Research and Lecture notes

Riesz transforms on the Vicsek set

Posted on September 27, 2026 by Fabrice Baudoin

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).

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Assistant Professor in Probability Theory & Stochastic Analysis

Posted on November 8, 2025 by Fabrice Baudoin

📍 Aarhus University – Department of Mathematics (Ny Munkegade 118, 8000 Aarhus C)
🗓 Position start: 1 August 2026 (3-year full-time fixed-term: 1 Aug 2026–31 Jul 2029)

Join the vibrant Stochastics Group at the Department of Mathematics, Aarhus University, at the faculty of Natural Sciences. Our research spans probability theory, stochastic analysis, stochastic differential geometry, Malliavin calculus, stochastic PDEs, random matrices, and related areas of analysis and geometry. The successful candidate will be affiliated with the “Stochastic Analysis in Aarhus” project, funded by the Villum Foundation and led by myself.

What you’ll bring:
A completed PhD in mathematics, with a strong research record in probability theory, stochastic analysis or closely related fields.
Demonstrated ability to conduct independent research and to collaborate as part of a research group.
Experience in teaching at university level and willingness to contribute to research-based teaching, including examination duties.

What we offer:
A stimulating research environment with an international profile and strong collaborations in stochastic analysis and related fields.
Excellent research infrastructure, including access to travel funds and visitor support.
A collegial, dynamic department with active seminar series, workshops and a broad range of research groups.
Support for international researchers and accompanying families, including relocation services and career counselling for expat partners.

Application deadline: 15 December 2025 – 23:59 CET
—
If you’re passionate about deepening your impact in stochastic analysis in a supportive, internationally engaged academic setting, we would love to hear from you.

💡 Feel free to share this opportunity with suitable colleagues or post-docs.

https://math.au.dk/om/ledige-stillinger/job/assistant-professor-in-probability-theory-and-stochastic-analysis

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Aarhus Summer School in Analysis and Probability

Posted on October 22, 2025 by Fabrice Baudoin

The 6–10 July, 2026, Aarhus Summer School in Analysis and Probability aims to bring together leading experts and young researchers in the areas of analysis, probability theory, and their interactions. The school will feature a series of five hours lecture courses delivered by distinguished speakers, offering participants the opportunity to engage with current developments at the forefront of the field. In addition to the main lectures, the program will include short talks by graduate students and early-career researchers. The summer school will provide a stimulating and friendly environment, ideal for fostering discussions, collaborations, and the exchange of ideas among researchers at all career stages.

Funding for Early Career Researchers

We are pleased to announce that limited funding will be available for early career researchers (PhD students and postdocs) who wish to participate in the summer school and propose to give a talk. Applicants interested in being considered for funding are invited to include a short abstract of their proposed presentation when registering or submitting their application to the organisers.

Registration will begin on December 1st.

https://conferences.au.dk/saa-summer-school-2026

The courses are the following.

Energy measures and applications by Mathav Murugan

I plan to cover some results on energy measures such as singularity/absolute continuity, its role in the attainment problem for conformal walk dimension (and Ahlfors regular conformal dimension), and also applications to martingale dimension.

The Sard conjecture in sub-Riemannian geometry by Luca Rizzi

Sard’s theorem asserts that the set of critical values of a smooth map between finite-dimensional manifolds has measure zero. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true. Counterexamples are known already in the class of polynomial maps from an infinite-dimensional Hilbert space to the Euclidean line. The problem is in particular relevant in sub-Riemannian geometry, where the validity of the above property for the endpoint map is the so-called Sard conjecture, and is one of the main open problems in the field. We present an overview of the Sard problem in sub-Riemannian geometry together with some recent results in collaboration with Lerario and Tiberio.

Gromov hyperbolicity, uniformization, and potential theory by Nageswari Shanmugalingam

In the non smooth setting of metric measure spaces, where related notions of interest are invariant under bi-Lipschitz transformations of a metric space, negative curvature has a counterpart as Gromov hyperbolicity. This is a coarser, more large-scale notion, but appears to be closely associated with bounded uniform domains via a transformation called uniformization. In this series of lectures we will discuss how such a transformation can link potential theory on Gromov hyperbolic spaces with potential theory on bounded uniform domains. We will begin the course by discussing analogs of Sobolev spaces in non-smooth setting.

Some aspects of parabolic Anderson models by Samy Tindel

In this mini-course I will introduce some basic elements allowing to understand parabolic Anderson models on geometric structures. The topics covered will be:

(1) Gaussian noises and their regularity.

(2) Existence and uniqueness results for the stochastic heat equation.

(3) Moment estimates.

(4) Extensions from Euclidian spaces to hypoelliptic settings, fractals and other geometric contexts.

Notice that I’m not assuming any knowledge of stochastic analysis from the audience. I will try to introduce the objects I’m manipulating in a self-contained way.

Posted in Uncategorized | Tagged education, Lectures, mathematics, science, Summer school, writing | Leave a comment

Postdoctoral Positions in Probability Theory and Analysis

Posted on August 19, 2025 by Fabrice Baudoin

Several postdoctoral positions at Aarhus are opening in my group. The earliest possible start date is February 1st 2026, with flexibility for later starts in the spring.

These positions are supported by a Villum Investigator Grant and the ERC, offering excellent opportunities for travels, research and international collaboration.

https://international.au.dk/about/profile/vacant-positions/job/postdoctoral-positions-in-probability-theory-and-analysis-aarhus-university

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Lecture notes: A short introduction to Dirichlet spaces

Posted on January 26, 2025 by Fabrice Baudoin

Here are the lecture notes of the minicourse given at NYU Abu Dhabi.

Dirichlet Download

Posted in Dirichlet forms at NYU, Lecture Notes | Tagged Dirichlet forms, Heat kernel estimates, Markov semigroups, Regular Dirichlet form, Sierpinski gasket, Sobolev inequality | Leave a comment

Lecture 7. Further topics

Posted on January 21, 2025 by Fabrice Baudoin

In this lecture let X be a locally compact and complete metric space equipped with a Radon measure μ supported on X. Let (ℰ, ℱ = dom(ℰ)) be a Dirichlet form on X. We assume throughout that the heat semigroup P_t is stochastically complete, i.e., P_t1 = 1 for every t ≥ 0.

Regular Dirichlet forms, Energy measures
Hunt process associated with a regular Dirichlet form
Intrinsic metric
Further reading

Regular Dirichlet forms, Energy measures

We denote by C_c(X) the vector space of all continuous functions with compact support in X and C₀(X) its closure with respect to the supremum norm.

A core for (X,μ,ℰ,ℱ) is a subset 𝒞 of C_c(X) ∩ ℱ which is dense in C_c(X) in the supremum norm and dense in ℱ in the norm

$(||f||^2_{L^2(X,\mu)} + \mathcal{E}(f,f))^{1/2}.$

Definition. The Dirichlet form ℰ is called regular if it admits a core.

Recall that for any f,g ∈ ℱ, we have

$\mathcal{E}(f,g)=\lim_{t\to 0}\frac1t\langle (I-P_t)f,g\rangle=\lim_{t\to 0}\frac1{t}\int_X \int_X(f(x)-f(y)) g(x) p_t(x,dy) d\mu(x),$

where p_t(x,·) are the heat kernel measures associated to the Dirichlet form (ℰ, ℱ).
From the symmetry property \eqref{heat kernel measure symmetry} of the heat kernel measure one also has

$\mathcal{E}(f, g)=\lim_{t\to 0}\frac1{2t}\int_X \int_X(f(x)-f(y))(g(x)-g(y)) p_t(x,dy) d\mu(x).$

Lemma. For f,g ∈ ℱ ∩ L^∞(X,μ), fg ∈ ℱ and

$\mathcal{E}(fg)^{1/2} \le ||f||_\infty \mathcal{E}(g)^{1/2} +||g||_\infty \mathcal{E}(f)^{1/2}.$

Proof. For f, g ∈ ℱ ∩ L^∞(X,μ) such that fg ∈ ℱ,

$\mathcal{E}(fg)=\lim_{t\to 0}\frac1{2t}\int_X \int_X(f(x)g(x)-f(y)g(y))^2 p_t(x,dy) d\mu(x).$

Write f(x)g(x) – f(y)g(y) = f(x)(g(x) – g(y)) + g(y)(f(x) – f(y)), then by Minkowski’s inequality

$\left(\int_X \int_X(f(x)g(x)-f(y)g(y))^2 p_t(x,dy) d\mu(x)\right)^{1/2}$

$\le ||f||_{\infty}\left(\int_X \int_X (g(x)-g(y))^2 p_t(x,dy) d\mu(x)\right)^{1/2}$

$+||g||_{\infty}\left(\int_X \int_X(f(x)-f(y))^2 p_t(x,dy) d\mu(x)\right)^{1/2}.$

We conclude the expected inequality by multiplying by 1/√2t and taking the limit t → 0 for both sides above.

