TY - JOUR
T1 - Calderón-Zygmund kernels and rectifiability in the plane
AU - Chousionis, V.
AU - Prat, L.
AU - Tolsa, X.
AU - Mateu Bennassar, Juan Eugenio
PY - 2012/9/10
Y1 - 2012/9/10
N2 - Let E⊂C be a Borel set with finite length, that is, 0<H 1(E)<∞. By a theorem of David and Léger, the L 2(H 1⌊E)-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts x/{pipe}z{pipe} 2,y/{pipe}z{pipe} 2,z=(x,y)∈C) implies that E is rectifiable. We extend this result to any kernel of the form x 2n-1/{pipe}z{pipe} 2n,z=(x,y)∈C,n∈N. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose L 2-boundedness implies rectifiability. © 2012 Elsevier Ltd.
AB - Let E⊂C be a Borel set with finite length, that is, 0<H 1(E)<∞. By a theorem of David and Léger, the L 2(H 1⌊E)-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts x/{pipe}z{pipe} 2,y/{pipe}z{pipe} 2,z=(x,y)∈C) implies that E is rectifiable. We extend this result to any kernel of the form x 2n-1/{pipe}z{pipe} 2n,z=(x,y)∈C,n∈N. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose L 2-boundedness implies rectifiability. © 2012 Elsevier Ltd.
KW - Calderón-Zygmund singular integrals
KW - Rectifiability
U2 - https://doi.org/10.1016/j.aim.2012.04.025
DO - https://doi.org/10.1016/j.aim.2012.04.025
M3 - Article
SN - 0001-8708
VL - 231
SP - 535
EP - 568
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -