arXiv subjects
Publications and source records attributed to Xavier Tolsa.
These are the lecture notes of a seminar held at the Universitat Autònoma de Barcelona where the Jones-Wolff theorem about the dimension of harmonic measure in the plane is explained in full detail, for non-expert readers.
Let $Ω\subset \mathbb{R}^{n+1}$, $n\geq 2$, be a bounded open and connected set satisfying the corkscrew condition with uniformly $n$-rectifiable boundary. In this paper we study the connection between the solvability of $(D_{p'})$, the Dirichlet problem for the Laplacian with boundary data in $L^{p'}(\partial Ω)$, and $(R_{p})$ (resp. $(\tilde R_{p})$), the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space $W^{1,p}(\partial Ω)$ (resp. $\tilde W^{1,p}(\partial Ω)$, the usual Sobolev space in terms of the tangential derivative), where $p \in (1,2+\varepsilon)$ and $1/p+1/p'=1$. Our main result shows that $(D_{p'})$ is solvable if and only if so is $(R_{p})$. Under additional geometric assumptions (two-sided local John condition or weak Poincaré inequality on the boundary), we prove that $(D_{p'}) \Rightarrow (\tilde R_{p})$. In particular, we deduce that in bounded chord-arc domains (resp. two-sided chord-arc domains) there exists $p_0 \in (1,2+\varepsilon)$ so that $(R_{p_0})$ (resp. $(\tilde R_{p_0})$) is solvable. We also extend the results to unbounded domains with compact boundary and show that in two-sided corkscrew domains with $n$-Ahlfors-David regular boundaries the single layer potential operator is invertible from $L^p(\partial Ω)$ to the inhomogeneous Sobolev space $ W^{1,p}(\partial Ω)$. Finally, we provide a counterexample of a chord-arc domain $Ω_0 \subset \mathbb{R}^{n+1}$, $n \geq 3$, so that $(\tilde R_p)$ is not solvable for any $p \in [1, \infty)$.
Inspired by recent work of Mourgoglou and the second named author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincaré inequalities in open sets $Ω\subset \mathbb{R}^{n+1}$, with codimension $1$ Ahlfors--David regular boundaries. First, we prove that if $Ω$ satisfies both the local John condition and the exterior corkscrew condition, then $Ω$ also satisfies the Harnack chain condition (and hence, is a chord-arc domain). Second, we show that if $Ω$ is a $2$-sided chord-arc domain, then the boundary $\partial Ω$ supports a Heinonen--Koskela type weak $1$-Poincaré inequality. We also construct an example of a set $Ω\subset \mathbb{R}^{n+1}$ such that the boundary $\partial Ω$ is Ahlfors--David regular and supports a weak boundary $1$-Poincaré inequality but $Ω$ is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincaré theories.
In this paper it is shown that if $E\subset\mathbb R^{n+1}$ is an $s$-AD regular compact set, with $s\in [n-\frac12,n)$, and $E$ is contained in a hyperplane or, more generally, in an $n$-dimensional $C^1$ manifold, then the Hausdorff dimension of the harmonic measure for the domain $\mathbb R^{n+1}\setminus E$ is strictly smaller than $s$, i.e., than the Hausdorff dimension of $E$.
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.
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 $$ 1+ \|T_μ\|_{L^2(μ)\to L^2(μ)}\approx 1+ \|\mathcal R_μ\|_{L^2(μ)\to L^2(μ)},$$ where $\mathcal R_μ$ indicates the $n$-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $\mathcal R_μ$, which were recently extended to $T_μ$ associated with $L_A$ with Hölder continuous coefficients. In particular, we show the following: 1) If $μ$ is an $n$-Ahlfors-David-regular measure on $\mathbb{R}^{n+1}$ with compact support, then $T_μ$ is bounded on $L^2(μ)$ if and only if $μ$ is uniformly $n$-rectifiable. 2) Let $E\subset \mathbb{R}^{n+1}$ be compact and $\mathcal H^n(E)<\infty$. If $T_{\mathcal H^n|_E}$ is bounded on $L^2(\mathcal H^n|_E)$, then $E$ is $n$-rectifiable. 3) If $μ\not\equiv 0$ satisfies $\limsup_{r\to 0}\tfrac{μ(B(x,r))}{(2r)^n}$ {is positive and finite} for $μ$-a.e. $x\in \mathbb{R}^{n+1}$ and $\liminf_{r\to 0}\tfrac{μ(B(x,r))}{(2r)^n}$ vanishes for $μ$-a.e. $x\in \mathbb{R}^{n+1}$, then $T_μ$ is not bounded on $L^2(μ)$. 4) If $μ$ is a compactly supported Radon measure satisfying a certain set of local conditions at the level of a ball $B$ with small radius, then a significant portion of $μ|_B$ can be covered by a UR set.
This work provides a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{dist(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset \mathbb R^{n+1}$. As a corollary, one obtains a characterization of the removable sets for Lipschitz harmonic functions in terms of a metric-geometric potential and one deduces that the class of removable sets for Lipschitz harmonic functions is invariant by bilipschitz mappings.