Theorem (Energy measures) Assume that ℰ is regular. For f ∈ ℱ ∩ L^∞(X,μ), there exists a unique Radon measure on X, denoted by dΓ(f), so that for every ϕ ∈ ℱ ∩ C_c(X),

$\int_X\phi\, d\Gamma(f)=\frac{1}{2}[2\mathcal{E}(\phi f,f)-\mathcal{E}(\phi, f^2)]$

$= \lim_{t \to 0} \frac{1}{2t} \int_X \int_X \phi(x) (f(x)-f(y))^2 p_t(x,dy) d\mu(x).$

The Radon measure dΓ(f) is called the energy measure of f (and is therefore the weak * limit of (1/(2t)) ∫_X (f(x) – f(y))² p_t(x,dy)).

Proof. Let f ∈ ℱ ∩ L^∞(X,μ). For any ϕ ∈ ℱ ∩ C_c(X),

$\frac1{2t} \int_X \int_X \phi(x) (f(x)-f(y))^2 p_t(x,dy) d\mu(x)=-\frac1{2t} \langle (I-P_t)f^2 ,\phi\rangle+\frac1{t} \langle (I-P_t)f ,f\phi\rangle.$

Letting t → 0, the right-hand side converges to

$\frac{1}{2}[2\mathcal{E}(\phi f,f)-\mathcal{E}(\phi, f^2)]$ .

On the other hand, observing that

$\frac1{2t} \int_X \int_X |\phi(x)| (f(x)-f(y))^2 p_t(x,dy) d\mu(x) \le \|\phi\|_{\infty} \frac1{2t} \int_X \int_X (f(x)-f(y))^2 p_t(x,dy) d\mu(x),$

we deduce

$\left|\frac{1}{2}[2\mathcal{E}(\phi f,f)-\mathcal{E}(\phi, f^2)]\right|\le \|\phi\|_{\infty}\mathcal E(f).$

Therefore we conclude the proof by applying the Riesz-Markov representation theorem.

One can actually define dΓ(f,f) for every f ∈ ℱ using the following lemmas.

Lemma. Let f ∈ ℱ. Then f_n = min(n, max(-n, f)) ∈ ℱ and ℰ(f – f_n) → 0.

Proof. Let f ∈ ℱ. For every x, y ∈ X, we have

$|f_n(x) - f_n(y)| \leq |f(x) - f(y)|$

and |f_n(x)| ≤ |f(x)|. So f_n is a normal contraction of f and ℰ(f_n) ≤ ℰ(f).

Recall that

$\mathcal{E}(f-f_n)=\lim_{t\to 0}\frac1{2t} \int_X \int_X(f(x)-f_n(x)-f(y)+f_n(y))^2 p_t(x,dy) d\mu(x).$

Expanding the square inside the integral gives that

$\mathcal{E}(f-f_n)=\mathcal E(f)+\mathcal E(f_n)-2\mathcal E(f_n,f)\le 2\mathcal E(f)-2\mathcal E(f_n,f).$

Letting n → ∞, we have ℰ(f_n,f) → ℰ(f). Therefore ℰ(f – f_n) converges to 0 as n → ∞.

Lemma. For f,g ∈ ℱ ∩ L^∞(X,μ) and nonnegative ϕ ∈ ℱ ∩ C_c(X),

$\left| \sqrt{\int_X\phi\, d\Gamma(f)} -\sqrt{\int_X\phi\, d\Gamma(g) } \right|^2 \le \int_X\phi\, d\Gamma(f-g) \le ||\phi||_{L^\infty(X,\mu)}\mathcal{E}(f-g)$

Proof. The second inequality follows from the proof in Theorem 16.3. For the first inequality, it suffice to show that for any f, g ∈ ℱ ∩ L^∞(X,μ) and any nonnegative ϕ ∈ ℱ ∩ C_c(X),

$\sqrt{\int_X\phi\, d\Gamma(f)} \le \sqrt{\int_X\phi\, d\Gamma(g) } +\sqrt{ \int_X\phi\, d\Gamma(f-g) }.$

Indeed, this inequality follows from a similar proof as in Lemma 16.2 by noting that f = f – g + g and using Minkowski’s inequality.

Thanks to the previous lemmas, by approximation, one can define dΓ(f) for every f ∈ ℱ by

dΓ(f) = sup{dΓ(f_n) : f_n = min(n, max(-n, f)), n = 1, 2, …}.

For f,g ∈ ℱ, one can define dΓ(f,g) by polarization

dΓ(f,g) = (1/4)(dΓ(f + g) – dΓ(f – g)).

The following representation theorem for regular Dirichlet forms then holds.

Theorem (Beurling-Deny) Assume that ℰ is regular. For u,v ∈ ℱ, ℰ(u,v) = ∫_X dΓ(u,v).

Hunt process associated with a regular Dirichlet form

Definition. A Hunt process with state space X is a family of stochastic process (X_t)_t≥0 and probability measures (ℙ_x)_x∈X defined on a measure space (Ω, ℱ), such that (X_t)_t≥0 is adapted w.r.t. the right-continuous minimal completed admissible filtration (ℱ_t)_t≥0, X₀ = x, ℙ_x-a.s. and the following hold:

(i) x → ℙ_x(X_t ∈ B) is measurable for all t > 0 and B ∈ ℬ(X),
(ii) X is a strong Markov process, i.e. for every stopping time T, X_T is ℱ_T-measurable and for every B ∈ ℬ(X)
$\mathbb{P}_x(X_{T+t} \in B | \mathcal{F}_T) = \mathbb{P}_{X_T}(X_t \in B) \quad \mathbb{P}_x\text{-a.s. on } \{T < \infty\},$
(iii) X is right-continuous, i.e.
$\lim_{s \downarrow t} X_s = X_t, \quad \forall t \quad \mathbb{P}_x\text{-a.s.}$
(iv) X is quasi left-continuous, i.e. for all stopping times T and (T_n)_n such that T_n ↑ T a.s.
$\lim_{n \to \infty} X_{T_n} = X_T, \quad \mathbb{P}_x\text{-a.s. on } \{T < \infty\}.$

Remark

Note that quasi left-continuity does not necessarily imply left-continuity, because the set
$A = \left\{ \lim_{n \to \infty} X_{s_n} = X_t \right\}$

might depend on the choice of sequence (s_n)_n, s_n ↑ t.
In more generality, one might consider situations where (ℙ_x(X_t ∈ ·))_x,t are sub-probability measures (i.e. all the axioms of probability measures are satisfied but ℙ_x(X_t ∈ Ω) ≤ 1). In that case we can perform a one-point compactification of X by introducing a cemetery state ∂ ∉ X and redefine ℙ_x to be a probability measure on X ∪ {∂}.

The following theorem can then be proved, see the book [FOT], Theorem 4.2.1.

Theorem (Fukushima) Assume that ℰ is regular, then there exists a Hunt process ((X_t)_t≥0, (ℙ_x)_x∈X) such that for μ-a.e. x ∈ X, A ∈ ℬ(X) and t ≥ 0,

$\mathbb{P}_x (X_t \in A)=p_t(x,A)$

where p_t(x, ·) are the heat kernel measures associated to the Dirichlet form (ℰ, 𝒟(ℰ)).

Intrinsic metric

Definition. The Dirichlet form ℰ is called strongly local if for any two functions f,g ∈ ℱ with compact supports such that f is constant in a neighborhood of the support of g, we have ℰ(f,g) = 0.

With respect to ℰ we can define the following intrinsic metric d_ℰ on X by

$d_{\mathcal{E}}(x,y)=\sup\{u(x)-u(y)\, :\, u\in\mathcal{F}\cap C_0(X)\text{ and } d\Gamma(u,u)\le d\mu\}.$

Here the condition dΓ(u,u) ≤ dμ means that Γ(u,u) is absolutely continuous with respect to μ with Radon-Nikodym derivative bounded by 1.

The term “intrinsic metric” is potentially misleading because in general there is no reason why d_ℰ is a metric on X (it could be infinite for a given pair of points x,y or zero for some distinct pair of points).

