Riesz transforms on the Vicsek set

This post is about a long paper with Aobo Chen and Li Chen, Riesz transforms on the Vicsek set. I want to explain, without too much technicalities, what we prove and put the results into context.

As a side note, I point out that artificial intelligence tools have contributed in a substantial way in the conception and the architecture of some the arguments. In particular, the proof of the failure of the strong $L^p$ bounds at the critical exponent is essentially due to ChatGPT 5.5. pro. (After research, it appears that the strategy relies on methods previously used by Fefferman). Concretely, 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.

1. Riesz transforms on manifolds

On $\mathbb{R}^n$ the Riesz transform is the operator $\nabla(-\Delta)^{-1/2}$ and its $L^p$ -boundedness for $1<p<\infty$ is related to the two inequalities,

$\displaystyle \|\nabla f\|_{p}\le C\,\|(-\Delta)^{1/2}f\|_{p}\qquad\text{and}\qquad \|(-\Delta)^{1/2}f\|_{p}\le C\,\|\nabla f\|_{p},$

the Riesz and the reverse Riesz inequality. For $p=2$ , by spectral theory, both are identities with $C=1$ on any complete Riemannian manifold, since $\int|\nabla f|^2=\int f(-\Delta f)$ . For $p\ne2$ geometry enters.

Coulhon and Duong proved in 1999 that under volume doubling and a Gaussian upper bound on the heat kernel, the Riesz transform is of weak type $(1,1)$ and bounded on $L^p$ for $1<p\le2$ [CD99], and conjectured that this holds on every complete manifold with no assumption at all [CD03, Conjecture 1.1]. After intermediate steps (notably [CCFR17], which only needs sub-Gaussian bounds and thus covers fractal-like manifolds), the conjecture was recently settled in September 2026 by Rui Chen, Renjin Jiang, Bo Li and Hong-Quan Li [CJLL26]: on every complete non-compact Riemannian manifold, the Riesz transform is of weak type $(1,1)$ , with constant $2$ for real-valued functions, hence bounded on $L^p$ for $1<p\le2$ .

On the other hand, for $p>2$ the Riesz inequality genuinely requires some geometry: it holds for all $p$ under non-negative Ricci curvature (Bakry), it is characterised under doubling and Poincaré by gradient bounds for the heat semigroup [ACDH04], it fails for large $p$ on connected sums of Euclidean spaces [CD99], and it fails for every $p>2$ on spaces with a slow, sub-Gaussian diffusion [Fen26a]. The reverse inequality for $p'$ follows from the Riesz inequality for $p$ by duality, so [CJLL26] gives it for all $p\ge2$ ; for $p<2$ it needs Poincaré-type hypotheses [AC05, DR22] and fails on Vicsek manifolds and graphs [CCFR17].

Therefore: on manifolds the exponent $1/2$ is universal, and the range $1<p\le2$ is free of any assumption. The situation turns out to be quite different on fractals: In particular the exponent 1/2 has to be replaced by an exponent that depends on $p$ .

2. The Vicsek set

The world of fractal sets is a mathematically interesting world in which some familiar functional analytic results (like Sobolev or isoperimetric inequalities) often take a very different form with the appearance of new critical exponents which reflect the rough nature of the underlying set.. With my co-authors we were interested in looking for analogues of the Riesz inequalities in this world. Thanks to recent works with Li Chen [BC23,BC24], we have a solid understanding of Sobolev theory in a specific fractal, the Vicsek set. It therefore was a convenient framework to understand what the results could look like. The Vicsek set is the self-similar cross made of five copies of itself scaled by $1/3$ ; we use its unbounded version $X$ , of Hausdorff dimension $d_f=\log_35$ , with its geodesic distance and Hausdorff measure $m$ . Notably, it is a tree: two points are joined by a unique arc, and the union of all arcs, the skeleton $\mathcal S$ , is a countable union of segments carrying a length measure $\nu$ . Functions that are absolutely continuous along segments have a derivative $\partial f$ on the skeleton, which gives Sobolev spaces $W^{1,p}(X)$ ( $f\in L^p(m)$ , $\partial f\in L^p(\nu)$ ) [BC23] and the Dirichlet form $\mathcal E(f,f)=\int_{\mathcal S}|\partial f|^2\,d\nu$ on $L^2(X,m)$ , with Laplacian $\Delta$ and heat semigroup $P_t$ . The heat kernel has sub-Gaussian estimates with walk dimension $d_w=d_f+1$ .

Two features make the situation different from the smooth setting . The measures $m$ and $\nu$ are mutually singular: the skeleton has $m$ -measure zero, so gradients live on a set of Hausdorff measure zero ! In Dirichlet form language there is no carré du champ, and $X$ falls outside the framework of [CJLL26].

