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Steve Hofmann

Publications and source records attributed to Steve Hofmann.

At least 37 records · Page 2Linked to original sources
arXiv2020

Coronizations and big pieces in metric spaces

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 David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when suitably interpreted, in metric spaces having a fixed positive (perhaps non-integer) dimension, equipped with a Borel regular measure and with arbitrary approximating sets. As a novel application we highlight how to utilize this general setting in the context of parabolic uniform rectifiability.

math.CA↗
arXiv2020

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition

The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. We split our proof on two main steps. In the first one we considered the case in which the desired Carleson measure condition on the coefficients holds with "sufficiently small constant", using a novel application of techniques developed in geometric measure theory. In the second step we establish the final result, that is, the "large constant case". The key elements are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, and a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.

math.AP↗
arXiv2020

Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains

In relatively nice geometric settings, in particular, on Lipschitz domains, absolute continuity of elliptic measure with respect to the surface measure is equivalent to Carleson measure estimates, to square function estimates, and to $\varepsilon$-approximability, for solutions to the second order divergence form elliptic partial differential equations $ Lu= -{\rm div\,} (A \nabla u)=0$. In more general situations, notably, in an open set $Ω$ with a uniformly rectifiable boundary, absolute continuity of elliptic measure with respect to the surface measure may fail, already for the Laplacian. In the present paper, the authors demonstrate that nonetheless, Carleson measure estimates, square function estimates, and $\varepsilon$-approximability remain valid in such $Ω$, for solutions of $Lu=0$, provided that such solutions enjoy these properties in Lipschitz subdomains of $Ω$. Moreover, we establish a general real-variable transference principle, from Lipschitz to chord-arc domains, and from chord-arc to open sets with uniformly rectifiable boundary, that is not restricted to harmonic functions or even to solutions of elliptic equations. In particular, this allows one to deduce the first Carleson measure estimates and square function bounds for higher order systems on open sets with uniformly rectifiable boundaries and to treat subsolutions and subharmonic functions.

math.AP↗
arXiv2020

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $ Ω\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $Ω$, with data in $L^p(\partialΩ)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.

math.CA↗
arXiv2020

Uniform rectifiability implies Varopoulos extensions

We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $Ω\subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial Ω$, and let $σ:= \mathcal{H}^n\lfloor_{\partial Ω}$ denote the surface measure on $\partial Ω$. We show that if $f \in \text{BMO}(\partial Ω,dσ)$ with compact support on $\partial Ω$, then there exists a smooth function $V$ in $Ω$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson norm controlled by the BMO norm of $f$, and such that $V$ converges in some non-tangential sense to $f$ almost everywhere with respect to $σ$. Our results should be compared to recent geometric characterizations of $L^p$-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all $f \in C_c(\partial Ω)$, a harmonic extension $u$, with $|\nabla u(Y)|^2 \text{dist}(Y,\partial Ω) \, dY $ a Carleson measure controlled by the BMO norm of $f$, only in the presence of an appropriate quantitative connectivity condition.

math.AP↗
arXiv2020

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part I: The small constant case

The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. This paper addresses the free boundary problem under the assumption of smallness of the Carleson measure of the coefficients. Part II of this work develops an extrapolation argument to bootstrap this result to the general case. The ideas in Part I constitute a novel application of techniques developed in geometric measure theory. They highlight the synergy between several areas. The ideas developed in this paper are well suited to study singularities arising in variational problems in a geometric setting.

math.AP↗
arXiv2020

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case

The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. The first step in this direction was taken in our previous paper [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant. In this paper we establish the final, general result, that is, the "large constant case". The key elements of our approach are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, as well as a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.

math.AP↗
arXiv2019

Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-{\rm div}(A\nabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix, imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_\infty$ class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator $L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition then $ω_L\in A_\infty$ if and only if $ω_{L^\top}\in A_\infty$. As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with $L$ to the class $A_\infty$ yields that the domain is indeed a chord-arc domain.

math.CA↗
arXiv2019

Uniform rectifiability and $\varepsilon$-approximability of harmonic functions in $L^p$

Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(Ω)$ for every $p \in (1,\infty)$, where $Ω:= \mathbb{R}^{n+1} \setminus E$. Together with results of many authors this shows that pointwise, $L^\infty$ and $L^p$ type $\varepsilon$-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension $1$ Ahlfors-David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda.