Definition. A strongly local regular Dirichlet space is called strictly local if d_ℰ is a metric on X and the topology induced by d_ℰ coincides with the topology on X.

Example (Riemannian manifolds) Let (M,g) be a complete n-dimensional Riemannian manifold with Riemannian volume measure μ. We consider the standard Dirichlet form ℰ on M, which is obtained by closing the bilinear form

$\mathcal{E}(f,g)=\int_\mathbb{M} \langle \nabla f ,\nabla g \rangle d\mu, \quad f,g \in C_0^\infty(\mathbb M).$

Then ℰ is a strictly local Dirichlet form such that

$d_\mathcal{E} (x,y)=d_g(x,y).$

Example (Carnot groups) Let G be a Carnot group with sub-Laplacian

$L=\sum_{i=1}^d V_i^2$

and Dirichlet form

$\mathcal{E}(f)=\int_{\mathbb G} \sum_{i=1} (V_if)^2 d\mu.$

Then ℰ is a strictly local Dirichlet form such that

$d_\mathcal{E} (x,y)=d_{CC} (x,y)$

where d_CC is the so-called Carnot-Carathéodory distance which is defined as follows.

An absolutely continuous curve γ : [0,T] → G is said to be subunit for the operator L if for every smooth function f : G → ℝ we have

$\left| \frac{d}{dt} f ( \gamma(t) ) \right| \le \sqrt{ (\Gamma f) (\gamma(t)) }.$

We then define the subunit length of γ as ℓ_s(γ) = T.

Given x,y ∈ G, we indicate with

S(x,y) = {γ : [0,T] → G | γ is subunit for Γ, γ(0) = x, γ(T) = y}.

It is a consequence of the Chow-Rashevskii theorem that

S(x,y) ≠ ∅, for every x, y ∈ G.

One defines then

$d_{CC}(x,y) = \inf\{ \ell_s(\gamma) \mid \gamma \in S(x,y)\}$

Example Consider on the Sierpinski gasket the standard Dirichlet form ℰ. Then ℰ is regular, but unless f is constant, for f ∈ ℱ, dΓ(f) is singular with respect to the Hausdorff measure μ, see [KM19]. As a consequence ℰ is not strictly local.

Lecture 6. Gagliardo-Nirenberg inequalities in Dirichlet spaces

Posted on January 19, 2025 by Fabrice Baudoin

Let (X,μ,ℰ,ℱ) be a Dirichlet space and {P_t}_t∈[0,∞) denote the associated Markovian semigroup. Throughout the lecture, we shall assume that P_t admits a measurable heat kernel p_t(x,y) satisfying, for some C > 0 and β > 0,

$p_{t}(x,y)\leq C t^{-\beta}$

for μ × μ-a.e. (x,y) ∈ X × X, and for each t ∈ (0,+∞). We also assume stochastic completeness, which means that P_t1 = 1 for every t ≥ 0.

The goal of the lecture is to prove the Gagliardo-Nirenberg inequalities in that setting. The techniques we use come from a paper by Bakry-Coulhon-Ledoux-Saloff-Coste. For another approach to the Sobolev inequality, we refer to the post.

Preliminary lemmas
Sobolev inequality
Gagliardo-Nirenberg inequalities

Preliminary lemmas

Lemma. For every f ∈ ℱ, t ≥ 0,

$|| P_t f -f ||_{L^2(X,\mu)} \le C \sqrt{t} \mathcal{E}(f,f)^{1/2}.$

Proof: Let Δ be the generator of the semigroup (P_t)_t≥0. We use spectral theorem to represent Δ as a multiplier in some L²(Ω,ν) space:

$U^{-1} A U g (x)=-\lambda(x) g(x),$

so that

$|| P_t f -f ||_{L^2(X,\mu)}^2=\int_{\Omega} \left( 1-e^{-t\lambda(x)}\right)^2(U^{-1} f)^2(x) d\nu(x)$

$\le C t \int_{\Omega} \lambda(x) (U^{-1} f)^2(x) d\nu(x)$

$= Ct \mathcal{E}(f,f).$

Lemma. For every f ∈ ℱ,

$\lim_{t \to 0} t^{-1}\int_X \int_X |f (x)-f(y)|^2 p_t(x,y) d\mu(x)d\mu(y) =2 \mathcal{E}(f,f).$

Proof: Using P_t1 = 1 we have

$\int_X \int_X |f (x)-f(y)|^2 p_t(x,y) d\mu(x)d\mu(y)$

$\int_X \int_X( f (x)^2-2f(x)f(y)+f(y)^2) p_t(x,y) d\mu(x)d\mu(y)$

$=2\int_X f(x)^2 d\mu(x)-2\int_X f(x)(P_t f) (x)d\mu(x)$

$=2 \int_X (f(x)-P_tf(x)) f(x) d\mu(x)$

and we conclude thanks to a previous result.

Sobolev inequality

Lemma. Let 1 ≤ q < +∞. There exists a constant C > 0 such that for every f ∈ ℱ ∩ L^q(X,μ) and s ≥ 0,

$\sup_{s \ge 0} s^{1 +\frac{q}{2\beta}} \mu \left( \{ x \in X\, :\, | f(x) | > s \} \right)^{\frac{1}{2}}\le C \mathcal{E}(f,f)^{1/2} || f ||_{L^q(X,\mu)}^{\frac{q}{2\beta}}.$

Proof: Let f ∈ ℱ and denote

$F(s)=\mu \left( \{ x \in X\, :\, | f(x) | > s \} \right).$

We have then

$F(s) \le \mu \left( \{ x \in X\, :\, | f(x) -P_t f (x) | > s/2 \} \right)+\mu \left( \{ x \in X\, :\, | P_t f (x) | > s/2 \} \right).$

Now, from the heat kernel upper bound p_t(x,y) ≤ C t^-β, t > 0, one deduces, for g ∈ L¹(X,μ), that

$|P_t g (x) | \le Ct^{-\beta} \| g \|_{L^1(X,\mu)}.$

Since P_t is a contraction in L^∞(X,μ), by the Riesz-Thorin interpolation one obtains

$| P_t f (x) | \le \frac{C^{1/q}}{t^{\beta /q}} \| f \|_{L^q(X,\mu)}.$

Therefore, for

$s= 2 \frac{C^{1/q}}{t^{\frac{\beta}{ q}}} \| f \|_{L^q(X,\mu)},$

one has

$\mu \left( \{ x \in X\, :\, | P_t f (x) | > s/2 \} \right)=0.$

On the other hand, from previous lemma,

$\mu \left( \{ x \in X\, :\, | f(x) -P_t f (x) | > s/2 \} \right) \le C s^{-2} t \mathcal{E}(f,f).$

We conclude that

$F(s)^{1/2} \le C s^{-1 -\frac{q}{2\beta}} \mathcal{E}(f,f)^{1/2} \| f \|_{L^q(X,\mu)}^{\frac{q}{2\beta}}.$

Lemma. Assume β > 1. There exists a constant C > 0 such that for every f ∈ ℱ,

$\sup_{s \ge 0}\, s\, \mu \left( \{ x \in X\, :\, | f(x) | \ge s \} \right)^{\frac{1}{q}} \le C \mathcal{E}(f,f)^{1/2},$

where q = 2β/(β – 1).

