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Xavier Tolsa

Publications and source records attributed to Xavier Tolsa.

At least 55 records · Page 3Linked to original sources
arXiv2018

On $C^1$-approximability of functions by solutions of second order elliptic equations on plane compact sets and $C$-analytic capacity

Criteria for approximability of functions by solutions of homogeneous second order elliptic equations (with constant complex coefficients) in the norms of the Whitney $C^1$-spaces on compact sets in $\mathbb R^2$ are obtained in terms of the respective $C^1$-capacities. It is proved that the mentioned $C^1$-capacities are comparable to the classic $C$-analytic capacity, and so have a proper geometric measure characterization.

math.CA↗
arXiv2018

A family of singular integral operators which control the Cauchy transform

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\frac{(\Re z)^{3}}{|z|^{4}}+t\cdot \frac{\Re z}{|z|^{2}}, \quad t\in \mathbb{R},\qquad k_\infty(z):=\frac{\Re z}{|z|^{2}}\equiv \Re \frac{1}{z},\quad z\in \mathbb{C}\setminus\{0\}. $$ It is shown that for any positive locally finite Borel measure with linear growth the corresponding $L^2$-norm of $T_{k_0}$ controls the $L^2$-norm of $T_{k_\infty}$ and thus of the Cauchy transform. As a corollary, we prove that the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of $T_{k_t}$ with a fixed $t\in (-t_0,0)$, where $t_0>0$ is an absolute constant, implies that $E$ is rectifiable. This is so in spite of the fact that the usual curvature method fails to be applicable in this case. Moreover, as a corollary of our techniques, we provide an alternative and simpler proof of the bi-Lipschitz invariance of the $L^2$-boundedness of the Cauchy transform, which is the key ingredient for the bi-Lipschitz invariance of analytic capacity.

math.CA↗
arXiv2018

Characterization of rectifiable measures in terms of $α$-numbers

We characterize Radon measures $μ$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $μ$-measure zero by countably many $d$-dimensional Lipschitz graphs and $μ\ll \mathcal{H}^{d}$. The characterization is in terms of a Jones function involving the so-called $α$-numbers. This answers a question left open in a former work by Azzam, David, and Toro.

math.CA↗
arXiv2018

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

Let $Ω\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $Ω$ satisfies the so-called weak-$A_\infty$ condition, then $Ω$ satisfies a suitable connectivity condition, namely the weak local John condition. Together with other previous results by Hofmann and Martell, this implies that the weak-$A_\infty$ condition for harmonic measure holds if and only if $\partialΩ$ is uniformly $n$-rectifiable and the weak local John condition is satisfied. This yields the first geometric characterization of the weak-$A_\infty$ condition for harmonic measure, which is important because of its connection with the Dirichlet problem for the Laplace equation.

math.AP↗
arXiv2018

Failure of $L^2$ boundedness of gradients of single layer potentials for measures with zero low density

Consider a totally irregular measure $μ$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{μ(B(x,r))}{(2r)^n}$ is positive $μ$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density $\liminf_{r\to0}\frac{μ(B(x,r))}{(2r)^n}$ vanishes $μ$-a.e. in $\mathbb{R}^{n+1}$. We show that if $T_μf(x)=\int K(x,y)\,dμ(y)$ is an operator whose kernel $K(\cdot,\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Hölder continuous coefficients, then $T_μ$ is not bounded in $L^2(μ)$. This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the $n$-dimensional Riesz transform.

math.CA↗
arXiv2018

Rectifiability of measures and the $β_p$ coefficients

In some former works of Azzam and Tolsa it was shown that $n$-rectifiability can be characterized in terms of a square function involving the David-Semmes $β_2$ coefficients. In the present paper we construct some counterexamples which show that a similar characterization does not hold for the $β_p$ coefficients with $p\neq2$. This is in strong contrast with what happens in the case of uniform $n$-rectifiability. In the second part of this paper we provide an alternative argument for a recent result of Edelen, Naber and Valtorta about the $n$-rectifiability of measures with bounded lower $n$-dimensional density. Our alternative proof follows from a slight variant of the corona decomposition in one of the aforementioned works of Azzam and Tolsa and a suitable approximation argument.

math.CA↗
arXiv2017

Characterization of $n$-rectifiability in terms of Jones' square function: Part I

