arXiv subjects

Sylvester Eriksson-Bique

Publications and source records attributed to Sylvester Eriksson-Bique.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2024

Analytically one-dimensional planes and the Combinatorial Loewner Property

It is a major problem in analysis on metric spaces to understand when a metric space is quasisymmetric to a space with strong analytic structure, a so-called Loewner space. A conjecture of Kleiner, recently disproven by Anttila and the second author, proposes a combinatorial sufficient condition. The counterexamples constructed are all topologically one dimensional, and the sufficiency of Kleiner's condition remains open for most other examples. A separate question of Kleiner and Schioppa, apparently unrelated to the problem above, asks about the existence of "analytically $1$-dimensional planes": metric measure spaces quasisymmetric to the Euclidean plane but supporting a $1$-dimensional analytic structure in the sense of Cheeger. In this paper, we construct an example for which the conclusion of Kleiner's conjecture is not known to hold. We show that either this conclusion fails in our example or there exists an "analytically $1$-dimensional plane". Thus, our construction either yields a new counterexample to Kleiner's conjecture, different in kind from those of Anttila and the second author, or a resolution to the problem of Kleiner--Schioppa.

math.MG↗

arXiv2023

Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

We give a sharp Hausdorff content estimate for the size of the accessible boundary of any domain in a metric measure space of controlled geometry, i.e., a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality for a fixed $1\le p<\infty$. This answers a question posed by Jonas Azzam. In the process, we extend the result to every doubling gauge in metric measure spaces which satisfies a codimension one bound.

math.MG↗

arXiv2023

Density of continuous functions in Sobolev spaces with applications to capacity

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if $(X,d,μ)$ is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space $N^{1,p}(X)$. Here the measure $μ$ is Borel and is finite and positive on all metric balls. In particular, we don't assume properness of $X$, doubling of $μ$ or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to locally complete spaces $X$ and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity.

math.MG↗

arXiv2023

Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces

We prove that for a quasiself-similar and arcwise connected compact metric space all three known versions of the conformal dimension coincide: the conformal Hausdorff dimension, conformal Assouad dimension and Ahlfors regular conformal dimension. This answers a question posed by Mathav Murugan. Quasisimilar spaces include all approximately self-similar spaces. As an example, the standard Sierpiński carpet is quasiself-similar and thus the three notions of conformal dimension coincide for it. We also give the equality of the three dimensions for combinatorially $p$-Loewner (CLP) spaces. Both proofs involve using a new notion of combinatorial modulus, which lies between two notions of modulus that have appeared in the literature. The first of these is the modulus studied by Pansu and Tyson, which uses a Carathéodory construction. The second is the one used by Keith and Laakso (and later modified and used by Bourdon, Kleiner, Carrasco-Piaggio, Murugan and Shanmugalingam). By combining these approaches, we gain the flexibility of giving upper bounds for the new modulus from the Pansu-Tyson approach, and the ability of getting lower bounds using the Keith-Laakso approach. Additionally the new modulus can be iterated in self-similar spaces, which is a crucial, and novel, step in our argument.

math.MG↗

arXiv2022

Tensorization of quasi-Hilbertian Sobolev spaces

The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $X\times Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W^{1,2}(X\times Y)=J^{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space $W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally for $p\in (1,\infty)$ we obtain the norm-one inclusion $\|f\|_{J^{1,p}(X,Y)}\le \|f\|_{W^{1,p}(X\times Y)}$ and show that the norms agree on the algebraic tensor product $W^{1,p}(X)\otimes W^{1,p}(Y)\subset W^{1,p}(X\times Y)$. When $p=2$ and $X$ and $Y$ are infinitesimally quasi-Hilbertian, standard Dirichlet form theory yields the density of $W^{1,2}(X)\otimes W^{1,2}(Y)$ in $J^{1,2}(X,Y)$ thus implying the equality of the spaces. Our approach raises the question of the density of $W^{1,p}(X)\otimes W^{1,p}(Y)$ in $J^{1,p}(X,Y)$ in the general case.

math.FA↗

arXiv2022

Density of Lipschitz functions in Energy

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space $N^{1,p}(X)$ holds for all $p\in [1,\infty)$ whenever the space $X$ is complete and separable and the measure is Radon and finite on balls. Emphatically, $p=1$ is allowed. We also give a few corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. It also yields a new approximation technique, which has not appeared in prior work. Notable with all of this is that we do not use any form of Poincaré inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of existing literature on the topic.

math.CA↗

arXiv2022

Tensorization of $p$-weak differentiable structures

We consider $p$-weak differentiable structures that were recently introduced by the first and last named authors, and prove that the product of $p$-weak charts is a $p$-weak chart. This implies that the product of two spaces with a $p$-weak differentiable structure also admits a $p$-weak differentiable structure. We make partial progress on the tensorization problem of Sobolev spaces by showing an isometric embedding result. Further, we establish tensorization when one of the factors is PI.

math.DG↗

arXiv2022

A New Hausdorff Content Bound for Limsup Sets

We give a new Hausdorff content bound for limsup sets, which is related to Falconer's sets of large intersection. Falconer's sets of large intersection satisfy a content bound for all balls in a space. In comparison, our main theorem only assumes a scale-invariant bound for the balls forming the limit superior set in question. We give four applications of these ideas and our main theorem: a new proof and generalization of the mass transference principle related to Diophantine approximations, a related result on random limsup sets, a new proof of Federer's characterization of sets of finite perimeter and a statement concerning generic paths and the measure theoretic boundary. The new general mass transference principle transfers a content bound of one collection of balls, to the content bound of another collection of sets -- however, this content bound must hold on all balls in the space. The benefit of our approach is greatly simplified arguments as well as new tools to estimate Hausdorff content. The new methods allow for us to dispense with many of the assumptions in prior work. Specifically, our general Mass Transference Principle, and bounds on random limsup sets, do not assume Ahlfors regularity. Further, they apply to any complete metric space. This generality is made possible by the fact that our general Hausdorff content estimate applies to limsup sets in any complete metric space.

math.MG↗

arXiv2022

On Limits at Infinity of Weighted Sobolev Functions

We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense in which the limit is taken. First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and radial weights, we give if and only if statements. These generalize and give new proofs for results of Fefferman and Uspenski\uı.

math.AP↗

arXiv2021

Nilpotent groups and biLipschitz embeddings into $L^1$

We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^1$ space, then it is abelian. We reach this conclusion by proving that every Carnot group that biLipschitz embeds into $L^1$ is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into $L^1$ and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on "generic" tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipschitz embeddings can not exist in the non-abelian settings.

math.MG↗

arXiv2021

Isoperimetric and Poincaré inequalities on non-self-similar Sierpiński sponges: the borderline case

In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincaré inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate such inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpiński carpets, and extend them to many more general shapes as well as higher dimensions.

math.MG↗

arXiv2021

Almost uniform domains and Poincaré inequalities

Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincaré inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio and Sarvas. Our condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components. In doing so, our results hold true for metric spaces equipped with doubling measures and Poincaré inequalities in general, and for the Heisenberg groups in particular. To our knowledge, these are the first examples of such subsets on any step-2 Carnot group. Such subsets also give, in general, new examples of Sobolev extension domains on doubling metric measure spaces. When specialized to the plane, we give general sufficient conditions for planar subsets, possibly with empty interior, to be Ahlfors 2-regular and to satisfy a (1,2)-Poincaré inequality. In the Euclidean case, our construction also covers the non-self-similar Sierpiński carpets of Mackay, Tyson, and Wildrick, as well as higher dimensional analogues not treated in the literature. The analysis of the Poincaré inequality with exponent p=1, for these carpets and their higher dimensional analogues, includes a new way of proving an isoperimetric inequality on a space without constructing Semmes families of curves.

math.MG↗

arXiv2021

Bi-Lipschitz embeddings of quasiconformal trees

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 in \cite{DV} (arXiv:2007.12297).

math.MG↗

arXiv2021

Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential

We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, by maximal directional derivatives along "generic" curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of $p$-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along $p$-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. The arising $p$-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger's structure in the presence of a Poincaré inequality. It is moreover compatible with, and gives a geometric interpretation of, Gigli's abstract differentiable structure, whenever it exists. The $p$-weak charts give rise to a finite dimensional $p$-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.

math.MG↗

arXiv2021

On the Sharp Lower Bound for Duality of Modulus

We establish a sharp reciprocity inequality for modulus in compact metric spaces $X$ with finite Hausdorff measure. In particular, when $X$ is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua $E$ and $F$ in $X$ and the modulus of the family of surfaces of finite Hausdorff measure that separate $E$ and $F$. The paper also develops approximation techniques, which may be of independent interest.

math.MG↗

arXiv2020

Regularity of Solutions to the Fractional Cheeger-Laplacian on Domains in Metric Spaces of Bounded Geometry

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,μ_X)$ satisfying a $2$-Poincaré inequality. Given a bounded domain $Ω\subset X$ with $μ_X(X\setminusΩ)>0$, and a function $f$ in the Besov class $B^θ_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^θ_{2,2}(X)$ such that $u=f$ in $X\setminusΩ$ and $\mathcal{E}_θ(u,u)\le \mathcal{E}_θ(h,h)$ whenever $h\in B^θ_{2,2}(X)$ with $h=f$ in $X\setminusΩ$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on $Ω$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.

math.AP↗