Proof: Let f ∈ ℱ be a non-negative function. For k ∈ ℤ, we denote

$f_k=(f-2^k)_+ \wedge 2^k.$

Observe that f_k ∈ L²(X,μ) and ||f_k||_L²(X,μ) ≤ ||f||_L²(X,μ). Moreover, for every x, y ∈ X, |f_k(x) – f_k(y)| ≤ |f(x) – f(y)| and so ℰ(f_k,f_k) ≤ ℰ(f,f). We also note that f_k ∈ L¹(X,μ), with

$||f_k||_{L^1(X,\mu)} =\int_X |f_k| d\mu \le 2^k \mu (\{ x \in X\, :\, f(x) \ge 2^k \}).$

We now use the previous lemma to deduce:

$\sup_{s \ge 0} s^{1 +\frac{1}{2\beta}} \mu \left( \{ x \in X\, :\, f_k(x) > s \} \right)^{\frac{1}{2}} \le C \mathcal{E}(f_k,f_k)^{1/2} || f_k \|_{L^1(X,\mu)}^{\frac{1}{2\beta}}$

$\le C \mathcal{E}(f_k,f_k)^{1/2} \left( 2^k \mu (\{ x \in X\, :\, f(x) \ge 2^k \})\right)^{\frac{1}{2\beta}}$

In particular, by choosing s = 2^k we obtain

$2^{k \left(1 +\frac{1}{2\beta} \right)} \mu \left( \{ x \in X\, :\, f(x) \ge 2^{k+1} \} \right)^{\frac{1}{2}} \le C \mathcal{E}(f_k,f_k)^{1/2} \left( 2^k \mu (\{ x \in X\, :\, f(x) \ge 2^k \})\right)^{\frac{1}{2\beta}}$

Let

$M(f)=\sup_{k \in \mathbb{Z}} 2^k \mu (\{ x \in X\, :\, f(x) \ge 2^k \})^{1/q}$

where q = 2β/(β – 1). Using the fact that 1/q = 1/2 – 1/(2β) and the previous inequality we obtain:

$2^{k} \mu \left( \{ x \in X\, :\, f(x) \ge 2^{k+1} \} \right)^{\frac{1}{2}} \le C 2^{ -\frac{kq}{2\beta}}\mathcal{E}(f,f)^{1/2} M(f)^{\frac{q}{2\beta}}$

and

$2^k \mu \left( \{ x \in X\, :\, f(x) \ge 2^{k+1} \} \right)^{\frac{1}{q}} \le C^{\frac{2}{q} } \mathcal{E}(f,f)^{1/q}M(f)^{\frac{1}{\beta}}.$

Therefore

$M(f)^{1-\frac{1}{\beta}} \le 2 C^{\frac{2}{q} } \mathcal{E}(f,f)^{1/q}$

and one concludes

$M(f) \le 2^{q/2} C \mathcal{E}(f,f)^{1/2}.$

This easily yields:

$\sup_{s \ge 0} s \mu \left( \{ x \in X\, :\, f(x) \ge s \} \right)^{\frac{1}{q}} \le 2^{1+q/2} C \mathcal{E}(f,f)^{1/2}$

Let now f ∈ ℱ, which is not necessarily non-negative. From the previous inequality applied to |f| we deduce

$\sup_{s \ge 0}\, s\, \mu \left( \{ x \in X\, :\, |f(x) | \ge s \} \right)^{\frac{1}{q}}$

$\le 2^{1+q/2} C \mathcal{E}(|f|,|f|)^{1/2}$

$\le 2^{1+q/2} C \mathcal{E}(f,f)^{1/2}.$

Theorem. (Sobolev inequality) Assume β > 1. There exists a constant C > 0 such that for every f ∈ ℱ,

$\| f \|_{L^q(X,\mu)} \le C \mathcal{E}(f,f)^{1/2}$

where q = 2β/(β – 1).

To show that the weak type inequality implies the desired Sobolev inequality, we will need another slicing argument and the following lemma is needed.

Lemma. For f ∈ ℱ, f ≥ 0, denote f_k = (f – 2^k)₊ ∧ 2^k, k ∈ ℤ. There exists a constant C > 0 such that for every f ∈ ℱ,

$\sum_{k \in \mathbb{Z}} \mathcal{E} (f_k,f_k) \le C \mathcal{E} (f,f).$

Proof: Let p_t(x,y) denote the heat kernel of the semigroup P_t. We first observe that, once we prove

$\sum_{k \in \mathbb{Z}} \int_X \int_X |f_{k} (x)-f_{k}(y)|^2 p_t(x,y) d\mu(x)d\mu(y) \le C\int_X \int_X |f (x)-f(y)|^2 p_t(x,y) d\mu(x)d\mu(y)$

where C > 0 is independent from t, then

$\liminf_{t \to 0^+} \sum_{k \in \mathbb{Z}} t^{-1 } \!\!\!\int_X \int_X |f_{\rho} (x)-f_{\rho}(y)|^2 p_t(x,y)d\mu(x)d\mu(y)$

$\le C\liminf_{t \to 0^+} t^{-1 } \!\!\!\int_X \int_X |f (x)-f(y)|^p p_t(x,y) d\mu(x)d\mu(y),$

and, using the superadditivity of the liminf, one concludes

$\sum_{k \in \mathbb{Z}} \liminf_{t \to 0^+} t^{-1 } \int_X \int_X |f_{k} (x)-f_{k}(y)|^2 p_t(x,y) d\mu(x)d\mu(y)$

$\le C \liminf_{t \to 0^+} t^{-1 } \int_X \int_X |f (x)-f(y)|^2 p_t(x,y) d\mu(x)d\mu(y)$

which yields

$\sum_{k \in \mathbb{Z}} \mathcal{E} (f_k,f_k) \le C \mathcal{E} (f,f).$

We therefore aim to prove the inequality above. For each k ∈ ℤ, set B_k = {x ∈ X : 2^k < f ≤ 2^k+1}. In this way, the external integral on the left-hand side is decomposed it into an integral over B_k and B_k^c. For the integrals over B_k, since the mapping f → f_k is a contraction, it follows that

$\sum_{k\in\mathbb{Z}}\int_{B_k} \int_X |f_{k} (x)-f_{k}(y)|^2 p_t(x,y)d\mu(x)d\mu(y)\leq \int_X \int_X |f (x)-f(y)|^2 p_t(x,y)d\mu(x)d\mu(y).$

To perform the integrals over B_k^c, we decompose them as

$\sum_{k\in\mathbb{Z}}\int_{B_k^c} \int_{B_k} |f_{k} (x)-f_{k}(y)|^2 p_t(x,y) d\mu(x)d\mu(y) +\sum_{k\in\mathbb{Z}}\int_{B^c_k} \int_{B_k^c}|f_{k} (x)-f_{k}(y)|^2 p_t(x,y)d\mu(x)d\mu(y)$

$=:\sum_{k\in\mathbb{Z}}J_1(k)+\sum_{k\in\mathbb{Z}}J_2(k).$

Again, the contraction property of f → f_k yields

$\sum_{k\in\mathbb{Z}}J_1(k)\leq \sum_{k\in\mathbb{Z}} \int_X \int_{B_k}|f_{k} (x)-f_{k}(y)|^2 p_t(x,y)d\mu(x)d\mu(y)$

$\leq \int_X \sum_{k\in\mathbb{Z}} \int_{B_k}|f_{k} (x)-f_{k}(y)|^2 p_t(x,y)d\mu(x)d\mu(y)\leq \int_X \int_X |f (x)-f(y)|^2 p_t(x,y) d\mu(x)d\mu(y).$

On the other hand, notice that for any (x,y) ∈ B_k^c × B_k^c we have |f_k(x) – f_k(y)| ≠ 0 only if

$(x,y)\in \{f(x)\leq 2^k<f(y)/2 \}\cup\{f(y)\leq 2^k<f(x)/2\}=:Z_k\cup Z_k^*.$

Also, |f_k(x) – f_k(y)| = 2^k for (x,y) ∈ Z_k ∪ Z_k^*. Thus,

$\sum_{k\in\mathbb{Z}}J_2(k)\leq \sum_{k\in \mathbb{Z}}\int_X\int_X\big(\mathbf{1}_{Z_k}(x,y)+\mathbf{1}_{Z^*_k}(x,y)\big)|f_k(x)-f_k(y)|^2 p_t(x,y)d\mu(x)d\mu(y)$

$=\int_X\int_X\sum_{k\in\mathbb{Z}}\big(\mathbf{1}_{Z_k}(x,y)+\mathbf{1}_{Z^*_k}(x,y)\big)2^{2k} p_t(x,y) d\mu(x)d\mu(y).$

One can see that

$\sum_{k\in\mathbb{Z}}\mathbf{1}_{Z_k}(x,y) 2^{2k}\leq 2 |f(x)-f(y)|^2$

and the same holds for Z_k^*, hence

$\sum\limits_{k\in\mathbb{Z}}J_1(k)+\sum\limits_{k\in\mathbb{Z}}J_2(k)\leq 5 \int_X\int_X|f(x)-f(y)|^2 p_t(x,y) d\mu(x)d\mu(y).$

Adding to these the term above finally yields the result

We can now conclude the proof of the Sobolev inequality.

