We show that, given a set E ⊂ ℝn+1 with finite n-Hausdorff measure Hn, if the n-dimensional Riesz transform (Equation Presented) is bounded in L2 (Hn90E), then E is n-rectifiable. From this result we deduce that a compact set E ⊂ ℝn+1 with Hn(E) < ∞ 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.
- Lipschitz harmonic functions
- Riesz transform
Nazarov, F., Tolsa, X., & Volberg, A. (2014). The riesz transform, rectifiability, and removability for Lipschitz harmonic functions. Publicacions Matematiques, 58(2), 517-532. https://doi.org/10.5565/PUBLMAT-58214-26