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Documents  Tolsa, Xavier | enregistrements trouvés : 3

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Research talks;Analysis and its Applications;Partial Differential Equations

The weak-$A_\infty$ condition is a variant of the usual $A_\infty$ condition which does not require any doubling assumption on the weights. A few years ago Hofmann and Le showed that, for an open set $\Omega\subset \mathbb{R}^{n+1}$ with $n$-AD-regular boundary, the BMO-solvability of the Dirichlet problem for the Laplace equation is equivalent to the fact that the harmonic measure satisfies the weak-$A_\infty$ condition. Aiming for a geometric description of the open sets whose associated harmonic measure satisfies the weak-$A_\infty$ condition, Hofmann and Martell showed in 2017 that if $\partial\Omega$ is uniformly $n$-rectifiable and a suitable connectivity condition holds (the so-called weak local John condition), then the harmonic measure satisfies the weak-$A_\infty$ condition, and they conjectured that the converse implication also holds.
In this talk I will discuss a recent work by Azzam, Mourgoglou and myself which completes the proof of the Hofman-Martell conjecture, by showing that the weak-$A_\infty$ condition for harmonic measure implies the weak local John condition.
The weak-$A_\infty$ condition is a variant of the usual $A_\infty$ condition which does not require any doubling assumption on the weights. A few years ago Hofmann and Le showed that, for an open set $\Omega\subset \mathbb{R}^{n+1}$ with $n$-AD-regular boundary, the BMO-solvability of the Dirichlet problem for the Laplace equation is equivalent to the fact that the harmonic measure satisfies the weak-$A_\infty$ condition. Aiming for a geometric ...

31B15 ; 28A75 ; 28A78 ; 35J15 ; 35J08

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- v; 130 p.
ISBN 978-1-4704-2252-3

Memoirs of the american mathematical society , 1158

Localisation : Collection 1er étage

mesure de Radon # théorie de la mesure # transformation de Cauchy # fonction carré # densité # rectifiabilité

28A75 ; 42B20 ; 28A78

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- xiii; 396 p.
ISBN 978-3-319-00595-9

Progress in mathematics , 0307

Localisation : Collection 1er étage

Transformation de Cauchy # conjecture de Vitushkin # capacité analytique # rectifiabilité

30-02 ; 31-02 ; 30C85 ; 31C15

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