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The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions

Fedor Nazarov, Xavier Tolsa, and Alexander Volberg

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We show that, given a set $E\subset{\mathbb R}^{n+1}$ with finite $n$-Hausdorff measure${\mathcal H}^n$, if the $n$-dimensional Riesz transform

$$R_{{\mathcal H}^n{\lfloor} E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}}\,f(y)\,{\mathcal H}^n(y)$$

is bounded in $L^2({\mathcal H}^n{\lfloor} E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\subset{\mathbb R}^{n+1}$ with ${\mathcal H}^n(E)<\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions.

Article information

Publ. Mat., Volume 58, Number 2 (2014), 517-532.

First available in Project Euclid: 21 July 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 31B25: Boundary behavior

Riesz transform rectifiability Lipschitz harmonic functions


Nazarov, Fedor; Tolsa, Xavier; Volberg, Alexander. The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions. Publ. Mat. 58 (2014), no. 2, 517--532.

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