New Riemannian manifolds with L^p-unbounded Riesz transform for p > 2

Abstract

We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on $L^p(M)$ for all $p > 2$. This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.

1 Introduction

Consider a Riemannian manifold M with gradient $\nabla$ and Laplace–Beltrami operator $\varDelta$. The Riesz transform $\nabla \varDelta ^{-1/2}$, with $\varDelta ^{-1/2}$ defined via the spectral theorem, maps $L^2(M)$ boundedly to the space of square integrable vector fields $L^2(M;TM)$. Much attention has been given to the question of whether this operator extends to a bounded map from $L^p(M)$ to $L^p(M;TM)$ for $p \ne 2$, or equivalently, whether the estimate

$$\begin{aligned} (R_p): \quad \Vert |\nabla f| \Vert _p \lesssim \Vert \varDelta ^{1/2} f \Vert _p \qquad \hbox {for all } f \in C_c^\infty (M) \end{aligned}$$

holds. It is conjectured that for $p \in (1,2)$ the estimate $(R_p)$ holds whenever M is complete, with implicit constant depending only on p; in [1] it is shown that the failure of this uniformity among manifolds of a fixed dimension would imply the existence of a manifold for which $(R_p)$ fails.

One is naturally led to consider also the ‘reverse’ estimate

$$\begin{aligned} (RR_p): \quad \Vert \varDelta ^{1/2} f \Vert _p \lesssim \Vert |\nabla f| \Vert _p \qquad \hbox {for all} f \in C_c^\infty (M). \end{aligned}$$

A duality argument shows that for $p \in (1,\infty )$, $(R_p)$ implies $(RR_{p^\prime })$, where $p^\prime = p/(p-1)$ is the Hölder conjugate of p. If $(R_p)$ and $(RR_p)$ both hold, then we have a norm equivalence

$$\begin{aligned} \Vert |\nabla f| \Vert _p \simeq \Vert \varDelta ^{1/2} f \Vert _p, \end{aligned}$$

which says that the homogeneous Sobolev space $\dot{W}^p_1(M)$ may be defined either via the gradient or via the square root of the Laplace–Beltrami operator.

Generally $(R_p)$ holds only for some interval of $p \in (1,\infty )$ including 2, and proving $(R_p)$ presents different difficulties depending on whether $p < 2$ or $p > 2$. When $1< p < 2$, $(R_p)$ is known to follow from the volume doubling property and Gaussian or sub-Gaussian heat kernel upper estimates [9, 10] (see also [15] for examples which do not satisfy such kernel estimates). The volume doubling property and an appropriately scaled $L^2$-Poincaré inequality imply $(R_p)$ for some $p > 2$ [2]. In [3] this is linked with gradient estimates for the heat kernel, and in [5] the $L^2$-Poincaré inequality is replaced by a relative Faber–Krahn inequality and a reverse Hölder inequality.

Some manifolds for which $(R_p)$ fails for some $p > 2$ are known. If M is the connected sum of two copies of $\mathbb {R}^n \setminus B(0,1)$ with $n \ge 3$ —or more generally, an n-dimensional manifold with at least two (and finitely many) Euclidean ends—$(R_p)$ holds if and only if $p \in (1,n)$ [7, 10]. Similar results are known for conical manifolds [14] and for 2-hyperbolic, p-parabolic manifolds with at least two ends [6].

The most relevant examples to this article are Vicsek manifolds, which are ‘thickenings’ of Vicsek graphs (pictured in the 2-dimensional case in Fig. 2). The Vicsek graph, being a graphical realisation of a Vicsek fractal, is a ‘fractal at infinity’. Locally a Vicsek manifold behaves like Euclidean space (it is, of course, a manifold), but at large scale it behaves like a fractal. In [9] it is shown that for a Vicsek manifold of any dimension, $(R_p)$ holds if and only if $p \in (1,2]$. The result for $p < 2$ is a consequence of volume doubling and sub-Gaussian heat kernel estimates. The proof that $(R_p)$ fails for $p > 2$ directly uses the definition of the Vicsek graph [9, Theorem 5.1].

In this article we construct a class of manifolds of arbitrary dimension for which $(R_p)$ fails for all $p > 2$.¹ These manifolds are thickenings of what we call spinal graphs, satisfying generalised dimension conditions defined in terms of the spinal structure along with a polynomial volume lower bound. The Vicsek graphs satisfy these conditions, but the proof of this exploits their fractal nature. We construct a large class of non-fractal spinal graphs with the desired dimension conditions and volume lower bounds, thus yielding manifolds of arbitrary dimension with no fractal structure that fail $(R_p)$ for all $p > 2$.

2 Notation

The graphs we consider are non-directed, with at most one edge per pair of vertices, and with no edges from a vertex to itself. The set of vertices of a graph G is denoted by V(G), and if two vertices $x,y \in V(G)$ are neighbours we write $x \sim y$. The set of edges of G is denoted by E(G). For a connected graph G we let $d_G(x,y)$ denote the combinatorial distance between x and y, given by the minimum length of a path from x to y, and for $x \in V(G)$, $r > 0$ let

$$\begin{aligned} B_G(x,r) := \{y \in V(G) : d_G(x,y) \le r\}. \end{aligned}$$

3 Spinal graphs

Definition 1

Let G be a connected graph, $\varSigma \subset V(G)$, and let $\pi :V(G) \rightarrow \varSigma$ be a function such that

• $\pi (x) = x$ for all $x \in \varSigma$,

• $\pi ^{-1}(x)$ is finite for all $x \in \varSigma$,

• if $a,b \in V(G)$ and $\pi (a) \ne \pi (b)$, then every path from a to b contains a subpath from $\pi (a)$ to $\pi (b)$.

