S. Bortz

activity

20152026

most citedNon-local self-improving properties: A functional analytic approach

26 citations · 52 across the 26 of their papers we have counts for

collaborators

32 papers

math.AP2026

Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions

Simon Bortz, Pablo Hidalgo-Palencia, Steve Hofmann +1

We show that the parabolic uniform rectifiability of a set can be characterized by an interior PDE property, namely Carleson measure estimates for bounded solutions to the heat equ…

math.CA2026

Big Pieces of Regular Parabolic (bi-)Lipschitz Images is Equivalent to Parabolic Uniform Rectifiability

Simon Bortz, Matthew Hyde, Mason Sharp

We define the notion of regular parabolic (bi-)Lipschitz images as the parabolic (bi-)Lipschitz maps from $n$ -dimensional space time which, up to translation, fix the $t$ variable…

math.AP2026

The parabolic Dini- $β$ condition and absolute continuity of surface and caloric measure

Simon Bortz, Moritz Egert, Sandra Ferris +1

We show if $\partial Ω$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $σ$ of $\partial Ω$ is absolutely continuous with respect to its caloric meas…

math.CA2026

Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates

Simon Bortz, Kabe Moen, Andrea Olivo +2

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This re…

math.AP2025

Solvability of the $L^{p}$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability

Simon Bortz, Steven Hofmann, José María Martell +1

Let $Ω \subset R^{n + 1}$ be an open set in space-time with boundary $Σ = \partial Ω$ . Under minimal and natural background assumptions - namely, that $Σ$ is time-symmetricall…

math.CA2025

Weighted Bounded Variation Revisited

Simon Bortz, Matthew Gossett, Joseph Kasel +1

In this article, we investigate the theory of weighted functions of bounded variation (BV), as introduced by Baldi [Ba01]. Depending on the theorem, we impose lower semicontinuity…

math.AP2025

A Variable Coefficient Free Boundary Problem for $L^{p}$ -solvability of Parabolic Dirichlet Problems in Graph Domains

Simon Bortz, Sandra Ferris, Pablo Hidalgo-Palencia +1

We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nyström [BHMN25]. In particular, we show that if $Ω$ is the region above the graph of a…

math.AP2024

A stability result for parabolic measures of operators with singular drifts

Simon Bortz, Moritz Egert, Olli Saari

We study the operator \[ \partial_t - \text{div} A \nabla + B \cdot \nabla \] in parabolic upper-half-space, where $A$ is an elliptic matrix satisfying an oscillation condition and…

math.AP2023★ 1 cited

Solvability of the $L^{p}$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

Simon Bortz, Steven Hofmann, José María Martell +1

We prove that if a parabolic Lipschitz (i.e., Lip(1,1/2)) graph domain has the property that its caloric measure is a parabolic $A_{\infty}$ weight with respect to surface measure (w…

math.AP2023

The $A_{\infty}$ condition, $ε$ -approximators, and Varopoulos extensions in uniform domains

Simon Bortz, Bruno Poggi, Olli Tapiola +1

Suppose that $Ω \subset R^{n + 1}$ , $n \geq 1$ , is a uniform domain with $n$ -Ahlfors regular boundary and $L$ is a (not necessarily symmetric) divergence form elliptic, real, bou…

math.AP2023

Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates

Simon Bortz, Steve Hofmann, José Luis Luna Garcia +2

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $- div A \nabla$ by first and zero order…

math.CA2021

Carleson Conditions for Weights: The quantitative small constant case

Simon Bortz, Moritz Egert, Olli Saari

We investigate the small constant case of a characterization of $A_{\infty}$ weights due to Fefferman, Kenig and Pipher. In their work, Fefferman, Kenig and Pipher bound the logarith…

math.AP2021

Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

Simon Bortz, Tatiana Toro, Zihui Zhao

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent stu…

math.AP2021★ 4 cited

Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

Simon Bortz, John Hoffman, Steve Hofmann +2

Let $E \subset R^{n + 1}$ be a parabolic uniformly rectifiable set. We prove that every bounded solution $u$ to $$\partial_tu- Δu=0, \quad \text{in} \quad \mathbb R^{n+1}\set…

math.CA2021

Parabolic Singular Integrals with Nonhomogeneous Kernels

Simon Bortz, John Hoffman, Steve Hofmann +2

We establish $L^{2}$ boundedness of all "nice" parabolic singular integrals on "Good Parabolic Graphs", aka {\em regular} Lip(1,1/2) graphs. The novelty here is that we include non-h…

