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Riku Anttila

Publications and source records attributed to Riku Anttila.

10 recordsLinked to original sources
arXiv2026

One-dimensional Dirichlet forms with prescribed Hölder regularity

We study a class of strongly local, regular Dirichlet forms on the standard unit interval. The aim of our work is to record some Hölder regularity properties that have not been noted in the prior literature. In particular, as our main result, we show that for every $δ\in (0,1]$ there exists a metric measure space $(X,d,μ)$ equipped with a strongly local, regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(μ)$ with the property that $δ$ is the supremum of $α\in (0,1]$ for which the domain $\mathcal{F}$ of the Dirichlet form contains a non-constant $α$-Hölder continuous function. To the best of our knowledge, such examples were previously known only for $δ= 1$. In our construction, the value $δ$ is characterized by the upper Hausdorff dimension of a certain Radon measure that is used to define $(\mathcal{E},\mathcal{F})$.

math.FA↗
arXiv2026

Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces

We construct self-similar $p$-energy forms $\mathscr{E}_p$ on a rich class of \emph{Laakso-type fractal spaces} and study the properties of the associated Sobolev spaces $\mathscr{F}_p$. The main result of the paper is the discovery of a new analytic phenomenon, which we refer to as \emph{singularity of Sobolev spaces}. This means that the associated Sobolev spaces $\mathscr{F}_{p_1}$ and $\mathscr{F}_{p_2}$ for distinct $p_1,p_2 \in (1,\infty)$ intersect only at constant functions. We show that the Laakso diamond space of Lang--Plaut is one such example, and explain why this does not contradict the inverse limit construction of Cheeger--Kleiner which proves Laakso Diamond to support Poincaré inequality of Heinonen--Koskela.

math.MG↗
arXiv2026

Cartesian products of Sierpiński carpets do not attain their conformal dimension

It is a long-standing open question to determine whether the Sierpiński carpet attains its conformal dimension or not. While this problem remains unresolved, we prove that Cartesian products $\mathbb{S}^k$, where $\mathbb{S}$ is the Sierpiński carpet and $k \geq 2$, do not attain their conformal dimension. Our approach is based on the Sobolev spaces and energy measures on $\mathbb{S}$ -- constructed by Shimizu, Kigami, and Murugan and Shimizu -- together with a certain singularity result of energy measures from the theory of analysis on fractals. This work formulates a general non-attainment result of conformal dimension for product metric spaces $X^k$ for $k \geq 2$ in terms of self-similarity and energy measures of the factor $X$. It applies, in particular, to the cases where $X$ is the Sierpiński carpet, the Sierpiński gasket, the Menger sponge, and the Laakso diamond.

math.MG↗
arXiv2026

Conformal dimension and its attainment on self-similar Laakso-type fractal spaces

A general construction of Laakso-type fractal spaces was recently introduced by the first two authors. In this paper, we establish a simple condition characterizing when the Ahlfors regular conformal dimension of a symmetric Laakso-type fractal space is attained. The attaining metrics are constructed explicitly. This gives new examples of attainment and clarifies the possible obstructions.

math.MG↗
arXiv2025

The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces

This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpiński carpet and the higher dimensional Menger sponges, but also a rich class of new examples. The main result of the work roughly speaking states: If the construction is sufficiently symmetric then the limiting object supports useful moduli estimates, namely the Combinatorial Loewner property of Bourdon--Kleiner and the super-multiplicativity inequalities. The latter are established on Menger sponges for which it had not been previously known. The main new technique the work offers is a general framework of flows and resistance estimates.

math.MG↗
arXiv2025

An approach to sub-Gaussian heat kernel estimates via analysis on metric spaces

In this work, we establish a new characterization of sub-Gaussian heat kernel estimates for strongly local regular Dirichlet forms on metric measure spaces. Our formulation is based on the newly introduced cutoff energy condition, which offers a simpler and more transparent alternative for earlier technical energy inequalities, in particular the cutoff Sobolev inequality. The main idea of our approach is to reinterpret the cutoff Sobolev inequality as a Poincaré type inequality, and analyze it using Hajłasz--Koskela techniques from analysis on metric spaces. Applications of the new characterization are also discussed.

math.PR↗
arXiv2024

On Constructions of Fractal Spaces Using Replacement and the Combinatorial Loewner Property

The combinatorial Loewner property was introduced by Bourdon and Kleiner as a quasisymmetrically invariant substitute for the Loewner property for general fractals and boundaries of hyperbolic groups. While the Loewner property is somewhat restrictive, the combinatorial Loewner property is very generic -- Bourdon and Kleiner showed that many familiar fractals and group boundaries satisfy it. If $X$ is quasisymmetric to a Loewner space, it has the combinatorial Loewner property. Kleiner conjectured in 2006 that the converse to this holds for self-similar fractals -- the hope being that this would lead to the existence of many exotic Loewner spaces. We disprove this conjecture and give the first examples of spaces which are self-similar, combinatorially Loewner and which are not quasisymmetric to Loewner spaces. In the process we introduce a self-similar replacement rule, called iterated graph systems (IGS), which is inspired by the work of Laakso. This produces a new rich class of fractal spaces, where closed form computations of potentials and their conformal dimensions are possible. These spaces exhibit a rich class of behaviors from analysis on fractals in regards to diffusions, Sobolev spaces, energy measures and conformal dimensions. These behaviors expand on the known examples of Cantor sets, gaskets, Vicsek sets, and the often too difficult carpet-like spaces. Especially the counterexamples to Kleiner's conjecture that arise from this construction are interesting, since they open up the possibility to study the new realm of combinatorially Loewner spaces that are not quasisymmetric to Loewner spaces.

math.MG↗