We refer to $(G,\varSigma ,\pi )$ as a spinal graph, and the set of vertices $\varSigma$ is called the spine.

Remark 1

One could formulate this definition without the finiteness condition, but it will be convenient for us to keep it.

Fig. 1

A spinal graph $(G,\varSigma ,\pi )$, with a few vertices labeled. The spine $\varSigma \subset V(G)$ consists of the shaded vertices

An example of a spinal graph $(G,\varSigma ,\pi )$ is pictured in Fig. 1. There the vertices of the spine $\varSigma$ are shaded black, while the other vertices are unshaded; for each vertex x, the point $\pi (x)$ is the (uniquely determined) point on $\varSigma$ of minimal distance to x. The dotted lines are not edges of G; if they were to be added to G, then the resulting graph would not be a spinal graph.

To help the reader familiarise themselves with the definition of a spinal graph we prove the following lemma (which will be useful later).

Lemma 1

Let $(G,\varSigma ,\pi )$ be a spinal graph, and suppose $a,b \in V(G)$ with $\pi (a) = \pi (b) =: x$. Then every minimal path from a to b is contained entirely in $\pi ^{-1}(x)$.

Proof

Suppose this is false. Then there exist $a,b \in V(G)$ with $\pi (a) = \pi (b) =: x$ and a path $\gamma$ from a to b of minimal length which passes through a vertex c with $\pi (c) \ne x$. Since $\pi (a) \ne \pi (c)$, by the third condition in the definition of a spinal graph, there exists a subpath $\heartsuit$ of $\gamma$ from $\pi (a) = x$ to $\pi ^2(c) = \pi (c)$, so we can write $\gamma$ as a concatenation of paths $\gamma = \alpha *\heartsuit *\beta$, where $\alpha$ is a path from a to x and $\beta$ is a path from $\pi (c)$ to b; viewed as a commutative diagram,

Similarly, there is a subpath $\heartsuit ^\prime$ of $\beta$ from $\pi (c)$ to $\pi (b) = x$, and we can write $\beta = \delta *\heartsuit ^\prime *\delta ^\prime$ as a concatenation of paths, summarised by the commutative diagram

Putting these commutative diagrams together we get

from which we can read that

$$\begin{aligned} \ell (\gamma ) = \ell (\alpha *\heartsuit *\delta *\heartsuit ^\prime *\delta ^\prime ) = \ell (\alpha ) + \ell (\heartsuit ) + \ell (\delta ) + \ell (\heartsuit ^\prime ) + \ell (\delta ^\prime ) \end{aligned}$$

where $\ell (\cdot )$ denotes the length of a path. Since $x \ne \pi (c)$, the paths $\heartsuit$ and $\heartsuit ^\prime$ have positive length, so we find that

$$\begin{aligned} \ell (\alpha * \delta ^\prime ) = \ell (\alpha ) + \ell (\delta ^\prime ) < \ell (\gamma ). \end{aligned}$$

Since $\alpha * \delta ^\prime$ is a path from a to b, this contradicts the assumption that $\gamma$ has minimal length. $\square$

Spinal graphs may be constructed by gluing a collection of finite graphs along another graph; this is made precise in the following example. In fact, this construction yields all spinal graphs (up to isomorphism, in the usual graph-theoretical sense), as will be shown in Proposition 1.

Example 1

Let $\varGamma$ be a connected graph and let $(G_x)_{x \in V(\varGamma )}$ be a collection of finite connected graphs indexed by the vertices of $\varGamma$. Suppose that for each $x \in V(\varGamma )$ a distinguished vertex $z_x \in V(G_x)$ is given. Then one can construct a graph G by gluing each $G_x$ to $\varGamma$ with the identification $z_x \sim x$. More precisely we have

$$\begin{aligned} V(G) := \bigsqcup _{x \in V(\varGamma )} V(G_x) = \{(x,z) : x \in V(\varGamma ), z \in G_x\}, \end{aligned}$$

(1)

and two vertices $(x_1,y_1)$, $(x_2,y_2)$ are neighbours if and only if either

$$\begin{aligned} x_1 = x_2 = x \hbox { and } y_1 \sim y_2 \hbox { in } G_x, \end{aligned}$$

or

$$\begin{aligned} y_1 = z_{x_1} \hbox { and } y_2 = z_{x_2} \hbox { and } x_1 \sim x_2 in \varGamma . \end{aligned}$$

The graph $\varGamma$, along with the graphs $(G_x)_{x \in V(\varGamma )}$, naturally embed into G. We set $\varSigma := V(\varGamma )$ in this embedding; in the disjoint union representation (1) we have

$$\begin{aligned} \varSigma := \{(x,z_x) : x \in V(\varGamma )\}. \end{aligned}$$

Every vertex $z \in V(G)$ belongs to precisely one of the embedded graphs $G_x$ with $x \in V(\varGamma )$, and we define $\pi :V(G) \rightarrow \varSigma$ by the relation $z \in G_{\pi (z)}$; in the disjoint union representation (1), we have $\pi (x,z) = (x,z_x)$.