In this work we obtain a geometric characterization of the measures $μ$ in $\mathbb{R}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $\mathcal{R}μ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$, under the assumption that $μ$ satisfies the following Wolff energy estimate, for any ball $B\subset\mathbb{R}^{n+1}$: $$\int_B \int_0^\infty \left(\frac{μ(B(x,r))}{r^{n-\frac38}}\right)^2\,\frac{dr}r\,dμ(x)\leq M\,\bigg(\frac{μ(2B)}{r(B)^{n-\frac38}}\bigg)^2\,μ(2B).$$ More precisely, we show that $μ$ satisfies the following estimate: $$\|\mathcal{R}μ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{μ,2}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{\mathrm{dist}(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset\mathbb{R}^{n+1}$. In a companion paper which relies on the results obtained in this work it is shown that the same result holds without the above assumption regarding the Wolff energy of $μ$. This result has important consequences for the Painlevé problem for Lipschitz harmonic functions.
Let $Ω\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $Ω$ and continuous in $\overline Ω$ which vanishes in a relatively open subset $Σ\subset\partialΩ$ and moreover the normal derivative $\partial_νu$ vanishes in a subset of $Σ$ with positive surface measure, then $u$ is identically $0$.
In this paper we study removable singularities for regular $(1,1/2)$-Lipschitz solutions of the heat equation in time varying domains. We introduce an associated Lipschitz caloric capacity and we study its metric and geometric properties and the connection with the $L^2$ boundedness of the singular integral whose kernel is given by the gradient of the fundamental solution of the heat equation.
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.
Let $Ω^+\subset\mathbb R^{n+1}$ be an NTA domain and let $Ω^-= \mathbb R^{n+1}\setminus \overline{Ω^+}$ be an NTA domain as well. Denote by $ω^+$ and $ω^-$ their respective harmonic measures. Assume that $Ω^+$ is a $δ$-Reifenberg flat domain for some $δ>0$ small enough. In this paper we show that $\log\frac{dω^-}{dω^+}\in VMO(ω^+)$ if and only if $Ω^+$ is vanishing Reifenberg flat, $Ω^+$ and $Ω^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $Ω^+$ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.
We assume that $Ω_1, Ω_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive harmonic measure. Then we show that in a fixed ball $B$ centered on $F$, if the harmonic measure of $Ω_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the harmonic measure of $Ω_2$ in $B$, then there exists a uniformly $n$-rectifiable set $Σ$ so that the harmonic measure of $Σ\cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable feature of this result is that the harmonic measures do not need to satisfy any doubling condition. In the particular case that $Ω_1$ and $Ω_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_\infty$ condition between the respective harmonic harmonic measures of $Ω_1$ and $Ω_2$.
In this paper the jump formulas for the double layer potential and other singular integrals are proved for arbitrary rectifiable sets, by defining suitable non-tangential limits. The arguments are quite straightforward and only require some Calderón-Zygmund techniques.
In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $\varepsilon^2$-square function.
Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with Hölder continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let $μ$ be a compactly supported $n$-AD-regular measure in $\mathbb R^{n+1}$ and consider the associated operator $$T_μf(x) = \int \nabla_x\mathcal E_A(x,y)\,f(y)\,dμ(y).$$ We show that if $T_μ$ is bounded in $L^2(μ)$, then $μ$ is uniformly $n$-rectifiable. This extends the solution of the codimension $1$ David-Semmes problem for the Riesz transform to the gradient of the single layer potential. Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given $E\subset\mathbb R^{n+1}$ with finite Hausdorff measure $\mathcal H^n$, if $T_{\mathcal H^n|_E}$ is bounded in $L^2(\mathcal H^n|_E)$, then $E$ is $n$-rectifiable. Further, as an application we show that if the elliptic measure associated to the above PDE is absolute continuous with respect to surface measure, then it must be rectifiable, analogously to what happens with harmonic measure.
Let $Ω\subset\mathbb R^{n+1}$ be open and let $E\subset \partialΩ$ with $0<H^s(E)<\infty$, for some $s\in(n,n+1)$, satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with $H^s$ on $E$. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that $Ω$ is a uniform domain.
In this paper we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if $E\subset \mathbb C$ is compact and $μ$ is a Borel measure supported on $E$, then the analytic capacity of $E$ satisfies $$ γ(E) \geq c\,\frac{μ(E)^2}{\int_I \|P_θμ\|_2^2\,dθ}, $$ where $c$ is some positive constant, $I\subset [0,π)$ is an arbitrary interval, and $P_θμ$ is the image measure of $μ$ by $P_θ$, the orthogonal projection onto the line $\{re^{iθ}:r\in\mathbb R\}$. This result is related to an old conjecture of Vitushkin about the relationship between the Favard length and analytic capacity. We also prove a generalization of the above inequality to higher dimensions which involves related capacities associated with signed Riesz kernels.