We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation results for $L^2$ square function estimates related to the analysis of integral operators that act on Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The inductive scheme is a natural application of the local $T(b)$ theorem and it implies the stability of $L^2$ square function estimates under the so-called big pieces functor. In particular, this analysis implies $L^p$ and Hardy space square function estimates for integral operators on uniformly rectifiable subsets of the Euclidean space.
It was recently shown that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a domain with an $n-1$ dimensional uniformly rectifiable boundary, in the presence of now well understood additional topological constraints. The topological restrictions, while mild, are necessary, as the counterexamples of C. Bishop and P. Jones show, and no analogues of these results have been available for higher co-dimensional sets. In the present paper we show that for any $d<n-1$ and for any domain with a $d$-dimensional uniformly rectifiable boundary the elliptic measure of…
We establish square function estimates for integral operators on uniformly rectifiable sets by proving a local $T(b)$ theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, we consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local $T(b)$ theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on t…
We consider the minimization problem of an anisotropic energy in classes of $d$-rectifiable varifolds in $\mathbb R^n$, closed under Lipschitz deformations and encoding a suitable notion of boundary. We prove that any minimizing sequence with density uniformly bounded from below converges (up to subsequences) to a $d$-rectifiable varifold. Moreover, the limiting varifold is integral, provided the minimizing sequence is made of integral varifolds with uniformly locally bounded anisotropic first variation.
In the present paper, we consider elliptic operators $L=-\textrm{div}(A\nabla)$ in a domain bounded by a chord-arc surface $Γ$ with small enough constant, and whose coefficients $A$ satisfy a weak form of the Dahlberg-Kenig-Pipher condition of approximation by constant coefficient matrices, with a small enough Carleson norm, and show that the elliptic measure with pole at infinity associated to $L$ is $A_\infty$-absolutely continuous with respect to the surface measure on $Γ$, with a small $A_\infty$ constant. In other words, we show that for relatively flat uniformly rectifiable sets and for …
It has been recently established by the first and third author that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the "flagship" degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators $L_{β,γ} =- {\rm div} D^{d+1+γ-n} \nabla$ associated to a domain $Ω\subset \mathbb R^n$ with a uniformly rectifiable boundary $Γ$ of dimensi…
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.
The present paper establishes equivalence between uniform rectifiability of the boundary of a domain and the property that the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). The results are novel in a variety of ways, in particular (1) this is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; the "direct" result established here is new even in the half space; (2) the results are …
We consider a uniformly rectifiable set $Γ\subset \mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $Ω= \mathbb R^n \setminus Γ$, Guy David, Svitlana Mayboroda, and the author introduced a notion of harmonic measure on $Γ$. We prove in the present article that this harmonic measure on $Γ$ satisfies the $A^\infty$-property, that is the harmonic measure and the $d$-dimension Hausdorff measure on $Γ$ are mutually absolutely continuous in a quantitative and scale invariant way. Thus, we give an alternate proof of a recent theorem of David and Mayboroda, wh…
These series of notes serve as an introduction to some of both the classical and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is about studying general sets or measures which can be, in one sense or another, approximated on all scales by well behaved spaces, typically just Euclidean space itself. Such sets and measures turn out not to be arbitrary, and often times come with special structure inherited from what they are being approximated by. We will begin by recalling and proving the standard Reifenberg theorem, which says that sets in Euclidean space which are well…
In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function $G$ behaves like a distance function to the boundary, in the sense that $\Big|\frac{\nabla G(X)}{G(X)}-\frac{\nabla D(X)}{D(X)}\Big|^2D(X) dX$ is the density of a Carleson measure, where $D$ is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's $α$-number of uniformly recti…
In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is $k^{th}$-stratum of the singular set of $f$, then it is well known that $\dim S^k\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable. In the case of minimizing harmonic maps we go further, and prove that the singular set $S(f)$…
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $Ω:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates, and are "$\varepsilon$-approximable". Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure, and surface measure.
The aim of this article is to study effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from $\mathbb{R}^n$ continue to hold with no additional restrictions. For a general Banach spaces we will see that the classical Reifenberg theorem holds, and that a weak version of the effective Reifenberg theorem holds in that if one assumes a summability estimate $\int_0^2 β^k(x,r)^1 \frac{dr}{r}<M$ {\it without power gain}, then $μ$ must again be rectifiable with measure estimates. Improving this estimate in order to obtain a power gain tur…
Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partialΩ$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded…
In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance $W_2$ which asserts that if $μ$ and $ν$ are probability measures in $R^n$, $ϕ$ is a radial bump function smooth enough so that $\intϕdμ\gtrsim1$, and $μ$ has a density bounded from above and from below on the support of ϕ, then $W_2(ϕμ,aϕν)\leq c W_2(μ,ν),$ where $a=\intϕdμ/ \intϕ\,dν$.
We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.
We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix $A$ with real, merely bounded, and possibly non-symmetric coefficients. If $$ω_A(r)=\sup_{x\in \mathbb{R}^{n+1}} \frac{1}{|B(x,r)|}\int_{B(x,r)}\Big|A(z)-\frac{1}{|B(x,r)|}\int_{B(x,r)}A\Big|\,dz,$$ then, under suitable Dini-type assumptions on $ω_A$, we prove the following: if $μ$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$, and $T_μf(x)=\int \nabla_xΓ_A (x,y)f(y)\, dμ(y)$ denotes the gradient of the single layer potential associated with $L_A$, then $$ …
Let $Ω\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partialΩ$ is uniformly $n$-rectifiable if every bounded harmonic function on $Ω$ is $\varepsilon$-approximable or if every bounded harmonic function on $Ω$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $Ω=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we …
We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_\infty$ operator norm of the semigroup. As an application we prove a new $L_\infty$-bound for the torsion function of a Euclidean domain that is close to optimal.