It is immediate that $\pi (x) = x$ for all $x \in \varSigma$, and that each $\pi ^{-1}(x)$ is finite. Now suppose $a,b \in V(G)$ with $\pi (a) \ne \pi (b)$. By construction, every path including a that does not pass through $\pi (a)$ must be entirely contained in $G_{\pi (a)}$. Since $b \notin G_{\pi (a)}$, every path from a to b must pass through $\pi (a)$, and by symmetry such a path must also pass through $\pi (b)$. That is, every path from a to b contains a subpath from $\pi (a)$ to $\pi (b)$. Therefore $(G,\varSigma ,\pi )$ is indeed a spinal graph.

Proposition 1

Suppose $(G,\varSigma ,\pi )$ is a spinal graph. Then there exists a connected graph $\varGamma$ and a collection $(G_x)_{x \in V(\varGamma )}$ of finite connected graphs such that there exists an isomorphism between G and the graph constructed in Example 1, under which $\varSigma$ corresponds to $V(\varGamma )$ and $\pi$ corresponds to the map $\pi ^\prime$ given by the relation $z \in G_{\pi '(z)}$.

Proof

Let $(G,\varSigma ,\pi )$ be a spinal graph. Let $\varGamma$ be the full subgraph determined by $\varSigma$, and for every $x \in \varSigma = V(\varGamma )$ let $G_x$ be the full subgraph determined by $\pi ^{-1}(x)$. Then $\varGamma$ is connected, each $G_x$ is finite and connected (by Lemma 1), and we have a bijection

$$\begin{aligned} \varphi :V(G) \rightarrow \bigsqcup _{x \in V(\varGamma )} V(G_x), \quad a \mapsto (\pi (a),a). \end{aligned}$$

By the construction in Example 1, it suffices to show that $a,b \in V(G)$ are neighbours if and only if either $\pi (a) = \pi (b)$ and $a \sim b$ in $G_{\pi (a)}$, or $a = \pi (a)$ and $b = \pi (b)$ and $\pi (a) \sim \pi (b)$ in $\varGamma$.

By Lemma 1, if $\pi (a) = \pi (b)$ then every shortest path from a to b is entirely contained in $G_{\pi (a)}$, so in this case a and b are neighbours in G if and only if $a \sim b$ in $G_{\pi (a)}$. On the other hand, if $\pi (a) \ne \pi (b)$, then every path from a to b contains a subpath from $\pi (a)$ to $\pi (b)$, and thus a and b are neighbours if and only if either $a = \pi (a)$ and $b = \pi (b)$ or $a = \pi (b)$ and $b = \pi (a)$. In the first case, since $\varGamma$ is the full subgraph determined by $\varSigma = \pi (V(G))$, we have $\pi (a) \sim \pi (b)$ in $\varGamma$, and we are done. The second case never occurs: since $\pi (b) \in \varSigma$, we would have $\pi (a) = \pi (\pi (b)) = \pi (b)$, which is a contradiction. $\square$

4 Dimensions of a spinal graph

For a spinal graph $(G,\varSigma ,\pi )$ we write $d_\varSigma$ and $B_\varSigma$ for the combinatorial distance and balls in the full subgraph determined by $\varSigma$.

Definition 2

Let $(G,\varSigma ,\pi )$ be a spinal graph. For all $x,y \in V(G)$ define the spinal distance [x, y] by

$$\begin{aligned}{}[x,y] := d_\varSigma (\pi (x),\pi (y)), \end{aligned}$$

and for $r > 0$ we define associated spinal sets by

$$\begin{aligned} D(x,r) := \{y \in G : [x,y] \le r\} = \pi ^{-1}(B_\varSigma (\pi (x),r)). \end{aligned}$$

The spinal distance is a pseudometric on V(G), and the quotient metric space is isometric to $(\varSigma ,d_\varSigma )$, but we will not use this fact in what follows.

Definition 3

Let $\delta _\varSigma , \delta _G \ge 1$. We say that a spinal graph $(G,\varSigma ,\pi )$ has dimensions $(\delta _\varSigma , \delta _G)$ if there exists a point $x_0 \in \varSigma$ and an increasing sequence $(n_k)_{k \in \mathbb {N}}$ of natural numbers such that for all $k \in \mathbb {N}$,

$$\begin{aligned} |D(x_0,{2n_k})|\lesssim & {} |D(x_0,n_k)|, \end{aligned}$$

(2)

$$\begin{aligned} |B_\varSigma (x_0,2n_k)|\lesssim & {} n_k^{\delta _\varSigma }, \end{aligned}$$

(3)

$$\begin{aligned} |D(x_0,n_k)|\simeq & {} n_k^{\delta _G}. \end{aligned}$$

(4)

Note that the dimenions of a spinal graph need not be uniquely determined, and may vary for different choices of $x_0$ and $(n_k)_{k \in \mathbb {N}}$.

Example 2

Let $n \in \mathbb {N}$ and consider the Vicsek graph $\mathcal {V}^n$ in $\mathbb {R}^n$, the construction of which is given in [4, Proof of Theorem 4.1], [8, Chapter 5], and [9, Sec. 5]. One can consider $\mathcal {V}^n$ as a graph with $V(\mathcal {V}^n) \subset \mathbb {Z}^n$, defined as an increasing union of subgraphs $\cup _{m=0}^\infty \mathcal {V}_m^n$. The subgraph $\mathcal {V}_0^n$ consists of $2^n + 1$ vertices: one at each corner of the unit n-cube, and a central vertex at the origin. Each corner vertex is connected to the central vertex. For $m \ge 1$, $\mathcal {V}_m^n$ is constructed inductively by connecting a copy of $\mathcal {V}_{m-1}^n$ to each ‘corner’ of $\mathcal {V}_{m-1}^n$. It follows that $|V(\mathcal {V}_m^n)| \simeq (2^n + 1)^m$ (see [4, equation (4.10)]).