Proof of the Sobolev inequality: Let f ∈ ℱ. We can assume f ≥ 0. As before, denote f_k = (f – 2^k)₊ ∧ 2^k, k ∈ ℤ. From Lemma 13.2 applied to f_k, we see that

$\sup_{s \ge 0} s \mu \left( \{ x \in X\, :\, | f_k(x) | \ge s \} \right)^{\frac{1}{q}} \le C \mathcal{E}(f_k,f_k)^{1/2}.$

In particular for s = 2^k, we get

$2^k \mu \left( \{ x \in X\, :\, f (x) \ge 2^{k+1} \} \right)^{\frac{1}{q}} \le C \mathcal{E}(f_k,f_k)^{1/2}$

Therefore,

$\sum_{k \in \mathbb{Z}} 2^{k q} \mu \left( \{ x \in X\, :\, f (x) \ge 2^{k+1} \} \right) \le C^q \sum_{k \in \mathbb{Z}} \mathcal{E}(f_k,f_k)^{q/2}.$

Since q ≥ 2, one has

$\sum_{k \in \mathbb{Z}} \mathcal{E}(f_k,f_k)^{q/2} \le \left( \sum_{k \in \mathbb{Z}} \mathcal{E}(f_k,f_k) \right)^{q/2}$

Thus, from the previous lemma

$\sum_{k \in \mathbb{Z}} 2^{k q} \mu \left( \{ x \in X\, :\, f (x) \ge 2^{k+1} \} \right)\le C \mathcal{E}(f,f)^{q/2}.$

Finally, we observe that

$\sum_{k \in \mathbb{Z}} 2^{k q} \mu \left( \{ x \in X\, :\, f (x) \ge 2^{k+1} \} \right) \ge \frac{q}{2^{q+1}-2^q} \sum_{k \in \mathbb{Z}}\int_{2^{k+1}}^{2^{k+2}} s^{q-1} \mu \left( \{ x \in X\, :\, f (x) \ge s \} \right)ds$

$\ge \frac{1}{2^{q+1}-2^q} \| f \|_{L^q(X,\mu)}^q.$

The proof is thus complete.

Gagliardo-Nirenberg inequalities

Using similar methods (see the paper) in the general case β > 0 one can get the family of Gagliardo-Nirenberg inequalities.

Theorem. Let q = 2β/(β – 1) with the convention that q = ∞ if β = 1. Let r,s ∈ (0,+∞] and θ ∈ (0,1] satisfying

$\frac{1}{r}=\frac{\theta}{q}+\frac{1-\theta}{s}.$

If β = 1, we assume r < +∞. Then, there exists a constant C > 0 such that for every f ∈ ℱ,

$\| f \|_{L^r(X,\mu)} \le C \mathcal{E}(f,f)^{\theta/2}\|f\|_{L^s(X,\mu)}^{1-\theta}.$

We explicitly point out some particular cases of interest.

Assume that β > 1. If r = s, then r = 2β/(β – 1) and above recovers the Sobolev inequality
$\| f \|_{L^r(X,\mu)} \le C \mathcal{E}(f,f)^{1/2}.$
Assume that β > 1. If s = +∞ and r ≥ 2β/(β – 1), then above yields
$\| f \|_{L^r(X,\mu)} \le C \mathcal{E}(f,f)^{\theta/2} \| f \|^{1-\theta}_{L^\infty(X,\mu)}$

with θ = 2β/(r(β – 1)).
If r = 2 and s = 1, then above yields the Nash inequality
$\| f \|_{L^2(X,\mu)} \le C \mathcal{E}(f,f)^{\theta/2} \| f \|^{1-\theta}_{L^1(X,\mu)}$

with θ = β/(1 + β).

In the case β = 1 one obtains the Trudinger-Moser inequalities.

Corollary . Assume that β = 1. Then, there exist constants c,C > 0 such that for every f ∈ ℱ with ℰ(f,f) = 1,

$\int_X \left( \exp \left( c |f|^2 \right)-1 \right) d\mu \le C \| f \|^2_{L^2(X,\mu)}.$

Posted in Dirichlet forms at NYU | Tagged Dirichlet forms, Gagliardo-Nirenberg inequality, Sobolev inequality | Leave a comment

Lecture 5. Examples of Dirichlet spaces

Posted on January 17, 2025 by Fabrice Baudoin

Riemannian manifolds
Carnot groups
Sierpinski gasket
Cheeger space

Riemannian manifolds

Let (M,g) be an n-dimensional Riemannian manifold with Riemannian volume measure μ and Riemannian distance d. We consider the quadratic form ℰ on M, which is obtained as the closure in L²(M,μ) of the quadratic form

∫_M ⟨∇f, ∇g⟩ dμ, f,g ∈ C_c^∞(M).

The domain of the closure ℰ is the Sobolev space W₀^1,2(M) and its generator Δ is a self-adjoint extension of the Laplace-Beltrami operator. If the manifold is complete (which is equivalent to the metric space (M,d) being complete) then ℰ is the unique closed extension and W₀^1,2(M) = W^1,2(M). If we assume further that the Ricci curvature of M is bounded from below then the domain of Δ is the Sobolev space W^2,2(M). If we even assume further that the Ricci curvature of M is non-negative, it is a well-known result by Li and Yau that the heat semigroup P_t admits a smooth heat kernel function p_t(x,y) on [0,∞) × M × M for which there are constants c₁, c₂, C > 0 such that whenever t > 0 and x,y ∈ X,

$\frac{C^{-1}}{\mu(B(x,\sqrt t))} \exp \left(-\frac{ c_1 d(x,y)^2}{t}\right)\le p_t(x,y)\le \frac{C}{\mu(B(x,\sqrt t))} \exp \left(-\frac{c_2d(x,y)^2}{(t}\right).$

Carnot groups

A Carnot group of step N is a simply connected Lie group G whose Lie algebra can be stratified as follows:

g = 𝒱₁ ⊕ … ⊕ 𝒱_N,

where

[𝒱_i , 𝒱_j] = 𝒱_i+j

and

𝒱_s = 0, for s > N.

From the above properties, Carnot groups are nilpotent. The number

$Q=\sum_{i=1}^N i \dim \mathcal{V}_{i}$

is called the homogeneous dimension of G.

Let V₁,…,V_d be a basis of the vector space 𝒱₁. The vectors V_i‘s can be seen as left invariant vector fields on G. The left invariant sub-Laplacian on G is the operator:

$L=\sum_{i=1}^d V_i^2$

It is hypoelliptic and essentially self-adjoint on the space of smooth and compactly supported function f : G → ℝ with the respect to the Haar measure μ of G. The heat semigroup (P_t)_t≥0 on G, defined through the spectral theorem, is then seen to be a Markov semigroup. By hypoellipticity of L, this heat semigroup admits a heat kernel denoted by p_t(g,g’). It is then known that p_t satisfies the double-sided Gaussian bounds:

$\frac{C^{-1}}{t^{Q/2}} \exp \left(-\frac{ c_1 d(x,y)^2}{t}\right)\le p_t(x,y)\le \frac{C}{t^{Q/2}} \exp \left(-\frac{c_2d(x,y)^2}{(t}\right).$

for some constants C,c₁,c₂ > 0. Here d(g,g’) denotes the Carnot-Carathéodory distance from g to g’ on G which is defined by

$d(g,g')=\sup \left\{ |f(g)-f(g')| , \quad \sum_{i=1}^d (V_i f)^2 \le 1 \right\}.$

Sierpiński gasket

A large class of examples for which Dirichlet form theory is useful is the class of p.c.f. fractals. For the sake of presentation we illustrate in detail the case of the Sierpiński gasket, which is one of the most popular examples of a p.c.f. fractal.

One of the classical ways to define the Sierpiński gasket is as follows:
let V₀ = {p₁, p₂, p₃} be a set of vertices of an equilateral triangle of side 1 in ℂ. Define

f_i(z) = (z-p_i)/2 + p_i , for i = 1,2,3.

The Sierpiński gasket K is the unique non-empty compact subset in ℂ such that

$K=f_1(K) \cup f_2(K) \cup f_3(K)$

Sierpinski gasket

The set V₀ is called the boundary of K, we will also denote it by ∂K. The Hausdorff dimension of K with respect to the Euclidean metric (denoted d(x,y) = |x – y| ) is given by d_h = ln(3)/ln(2). A (normalized) Hausdorff measure on K is given by the Borel measure μ on K such that for every i₁, …, i_n ∈ {1,2,3} ,

μ(f_i₁ ∘ … ∘ f_{i_n} (K)) = 3^-n.

This measure μ is d_h-Ahlfors regular, i.e., there exist constants c,C > 0 such that for every x ∈ K and r ∈ [0, diam(K)],

cr^d_h ≤ μ(B(x,r)) ≤ C r^d_h.

