin calculus, Part III, J. Fac. Sci. Tokyo Univ., Sect. 1A, Math., 34 (1987) 391–442.
[21] Yu. T. Kuz’menko and S. A. Molchanov, Counterexamples of Liouville type theorems, (in Russian) Vestnik Moscov. Univ. Ser. I Mat. Mekh., (1976) no. 6, 39–43; Engl. transl. Moscow Univ. Math. Bull., 34 (1979) 35–39.
[22] P. Li and L-F Tam, Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set, Ann. Math., 125 (1987), 171-207.
[23] P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986) no. 3-4, 153–201.
[24] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964) 101–134. Correction: Comm. Pure Appl. Math. 20 (1967) 231-236.
[25] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices (1992), no. 2, 27–38.
[26] L. Saloff-Coste, Aspects of Sobolev type inequalities. London Math. Soc. Lecture Notes Series 289, Cambridge Univ. Press, 2002.

Geometric analysis on manifolds with ends

Key words and phrases:

2010 Mathematics Subject Classification:

Contents

1. Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

2. The state of the art

2.1. Setting

Definition 2.1.

Definition 2.2 ((RCA)).

2.2. Heat kernel estimates

2.2.1. Off-diagonal estimates

Theorem 2.3 (Grigor’yan and Saloff-Coste [15, Theorem 3.5]).

2.2.2. Non-parabolic case

Theorem 2.4.

2.2.3. Parabolic case

Definition 2.5 (c.f. [10]).

Theorem 2.6 (Grigor’yan, Ishiwata, Saloff-Coste [10]).

Remark 2.7.

Remark 2.8.

2.3. Poincaré constant estimates

Theorem 2.9 (Grigor’yan, Ishiwata, Saloff-Coste [10]).

Definition 2.10 ((COE)).

Theorem 2.11 ([10]).

3. Manifold with ends with oscillating volume functions

3.1. Preliminaries

Theorem 3.1 (Grigor’yan and Saloff-Coste [14, Theorem 5.7]).

3.2. Example 1

Lemma 3.2.

Proof.

3.3. Example 2

Lemma 3.3.

Proof.

Acknowledgments

References