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Simon Bortz

Publications and source records attributed to Simon Bortz.

At least 19 recordsLinked to original sources
arXiv2026

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

We show if $\partial \Omega$ is the graph of a parabolic Lipschitz function, then parabolic surface measure $\sigma$ of $\partial \Omega$ is absolutely continuous with respect to its caloric measure if and only if a (square) Dini-$\beta$ condition is satisfied. More specifically, the (square) Dini-$\beta$ condition is that \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty, \quad \text{$\sigma$-a.e. } (X,t) \in \partial \Omega.\] Here $\hat{\beta}$ is a parabolic version of the Jones ($L^2$) $\beta$-numbers. We show that these conditions are satisfied if and only if the graph is covered by a countable collection of {\it regular} Lipschitz graphs, that is, graphs with additional in-time regularity in the form of a half order time derivative in the parabolic BMO space. This supports the view that covering by {\it regular} Lipschitz graphs is the right notion for qualitative parabolic rectifiability in the context of parabolic PDEs. We also show that if \[\int_0^1 \hat{\beta}(X,t,r)^2 \frac{dr}{r} < \infty\] up to a set of caloric measure zero then the caloric measure is absolutely continuous with respect to surface measure.

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arXiv2026

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

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 and whose spatial components are each regular parabolic Lipschitz functions. We show that any parabolic Ahlfors-David regular is parabolic uniformly rectifiable if and only if it has big pieces of parabolic Lipschitz images of $n$-dimensional space time if and only if it has big pieces of parabolic bi-Lipschitz images of $n$-dimensional space time. This further extends the David-Semmes theory to the parabolic setting. Our proof combines the ideas of the first authors previous work [BH12,BHH+22] and some ideas of Azzam and Schul [AS12]. The proof easily adapts (and is far less complicated) to the Euclidean case to give an alternative proof of the analogous fact.

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arXiv2026

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

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 result has several significant consequences. It extends naturally to functions of weighted bounded variation and yields corresponding capacity and isoperimetric inequalities. The inequality is also closely connected to endpoint estimates for fractional operators, including bounds for fractional maximal functions and Hardy space endpoint estimates for the Riesz potential. Our main inequality yields a family of endpoint inequalities, characterized in terms of subrepresentation formulas, Lorentz space improvements, and isoperimetric inequalities for measures and bounded open sets. When one moves away from the endpoint to $p>1$, the analogous inequalities no longer hold in general; however, we identify a sharp bumped maximal function for which the corresponding non-endpoint inequality is valid. Finally, we show that this framework yields new $(p,p)$ two-weight Sobolev inequalities.

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arXiv2025

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

Let $\Omega \subset \mathbb{R}^{n+1}$ be an open set in space-time with boundary $\Sigma = \partial \Omega$. Under minimal and natural background assumptions - namely, that $\Sigma$ is time-symmetrically parabolic Ahlfors--David regular and that $\Omega$ satisfies an interior corkscrew condition - we treat a one-phase parabolic free boundary problem which establishes the necessity of parabolic uniform rectifiability for $L^p(d\sigma)$ solvability of the Dirichlet problem for the heat equation. More precisely, we prove that if the caloric measure associated with $\Omega$ satisfies a weak-$A_\infty$ condition with respect to the surface measure $\sigma = \mathcal{H}_{\mathrm{par}}^{n+1}\!\lfloor_{\Sigma}$, then $\Sigma$ is parabolically uniformly rectifiable, hence equivalently, that solvability of the Dirichlet problem for the heat (or adjoint heat) equation in $\Omega$ with boundary data in $L^p(d\sigma)$, for some $p \in (1,\infty)$, implies parabolic uniform rectifiability. Our main theorem thus identifies parabolic uniform rectifiability as the correct geometric framework for boundary regularity, and $L^p$ solvability, in the parabolic setting.

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arXiv2025

Weighted Bounded Variation Revisited

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 and/or a pointwise A1 condition on the weight. Our motivation is twofold: to establish weighted Gagliardo-Nirenberg-Sobolev (GNS) inequalities for BV functions, and to clarify and extend earlier results on weighted BV spaces. Our main contributions include a structure theorem under minimal assumptions (lower semicontinuity), a smooth approximation result, an embedding theorem, a weighted GNS inequality for BV functions, and a corresponding weighted isoperimetric inequality.

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arXiv2025

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

We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nystr\"om [BHMN25]. In particular, we show that if $\Omega$ is the region above the graph of a Lip(1,1/2) (parabolic Lipschitz) function and $L$ is a parabolic operator in divergence form \[L = \partial_t - \text{div} A \nabla\] with $A$ satisfying an $L^1$ Carleson condition on its spatial and time derivatives, then the $L^p$-solvability of the Dirichlet problem for $L$ and $L^*$ implies that the graph function has a half-order time derivative in BMO. Equivalently, the graph is parabolic uniformly rectifiable. In the case of $A$ symmetric, we only require that the Dirichlet problem for $L$ is solvable, which requires us to adapt a clever integration by parts argument by Lewis and Nystr\"om. A feature of the present work is that we must overcome the lack of translation invariance in our equation, which is a fundamental tool in similar works, including [BHMN25].

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arXiv2024

A stability result for parabolic measures of operators with singular drifts

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 $B$ is a singular drift with a Carleson control. Our main result establishes quantitative $A_{\infty}$-estimates for the parabolic measure in terms of oscillation of $A$ and smallness of $B$. The proof relies on new estimates for parabolic Green functions that quantify their deviations from linear functions of the normal variable and on a novel, quantitative Carleson measure criterion for anisotropic $A_{\infty}$-weights.

