arXiv subjects
Publications and source records attributed to Xavier Tolsa.
We show that if $Ω\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(σ)$, where $σ$ is the surface measure of $Ω$, then the outer unit normal to $\partialΩ$ belongs to $\textrm{VMO}(σ)$ too. The analogous result fails for dimensions larger than $3$. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.
Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $μ$ in $\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(μ)$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Δ)^{α/2}$, $α\in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
We prove in the context of local $Tb$ theorems with $L^p$ type testing conditions an improved version of Cotlar's inequality. This is related to the problem of removing the so called buffer assumption of Hytönen-Nazarov, which is the final barrier for the full solution of S. Hofmann's problem. We also investigate the problem of extending the Hytönen-Nazarov result to non-homogeneous measures. We work not just with the Lebesgue measure but with measures $μ$ in $\mathbb{R}^d$ satisfying $μ(B(x,r)) \le Cr^n$, $n \in (0, d]$. The range of exponents in the Cotlar type inequality depend on $n$. Without assuming buffer we get the full range of exponents $p,q \in (1,2]$ for measures with $n \le 1$, and in general we get $p, q \in [2-ε(n), 2]$, $ε(n) > 0$. Consequences for (non-homogeneous) local $Tb$ theorems are discussed.
In the present paper we prove that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.
In the present paper we sketch the proof of the fact that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $0<\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable.
In the present paper we prove that for any open connected set $Ω\subset{\mathbb R}^{n+1}$, $n\geq 1$, and any $E\subset \partialΩ$ with $0<{\mathcal H}^n(E)<\infty$ absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable.
We show that if $n\geq 1$, $Ω\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partialΩ$ is a set of finite and positive Hausdorff $H^{n}$-measure upon which the harmonic measure $ω$ is absolutely continuous with respect to $H^{n}$, then $ω|_E$ is concentrated on an $n$-rectifiable set.
In this paper it is shown that for anymeasure $μ$ in $\mathbb{R}^d$ and for a non-integer $0<s<d$, the Wolff energy $\displaystyle{\iint_0^\infty(\frac{μ(B(x,r))}{r^s})^2\,\frac{dr}{r}dμ(x)}$ is comparable to $$\iint_0^\infty(\frac{μ(B(x,r))}{r^s} - \frac{μ(B(x,2r))}{(2r)^s})^2\,\frac{dr}rdμ(x),$$ unlike in the case when $s$ is an integer. We also study the relation with the $L^2-$norm of $s$-Riesz transforms, $0<s<1$, and we provide a counterexample in the integer case.
A measure $μ$ on $\mathbb{R}^d$ is called reflectionless for the $s$-Riesz transform if the singular integral $R^sμ(x)=\int \frac{y-x}{|y-x|^{s+1}}\,dμ(y)$ is constant on the support of $μ$ in some weak sense and, moreover, the operator defined by $R^s_μ(f)=R^s(f\,μ)$ is bounded in $L^2(μ)$. In this paper we show that the only reflectionless measure for the $s$-Riesz transform is the zero measure when $0<s<1$.
This paper is devoted to the proof of two related results. The first one asserts that if $μ$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{μ(B(x,r))}{r}>0\quad \text{ and }\quad \int_0^1\left|\frac{μ(B(x,r))}{r} - \frac{μ(B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty$$ for $μ$-a.e. $x\in\mathbb R^d$, then $μ$ is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set $E\subset\mathbb R^d$ with finite $1$-dimensional Hausdorff measure $H^1$ is rectifiable if and only $$\int_0^1\left|\frac{H^1(E\cap B(x,r))}{r} - \frac{H^1(E\cap B(x,2r))}{2r}\right|^2\,\frac{dr}r< \infty \quad\mbox{ for $H^1$-a.e. $x\in E$.}$$ The second result of the paper deals with the relationship between a similar square function in the complex plane and the Cauchy transform $C_μf(z) = \int \frac1{z-ξ}\,f(ξ)\,dμ(ξ)$. Suppose that $μ$ has linear growth, that is, $μ(B(z,r))\leq c\,r$ for all $z\in\mathbb C$ and all $r>0$. It is proved that $C_μ$ is bounded in $L^2(μ)$ if and only if $$ \int_{z\in Q}\int_0^\infty\left|\frac{μ(Q\cap B(z,r))}{r} - \frac{μ(Q\cap B(z,2r))}{2r}\right|^2\,\frac{dr}r\,dμ(z)\leq c\,μ(Q) \quad\mbox{ for every square $Q\subset\mathbb C$.} $$
Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given $0 2$ and a Lipschitz domain $Ω\subset \mathbb{C}$, the Beurling transform $Bf=- {\rm p.v.}\frac1{πz^2}*f$ is bounded in the Sobolev space $W^{s,p}(Ω)$ if and only if $Bχ_Ω\in W^{s,p}(Ω)$. In this paper we obtain a generalized version of the former result valid for any $s\in \mathbb{N}$ and for a larger family of Calderón-Zygmund operators in any ambient space $\mathbb{R}^d$ as long as $p>d$. In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for $p\leq d$. In the particular case $s=1$, this condition is in fact necessary, which yields a complete characterization.
