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The Riesz Transform of Codimension Smaller Than One and the Wolff Energy, PDF eBook

The Riesz Transform of Codimension Smaller Than One and the Wolff Energy PDF

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Fix $d\geq 2$, and $s\in (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $\mu $ in $\mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\mu )$ in terms of the Wolff energy.

This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known.

As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\Delta )^\alpha /2$, $\alpha \in (1,2)$, in terms of a well-known capacity from non-linear potential theory.

This result contrasts sharply with removability results for Lipschitz harmonic functions.

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