0000000000775635

AUTHOR

Xavier Tolsa

showing 3 related works from this author

A proof of Carleson's $\varepsilon^2$-conjecture

2019

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.

Pure mathematicsConjectureMathematics::Classical Analysis and ODEsTangentMetric Geometry (math.MG)Jordan curve theoremsymbols.namesakeMathematics (miscellaneous)Mathematics - Analysis of PDEsMathematics - Metric GeometryMathematics - Classical Analysis and ODEssymbolsClassical Analysis and ODEs (math.CA)FOS: MathematicsStatistics Probability and Uncertainty28A75 42B20MathematicsAnalysis of PDEs (math.AP)
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Analytic capacity and quasiconformal mappings with $W^{1,2}$ Beltrami coefficient

2008

We show that if $\phi$ is a quasiconformal mapping with compactly supported Beltrami coefficient in the Sobolev space $W^{1,2}$, then $\phi$ preserves sets with vanishing analytic capacity. It then follows that a compact set $E$ is removable for bounded analytic functions if and only if it is removable for bounded quasiregular mappings with compactly supported Beltrami coefficient in $W^{1,2}$.

Sobolev spaceQuasiconformal mappingComputer Science::GraphicsCompact spaceMathematics::Complex VariablesGeneral MathematicsBounded functionMathematical analysisAnalytic capacityAnalytic functionMathematicsMathematical Research Letters
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A proof of Carleson's 𝜀2-conjecture

2021

In this paper we provide a proof of the Carleson 𝜀2-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 𝜀2-square function. peerReviewed

Mathematics::Complex Variablessquare functiontangentJordan curveMathematics::Classical Analysis and ODEsrectifiabilitymittateoriaharmoninen analyysi
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