arXiv subjects

Behnam Esmayli

Publications and source records attributed to Behnam Esmayli.

8 recordsLinked to original sources

arXiv2026

Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas

We give a self-contained exposition of metric differentiability for Lipschitz mappings from sets in Euclidean spaces into arbitrary metric spaces, together with the corresponding area and co-area formulas. The novelty is a new metric implicit function theorem for mappings into metric spaces, which is then used to give a direct and geometric proof of the co-area formula for Lipschitz mappings into metric spaces.

math.AP↗

arXiv2026

Exceptional Sets for Quasiconformal Mappings in General Metric Spaces II

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved that it will suffice to verify the uniform bound up to a set of measure zero as long as we can show that the dilatation is finite outside a subset of finite Hausdorff--$(n-1)$ measure. In short, we say that we can allow an exceptional codimension $1$ subset. In the metric setting, it has been proved, roughly speaking, that one can allow an exceptional codimension $p$ subset, $p \ge 1$, if the source space satisfies a $p$-Poincaré inequality. We prove, effectively, the sharpness of the latter claim.

math.FA↗

arXiv2025

On Removable Sets for Weighted Sobolev Functions

We give sufficient geometric conditions, not involving capacities, for a compact null set to be removable for the Sobolev functions on weighted $\mathbb R^n$, defined as the closure of smooth functions in the weighted Sobolev norm. Our porosity conditions are in terms of suitable coverings by cubes. The weights are assumed to be doubling and satisfy a Poincaré inequality, which includes, but is not equal to, the famous class of Muckenhoupt weights. Our proofs use ideas and techniques from the theory of analysis on metric spaces.

math.FA↗

arXiv2025

Conformal Uniformization of Domains Bounded by Quasitripods

We prove Koebe's conjecture and a version of Schramm's cofat uniformization theorem for domains $Ω\subset \mathbb C$ satisfying conditions involving quasitripods, i.e., quasisymmetric images of the standard tripod. If the non-point complementary components of $Ω$ contain uniform quasitripods with large diameters and satisfy a packing condition, then there exists a conformal map $f\colonΩ\to D$ onto a circle domain $D$. Moreover, $f$ preserves the classes of point-components and non-point components. The packing condition is satisfied if $Ω$ is cospread, i.e., if the complementary components contain uniform quasitripods in all scales.

math.CV↗

arXiv2024

On Carrasco Piaggio's theorem connecting combinatorial modulus and Ahlfors regular conformal dimension

The goal of this paper is to provide an expository description of a result of Carrasco Piaggio connecting the Ahlfors regular conformal dimension of a compact uniformly perfect doubling metric space with the combinatorial $p$-moduli of the metric space. We give detailed construction of a metric associated with the $p$-modulus of the space when the $p$-modulus is zero, so that the constructed metric is in the Ahlfors regular conformal gauge of the metric space. To do so, we utilize the tools of hyperbolic filling, developed first by Gromov and by Bourdon and Pajot.

math.MG↗

arXiv2022

Coarea Inequality for Monotone Functions on Metric Surfaces

We study coarea inequalities for metric surfaces -- metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure $\mathcal{H}^2$. For monotone Sobolev functions $u\colon X \to \mathbb{R} $, we prove the inequality \begin{equation*} \int_{ \mathbb{R} }^{*} \int_{ u^{-1}(t) } g \,d\mathcal{H}^{1} \,dt \leq κ \int_{ X } g ρ \,d\mathcal{H}^{2} \quad\text{for every Borel $g \colon X \rightarrow \left[0,\infty\right]$,} \end{equation*} where $ρ$ is any integrable upper gradient of $u$. If $ρ$ is locally $L^2$-integrable, we obtain the sharp constant $κ=4/π$. The monotonicity condition cannot be removed as we give an example of a metric surface $X$ and a Lipschitz function $u \colon X \to \mathbb{R}$ for which the coarea inequality above fails.

math.MG↗

arXiv2022

Lipschitz mappings, metric differentiability, and factorization through metric trees

Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if $f$ is a Lipschitz mapping from an open set in $\mathbb{R}^n$ onto a metric space $X$, then the topological dimension of $X$ equals $n$ if and only if $X$ has positive $n$-dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, in particular, a new formula for integration on countably rectifiable sets in the Heisenberg group.

math.MG↗

arXiv2020

The Coarea Inequality

The aim of this paper is to provide a self-contained proof of a general case of the coarea inequality, also known as the Eilenberg inequality. The result is known, but we are not aware of any place that a proof would be written with all details. The known proof is based on a difficult result of Davies. Our proof is elementary and does not use Davies' theorem. Instead we use an elegant argument that we learned from Nazarov through MathOverflow. We also obtain some generalizations of the coarea inequality.

math.CA↗