In this paper it is shown that if $μ$ is a finite Radon measure in $\mathbb R^d$ which is $n$-rectifiable and $1\leq p\leq 2$, then $$\int_0^\infty β_{μ,p}^n(x,r)^2\,\frac{dr}r<\infty \quad {for $μ$-a.e. $x\in\mathbb R^d$,}$$ where $$β_{μ,p}^n(x,r) = \inf_L (\frac1{r^n} \int_{\bar B(x,r)} (\frac{\mathrm dist(y,L)}{r})^p\,dμ(y))^{1/p},$$ with the infimum taken over all the $n$-planes $L\subset \mathbb R^d$. The $β_{μ,p}^n$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform $n$-rectifiability. An analogous necessary condition for $n$-rectifiability in terms of other coefficients involving some variant of the Wasserstein distance $W_1$ is also proved.

math.CA↗
arXiv2017

The Riesz transform and quantitative rectifiability for general Radon measures

In this paper we show that if $μ$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_μ$ is bounded in $L^2(μ)$, and $B\subset\mathbb R^{n+1}$ is a ball with $μ(B)\approx r(B)^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $δ>0$ small enough, it holds $\int_B \frac{dist(x,L)}{r(B)}\,dμ(x)\leq δ\,μ(B),$ (b) for some constant $ε>0$ small enough, $\int_B |R_\mu1(x) - m_{μ,B}(R_\mu1)|^2\,dμ(x) \leq ε\,μ(B)$, where $m_{μ,B}(R_\mu1)$ stands for the mean of $R_\mu1$ on $B$ with respect to $μ$; then there exists a uniformly $n$-rectifiable subset $Γ$, with $μ(Γ\cap B)\gtrsim μ(B)$, and so that $μ|_Γ$ is absolutely continuous with respect to $H^n|_Γ$. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in a subsequent paper by Azzam, Mourgoglou and Tolsa.

math.CA↗
arXiv2017

Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula

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 solution of $L^*v=0$ in $Ω$ is $\varepsilon$-approximmable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $Ω$ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called $S<N$ estimates, and another in terms of a suitable corona decomposition involving $L$-harmonic and $L^*$-harmonic measures. We also prove that if $L$-harmonic measure and $L^*$-harmonic measure satisfy a weak $A_\infty$-type condition, then $\partial Ω$ is $n$-uniformly rectifiable. In the process we obtain a version of Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.

math.CA↗
arXiv2016

The measures with an associated square function operator bounded in $L^2$

In this paper we provide an extension of a theorem of David and Semmes ('91) to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators, or singular integral operators, are bounded in $L^2(μ)$. The description is given in terms of a modification of Jones' $β$-coefficients.

math.CA↗
arXiv2016

On Tsirelson's theorem about triple points for harmonic measure

A theorem of Tsirelson from 1997 asserts that given three disjoint domains in $\mathbb R^{n+1}$, the set of triple points belonging to the intersection of the three boundaries where the three corresponding harmonic measures are mutually absolutely continuous has null harmonic measure. The original proof by Tsirelson is based on the fine analysis of filtrations for Brownian and Walsh-Brownian motions and can not be translated into potential theory arguments. In the present paper we give a purely analytical proof of the same result.

math.CA↗
arXiv2016

Uniform rectifiability from Carleson measure estimates and $\varepsilon$-approximability of bounded harmonic functions

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 also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "$S<N$" estimates, and another in terms of a suitable corona decomposition involving harmonic measure.

math.CA↗
arXiv2016

On a two-phase problem for harmonic measure in general domains

We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition.

math.CA↗
arXiv2016

Harmonic measure and Riesz transform in uniform and general domains

Let $Ω\subsetneq\mathbb R^{n+1}$ be open and let $μ$ be some measure supported on $\partialΩ$ such that $μ(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the harmonic measure in $Ω$ satisfies some scale invariant $A_\infty$ type conditions with respect to $μ$, then the $n$-dimensional Riesz transform $$R_μf(x) = \int \frac{x-y}{|x-y|^{n+1}}\,f(y)\,dμ(y)$$ is bounded in $L^2(μ)$. We do not assume any doubling condition on $μ$. We also consider the particular case when $Ω$ is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.

math.CA↗
arXiv2016

Singular integrals unsuitable for the curvature method whose $L^2$-boundedness still implies rectifiability

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the $L^2(μ)$-boundedness of certain singular integral operators to the geometric properties of the support of measure $μ$, e.g. rectifiability. It can be applied however only if Menger curvature-like permutations, directly associated with the kernel of the operator, are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the $L^2(μ)$-boundedness of the operator still implies that the support of $μ$ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors-David regularity conditions.

math.CA↗
arXiv2016

Singular sets for harmonic measure on locally flat domains with locally finite surface measure

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $Ω\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partialΩ)<\infty$ and a subset $E\subset \partial Ω$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimensions for this type of domains.

math.CA↗