Self-similarity decides the critical exponent for the Riesz inequalities. Dilating a function $f$ with $f_n(x)=f(3^{-n}x)$ multiplies $\|\partial f_n\|_{L^p(\nu)}$ by $3^{n(1/p-1)}$ and $\|(-\Delta)^\gamma f_n\|_{L^p(m)}$ by $3^{n(d_f/p-d_w\gamma)}$ (the measure scales by $5^n$ , the Laplacian by $3^{-nd_w}$ ). An inequality between the two, uniform in $n\in\mathbb Z$ , forces

$\displaystyle \gamma=\gamma_p:=\frac{d_f+p-1}{p\,d_w},\qquad\text{i.e.}\qquad \gamma_p-\frac12=\frac{d_f-1}{d_w}\Big(\frac1p-\frac12\Big).$

So the natural Riesz transform is $\mathcal R_p=\partial(-\Delta)^{-\gamma_p}$ , with an exponent that depends on $p$ and which deviates from the manifold universal exponent $1/2$ . We note that the same exponent appears as a Besov critical exponent, in the heat kernel gradient bounds of [BC23, BC24], on the Vicsek cable system [DR26] and on Vicsek graphs [Fen26b].

3. Our results

Off the critical exponent. For the heat regularisation similar to Li Chen’s quasi-Riesz inequalities [Che15], we first prove that the reverse inequality $\|(-\Delta)^\gamma P_1f\|_{L^p(m)}\le C\|\partial f\|_{L^p(\nu)}$ holds for every $p\in[1,\infty)$ and $\gamma\in(\gamma_p,1)$ , and fails, even in weak $L^p$ , for $\gamma\in(0,\gamma_p)$ .

At the critical exponent. For $p\in[1,2)$ the reverse inequality holds in weak $L^p$ , uniformly in the regularisation,

$\displaystyle \sup_{t>0}\|(-\Delta)^{\gamma_p}P_tf\|_{L^{p,\infty}(m)}\le C_p\,\|\partial f\|_{L^p(\nu)},$

so that $(-\Delta)^{\gamma_p}P_tf$ has a weak- $L^p$ limit as $t\downarrow0$ for every $f\in W^{1,p}(X)$ , but the strong $L^p$ inequality fails. For $p>2$ the inequality fails even in weak $L^p$ .

The Riesz transform. $\mathcal R_1$ is not of weak type $(1,1)$ . For $1<p<2$ , $\mathcal R_p$ is bounded from $L^p(m)$ to $L^{p,\infty}(\nu)$ but not from $L^{p,1}(m)$ to $L^p(\nu)$ . For $p=2$ it is of course an isometry. For $p>2$ it is bounded from the Lorentz space $L^{p,1}(m)$ to $L^p(\nu)$ but not from $L^p(m)$ to $L^p(\nu)$ . In short, $\mathcal R_p$ is bounded $L^p(m)\to L^p(\nu)$ only for $p=2$ .

References

[AC05] P. Auscher, T. Coulhon, Riesz transform on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 531–555.

[ACDH04] P. Auscher, T. Coulhon, X. T. Duong, S. Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), 911–957.

[BC23] F. Baudoin, L. Chen, Sobolev spaces and Poincaré inequalities on the Vicsek fractal, Ann. Fenn. Math. 48 (2023), 3–26.

[BC24] F. Baudoin, L. Chen, Heat kernel gradient estimates for the Vicsek set, Math. Nachr. 297 (2024), 4450–4477.

[CCFR17] L. Chen, T. Coulhon, J. Feneuil, E. Russ, Riesz transform for $1\le p\le2$ without Gaussian heat kernel bound, J. Geom. Anal. 27 (2017), 1489–1514.

[CD99] T. Coulhon, X. T. Duong, Riesz transforms for $1\le p\le2$ , Trans. Amer. Math. Soc. 351 (1999), 1151–1169.

[CD03] T. Coulhon, X. T. Duong, Riesz transform and related inequalities on noncompact Riemannian manifolds, Comm. Pure Appl. Math. 56 (2003), 1728–1751.

[Che15] L. Chen, Sub-Gaussian heat kernel estimates and quasi Riesz transforms for $1\le p\le2$ , Publ. Mat. 59 (2015), 313–338.

[CJLL26] R. Chen, R. Jiang, B. Li, H.-Q. Li, The Coulhon–Duong conjecture for the Riesz transform on complete Riemannian manifolds, arXiv:2609.23503 (2026).

[DR22] B. Devyver, E. Russ, Reverse inequality for the Riesz transforms on Riemannian manifolds, arXiv:2209.05083 (2022).

[DR26] B. Devyver, E. Russ, Reverse inequality for quasi-Riesz transforms on cable systems, J. Geom. Anal. 36 (2026), Paper No. 2.

[DRY23] B. Devyver, E. Russ, M. Yang, Gradient estimate for the heat kernel on some fractal-like cable systems and quasi-Riesz transforms, Int. Math. Res. Not. IMRN (2023), 15537–15583.

[Fen26a] J. Feneuil, In spaces with a slow diffusion, the Riesz transform is unbounded on $L^p$ , $p\in(2,\infty)$ , J. Geom. Anal. 36 (2026), Paper No. 61.

[Fen26b] J. Feneuil, Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion, arXiv:2606.05475 (2026).