It will be useful to approximate the gasket K by a sequence of discrete objects. Namely, starting from the set V₀ = {p₁, p₂, p₃}, we define a sequence of sets {V_m}_m≥0 inductively by

$V_{m+1}=\bigcup_{i=1}^3 f_i(V_m)$

Then we have a natural sequence of Sierpiński gasket graphs (or pre-gaskets) {G_m}_m≥0 whose edges have length 2^-m and whose set of vertices is V_m. Notice that #V_m = 3(3^m + 1)/2. We will use the notations V_* = ⋃_{m ≥ 0} V_m and V_*⁰ = ⋃_m≥0 V_m \ V₀.

The Dirichlet form on the metric space K is defined by approximation. Let m ≥ 1. For any f ∈ ℝ^V_m, we consider the quadratic form

$\mathcal E_m(f,f)= \left(\frac53\right)^m\sum_{p,q\in V_m, p\sim q} (f(p)-f(q))^2$

where $p \sim q$ means that $p,q$ are neighbors in the graph V_m.

We can then define a pre-Dirichlet form (ℰ, ℱ_*) on V_* by setting

ℱ_* = {f ∈ ℝ^V_* , lim_{m → ∞} ℰ_m(f,f) < ∞}

and for f ∈ ℱ_*

ℰ(f,f) = lim_{m → ∞} ℰ_m(f,f).

Each function f ∈ ℱ_* can be uniquely extended into a continuous function defined on K. We denote by ℱ the set of functions with such extensions. (ℰ, ℱ) is a Dirichlet form on L²(K,μ). The generator of the Dirichlet form (ℰ, ℱ), denoted by Δ, corresponds to the Laplacian with Neumann boundary condition.

In this example we have a continuous heat kernel p_t(x,y) satisfying, for some c₁, c₂, c₃, c₄ ∈ (0,∞) and d_H ≥ 1, d_W ∈ [2,+∞),

$c_{1}t^{-d_{H}/d_{W}}\exp\biggl(-c_{2}\Bigl(\frac{d(x,y)^{d_{W}}}{t}\Bigr)^{\frac{1}{d_{W}-1}}\biggr) \le p_{t}(x,y)\leq c_{3}t^{-d_{H}/d_{W}}\exp\biggl(-c_{4}\Bigl(\frac{d(x,y)^{d_{W}}}{t}\Bigr)^{\frac{1}{d_{W}-1}}\biggr)$

for μ × μ-a.e. (x,y) ∈ X × X and each t ∈ (0,+∞). Here, d_W = ln(5)/ln(3) is the so-called walk dimension of the Sierpiński gasket. A standard reference are the lecture notes by M. Barlow.

Cheeger metric measure spaces

Consider a locally compact, complete, metric measure space (X,d,μ) where μ is a Radon measure. Any open metric ball centered at x ∈ X with radius r > 0 will be denoted by

B(x,r) = {y ∈ X, d(x,y) < r}.

Definition. The measure μ is said to be doubling (VD) if there exists a constant C > 0 such that for every x ∈ X, r > 0,

0 < μ(B(x,2r)) ≤ C μ(B(x,r)) < +∞.

The Lipschitz constant of a function f ∈ Lip(X) is defined as

$(\mathrm{Lip} f )(y):=\limsup_{r \to 0^+} \sup_{x \in X, d(x,y) \le r} \frac{|f(x)-f(y)|}{r}$

Definition. The metric measure space (X,d,μ) is said to satisfy the 2-Poincaré inequality (P) if for any f ∈ Lip(X) and any ball B(x,R) of radius R > 0,

$\int_{B(x,R)} | f(y) -f_{B(x,R)}|^2 d\mu (y) \le C R^2 \int_{B(x,\lambda R)} (\mathrm{Lip} f )(y)^2 d\mu (y)$

where

$f_{B(x,R)}:=\frac{1}{\mu(B(x,r))} \int_{B(x,R)} f(y) d\mu(y)$
The constants C > 0 and λ ≥ 1 are independent from x, R and f.

Definition. A metric measure space satisfying (VD) and (P) is often called a Cheeger space (or PI space).

One can construct a “nice” Dirichlet form and Laplacian on any Cheeger space by the using the technique of Γ-convergence.

Definition. A sequence of forms {ℰ_n}_{n ≥ 1} is said to Mosco-converge to ℰ if

For any sequence {f_n}_n≥1 ⊂ L²(X,μ) that converges *weakly* to f ∈ L²(X,μ) in L²(X,μ),
liminf_n→∞ ℰ_n(f_n, f_n) ≥ ℰ(f,f).
For any f ∈ L²(X,μ) there exists a sequence {f_n}_n≥1 ⊂ L²(X,μ) that converges strongly to f in L²(X,μ) and
limsup_n→∞ ℰ(f_n, f_n) ≤ ℰ(f,f).

The idea is to consider Korevaar-Schoen type energy functionals defined for any f ∈ L²(X,μ) as

$E(f,r):= \int_X\frac{1}{\mu(B(x,r))}\int_{B(x,r)} \frac{|f(y)-f(x)|^2}{r^{2}} d\mu(y) d\mu(x)$

and the associated Korevaar-Schoen space

KS^1,2(X) := {f ∈ L²(X,μ), limsup_{r → 0⁺} E(f,r) < +∞}.

One has then the following result:

Theorem. There exists a Dirichlet form (ℰ, ℱ) on L²(X,μ) such that:

ℰ has domain ℱ = KS^1,2(X);
ℰ is a Γ-limit of E(f,r_n), where r_n is a positive sequence such that r_n → 0;
ℰ has a continuous heat kernel p_t(x,y) that satisfies for t > 0 and x,y ∈ X, $\frac{C^{-1}}{\mu(B(x,\sqrt t))} \exp \left(-\frac{ c_1 d(x,y)^2}{t}\right)\le p_t(x,y)\le \frac{C}{\mu(B(x,\sqrt t))} \exp \left(-\frac{c_2d(x,y)^2}{(t}\right).$

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Lecture 4. The Lp theory of semigroups and diffusion operators as generators of Dirichlet forms

Posted on January 15, 2025 by Fabrice Baudoin

Lp theory of semigroups
Diffusion operators as generators of Dirichlet forms

The Lp theory of heat semigroups

Our goal, in this section, is to define, for 1 ≤ p ≤ +∞, P_t on L^p := L^p(X,μ). This may be done in a natural way by using the Riesz-Thorin interpolation theorem that we recall below.

Theorem (Riesz-Thorin interpolation theorem)

Let 1 ≤ p₀, p₁, q₀, q₁ ≤ ∞, and θ ∈ (0,1). Define 1 ≤ p,q ≤ ∞ by

1/p = (1-θ)/p₀ + θ/p₁, 1/q = (1-θ)/q₀ + θ/q₁.

If T is a linear map such that

T : L^p₀ → L^q₀, ||T||_{L^p₀→L^q₀} = M₀

T : L^p₁ → L^q₁, ||T||_{L^p₁→L^q₁} = M₁,

then, for every f ∈ L^p₀ ∩ L^p₁,

||Tf||_q ≤ M₀^1-θM₁^θ ||f||_p.

Hence T extends uniquely as a bounded map from L^p to L^q with

||T||_L^p→L^q ≤ M₀^1-θM₁^θ.

Remark The statement that T is a linear map such that

T : L^p₀ → L^q₀, ||T||_{L^p₀→L^q₀} = M₀

T : L^p₁ → L^q₁, ||T||_{L^p₁→L^q₁} = M₁

means that there exists a map T : L^p₀ ∩ L^p₁ → L^q₀ ∩ L^q₁ with

sup_{f ∈ L^p₀ ∩ L^p₁ , ||f||_p₀ ≤ 1} ||Tf||_q₀ = M₀

and

sup_{f ∈ L^p₀ ∩ L^p₁ , ||f||_p₁ ≤ 1} ||Tf||_q₁ = M₁.

In such a case, T can be uniquely extended to bounded linear maps T₀ : L^p₀ → L^q₀ , T₁ : L^p₁ → L^q₁. With a slight abuse of notation, these two maps are both denoted by T in the theorem.

Remark If f ∈ L^p₀ ∩ L^p₁ and p is defined by

1/p = (1-θ)/p₀ + θ/p₁,

then by Hölder’s inequality, f ∈ L^p and

||f||_p ≤ ||f||_p₀^1-θ ||f||_p₁^θ.

We now are in position to state the following theorem:

Theorem Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ). The space L¹ ∩ L^∞ is invariant under P_t and P_t may be extended from L¹ ∩ L^∞ to a contraction semigroup (P_t^(p))_t≥0 on L^p for all 1 ≤ p ≤ ∞: For f ∈ L^p,

||P_tf||_L^p ≤ ||f||_L^p.