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arXiv2023

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

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 (which in turn is equivalent to $L^p$ solvability of the Dirichlet problem for some finite $p$), then the function defining the graph has a half-order time derivative in the space of (parabolic) bounded mean oscillation. Equivalently, we prove that the $A_\infty$ property of caloric measure implies, in this case, that the boundary is parabolic uniformly rectifiable. Consequently, by combining our result with the work of Lewis and Murray we resolve a long standing open problem in the field by characterizing those parabolic Lipschitz graph domains for which one has $L^p$ solvability (for some $p <\infty$) of the Dirichlet problem for the heat equation. The key idea of our proof is to view the level sets of the Green function as extensions of the original boundary graph for which we can prove (local) square function estimates of Littlewood-Paley type.

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arXiv2023

The $A_\infty$ condition, $\varepsilon$-approximators, and Varopoulos extensions in uniform domains

Suppose that $Ω\subset\mathbb R^{n+1}$, $n\geq1$, is a uniform domain with $n$-Ahlfors regular boundary and $L$ is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in $Ω$. We show that the corresponding elliptic measure $ω_L$ is quantitatively absolutely continuous with respect to surface measure of $\partialΩ$ in the sense that $ω_L \in A_\infty(σ)$ if and only if any bounded solution $u$ to $Lu = 0$ in $Ω$ is $\varepsilon$-approximable for any $\varepsilon \in (0,1)$. By $\varepsilon$-approximability of $u$ we mean that there exists a function $Φ= Φ^\varepsilon$ such that $\|u-Φ\|_{L^\infty(Ω)} \le \varepsilon\|u\|_{L^\infty(Ω)}$ and the measure $\widetildeμ_Φ$ with $d\widetildeμ = |\nabla Φ(Y)| \, dY$ is a Carleson measure with $L^\infty$ control over the Carleson norm. As a consequence of this approximability result, we show that boundary $\operatorname{BMO}$ functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy $L^1$-type Carleson measure estimates with $\operatorname{BMO}$ control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.

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arXiv2023

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

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 us to conclude that if $Σ\subset\mathbb{R}^{n+1}$ is parabolic Ahlfors-David regular, then the following statements are equivalent. (1) $Σ$ is parabolic uniformly rectifiable. (2) $Σ$ admits a corona decomposition with respect to regular Lip(1,1/2) graphs. (3) $Σ$ admits a bilateral corona decomposition with respect to regular Lip(1,1/2) graphs. (4) $Σ$ is big pieces squared of regular Lip(1,1/2) graphs.

math.MG↗
arXiv2023

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

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi-Nash-Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-$L^p$ ''$N<S$'' estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full $L^2$ bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. As a corollary, we claim the first results in an unbounded domain concerning the $L^p$-solvability of boundary value problems for the magnetic Schrödinger operator $-(\nabla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space.

math.AP↗
arXiv2023

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

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness of the Dirichlet, Neumann and Regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. For instance, this allows us to claim the first results in the setting of an unbounded domain concerning the solvability of boundary value problems for the magnetic Schrödinger operator $-(\nabla-i{\bf a})^2+V$ when the magnetic potential ${\bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space. In the present paper, we establish $L^2$ control of the square function via a vector-valued $Tb$ theorem and abstract layer potentials, and use these square function bounds to obtain uniform slice bounds for solutions. The existence and uniqueness of solutions, as well as bounds for the non-tangential maximal operator, are considered in the upcoming paper.

math.AP↗
arXiv2021

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

Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of an operator with coefficients satisfying a vanishing Carleson condition in the upper half space is an asymptotically optimal $A_\infty$ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation. To achieve this, we prove local, quantitative estimates on a quantity (introduced by Fefferman, Kenig and Pipher) that controls the $A_\infty$ constant. Our work uses recent results obtained by David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems.

math.AP↗
arXiv2021

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

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 every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.

math.AP↗
arXiv2021

Carleson Conditions for Weights: The quantitative small constant case

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 logarithm of the $A_\infty$ constant by the Carleson norm of a measure built out of the heat extension, up to a multiplicative and additive constant (as well as the converse). We prove, qualitatively, that when one of these quantities is small so is the other. In fact, we show that these quantities are bounded by a constant times the square root of the other, provided at least one of them is sufficiently small. We also give an application of our result to the study of elliptic measures associated to elliptic operators with coefficients satisfying the ``Dahlberg-Kenig-Pipher" condition. We suspect that the square root dependence in the bound used in this application is sharp and give some justification for this in the last section.

math.CA↗
arXiv2021

Parabolic Singular Integrals with Nonhomogeneous Kernels

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-homogeneous kernels, which are relevant to the theory of parabolic uniform rectifiability. Previously, the third named author had treated the case of homogeneous kernels. The present proof combines the methods of that work (which in turn was based on methods described in Christ's CBMS lecture notes), with the techniques of Coifman-David-Meyer.

math.CA↗
arXiv2021

Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets

Let $E \subset \mathbb R^{n+1}$ be a parabolic uniformly rectifiable set. We prove that every bounded solution $u$ to $$\partial_tu- \Delta u=0, \quad \text{in} \quad \mathbb R^{n+1}\setminus E$$ satisfies a Carleson measure estimate condition. An important technical novelty of our work is that we develop a corona domain approximation scheme for $E$ in terms of regular Lip(1/2,1) graph domains. This approximation scheme has an analogous elliptic version which is an improvement of the known results in that setting.

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