Let $\varSigma \in V(\mathcal {V}^n)$ be the set of vertices along the $2^n$ diagonals: with $V(\mathcal {V}^n) \subset \mathbb {Z}^n$, we have

$$\begin{aligned} \varSigma = \{(\varepsilon _1 m,\varepsilon _2 m,\ldots ,\varepsilon _n m) \in \mathbb {Z}^n : m \in \mathbb {N}, \varepsilon _j \in \{1,-1\}\}. \end{aligned}$$

For every vertex $x \in V(\mathcal {V}^n)$ there is a unique $y \in \varSigma$ such that x and y are connected by a path which intersects $\varSigma$ only at y. Setting $\pi (x) := y$ makes $(\mathcal {V}^n,\varSigma )$ a spinal graph. Pictured in Fig. 2 are the first few steps of the construction of $\mathcal {V}^2$, with the spine $\varSigma$ emphasised.

Let $o \in V(\mathcal {V})$ be the ‘center vertex’ of $\mathcal {V}_0^n$, and for $k \in \mathbb {N}$ let $n_k := 3^k$. Then $D(o,n_k) = \mathcal {V}_k^n$, and so

$$\begin{aligned} |D(o,n_k)| = (2^n + 1)^k = n_k^{\log _3(2^n + 1)}. \end{aligned}$$

We also have

$$\begin{aligned} |D(o,2n_k)| \le |D(o,n_{k+1})| = n_{k+1}^{\log _3(2^n + 1)} \simeq n_k^{\log _3(2^n + 1)} = |D(o,n_k)| \end{aligned}$$

and

$$\begin{aligned} |B_\varSigma (o, 2n_k)| = 2^n(2n_k) - 1 < 2^{n+1} 3^k \simeq n_k, \end{aligned}$$

which tells us that the spinal graph $(\mathcal {V}^n,\varSigma )$ has dimensions $(1,\log _3(2^n + 1))$. In addition, $\mathcal {V}^n$ has polynomial volume growth of dimension $\log _3(2^n + 1)$, that is

$$\begin{aligned} |B_\mathcal {V}(x,r)| \simeq r^{\log _3(2^n + 1)} \end{aligned}$$

for all $x \in V(\mathcal {V}^n)$ and $r \in \mathbb {N}$. (see [4, page 632]). The lower estimate will allow us to apply Corollary 2 to $\mathcal {V}^n$.

Fig. 2

The first three steps of the construction of the Vicsek graph $\mathcal {V}^2$, with spine

In Section 6 we construct spinal graphs with global volume lower bounds and dimensions (1, D) with $D > 1$ that do not arise from fractals.

5 Nash-type inequalities and spinal dimensional consequences

Now we assume that G is locally finite. For each vertex $x \in V(G)$ let $m(x) < \infty$ denote the number of neighbours of x, and for each $f :V(G) \rightarrow \mathbb {C}$ define the length of the gradient $|\nabla f(x)|$ by

$$\begin{aligned} |\nabla f(x)| := \left( \frac{1}{2} \sum _{\begin{array}{c} y \in V(G) \\ y \sim x \end{array}} \frac{1}{m(x)} |f(y) - f(x)|^2 \right) ^{1/2}. \end{aligned}$$

For $1 < p \le \infty$ and $\beta > 0$, we consider the Nash-type inequality

$$\begin{aligned} S(p,\beta ): \qquad \Vert f \Vert _p^{1 + \frac{p^\prime }{\beta }} \lesssim \Vert f \Vert _1^{\frac{p^\prime }{\beta }} \Vert |\nabla f| \Vert _p \qquad (f :V(G) \rightarrow \mathbb {C} \hbox { finitely supported}), \end{aligned}$$

which G may or may not satisfy.

In the presence of a spinal structure, the inequality $S(p,\beta )$ gives quantitative information connecting the ‘spinal volume growth’ of G with the volume growth of $\varSigma$. This is shown by constructing test functions, defined in terms of the spinal distance, which are constant on the fibres $\pi ^{-1}(x)$. The gradients of these test functions are supported on the spine $\varSigma$, while the functions themselves are supported on spinal sets.

Lemma 2

Let $(G,\varSigma ,\pi )$ be a spinal graph. Fix $p \in (1,\infty )$ and suppose that G satisfies $S(p,\beta )$. Then for every $x_0 \in \varSigma$ and $n \in \mathbb {N}$ we have

$$\begin{aligned} |D(x_0,n)|^{\frac{1}{p} \left( 1 + \frac{p^\prime }{\beta } \right) } \lesssim n^{-1} |D(x_0,2n)|^{\frac{p^\prime }{\beta }} |B_\varSigma (x_0,2n)|^{\frac{1}{p}}. \end{aligned}$$

(5)

Proof

For each $x_0 \in \varSigma$ and $n \in \mathbb {N}$ define $g_n :V(G) \rightarrow [0,1]$ by

$$\begin{aligned} g_{n}(x) := \frac{\max (0,n-[x,x_0])}{n}. \end{aligned}$$

Note that $g_{n}(x) = 0$ if and only if $[x,x_0] \ge n$, so that ${\text {supp}}g_{n} = D(x_0,n-1)$. Furthermore note that $g_{n}$ is constant on each $\pi ^{-1}(x)$.

