arXiv subjects

Sylvester Eriksson-Bique

Publications and source records attributed to Sylvester Eriksson-Bique.

At least 19 recordsLinked to original sources

arXiv2026

Fragment-wise differentiable structures

The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for $p$-duality in terms of Alberti representations and $\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds.

math.CA↗

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

On universal elements for doubling geodesic trees

For $n\ge 3$ and $c\in(0,1)$, let $\mathcal{GT}(n,c)$ denote the class of geodesic metric trees of valence at most $n$ whose branch points are uniformly relatively separated with constant $c$. We prove that $\mathcal{GT}(n,c)$ has no bi-Lipschitz universal element. More precisely, we construct a family $(T_a)_{a\in[1/4,1/3]}\subset\mathcal{GT}(n,c)$ such that, for every $n_M\geq 3, c_M\in(0,1)$ and every $M\in\mathcal{GT}(n_M, c_M)$, there are at most countably many parameters $a$ for which $T_a$ admits a bi-Lipschitz embedding into $M$, whereas each $T_a$ admits a bi-Lipschitz embedding into $\mathbb R^2$. Thus the obstruction is neither dimensional nor caused by a failure of planar embeddability. This gives a negative answer to a question of Chrontsios-Garitsis, Ioannidis, and Vellis~\cite[Question~1.11]{CGIV2024}. Furthermore, we show a complementary positive result for ultrametric spaces: every bounded ultrametric space $X$ admits a bi-Lipschitz embedding into every complete metric space $Y$ satisfying $\dim_A X<\dim_{LA} Y$ where $\dim_A X$ and $\dim_{LA} X$ are Assouad and lower Assouad dimensions, respectively.

math.MG↗

arXiv2026

Embeddability and rectifiability of Lipschitz differentiability spaces

We prove that Lipschitz differentiability spaces which bi-Lipschitz embed into an RNP-space are countably rectifiable. In contrast to earlier methods of Cheeger and Kleiner, our approach does not rely on differentiating RNP-targets, and uses instead decomposability bundles and a careful blow-up analysis. We also present decomposability bundles in a way which avoids the mention of Alberti representations and generalizes the approach of Alberti--Marchese to measures in RNP-spaces. We moreover study fragment-wise differentiability into RNP-targets, give a new ${\rm Lip}-{\rm lip}$-type characterization of RNP-differentiability spaces, and address a question of Le Donne asking for a characterization of spaces $(X,μ)\subset\ell^2$ whose Gromov--Hausdorff tangents are Hausdorff limits of $r^{-1} (X-x)$ in $\ell^2$ as $r\to 0$.

math.MG↗

arXiv2026

Quantitative non-embeddability theorems and metric embeddings of slit carpets

We study the bi-Lipschitz embedding problem for a class of metric spaces called slit carpets. First we show that the $n$th stage $\mathbb{M}_n$ of the standard slit carpet of Merenkov admits a bi-Lipschitz embedding into Euclidean space with distortion $ O(\sqrt{n})$. Then, we show a nearly sharp lower bound of $Ω\left(\sqrt{\frac{n}{\log(n)}}\right)$. This result quantifies the recent result by David and Eriksson-Bique, and thus gives a quantified answer to the question 8 in the paper by Heinonen and Semmes by showing that $\mathbb{M}_\infty$ does not bi-Lipschitz embed into Euclidean spaces. Then, we study the $L^1$ embeddability of the standard slit carpet. We show that the standard slit carpet has Lipschitz dimension $1$ in the sense of Cheeger and Kleiner, and consequently prove that it admits a bi-Lipschitz embedding into $ L^1 $. Third, we generalize the results in terms of targets and domains. First, we give a qualitative and Lebesgue differentiation based argument which shows that general slit carpets do not bi-Lipschitz embed into any Banach space with the RNP property. As a consequence, $\mathbb{M}_\infty$ does not bi-Lipschitz embed to $\ell_1$. We then consider carpets $\mathbb{M}^a$ where the relative sizes of slits decrease according to a sequence $a\in c_0$. We give a quantitative $β$-number based argument which shows that the carpets $\mathbb{M}^a$ do not bi-Lipschitz embed into Hilbert space if $a\not\in \ell_{1+ε}$.

