Riesz transform on eventually Gaussian local trees

With Aobo Chen and Li Chen, we have posted a new paper, Riesz transform on eventually Gaussian local trees which is a companion paper to the paper discussed in the previous post to which we refer for background about Riesz transform on manifolds. In this work, we study spaces which have a fractal structure at small scales but Gaussian diffusion at large scales. On a family of Vicsek fractafolds, we obtain a complete description of the Riesz and reverse Riesz inequalities. The interesting feature is that the ranges of exponents are exactly reversed compared with those on Vicsek manifolds.

More precisely, the starting point for us was the work of Li Chen, Thierry Coulhon, Joseph Feneuil and Emmanuel Russ. They prove weak type $(1,1)$ and boundedness for $1<p\leq2$ under volume doubling and heat kernel upper bounds which are Gaussian at small times and sub-Gaussian at large times.

Their examples include Vicsek manifolds: smooth manifolds constructed by replacing the edges of a Vicsek graph by tubes. Locally, diffusion has the usual Gaussian behavior, but it has a sub-Gaussian behavior at large scales. On these manifolds, the Riesz inequality holds exactly for $1<p\leq2$ , and its reverse exactly for $2\leq p<\infty$ .

Our spaces have the opposite arrangement of scales. They retain genuine fractal structure at arbitrarily small distances, while their large-scale organization produces Gaussian diffusion. To describe this, let $\Psi(r)$ denote the characteristic diffusion time at distance $r$ . A model case is

$\displaystyle \Psi(r)\asymp\begin{cases}r^\beta,&0<r\leq1,\\r^2,&r\geq1,\end{cases}\qquad\beta>2.$

The word eventually refers to large distances, or equivalently to long times. The space itself remains singular at small scales.

We work on uniform local trees: geodesic spaces whose sufficiently small balls are real trees, with a radius that can be chosen uniformly over the space. Global loops are allowed. The tree structure gives a concrete derivative along arcs.

There are two measures. The reference measure $m$ (the Hausdorff measure in most examples) describes volume and is used to integrate functions. The length measure $\nu$ is carried by the skeleton $S$ , the union of the open geodesic arcs, and is used to integrate their weak derivatives $\partial f$ . The Dirichlet form is

$\displaystyle \mathcal E(f,f)=\int_S|\partial f|^2\,d\nu.$

This energy defines the Laplacian on $L^2(m)$ . The Riesz transform $\partial(-\Delta)^{-1/2}$ is initially defined on $L^2(m)$ with values in $L^2(\nu)$ . The boundedness question asks for which exponents it extends to a bounded map

$\displaystyle \mathcal R=\partial(-\Delta)^{-1/2}:L^p(m)\longrightarrow L^p(\nu).$

In the fractal examples, $m$ and $\nu$ are mutually singular.

Our result assumes uniform volume growth, two-sided heat kernel estimates with scale function $\Psi$ , and a compatible pointwise gradient estimate for the heat kernel. Under eventual Gaussian scaling, together with a local Dini condition, we prove

$\displaystyle \|\mathcal R f\|_{L^p(\nu)}\leq C_p\|f\|_{L^p(m)},\qquad 2\leq p<\infty.$

The Dini condition requires, for some $q>2$ ,

$\displaystyle \int_0^1\left(\frac{\Psi(r)}{r^2}\right)^{1/2-1/q}\frac{dr}{r}<\infty.$

It holds whenever $\Psi(r)\lesssim r^\beta$ near zero for some $\beta>2$ , so it includes the model above. Duality then gives the reverse inequality

$\displaystyle \|(-\Delta)^{1/2}f\|_{L^p(m)}\leq C_p\|\partial f\|_{L^p(\nu)},\qquad 1<p\leq2,$

for functions in the energy domain with finite right-hand side.

The obstruction on the other side of $2$ is expressed by a rigidity theorem. Assuming uniform volume growth and two-sided heat kernel estimates, boundedness of $\mathcal R$ for even one exponent $1<p<2$ , or a reverse inequality for even one exponent $p>2$ , forces

$\displaystyle m(B(x,r))\asymp r,\qquad\Psi(r)\asymp r^2$

uniformly at small radii. It also forces $m$ and $\nu$ to be comparable as measures. Thus these inequalities rule out the genuinely fractal local geometry of our examples. This necessity result does not require the pointwise heat kernel gradient estimate used in the positive theorem.

We study a concrete family of examples in the paper. The alternating Vicsek fractafold is obtained by joining compact Vicsek sets in a periodic arrangement. For each integer $d\geq2$ , it is locally fractal and has the large-scale geometry of $\mathbb Z^d$ . Write

$\displaystyle \alpha_d=\frac{\log(2^d+1)}{\log3}.$

Its volume growth and diffusion scale satisfy

$\displaystyle m(B(x,r))\asymp\begin{cases}r^{\alpha_d},&0<r\leq1,\\r^d,&r\geq1,\end{cases}$

$\displaystyle \Psi(r)\asymp\begin{cases}r^{\alpha_d+1},&0<r\leq1,\\r^2,&r\geq1.\end{cases}$

Since $\alpha_d>1$ , the positive theorem and the rigidity theorem give the complete answer. The comparison with the Vicsek manifolds is:

Property	Vicsek manifolds	Vicsek fractafolds
Small-scale diffusion	Gaussian	Sub-Gaussian
Large-scale diffusion	Sub-Gaussian	Gaussian
Riesz inequality	$1<p\leq2$	$2\leq p<\infty$
Reverse Riesz inequality	$2\leq p<\infty$	$1<p\leq2$

The main idea of the proof is to separate the two scales. With $P_t=e^{t\Delta}$ , we write

$\displaystyle \mathcal R=\mathcal R(I-P_1)+\mathcal RP_1.$

The first term contains the high frequencies and is controlled using the local gradient estimates and the Dini condition. The second term contains the low frequencies. For this part, we discretize the space by a graph and apply a good-lambda argument to local energy norms of the gradient. This is where the large-scale Gaussian behavior enters.

Similarly to the paper, about Riesz transforms on the Vicsek set, AI tools were used during the preliminary phase of the work to explore different directions and strategies and again in the final phase for polishing and proofreading. The writing took a few months. The authors are responsible for the content.

References

F. Baudoin, A. Chen and L. Chen, Riesz transform on eventually Gaussian local trees, arXiv:2609.34331 (2026).
L. Chen, T. Coulhon, J. Feneuil and E. Russ, Riesz transform for $1\leq p\leq2$ without Gaussian heat kernel bound, Journal of Geometric Analysis 27 (2017), 1489-1514.