Xavier Tolsa

Bio: Xavier Tolsa is an academic researcher from Autonomous University of Barcelona. The author has contributed to research in topics: Bounded function & Lipschitz continuity. The author has an hindex of 30, co-authored 160 publications receiving 3658 citations. Previous affiliations of Xavier Tolsa include Chalmers University of Technology & Catalan Institution for Research and Advanced Studies.

Painlevé's problem and the semiadditivity of analytic capacity

Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

BMO, H^1, and Calderon-Zygmund operators for non doubling measures

Abstract: Given a Radon measure $\mu$ on ${\mathbb R}^d$ , which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space $BMO(\mu)$ when $\mu$ is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming $\mu$ doubling. For instance, Calderon-Zygmund operators which are bounded on $L^2(\mu)$ are also bounded from $L^\infty(\mu)$ into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space $H^1$ . Using a sharp maximal operator it is shown that operators which are bounded from $L^\infty(\mu)$ into the new BMO space and from its predual $H^1$ into $L^1(\mu)$ must be bounded on $L^p(\mu)$ , $1< p< infty$ . From this result one can obtain a new proof of the T(1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderon-Zygmund operators bounded on $L^2(\mu)$ with functions of the new BMO are bounded on $L^p(\mu), 1< p < \infty$ .

Painleve's problem and the semiadditivity of analytic capacity

Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory

Abstract: Introduction.- Basic notation.- Chapter 1. Analytic capacity.- Chapter 2. Basic Calderon-Zygmund theory with non doubling measures.- Chapter 3. The Cauchy transform and Menger curvature.- Chapter 4. The capacity gamma+.- Chapter 5. A Tb theorem of Nazarov, Treil and Volberg.- Chapter 6. The comparability between gamma and gamma +, and the semiadditivity of analytic capacity.- Chapter 7. Curvature and rectifiability.- Chapter 8. Principal values for the Cauchy transform and rectifiability.- Chapter 9. RBMO(mu) and H1 atb(mu).- Bibliography.- Index.

On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1

Abstract: We prove that if μ is a d-dimensional Ahlfors-David regular measure in \({\mathbb{R}^{d+1}}\) , then the boundedness of the d-dimensional Riesz transform in L2(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.

Phd by thesis

Abstract: Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Finding a "Kneedle" in a Haystack: Detecting Knee Points in System Behavior

Abstract: Computer systems often reach a point at which the relative cost to increase some tunable parameter is no longer worth the corresponding performance benefit. These ``knees'' typically represent beneficial points that system designers have long selected to best balance inherent trade-offs. While prior work largely uses ad hoc, system-specific approaches to detect knees, we present Kneedle, a general approach to on line and off line knee detection that is applicable to a wide range of systems. We define a knee formally for continuous functions using the mathematical concept of curvature and compare our definition against alternatives. We then evaluate Kneedle's accuracy against existing algorithms on both synthetic and real data sets, and evaluate its performance in two different applications.

The Tb-theorem on non-homogeneous spaces

Abstract: 0 Introduction: main objects and results 3 0.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.2 An application of T1-heorem: electric intensity capacity . . . . . . . . . . . . 7 0.3 How to interpret Calderon–Zygmund operator T? . . . . . . . . . . . . . . . 9 0.3.1 Bilinear form is defined on Lipschitz functions . . . . . . . . . . . . . 10 0.3.2 Bilinear form is defined for smooth functions . . . . . . . . . . . . . . 11 0.3.3 Apriori boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.4 Plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

Abstract: Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincare inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincare inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincare inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincare inequalities.