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

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18 papers · 1 filter

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.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.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.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.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.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.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.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…

math.AP2018

Quantitative Fatou Theorems and Uniform Rectifiability

Simon Bortz, Steve Hofmann

We show that a suitable quantitative Fatou Theorem characterizes uniform rectifiability in the codimension 1 case.

math.AP2017★ 26 cited

Non-local self-improving properties: A functional analytic approach

Pascal Auscher, Simon Bortz, Moritz Egert +1

A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short…

math.AP2017

On regularity of weak solutions to linear parabolic systems with measurable coefficients