These semigroups are consistent in the sense that for f ∈ L^p ∩ L^q,

P_t^(p)f = P_t^(q)f.

Proof

If f,g ∈ L¹ ∩ L^∞ which is a subset of L¹ ∩ L^∞, then,

|∫_X (P_tf)g dμ| = |∫_X f (P_tg) dμ| ≤ ||f||_L¹ ||P_tg||_L^∞ ≤ ||f||_L¹ ||g||_L^∞.

This implies

||P_tf||_L¹ ≤ ||f||_L¹.

The conclusion follows then from the Riesz-Thorin interpolation theorem.

Exercise Show that if f ∈ L^p and g ∈ L^q with 1/p + 1/q = 1 then,

∫_ℝⁿ f P_t^(q) g dμ = ∫_ℝⁿ g P_t^(p) f dμ.

Exercise

Show that for each f ∈ L¹, the L¹-valued map t → P_t⁽¹⁾f is continuous.
Show that for each f ∈ L^p, 1 < p < 2, the L^p-valued map t → P_t^(p)f is continuous.
Finally, by using the reflexivity of L^p, show that for each f ∈ L^p and every p ≥ 1, the L^p-valued map t → P_t^(p)f is continuous.

We mention, that in general, the L^∞ valued map t → P_t^(∞)f is not continuous.

Diffusion operators as generators of Dirichlet forms

Consider a diffusion operator

L = ∑_i,j=1ⁿ σ_ij(x) ∂²/∂x_i∂x_j + ∑_i=1ⁿ b_i(x) ∂/∂x_i,

where b_i and σ_ij are continuous functions on ℝⁿ and for every x ∈ ℝⁿ, the matrix (σ_ij(x))_{1 ≤ i,j ≤ n} is a symmetric and non-negative matrix.

Assume that there is Borel measure μ on ℝⁿ which is equivalent to the Lebesgue measure and that symmetrizes L in the sense that for every smooth and compactly supported functions f, g : ℝⁿ → ℝ,

∫_ℝⁿ gLf dμ = ∫_ℝⁿ fLg dμ.

For instance, if one can write

Lf = -div(a ∇f),

where a is a smooth field of positive and symmetric matrices, then the Lebesgue measure symetrizes L. From a previous lemma the quadratic form

ℰ(f,g) = -∫_ℝⁿ g Lf dμ, f,g ∈ C_c^∞(ℝⁿ)

is closable. Let ℰ̄ denotes its closure in L²(ℝⁿ,μ).

Proposition The quadratic form ℰ̄ is a Dirichlet form.

Proof

We need to prove that ℰ̄ is Markovian. It is enough to prove that if u ∈ ℱ = 𝒟(ℰ), then |u| ∈ ℱ with ℰ̄(|u|, |u|) ≤ ℰ̄(u,u) and that if u ∈ ℱ with u ≥ 0, then u ∧ 1 ∈ ℱ with ℰ̄(|u|,|u|) ≤ ℰ̄(u,u). We prove the first requirement, the second being established in a similar manner is let as an exercise to the reader.

Let u ∈ C_c^∞(ℝⁿ) and consider

ϕ_ε(x) = (x² + ε²)^{1/2}, ε > 0.

One can check that ϕ_ε(u) → |u| in L²(ℝⁿ,μ) and that ϕ_ε(u) is a Cauchy sequence for the norm

||f||_ℱ² = ||f||²_{L²(ℝⁿ,μ)} + ℰ̄(f,f).

Since ℰ̄ is closed this implies that |u| ∈ ℱ and that ϕ_ε(u) → |u| converges to u in the above norm.

Now, using chain rule we see that for every smooth function u ∈ C_c^∞(ℝⁿ),

Lϕ_ε(u)(x) ≥ u(x)/(u(x)² + ε²)^{1/2} Lu(x).

Multiplying by ϕ_ε(u) and integrating we get

ℰ(ϕ_ε(u), ϕ_ε(u)) ≤ ℰ(u,u)

Taking the limit ε → 0 yields

ℰ̄(|u|, |u|) ≤ ℰ̄(u,u)

The above inequality extends then to all u ∈ ℱ by using the density of C_c^∞(ℝⁿ) in the || · ||_ℱ norm and the closedness of ℰ.

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Lecture 3. Markov semigroups and Dirichlet forms

Posted on January 11, 2025 by Fabrice Baudoin

Let (X, ℬ) be a measurable space. We say that (X, ℬ) is a good measurable space if there is a countable family generating ℬ and if every finite measure γ on (X × X, ℬ ⊗ ℬ) can be decomposed as

γ(dx dy) = k(x,dy) γ₁(dx)

where γ₁ is the projection of γ on the first coordinate and k is a kernel, i.e. k(x,·) is a finite measure on (X, ℬ) and x → k(x,A) is measurable for every A ∈ ℬ.

For instance, if X is a Polish space (or a Radon space) equipped with its Borel σ-field, then it is a good measurable space.

Throughout the lecture, we will consider (X, ℬ, μ) to be a good measurable space equipped with a σ-finite measure μ.

Markovian semigroups
Dirichlet forms

Markovian semigroups

Definition Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction semigroup on L²(X,μ). The semigroup (P_t)_t≥0 is called Markovian if and only if for every f ∈ L²(X,μ) and t ≥ 0:

f ≥ 0, a.e. ⇒ P_tf ≥ 0, a.e.
f ≤ 1, a.e. ⇒ P_tf ≤ 1, a.e.

We note that if (P_t)_t≥0 is Markovian, then for every f ∈ L²(X,μ) ∩ L^∞(X,μ),

||P_tf||_L^∞(X,μ) ≤ ||f||_L^∞(X,μ).

As a consequence (P_t)_t≥0 can be extended to a contraction semigroup defined on all of L^∞(X,μ).

Definition A transition function {p_t,t ≥ 0} on X is a family of kernelsnp_t : X × ℬ → [0,1]

such that:

For t ≥ 0 and x ∈ X, p_t(x, ·) is a finite measure on X;
For t ≥ 0 and A ∈ ℬ the application x → p_t(x,A) is measurable;
For s,t ≥ 0, a.e. x ∈ X and A ∈ ℬ,
p_t+s(x,A) = ∫_X p_t(y,A) p_s(x,dy).

The relation above is often called the Chapman-Kolmogorov relation.

Theorem (Heat kernel measure)nLet (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ).nThere exists a transition function {p_t,t ≥ 0} on X such that for every f ∈ L^∞(X,μ) and a.e. x ∈ X

P_tf(x) = ∫_X f(y) p_t(x,dy) , t > 0.

This transition function is called the heat kernel measure associated to (P_t)_t≥0.

The proof relies on the following lemma sometimes called the bi-measure theorem. A set function ν : ℬ ⊗ ℬ → [0,+∞) is called a bi-measure, if for every A ∈ ℬ, ν(A, ·) and ν(·, A) are measures.

Lemma If ν : ℬ ⊗ ℬ → [0,+∞) is a bi-measure, then there exists a measure γ on ℬ ⊗ ℬ such that for every A,B ∈ ℬ,

γ(A × B) = ν(A,B).

Proof of Theorem 7.3

We assume that μ is finite and let as an exercise the extension to σ-finite measures. For t > 0, we consider the set function

ν_t(A,B) = ∫_X 1_A P_t 1_B dμ.

Since P_t is supposed to be Markovian, it is a bi-measure. From the bi-measure theorem, there exists a measure γ_t on ℬ ⊗ ℬ such that for every A,B ∈ ℬ,

γ_t(A × B) = ν_t(A,B) = ∫_X 1_A P_t 1_B dμ.

The projection of γ_t on the first coordinate is (P_t1) dμ, thus from the measure decomposition theorem, γ_t can be decomposed as

γ_t(dx dy) = p_t(x,dy) μ(dx)

for some kernel p_t. One has then for every A,B ∈ ℬ

∫_X 1_A P_t 1_B dμ = ∫_A ∫_B p_t(x,dy) μ(dx),

from which it follows that for every f ∈ L^∞(X,μ), and a.e. x ∈ X

P_tf(x) = ∫_X f(y) p_t(x,dy).

The relation

p_t+s(x,A) = ∫_X p_t(y,A) p_s(x,dy)

follows from the semigroup property.

Exercise Prove Theorem 7.3 if μ is σ-finite.