Since $|g_{2n}| \le 1$ and ${\text {supp}}g_{2n} \subset D(x_0,2n)$ we have

$$\begin{aligned} \Vert g_{2n} \Vert _1 \lesssim |D(x_0,2n)|. \end{aligned}$$

(6)

Next, since $g_{2n}(x) \ge 1/2$ for $x \in D(x_0,n)$, we have

$$\begin{aligned} \Vert g_{2n} \Vert _p \ge \left( \sum _{x \in D(x_0,n)} 2^{-p} \right) ^{1/p} \simeq |D(x_0,n)|^{1/p}. \end{aligned}$$

(7)

Finally, note that $|\nabla g_{2n}(x)|= 0$ whenever $x \in G {\setminus } \varSigma$ (since $g_{2n}$ is constant on each connected component of $G \setminus \varSigma$) or $x \in G \setminus D(x_0,2n)$ (since ${\text {supp}}g_{2n} = D(x_0,2n-1)$). When $x \in \varSigma \cap D(x_0,2n)$ and $y \sim x$, we have

$$\begin{aligned} g_{2n}(x) - g_{2n}(y) = \left\{ \begin{array}{ll} (2n)^{-1} &{} \text {if } y \in \varSigma \cap D(x_0,2n) \\ 0 &{} \text {otherwise.} \end{array} \right. \end{aligned}$$

Therefore

$$\begin{aligned} \Vert |\nabla g_{2n}| \Vert _p&= \left( \sum _{x \in \varSigma \cap D(x_0,2n)} \left( \frac{1}{2} \sum _{y \sim x, y \in \varSigma \cap D(x_0,2n)} \frac{1}{m(x)} (2n)^{-2} \right) ^{p/2} \right) ^{1/p} \\&\lesssim n^{-1} |\varSigma \cap D(x_0,2n)|^{1/p} = n^{-1} |B_\varSigma (x_0,2n)|^{1/p} \end{aligned}$$

Therefore, applying $S(p,\beta )$ to $g_{2n}$, we get (5). $\square$

The previous lemma can be used to show that the Nash-type inequalities $S(p,\beta )$ restrict the possible dimensions of a spinal graph.

Lemma 3

Let $(G,\varSigma ,\pi )$ be a spinal graph with dimensions $(\delta _\varSigma , \delta _G)$. Fix $p > 1$ and $\beta > 0$, and suppose G satisfies $S(p,\beta )$. Then

$$\begin{aligned} \frac{\delta _G - \delta _\varSigma }{p} - \frac{\delta _G}{\beta } \le -1. \end{aligned}$$

(8)

Proof

Fix $x_0 \in \varSigma$ and a sequence $(n_k)_{k \in \mathbb {N}}$ as in Definition 3. From Lemma 2 and assumptions (3) and (2), for all $k \in \mathbb {N}$ we have

$$\begin{aligned} |D(x_0,n_k)|^{\frac{1}{p} \left( 1 + \frac{p^\prime }{\beta }\right) }&\lesssim n_k^{-1} |D(x_0,2n_k)|^{\frac{p^\prime }{\beta }} |B_\varSigma (x_0,2n_k)|^{\frac{1}{p}} \\&\lesssim n_k^{-1 + \frac{\delta _\varSigma }{p}} |D(x_0,n_k)|^{\frac{p^\prime }{\beta }}. \end{aligned}$$

Rearranging yields

$$\begin{aligned} |D(x_0,n_k)|^{\frac{1}{p} - \frac{1}{\beta }} \lesssim n_k^{-1 + \frac{\delta _\varSigma }{p}}, \end{aligned}$$

and then

$$\begin{aligned} n_k^{\delta _G\left( \frac{1}{p} - \frac{1}{\beta }\right) } \lesssim n_k^{-1 + \frac{\delta _\varSigma }{p}}. \end{aligned}$$

follows by (4). Since $n_k$ is increasing, taking the limit as $k \rightarrow \infty$ tells us that

$$\begin{aligned} \delta _G\left( \frac{1}{p} - \frac{1}{\beta }\right) \le -1 + \frac{\delta _\varSigma }{p}, \end{aligned}$$

which is equivalent to (8).$\square$

Corollary 1

Suppose the conditions of Lemma 3 are satisfied, with $\delta _G > \delta _\varSigma$. Then $\delta _G > \beta$ and

$$\begin{aligned} p \ge \beta \frac{\delta _G - \delta _\varSigma }{\delta _G - \beta }. \end{aligned}$$

(9)

Proof

Rearranging (8) gives

$$\begin{aligned} \delta _G \ge \beta \left( 1 + \frac{\delta _G - \delta _\varSigma }{p}\right) . \end{aligned}$$

Since $\delta _G - \delta _\varSigma > 0$, we get $\delta _G > \beta$. We can then rearrange further to get (9). $\square$

6 Riesz transform unboundedness for thickened spinal graphs

Definition 4

Let G be a uniformly locally finite graph (i.e. $\sup _{x \in V(G)} m(x) < \infty$) and $n \in \mathbb {N}$. Then an n-dimensional thickening of G is a smooth Riemannian manifold M constructed by replacing each vertex $x \in V(G)$ by an n-sphere, each edge $e \in E(G)$ by an n-cylinder, and welding the cylinders smoothly to the balls according to the graph structure of G, in such a way that M has bounded geometry (i.e. M has positive injectivity radius, and Ricci curvature bounded from below).