math.CA↗
arXiv2019

BMO Solvability and Absolute Continuity of Caloric Measure

We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-$A_\infty$ property) of caloric measure with respect to surface measure, for an open set $Ω\subset \mathbb{R}^{n+1}$, assuming as a background hypothesis only that the essential boundary of $Ω$ satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness. Since the weak-$A_\infty$ property of the caloric measure is equivalent to $L^p$ solvability of the initial-Dirichlet problem, we may then deduce that $BMO$-solvability implies $L^p$ solvability for some finite $p$.

math.AP↗
arXiv2018

Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{div}(A \nabla u) = 0$ in the upper half-space $(x,t)\in \mathbb{R}^{n+1}_+$ is solvable when $n\geq2$ and the boundary data is in $L^p_μ(\mathbb{R}^n)$ for some $p<\infty$. The coefficient matrix $A$ is only assumed to be measurable, real-valued and $t$-independent with a degenerate bound and ellipticity controlled by an $A_2$-weight $μ$. It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is in $A_\infty$ with respect to the $μ$-weighted Lebesgue measure on $\mathbb{R}^n$. The Carleson measure estimate allows us to avoid applying the method of $ε$-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.

math.AP↗
arXiv2018

Rectifiability, interior approximation and Harmonic Measure

We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $H^n|_E$ is "absolutely continuous" with respect to harmonic measure in the sense that any Borel subset of $E$ with strictly positive $H^n$ measure has strictly positive harmonic measure in some connected component of $\mathbb{R}^{n+1}\setminus E$. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set $E$ as above is the boundary of a connected domain $Ω\subset \mathbb{R}^{n+1}$ which satisfies an infinitesimal interior thickness condition, then $H^n|_{\partialΩ}$ is absolutely continuous (in the usual sense) with respect to harmonic measure for $Ω$. Local versions of these results are also proved: if just some piece of the boundary is $n$-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results by Azzam-Hofmann-Martell-Mayboroda-Mourgoglou-Tolsa-Volberg, we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is $n$-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely $n$-unrectifiable piece having vanishing harmonic measure.

math.CA↗
arXiv2018

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem. Part I

Let $Ω\subset \mathbb{R}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partialΩ$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc subdomains of $Ω$, and hence that harmonic measure for $Ω$ is weak-$A_\infty$ with respect to surface measure on $\partialΩ$, provided that $Ω$ satisfies a certain weak version of a local John condition. Under the further assumption that $Ω$ satisfies an interior Corkscrew condition, and combined with our previous work, and with recent work of Azzam, Mourgoglou and Tolsa, this yields a geometric characterization of domains whose harmonic measure is quantitatively absolutely continuous with respect to surface measure and hence a haracterization of the fact that the associated $L^p$-Dirichlet problem is solvable for some finite $p$.

math.CA↗
arXiv2017

Kato square root problem with unbounded leading coefficients

We prove the Kato conjecture for elliptic operators, $L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$, with $\mathbf A$ a complex measurable bounded coercive matrix and $\mathbf D$ a measurable real-valued skew-symmetric matrix in $\mathbb{R}^n$ with entries in $BMO(\mathbb{R}^n)$;\, i.e., the domain of $\sqrt{L}\,$ is the Sobolev space $\dot H^1(\mathbb{R}^n)$ in any dimension, with the estimate $\|\sqrt{L}\, f\|_2\lesssim \| \nabla f\|_2$.

math.AP↗
arXiv2017

Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

Let $Ω\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $\partialΩ$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators and let $ω_{L_0}$, $ω_L$ be the associated elliptic measures. We show that if $ω_{L_0}\in A_\infty(σ)$, where $σ=H^n|_{\partialΩ}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $ω_L\in A_\infty(σ)$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse Hölder classes, that is, if for some $1<p<\infty$, one has $ω_{L_0}\in RH_p(σ)$ then $ω_{L}\in RH_p(σ)$. Equivalently, if the Dirichlet problem with data in $L^{p'}(σ)$ is solvable for $L_0$ then so it is for $L$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $A_\infty(σ)$ then necessarily $Ω$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable.

math.CA↗