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

Coarse and pointwise tangent fields

Alberti, Csörnyei and Preiss introduced a notion of a "pointwise (weak) tangent field" for a subset of Euclidean space -- a field that contains almost every tangent line of every curve passing through the set -- and showed that all area-zero sets in the plane admit one-dimensional tangent fields. We extend their results in two distinct directions. First, a special case of our pointwise result shows that each doubling subset of Hilbert space admits a pointwise tangent field in this sense, with dimension bounded by the Nagata (or Assouad) dimension of the set. Second, inspired by the Analyst's Traveling Salesman Theorem of Jones, we introduce new, "coarse" notions of tangent field for subsets of Hilbert space, which take into account both large and small scale structure. We show that doubling subsets of Hilbert space admit such coarse tangent fields, again with dimension bounded by the Nagata (or Assouad) dimension of the set. For porous sets in the plane, this result can be viewed as a quantitative version of the Alberti--Csörnyei--Preiss result, though our results hold in all (even infinite) dimensions.

math.CA↗

arXiv2026

On the Resistance Conjecture

We give an affirmative answer to the resistance conjecture on characterization of parabolic Harnack inequalities in terms of volume doubling, upper capacity bounds and a Poincaré inequalities. The key step is to show that these three assumptions imply the so called cutoff Sobolev inequality, an important inequality in the study of anomalous diffusions, Dirichlet forms and re-scaled energies in fractals. This implication is shown in the general setting of $p$-Dirichlet Spaces introduced by the author and Murugan, and thus a unified treatment becomes possible to proving Harnack inequalities and stability phenomena in both analysis on metric spaces and fractals and for graphs and manifolds for all exponents $p\in (1,\infty)$. As an application, we also show that a Dirichlet space satisfying volume doubling, Poincaré and upper capacity bounds has finite martingale dimension and admits a type of differential structure similar to the work of Cheeger. In the course of the proof, we establish methods of extension and characterizations of Sobolev functions by Poincaré-inequalities, and extend the methods of Jones and Koskela to the general setting of $p$-Dirichlet spaces.

math.PR↗

arXiv2026

Traces of Newton-Sobolev functions on the visible boundary of domains in doubling metric measure spaces supporting a $p$-Poincaré inequality

We consider the question of whether a domain with uniformly thick boundary at all locations and at all scales has a large portion of its boundary visible from the interior; here, "visibility" indicates the existence of John curves connecting the interior point to the points on the "visible boundary". In this paper, we provide an affirmative answer in the setting of a doubling metric measure space supporting a $p$-Poincaré inequality for $1<p<\infty$, thus extending the results of [20,2,9] to non-Ahlfors regular spaces. We show that $t$-codimensional thickness of the boundary for $0<t<p$ implies $p$-codimensional thickness of the visible boundary. For such domains we prove that traces of Sobolev functions on the domain belong to the Besov class of the visible boundary.

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

On the energy image density conjecture of Bouleau and Hirsch

We affirmatively resolve the energy image density conjecture of Bouleau and Hirsch (1986). Beyond the original framework of Dirichlet structures, we establish the energy image density property in several related settings. In particular, we formulate a version of the property that encompasses strongly local, regular Dirichlet forms, Sobolev spaces defined via upper gradients, and self-similar energies on fractals, thereby unifying these under a single framework. As applications, we prove the finiteness of the martingale dimension for diffusions satisfying sub-Gaussian heat kernel bounds, and we obtain a new proof of a conjecture of Cheeger concerning the Hausdorff dimension of the images of differentiability charts in PI spaces. The proof of the energy image density property is based on a structure theorem for measures and normal currents in $\mathbb{R}^n$ due to De Philippis--Rindler, together with the notions of decomposability bundles due to Alberti--Marchese and cone null sets due to Alberti--Csörnyei--Preiss and Bate.