More precisely: define

$$\begin{aligned} \widetilde{M} := \bigsqcup _{x \in V(G)} B_x \sqcup \bigsqcup _{e \in E(G)} C_e, \end{aligned}$$

where $B_x$ is isometric to a round n-sphere $S^n$ with m(x) disjoint open balls of fixed small radius $\varepsilon$ removed, and where each $C_e$ is isometric to a cylinder $S_\varepsilon ^{n-1} \times [0,1]$, with $S_\varepsilon ^{n-1} = \partial B(0,\varepsilon ) \subset \mathbb {R}^n$. A $C^0$ Riemannian manifold $M^\prime$ is constructed as a quotient of $\widetilde{M}$ by gluing a cylinder $C_e$ to two spheres $B_x$ and $B_y$ if and only if $x \sim y$ in G (in such a way that every ‘hole’ in $B_x$ has a cylinder attached to it). A thickening M with bounded geometry may then be defined by smoothing the interface between spheres and the cylinders in $M^\prime$ arbitrarily (but uniformly among all the interfaces).

Remark 2

In what follows, we may replace a thickening of G (in the sense above) by any Riemannian manifold M of bounded geometry that is isometric to G at infinity in the sense of Coulhon–Saloff-Coste [12] (following Kanai [13]); our discretisation/thickening procedures only depend on results in [12].

The following proposition can be proven by directly following the proof of [11, Proposition 6.2] (see also the first part of [9, Theorem 5.1]). The proof involves the discretisation results of [12, Sec. 6].

Proposition 2

Let G be a locally uniformly finite graph, and let M be a thickening of G of any dimension. Fix $p \in (1,\infty )$ and suppose that M satisfies $(RR_p)$. Furthermore, suppose that the heat kernel h of M satisfies

$$\begin{aligned} h_t(x,x) \lesssim t^{-\alpha /2} \qquad \textit{for all}\ t > 1, x \in M. \end{aligned}$$

Then G satisfies $S(p,\alpha )$.

Since $S(p,\alpha )$ restricts the possible dimensions of a spinal graph, we may argue by contraposition to show that dimension and volume information on a spinal graph implies unboundedness of the Riesz transform on $L^p(M)$ for sufficiently large $p > 2$ (in fact, we prove that $(RR_{p'})$ does not hold for sufficiently large $p > 2$, which is strictly stronger).

Theorem 1

Let $(G,\varSigma ,\pi )$ be a locally uniformly finite spinal graph with dimensions $(\delta _\varSigma , \delta _G)$, with $\delta _G > \delta _\varSigma$. Furthermore, suppose that $B_G(x,r) > rsim r^\nu$ for all $r \ge 1$. Let M be a thickening of G of any dimension. Then for all

$$\begin{aligned} p > 2\frac{\delta _G-\delta _\varSigma }{\frac{\delta _G}{\nu ^\prime } - 2\delta _\varSigma + 2} =: p_c(\delta _\varSigma , \delta _G, \nu ), \end{aligned}$$

M does not satisfy $(R_p)$.

Proof

The volume assumption on G implies a corresponding large-scale volume estimate

$$\begin{aligned} V(x,r) > rsim r^{\nu } \qquad (\hbox {for all } r > 1, x \in M) \end{aligned}$$

on M (this may be derived from the results of [12, Sec. 6]). Since M has bounded geometry, [4, Theorem 1.1] implies the heat kernel estimate

$$\begin{aligned} h_t(x,x) \lesssim t^{-\nu /(\nu + 1)} \qquad \hbox {for all } t > 1 \end{aligned}$$

on M. Fix $q > 1$ and suppose that M satisfies $(R_q)$, hence also $(RR_{q^\prime })$. Proposition 2 then implies that G satisfies $S(q^\prime , 2\nu /(\nu + 1))$, and Corollary 1 then yields

$$\begin{aligned} q^\prime \ge \frac{2\nu }{\nu +1} \left( \frac{\delta _G - \delta _\varSigma }{\delta _G - \frac{2\nu }{\nu +1}} \right) = p_c^\prime \end{aligned}$$

or equivalently that $q \le p_c$. Therefore M does not satisfy $(R_p)$ for any $p > p_c$. $\square$

Taking $\delta _\varSigma = 1$ and $\nu = \delta _G$ gives the following corollary.

Corollary 2

Let $(G,\varSigma ,\pi )$ be a locally uniformly finite spinal graph with dimensions $(1,\delta _G)$, with $\delta _G > 1$, and suppose that $B_G(x,r) > rsim r^{\delta _G}$ for all $r \in \mathbb {N}$. Let M be a thickening of G of any dimension. Then the Riesz transform bound $(R_p)$ for M fails for all $p > 2$.

As remarked in Example 2, these assumptions are satisfied by the Vicsek graphs $\mathcal {V}^n$, reproving [9, Theorem 5.1].

