Riesz Transform for $1\le p \le 2$ Without Gaussian Heat Kernel Bound

Abstract

We study the $L^p$ boundedness of the Riesz transform as well as the reverse inequality on Riemannian manifolds and graphs under the volume doubling property and a sub-Gaussian heat kernel upper bound. We prove that the Riesz transform is then bounded on $L^p$ for $1<p<2$, which shows that Gaussian estimates of the heat kernel are not a necessary condition for this. In the particular case of Vicsek manifolds and graphs, we show that the reverse inequality does not hold for $1<p<2$. This yields a full picture of the ranges of $p\in (1,+\infty )$ for which respectively the Riesz transform is $L^p$-bounded and the reverse inequality holds on $L^p$ on such manifolds and graphs. This picture is strikingly different from the Euclidean one.

Appendix: Estimates for the Heat Kernel in Fractal Manifolds

Let $(\Gamma ,\mu )$ be a graph as in Sect. 4. For all $A\subset \Gamma$, let $\partial A$ denote the exterior boundary of A, defined as $\left\{ x\in \Gamma {\setminus } A; \text{ there } \text{ exists } y\sim x \text{ with } y\in A\right\}$, and let $\overline{A}:=A\cup \partial A$.

A function $h:\overline{A}\rightarrow \mathbb {R}$ is said to be harmonic in A if and only if, for all $x\in A, \Delta h(x)=0$.

Say that $(\Gamma ,\mu )$ satisfies a Harnack elliptic inequality if and only if, for all $x\in \Gamma$, all $R\ge 1$ and all nonnegative harmonic functions u in B(x, 2R),

Let $D>0$. Say that $\Gamma$ satisfies $(V_{D})$ if and only if, for all $x\in \Gamma$ and all $r>0$,

Denote by $(X_n)_{n\ge 1}$ a random walk on $\Gamma$, that is a Markov chain with transition probability given by p. For all $A\subset \Gamma$, define

$$\begin{aligned} T_A:=\min \left\{ n\ge 1;\ X_n\in A\right\} \end{aligned}$$

and, for all $x\in \Gamma$ and all $r>0$, let

$$\begin{aligned} \tau _{x,r}:=T_{B^c(x,r)}. \end{aligned}$$

If $m>0$, say that $\Gamma$ satisfies ($E_{m}$) if and only if, for all $x\in \Gamma$ and all $r>0$,

Theorem 2 in [3] claims that, for all $m\in [2,D+1]$, there exists a graph $\Gamma$ satisfying ($V_{D}$) (hence the doubling volume property), (EHI) and ($E_{m}$). Therefore, Theorem 2.15 in [5] (see also [29, Theorem 3.1]) implies that $\Gamma$ satisfies the following parabolic Harnack inequality: for all $x_0\in \Gamma$, all $R\ge 1$ and all non-negative functions $u:\llbracket 0,4N\rrbracket \times \overline{B(x_0,2R)}$ solving $u_{n+1}-u_n=\Delta u_n$ in $\llbracket 0,4N-1\rrbracket \times B(x_0,2R)$, one has

$$\begin{aligned} \max _{n\in \llbracket N,2N-1\rrbracket ,\ y\in B(x,R)} u_n(y)\lesssim \min _{n\in \llbracket 3N,4N-1\rrbracket ,\ y\in B(x,R)} (u_n(y)+u_{n+1}(y)), \end{aligned}$$

where N is an integer satisfying $N\sim R^{m}$ and $N\ge 2R$.

Consider now the manifold M built from $\Gamma$ with a self-similar structure at infinity by replacing the edges of the graph with tubes of length 1 and then gluing the tubes together smoothly at the vertices. Since M and $\Gamma$ are roughly isometric, Theorem 2.21 in [5] yields that M satisfies the following parabolic Harnack inequality: for all $x_0\in M$ and all $R>0$, for all non-negative solutions u of $\partial _tu=\Delta u$ in $(0,4R^{m})\times B(x_0,R)$, one has

$$\begin{aligned} \sup _{Q_-}u\lesssim \inf _{Q_+}u, \end{aligned}$$

where

$$\begin{aligned} Q_-=(R^{m},2R^{m})\times B(x_0,R) \end{aligned}$$

and

$$\begin{aligned} Q_+=(3R^{m},4R^{m})\times B(x_0,R). \end{aligned}$$

In turn the parabolic Harnack inequality implies (${\textit{UE}}_m$) (see [30, Theorem 5.3]). See also [31, Section 1.2.7].

Finally, let us explain why (${\textit{UE}}_m$) for $m>2$ is incompatible with (${\textit{UE}}$). It is classical (see, for instance, [13, Section 3]) that (${\textit{UE}}_m$) together with (D) implies the on-diagonal lower bound

$$\begin{aligned} h_{t}(x,x) \gtrsim \frac{1}{V(x,t^{1/m})}, \quad t\ge 1, \end{aligned}$$

which is clearly incompatible with

$$\begin{aligned} h_{t}(x,x) \lesssim \frac{1}{V(x,t^{1/2})}, \quad t\ge 1 \end{aligned}$$

because of the so-called reverse volume doubling property (see, for instance, [28, Proposition 5.2]).