math.MG2021

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

Simon Bortz, John Hoffman, Steve Hofmann +2

We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows u…

math.AP2021

On Big Pieces approximations of parabolic hypersurfaces

Simon Bortz, John Hoffman, Steve Hofmann +2

Let $Σ$ be a closed subset of $R^{n + 1}$ which is parabolic Ahlfors-David regular and assume that $Σ$ satisfies a 2-sided corkscrew condition. Assume, in addition, that $Σ…

math.AP2020

Optimal Poisson kernel regularity for elliptic operators with Hölder-continuous coefficients in vanishing chord-arc domains

Simon Bortz, Tatiana Toro, Zihui Zhao

We show that if $Ω$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with Hölder-continuous coefficient matrix, then $lo g k_{L} \in V M O$ , where $k_{L}$ i…

math.CA2020

Coronizations and big pieces in metric spaces

Simon Bortz, John Hoffman, Steve Hofmann +2

We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to Davi…

math.AP2020★ 3 cited

Critical Perturbations for Second Order Elliptic Operators. Part I: Square function bounds for layer potentials

Simon Bortz, Steve Hofmann, José Luis Luna Garcia +2

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $- div A \nabla$ by first an…

math.AP2019

Note on time-regularity for weak solutions to parabolic systems of p-Laplace type

Simon Bortz, Moritz Egert, Olli Saari

We show that local weak solutions to parabolic systems of p-Laplace type are H{ö}lder continuous in time with values in a spatial Lebesgue space and H{ö}lder continuous on almost e…

math.CA2019★ 2 cited

Sobolev contractivity of gradient flow maximal functions

Simon Bortz, Moritz Egert, Olli Saari

We prove that the energy dissipation property of gradient flows extends to the semigroup maximal operators in various settings. In particular, we show that the vertical maximal fun…

math.CA2019★ 1 cited

Two Phase Free Boundary Problem for Poisson Kernels

Simon Bortz, Max Engelstein, Max Goering +2

We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, if a domain has Ahlfors regular boundary, th…

math.AP2018

$ε$ -Approximability of Harmonic Functions in $L^{p}$ Implies Uniform Rectifiability

Simon Bortz, Olli Tapiola

Suppose that $Ω \subset R^{n + 1}$ , $n \geq 2$ , is an open set satisfying the corkscrew condition with an $n$ -dimensional ADR boundary, $\partial Ω$ . In this note, we show tha…

Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions

Big Pieces of Regular Parabolic (bi-)Lipschitz Images is Equivalent to Parabolic Uniform Rectifiability

The parabolic Dini-β condition and absolute continuity of surface and caloric measure

Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates

Solvability of the Lp Dirichlet problem for the heat equation implies parabolic uniform rectifiability

Weighted Bounded Variation Revisited

A Variable Coefficient Free Boundary Problem for Lp-solvability of Parabolic Dirichlet Problems in Graph Domains

A stability result for parabolic measures of operators with singular drifts

Solvability of the Lp Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

The A∞​ condition, ε-approximators, and Varopoulos extensions in uniform domains

Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates

Carleson Conditions for Weights: The quantitative small constant case

Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

Parabolic Singular Integrals with Nonhomogeneous Kernels

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

On Big Pieces approximations of parabolic hypersurfaces

Optimal Poisson kernel regularity for elliptic operators with Hölder-continuous coefficients in vanishing chord-arc domains

Coronizations and big pieces in metric spaces

Critical Perturbations for Second Order Elliptic Operators. Part I: Square function bounds for layer potentials

Note on time-regularity for weak solutions to parabolic systems of p-Laplace type

Sobolev contractivity of gradient flow maximal functions

Two Phase Free Boundary Problem for Poisson Kernels

ε-Approximability of Harmonic Functions in Lp Implies Uniform Rectifiability

The parabolic Dini- $β$ condition and absolute continuity of surface and caloric measure

Solvability of the $L^{p}$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability

A Variable Coefficient Free Boundary Problem for $L^{p}$ -solvability of Parabolic Dirichlet Problems in Graph Domains

Solvability of the $L^{p}$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph

The $A_{\infty}$ condition, $ε$ -approximators, and Varopoulos extensions in uniform domains

$ε$ -Approximability of Harmonic Functions in $L^{p}$ Implies Uniform Rectifiability