7 Non-fractal spinal graphs with volume lower bounds

Fix $D > 1$. In this section we show how to construct locally uniformly finite spinal graphs $(G,\varSigma ,\pi )$ with dimensions (1, D) and the volume lower bound

$$\begin{aligned} |B_G(x,r)| > rsim r^D \qquad (x \in V(G), r \in \mathbb {N}), \end{aligned}$$

(10)

but which need not possess any ‘fractal’ structure (in contrast with the Vicsek graph example). Corollary 2 applies to such spinal graphs, thus yielding many manifolds M for which $(R_p)$ fails for all $p > 2$.

First we need a technical lemma on volumes of intersections of balls in doubling graphs. We defer the proof to Sec. 7. Recall that a graph G is doubling if there exist constants $C_d,\nu > 0$ such that for all $0< r< R < \infty$ and $x \in V(G)$,

$$\begin{aligned} |B_G(x,R)| \le C_d(R/r)^\nu |B_G(x,r)|. \end{aligned}$$

Taking $R = 1$ and $r = 1-\varepsilon$ for $\varepsilon$ arbitrarily small shows that a doubling graph is locally uniformly finite, with $m(x) \le C_d$ for all $x \in V(G)$.

Lemma 4

Let G be a doubling graph. Then there exists $C > 0$, depending only on the doubling constants of G, such that for all $y \in V(G)$ and $R > 0$, and for all $x \in B(y,R)$ and $r \le 2R$, we have

$$\begin{aligned} |B_G(x,r) \cap B_G(y,R)| \ge C|B_G(x,r)|. \end{aligned}$$

Now we move on to our construction. This is inspired by the ‘plate’ construction in [4, Theorem 5.1].

Example 3

Fix $\delta > D$, and let $(P_n)_{n \in \mathbb {N}}$ be a family of graphs satisfying

$$\begin{aligned} |B_{P_n}(x,r)| \simeq r^\delta \qquad (x \in V(P_n), r \in \mathbb {N}) \end{aligned}$$

with implicit constants independent of n. For simplicity one can take each $P_n$ to be equal to a fixed graph P; one could even take $\delta \in \mathbb {N}$ and $P = \mathbb {Z}^\delta$. We allow for arbitrary choices to emphasise that self-similarity is not necessary. Let $\alpha = (D-1)/\delta$ (so that $\alpha \delta + 1 = D$ and $\alpha < 1$) and for each $n \in \mathbb {N}$ choose an arbitrary vertex $o_n \in V(P_n)$. Construct a spinal graph $(G,\varSigma ,\pi )$ with $\varSigma = \mathbb {N}$ as in Example 1 by taking $G_n$ to be the full subgraph of $P_n$ determined by $B_{P_n}(o_n,n^\alpha )$, and choosing as distinguished points $z_n = o_n$.

To show that this spinal graph has dimensions (1, D), take the sequence $n_k = k$ and observe that

$$\begin{aligned} |D(1,k)| = \sum _{n=1}^k |B_{P_n}(o_n,n^\alpha )| \simeq \sum _{n=1}^k n^{\alpha \delta } \simeq k^{\alpha \delta + 1} = k^D \end{aligned}$$

(the second sum may be estimated by comparing with integrals of the function $t \mapsto t^{\alpha \delta }$). In particular we have $|D(1,2k)| \simeq (2k)^D \simeq |D(1,k)|$, and furthermore it is clear that $|B_\mathbb {N}(x,r)| \simeq r$. Therefore the spinal graph has dimensions (1, D).

It is more difficult to show the global volume lower bound (10), but luckily the proof of [4, Theorem 5.1] already does this for a similar problem. Note that it suffices to assume $r \ge 2$.

First we show that $|B_G(n,r)| > rsim r^D$ for all $n \in \mathbb {N}$. To see this, write

$$\begin{aligned} |B_G(n,r)|&\ge \sum _{k=0}^{r} |B_{P_{n+k}}(o_{n+k},\min (r-k,(n+k)^\alpha ))| \\&\ge \sum _{k=0}^{\lfloor r/2 \rfloor } |B_{P_{n+k}}(o_{n+k},\min (r-k,k^\alpha ))| \\&\simeq \sum _{k=0}^{\lfloor r/2 \rfloor } k^{\alpha \delta } \simeq \lfloor r/2 \rfloor ^{\alpha \delta + 1} \simeq r^D, \end{aligned}$$

using that $k^\alpha < r-k$ for $k \le r/2$ in the third line.

Now suppose $x \in V(G)$ with $\pi (x) = n$. After identifying x with the appropriate vertex $z \in B_{P_n}(o_n, n^\alpha )$ (which, recall, is identified with $\pi (x)$), we get an identification of $B_G(x,r) \cap \pi ^{-1}(n)$ with $B_{P_n}(z,r) \cap B_{P_n}(o_n, n^\alpha )$. Thus for $r \le 2n^\alpha$ we have

$$\begin{aligned} |B_G(x,r)|&\ge |B_G(x,r) \cap \pi ^{-1}(n)| \\&= |B_{P_n}(z,r) \cap B_{P_n}(o_n,n^\alpha )| \\& > rsim |B_{P_n}(z,r)| \simeq r^\delta > r^D \end{aligned}$$

using Lemma 4 in the third line. On the other hand, if $r > 2n^\alpha$, then $B_G(x,r)$ contains both $\pi ^{-1}(n)$ and $B_G(n+1,r-1-n^\alpha )$, so

$$\begin{aligned} |B_G(x,r)|&\ge \max (|\pi ^{-1}(n)|, |B_G(n+1,r-1-n^\alpha )|) \\& > rsim |\pi ^{-1}(n)| + |B_G(n+1, r-1-n^\alpha )| \\& > rsim n^{\alpha D} + (r-1-n^\alpha )^D \\&\simeq (n^\alpha + r - 1 - n^\alpha )^D \\&\simeq r^D. \end{aligned}$$

This completes the proof of (10).

