TY - JOUR
T1 - The riesz transform, rectifiability, and removability for lipschitz harmonic functions
AU - Nazarov, Fedor
AU - Tolsa, Xavier
AU - Volberg, Alexander
PY - 2014/1/1
Y1 - 2014/1/1
N2 - We show that, given a set E ⊂ R n+1 with ffnite n-Hausdorff measure H n , if the n-dimensional Riesz transform RH n bEf(x) =/E/x-y/x-y/ n+1 f(y)H n (y)is bounded in L2(HnbE), then E is n-rectiáble. From this result we deduce that a compact set E ⊂ R n+1 with H n (E) < ∞ is removable for Lipschitz harmonic functions if and only if it is purely n-unrectiáble, thus proving the analog of Vitushkin's conjecture in higher dimensions.
AB - We show that, given a set E ⊂ R n+1 with ffnite n-Hausdorff measure H n , if the n-dimensional Riesz transform RH n bEf(x) =/E/x-y/x-y/ n+1 f(y)H n (y)is bounded in L2(HnbE), then E is n-rectiáble. From this result we deduce that a compact set E ⊂ R n+1 with H n (E) < ∞ is removable for Lipschitz harmonic functions if and only if it is purely n-unrectiáble, thus proving the analog of Vitushkin's conjecture in higher dimensions.
KW - Lipschitz harmonic functions
KW - Rectiá bility
KW - Riesz transform
UR - https://ddd.uab.cat/record/119046
U2 - https://doi.org/10.5565/PUBLMAT_58214_26
DO - https://doi.org/10.5565/PUBLMAT_58214_26
M3 - Article
VL - 58
SP - 517
EP - 532
JO - Publicacions Matematiques
JF - Publicacions Matematiques
SN - 0214-1493
IS - 2
ER -