Is there an analogue of the classical Bochner formula $\frac{1}{2} \Delta |\nabla u|^2 = |\nabla^2 u|^2$ for harmonic functions that holds for $s$-harmonic functions?
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$\begingroup$ The case of manifolds would also be interesting, when the classical Bochner formula also has a correction taking into account the Ricci curvature. In fact I realize now that I don't even know if PDE formulations (integral representations?) of fractional operators are well studied in manifolds (although the general theory of Lévy processes must cover this case, I presume) $\endgroup$– leo monsaingeonCommented Mar 24, 2021 at 10:53
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1$\begingroup$ According to Nicola Garofalo's Fractional thoughts, this is not known. See the end of Section 20 in arXiv:1712.03347. This may have changed recently, though, as Nicola Garofalo ends this section with "We plan to come back to these questions in future works." :-) $\endgroup$– Mateusz KwaśnickiCommented Mar 24, 2021 at 11:03
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