The following corollary is then an immediate consequence of Corollary 2.

Corollary 3

Suppose M is a thickening of a spinal graph $(G,\varSigma ,\pi )$ as constructed as in Example 3. Then the Riesz transform bound $(R_p)$ for M fails for all $p > 2$.

Remark 3

It is probably possible to construct spinal graphs with dimensions $(\delta _\varSigma , \delta _G)$ with $1< \delta _\varSigma < \delta _G$ and with a polynomial volume lower bound of dimension $\delta _G$, thus yielding manifolds M for which $(R_p)$ fails for all $p> p_c > 2$. Since our construction exploits taking $\varSigma = \mathbb {N}$, this is beyond the scope of this article. It may even be possible to show that $(R_p)$ holds on such manifolds for $p \in (2,p_c)$, but this is very much beyond the scope of this article.

8 Proof of Lemma 4

Here we prove the technical lemma needed in the construction of the previous section. We write $B(x,r) := B_G(x,r)$ and $d(x,y) := d_G(x,y)$.

Proof

First note that if the result is true for $r \le R$, then it holds for $r \le 2R$, because in this case for $r > R$ we have

$$\begin{aligned} |B(x,r) \cap B(y,R)| \ge |B(x,R) \cap B(y,R)|&\ge C|B(x,R)|\\&\ge CC_d^{-1}(R/r)^\nu |B(x,r)| \\&\ge CC_d^{-1}2^{-\nu } |B(x,r)| \end{aligned}$$

using the doubling condition and $R \ge r/2$ in the last step. Thus we assume that $r \le R$, and split the proof into two cases.

Case 1: $r > 2d(x,y)$. By definition of the combinatorial distance, there exists a vertex z such that $d(y,z) + d(z,x) = d(y,x)$ and $d(y,z) = \lceil d(y,x)/2 \rceil$. Suppose that $z^\prime \in B(z,r-d(y,z))$. Then

$$\begin{aligned} d(z^\prime ,x)&\le r - d(y,z) + d(z,x) \\&= r - 2\lceil d(y,x)/2 \rceil + d(y,x) \\&\le r - 2d(y,x) \le r \end{aligned}$$

and

$$\begin{aligned} d(z^\prime ,y) \le r - d(y,z) + d(z,y) = r \le R, \end{aligned}$$

so $B(z,r-d(y,z)) \subset B(x,r) \cap B(y,R)$. Therefore

$$\begin{aligned} |B(x,r) \cap B(y,R)|&\ge |B(z,r-d(y,z))| \\& > rsim \left( \frac{r-d(y,z)}{r+d(x,z)} \right) ^\nu |B(z,r+d(x,z))| \\&\ge \left( \frac{r-d(y,z)}{r+d(x,z)} \right) ^\nu |B(x,r)| \end{aligned}$$

for some $\nu > 0$ determined by the doubling constant of G. To see that the bracketed expression above is uniformly bounded below, estimate its reciprocal from above by

$$\begin{aligned} \frac{r+d(x,z)}{r-d(y,z)}&= \frac{r-d(y,z)+d(x,y)}{r-d(y,z)} \\&= 1 + \frac{d(x,y)}{r - d(y,z)} \\&< 1 + \frac{d(x,y)}{2d(x,y) - d(y,z)} \\&= 1 + \frac{d(x,y)}{d(x,y) + d(z,x)} \\&\le 2. \end{aligned}$$

using $r > 2d(x,y)$ in the third line and $d(z,x) = d(x,y) - d(y,z)$ in the fourth line.

Case 2: $r \le 2d(x,y)$. Note that the estimate for $r < 2$ follows from the fact that G is locally uniformly finite, so it suffices to consider $r \ge 2$. As in the previous case, there exists a vertex z such that $d(y,z) + d(z,x) = d(y,x)$ and $d(x,z) = \lfloor r/2 \rfloor$ (here we use that $r/2 \le d(x,y)$). If $z^\prime \in B(z,\lfloor r/2 \rfloor )$, then

$$\begin{aligned} d(z^\prime ,x) < 2\left\lfloor \frac{r}{2} \right\rfloor \le r \end{aligned}$$

and

$$\begin{aligned} d(z^\prime ,y) < \left\lfloor \frac{r}{2} \right\rfloor + d(z,y) = \left\lfloor \frac{r}{2}\right\rfloor + d(x,y) - \left\lfloor \frac{r}{2} \right\rfloor = d(x,y) \le R, \end{aligned}$$

using that $x \in B(y,R)$ by assumption, and so $B(z,\lfloor r/2 \rfloor ) \subset B(x,r) \cap B(y,R)$. Therefore

$$\begin{aligned} |B(x,r) \cap B(y,R)|&\ge |B(z,\lfloor r/2 \rfloor )| \\&\ge |B(z,r/4)| \\&\ge (8^\nu C_d)^{-1} |B(z,2r)| \\&\ge (8^\nu C_d)^{-1} |B(z,r+d(z,x))| \\&\ge (8^\nu C_d)^{-1} |B(x,r)| \end{aligned}$$

since $\lfloor r/2 \rfloor \ge r/4$ holds whenever $r \ge 2$, and using the doubling condition. $\square$