# Search results for: Xavier Tolsa

Calculus of Variations and Partial Differential Equations > 2019 > 58 > 1 > 1-11

Calculus of Variations and Partial Differential Equations > 2018 > 57 > 1 > 1-63

*n*, such that the

*n*-dimensional Riesz transform $${{\mathcal {R}}}_\mu $$ Rμ is bounded in $$L^2(\mu )$$ L2(μ) , and $$B\subset {{\mathbb {R}}}^{n+1}$$ B⊂Rn+1 is a ball with $$\mu (B)\approx r(B)^n$$ μ(B)≈r(B)n such that:(a)there is some

*n*-plane

*L*passing through the center of

*B*such...

Communications on Pure and Applied Mathematics > 70 > 11 > 2121 - 2163

*G*⊂ ℂ which admit the so-called estimate of the Cauchy integral, that is to say, $$\left| {\smallint _{\partial G} {\text{f}}\left( z \right)dz} \right| \leqslant C\left( G \right)\left\| {\text{f}} \right\|\infty \gamma \left( E \right)$$ for all

*E*⊂

*G*and

*f*∊

*H*

^{∞}(

*G*

*E*), where γ\(

*E*) is the analytic capacity of

*E*. The corresponding...

Comptes Rendus Mathematique > 2016 > 354 > 4 > 351-355

^{ d }is defined by the singular integral $$ \int {\frac{{x - y}} {{\left| {x - y} \right|^{n + 1} }}d\mu (y),} x \in \mathbb{R}^d . $$ Let

*E*⊂ ℝ

^{ d }with

*H*

^{ n }(

*E*) > ∞, where

*H*

^{ n }stands for the

*n*-dimensional Hausdorff measure. In...

Geometric and Functional Analysis > 2016 > 26 > 3 > 703-728

Calculus of Variations and Partial Differential Equations > 2015 > 54 > 4 > 3643-3665

*n*-rectifiable and $$1\le p\le 2$$ 1 ≤ p ≤ 2 , then $$\begin{aligned} \displaystyle \int _0^\infty \beta _{\mu ,p}^n(x,r)^2\,\frac{dr}{r}<\infty \quad \mathrm{for}\,\, \mu {\text {-}}\mathrm{a.e.}\,\, x\in {\mathbb R}^d, \end{aligned}$$ ∫ 0 ∞ β ...

Journal of Functional Analysis > 2015 > 268 > 10 > 2946-2989

Geometric and Functional Analysis > 2015 > 25 > 5 > 1371-1412

*n*-dimensional Hausdorff measure $${\mathcal{H}^n}$$ H n is

*n*-rectifiable if the so called Jones’ square function is finite $${\mu}$$ μ -almost everywhere. The converse of this result is proven in a companion paper by the second author, and...

Acta Mathematica > 2014 > 213 > 2 > 237-321

*μ*is a

*d*-dimensional Ahlfors-David regular measure in $${\mathbb{R}^{d+1}}$$ R d + 1 , then the boundedness of the

*d*-dimensional Riesz transform in

*L*

^{2}(

*μ*) implies that the non-BAUP David–Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of

*μ*.

Journal de mathematiques pures et appliquees > 2013 > 100 > 2 > 137-165

Potential Analysis > 2013 > 38 > 3 > 913-949

*z*and the

*L*

^{∞}-norm. For

*n*∈ ℕ, one has likewise capacities related to the kernels $K_i(x)=x_i^{2n-1}/|x|^{2n}$ , 1 ≤

*i*≤ 2, $x=(x_1,x_2)\in{\mathbb R}^2$ . The main result of this paper states that the capacities associated with the vectorial kernel (

*K*

_{1},

*K*

_{2}) are comparable to analytic capacity.

Journal of Functional Analysis > 2012 > 262 > 10 > 4423-4457

Advances in Mathematics > 2012 > 229 > 2 > 1313-1328

Geometric and Functional Analysis > 2012 > 22 > 2 > 478-527

*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 $${{\mathbb{R}^n}}$$ , $${{\varphi}}$$ is a radial bump function smooth enough so that ...

The Journal of Geometric Analysis > 2011 > 21 > 1 > 195-223

^{ d }, the capacity

*γ*

_{ s }associated with the

*s*-dimensional 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. 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.

Journal of Functional Analysis > 2008 > 254 > 7 > 1811-1863

Geometric and Functional Analysis > 2007 > 17 > 2 > 605-643

*μ*is a finite Borel measure on the complex plane such that $$ {\mathcal{C}}_{*} \mu (z) = {\mathop {\sup }\limits_{\varepsilon > 0} }{\left| {{\mathcal{C}}_{\varepsilon } \mu (z)} \right|} = {\mathop {\sup }\limits_{\varepsilon > 0} }{\left| {{\int_{|\xi - z| > \varepsilon } {\frac{1}{{\xi - z}}d\mu (\xi )} }} \right|} < \infty $$ for μ-a.e. $${z \in \mathbb{C}}$$...

Advances in Mathematics > 2001 > 164 > 1 > 57-116

^{d}which may be non-doubling. The only condition that μ must satisfy is μ(B(x, r))=<Cr

^{n}, for all x R

^{d}, r>0, and for some fixed 0<n=<d. In this paper, Littlewood-Paley theory for functions in L

^{p}(μ) is developed. One of the main difficulties to be solved is the construction of ''reasonable'' approximations of the identity...