Exercise Show that for every non-negative measurable function F : X × X → ℝ,

∫_X ∫_X F(x,y) p_t(x,dy) dμ(x) = ∫_X ∫_X F(x,y) p_t(y,dx) dμ(y).

Definition Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ). We say that the semigroup {P_t}_t∈[0,∞) admits a heat kernel if the heat kernel measures have a density with respect to μ, i.e. there exists a measurable function p : ℝ_>0 × X × X → ℝ_≥0, such that for every t > 0, a.e. x,y ∈ X, f ∈ L^∞(X,μ),

P_tf(x) = ∫_X p_t(x,y) f(y) dμ(y).

If the heat kernel exists, we will often denote p(t,x,y) as p_t(x,y) for t > 0 and a.e. x,y ∈ X.

Dirichlet forms

Definition A function v on X is called a normal contraction of the function u if for almost every x, y ∈ X,

|v(x)-v(y)| ≤ |u(x) – u(y)| and |v(x)| ≤ |u(x)|.

Definition Let (ℰ,ℱ = dom(ℰ)) be a densely defined closed quadratic form on L²(X,μ). The form ℰ is called a Dirichlet form if it is Markovian, that is, has the property that if u ∈ ℱ and v is a normal contraction of u then v ∈ ℱ and

ℰ(v,v) ≤ ℰ(u,u).

Exercise Show that a densely defined closed quadratic form on L²(X,μ) is Markovian if and only if for every u ∈ ℱ, (0 ∨ u) ∧ 1 ∈ ℱ and ℰ( (0 ∨ u) ∧ 1, (0 ∨ u) ∧ 1 ) ≤ ℰ(u,u).

Theorem Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction semigroup on L²(X,μ). Then, (P_t)_t≥0 is a Markovian semigroup if and only if the associated closed symmetric form on L²(X,μ) is a Dirichlet form.

Proof

Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ). There exists a transition function {p_t, t ≥ 0} on X such that for every u ∈ L^∞(X,μ) and a.e. x ∈ X

P_tu(x) = ∫_X u(y) p_t(x,dy), t > 0.

Denote

k_t(x) = P_t1(x) = ∫_X p_t(x,dy).

We observe that from the Markovian property of P_t, we have 0 ≤ k_t ≤ 1 a.e.
We have then

1/2 ∫_X ∫_X (u(x) – u(y))² p_t(x,dy) dμ(x) = ∫_X u(x)² k_t(x)dμ(x) – ∫_X u(x) P_tu(x) dμ(x).

Therefore,

⟨u – P_tu, u⟩ = 1/2 ∫_X ∫_X (u(x) – u(y))² p_t(x,dy) dμ(x) + ∫_X u(x)² (1 – k_t(x)) dμ(x).

Let us now assume that u ∈ ℱ and that v is a normal contraction of u. One has

∫_X ∫_X (v(x) – v(y))² p_t(x,dy) dμ(x) ≤ ∫_X ∫_X (u(x) – u(y))² p_t(x,dy) dμ(x)

and

∫_X v(x)² (1 – k_t(x)) dμ(x) ≤ ∫_X u(x)² (1 – k_t(x)) dμ(x).

Therefore,

⟨v – P_tv,v⟩ ≤ ⟨u – P_tu,u⟩

Since u ∈ ℱ, one knows that (1/t)⟨u – P_tu,u⟩ converges to ℰ(u) when t → 0. Since (1/t)⟨v – P_tv,v⟩ is non-increasing and bounded it does converge when t → 0. Thus v ∈ ℱ and

ℰ(v) ≤ ℰ(u).

One concludes that ℰ is Markovian.

Now, consider a Dirichlet form ℰ and denote by P_t the associated semigroup in L²(X,μ) and by A its generator.
The main idea is to first prove that for λ > 0, the resolvent operator (λId – A)^-1 preserves the positivity of function. Then, we may conclude by the fact that for f ∈ L²(X,μ), in the L²(X,μ) sense

P_tf = lim_{n → +∞} (Id – t/n A)^-nf.

Let λ > 0. We consider on ℱ the norm

||f||²_λ = ||f||²_L²(X,μ) + λℰ(f,f).

From the Markovian property of ℰ, if u ∈ ℱ, then |u| ∈ ℱ and

ℰ(|u|, |u|) ≤ ℰ(u, u).

We consider the bounded operator

R_λ = (Id – λA)^-1

that goes from L²(X,μ) to 𝒟(A) ⊂ ℱ. For f ∈ ℱ and g ∈ L²(X,μ) with g ≥ 0, we have

⟨|f| , R_λ g⟩_λ = ⟨|f| , R_λg⟩_L²(X,μ) – λ⟨|f| , AR_λ g⟩_L²(X,μ)

= ⟨|f|, (Id – λA) R_λ g⟩_L²(X,μ)

=⟨|f|, g⟩_L²(X,μ)

≥ |⟨f, g⟩_L²(X,μ)|

≥ |⟨f , R_λg⟩_λ|.

Moreover, from inequality above, for f ∈ ℱ,

|||f|||_λ² = |||f| ||²_L²(X,μ) + λℰ(|f|,|f|)

≤ ||f||²_L²(X,μ) + λℰ(f,f)

≤ ||f||_λ².

By taking f = R_λ g in the two above sets of inequalities, we draw the conclusion

|⟨R_λg, R_λg⟩_λ| ≤ ⟨|R_λg| , R_λg⟩_λ ≤ |||R_λg|||_λ ||R_λg||_λ ≤ |⟨R_λg, R_λg⟩_λ|.

The above inequalities are therefore equalities which implies

R_λg = |R_λg|.

As a conclusion if g ∈ L²(X,μ) is a.e. ≥ 0, then for every λ > 0, (Id – λA)^-1g ≥ 0 a.e.. Thanks to the spectral theorem, in L²(X,μ),

P_tg = lim_{n → +∞} (Id – t/n A)^-ng.

By passing to a subsequence that converges pointwise almost surely, we deduce that P_tg ≥ 0 almost surely.

The proof of

f ≤ 1, a.e. ⇒ P_tf ≤ 1, a.e.

follows the same lines:

The first step is to observe that if 0 ≤ f ∈ ℱ, then 1 ∧ f ∈ ℱ and moreover

ℰ(1 ∧ f, 1 ∧ f) ≤ ℰ(f,f).

2. Let f ∈ L²(X,μ) satisfy 0 ≤ f ≤ 1 and set g = R_λf = (Id – λA)^-1f ∈ ℱ and h = 1 ∧ g. According to the first step, h ∈ ℱ and ℰ(h,h) ≤ ℰ(g,g). Now, we observe that:

||g – h||_λ² = ||g||_λ² – 2⟨g,h⟩_λ + ||h||_λ²

= ⟨R_λf,f⟩_L²(X,μ) – 2⟨f,h⟩_L²(X,μ) + ||h||²_L²(X,μ) + λℰ(h,h)

= ⟨R_λf,f⟩_L²(X,μ) – ||f||²_L²(X,μ) + ||f – h||²_L²(X,μ) + λℰ(h,h)

≤ ⟨R_λf,f⟩_L²(X,μ) – ||f||²_L²(X,μ) + ||f – g||²_L²(X,μ) + λℰ(g,g) = 0.

As a consequence g = h, that is 0 ≤ g ≤ 1.

3. The previous step shows that if f ∈ L²(X,μ) satisfies 0 ≤ f ≤ 1 then for every λ > 0, 0 ≤ (Id – λL)^-1 f ≤ 1. Thanks to the spectral theorem, in L²(X,μ),

P_tf = lim_{n → +∞} (Id – t/n L)^-nf.

By passing to a subsequence that converges pointwise almost surely, we deduce that 0 ≤ P_tf ≤ 1 almost surely.

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1. Riesz transforms on manifolds

2. The Vicsek set

3. Our results

References

Funding for Early Career Researchers

Energy measures and applications by Mathav Murugan

The Sard conjecture in sub-Riemannian geometry by Luca Rizzi

Gromov hyperbolicity, uniformization, and potential theory by Nageswari Shanmugalingam

Some aspects of parabolic Anderson models by Samy Tindel

Contents

Regular Dirichlet forms, Energy measures

Hunt process associated with a regular Dirichlet form

Intrinsic metric

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Preliminary lemmas

Sobolev inequality

Gagliardo-Nirenberg inequalities

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Riemannian manifolds

Carnot groups

Sierpiński gasket

Cheeger metric measure spaces

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The Lp theory of heat semigroups

Diffusion operators as generators of Dirichlet forms

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Markovian semigroups

Dirichlet forms

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