We show that a Radon measure $μ$ in $\mathbb R^d$ which is absolutely continuous with respect to the $n$-dimensional Hausdorff measure $H^n$ is $n$-rectifiable if the so called Jones' square function is finite $μ$-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all $n$-rectifiable measures which are absolutely continuous with respect to $H^{n}$. Further, in this paper we also investigate the relationship between the Jones' square function and the so called Menger curvature of a measure with linear growth.
In this paper it is shown that if $μ$ is an n-dimensional Ahlfors-David regular measure in $R^d$ which satisfies the so-called weak constant density condition, then $μ$ is uniformly rectifiable. This had already been proved by David and Semmes in the cases n=1, 2 and d-1, and it was an open problem for other values of n. The proof of this result relies on the study of the n-uniform measures in $R^d$. In particular, it is shown here that they satisfy the "big pieces of Lipschitz graphs" property.
In this paper we obtain precise estimates for the $L^2$ norm of the $s$-dimensional Riesz transforms on very general measures supported on Cantor sets in $\mathbb R^d$, with $d-1<s<d$. From these estimates we infer that, for the so called uniformly disconnected compact sets, the capacity $γ_s$ associated with the Riesz kernel $x/|x|^{s+1}$ is comparable to the capacity $\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}}$ from non-linear potential theory.
Let $E$ be a set in $\mathbb R^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that $\liminf_{r\to0}r^{-n} H^n(B(x,r)\cap E)>0$ for $H^n$-a.e. $x\in E$. In this paper it is shown that $E$ is $n$-rectifiable if and only if $$\int_0^1 \left|\frac{H^n(B(x,r)\cap E)}{r^n} - \frac{H^n(B(x,2r)\cap E)}{(2r)^n}\right|^2\,\frac{dr}r < \infty$$ for $H^n$-a.e. $x\in E$; and also if and only if $$ \lim_{r\to0}\left(\frac{H^n(B(x,r)\cap E)}{r^n} - \frac{H^n(B(x,2r)\cap E)}{(2r)^n}\right) = 0$$ for $H^n$-a.e. $x\in E$. Other more general results involving Radon measures are also proved.
In this paper it is shown that an Ahlfors-David $n$-dimensional measure $μ$ on $\mathbb{R}^d$ is uniformly $n$-rectifiable if and only if for any ball $B(x_0,R)$ centered at $\operatorname{supp}(μ)$, $$ \int_0^R \int_{x\in B(x_0,R)} \left|\frac{μ(B(x,r))}{r^n} - \frac{μ(B(x,2r))}{(2r)^n} \right|^2\,dμ(x)\,\frac{dr}r \leq c\, R^n.$$ Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.
We show that, given a set $E\subset \mathbb R^{n+1}$ with finite $n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in $L^2(H^n|E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\subset\mathbb R^{n+1}$ with $H^n(E)<\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions.
We show that, for some Cantor sets in R^d, the capacity g_s associated to the s-dimensional Riesz kernel x/|x|^{s+1} is comparable to the capacity C_{2(d-s)/3,3/2} from non linear potential theory. It is an open problem to show that, when s is positive and non integer, they are comparable for all compact sets in R^d. We also discuss other open questions in the area.