math.PR↗

arXiv2025

Quasisymmetric mappings on two variants of fractal percolation

We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-extinction, the Hausdorff dimension of the fat fractal percolation cannot be lowered with a quasisymmetry and the Hausdorff dimension of the dense fractal percolation cannot be lowered with a power quasisymmetry.

math.MG↗

arXiv2025

Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted $\mathbf{R}^n$ and metric spaces

We study uniqueness of $p$-harmonic Green functions in domains $Ω$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, with $1<p<\infty$. For bounded domains in unweighted $\mathbf{R}^n$, the uniqueness was shown for the $p$-Laplace operator $Δ_p$ and all $p$ by Kichenassamy--Véron (Math. Ann. 275 (1986), 599-615), while for $p=2$ it is an easy consequence of the linearity of the Laplace operator $Δ$. Beyond that, uniqueness is only known in some particular cases, such as in Ahlfors $p$-regular spaces, as shown by Bonk--Capogna--Zhou (arXiv:2211.11974). When the singularity $x_0$ has positive $p$-capacity, the Green function is a particular multiple of the capacitary potential for $\text{cap}_p(\{x_0\},Ω)$ and is therefore unique. Here we give a sufficient condition for uniqueness in metric spaces, and provide an example showing that the range of $p$ for which it holds (while $x_0$ has zero $p$-capacity) can be a nondegenerate interval. In the opposite direction, we give the first example showing that uniqueness can fail in metric spaces, even for $p=2$.

math.AP↗

arXiv2025

Duality for the gradient of a $p$-harmonic function and the existence of gradient curves

Every convex optimization problem has a dual problem. The $p$-Dirichlet problem in metric measure spaces is an optimization problem whose solutions are $p$-harmonic functions. What is its dual problem? In this paper, we give an answer to this problem in the following form. We give a generalized modulus problem whose solution is the gradient of the $p$-harmonic function for metric measure spaces. Its dual problem is an optimization problem for measures on curves and we show exact duality and the existence of minimizers for this dual problem under appropriate assumptions. When applied to $p$-harmonic functions the minimizers of this dual problem are supported on gradient curves, yielding a natural concept associated to such functions that has yet to be studied. This process defines a natural dual metric current and proves the existence of gradient curves. These insights are then used to construct a counter example answering the old ``sheaf problem'' on metric spaces: in contrast to Euclidean spaces, in general metric spaces being $p$-harmonic is not strictly speaking a local property.

math.MG↗

arXiv2025

Universal Differentiability Sets in Laakso Space

We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincaré inequality.

math.FA↗

arXiv2024

Self-improvement of fractional Hardy inequalities in metric measure spaces via hyperbolic fillings

In this paper, we prove a self-improvement result for $(θ,p)$-fractional Hardy inequalities, in both the exponent $1<p<\infty$ and the regularity parameter $0<θ<1$, for bounded domains in doubling metric measure spaces. The key conceptual tool is a Caffarelli-Silvestre-type argument, which relates fractional Sobolev spaces on $Z$ to Newton-Sobolev spaces in the hyperbolic filling $\overline{X}_{\varepsilon}$ of $Z$ via trace results. Using this insight, it is shown that a fractional Hardy inequality in an open subset of $Z$ is equivalent to a classical Hardy inequality in the filling $\overline{X}_{\varepsilon}$. The main result is then obtained by applying a new weighted self-improvement result for $p$-Hardy inequalities. The exponent $p$ can be self-improved by a classical Koskela-Zhong argument, but a new theory of regularizable weights is developed to obtain the self-improvement in the regularity parameter $θ$. This generalizes a result of Lehrbäck and Koskela on self-improvement of $d_Ω^β$-weighted $p$-Hardy inequalities by allowing a much broader class of weights. Using the equivalence of fractional Hardy inequalities with Hardy inequalities in the fillings, we also give new examples of domains satisfying fractional Hardy inequalities.

math.AP↗

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↗