TY - JOUR
T1 - Singular integrals unsuitable for the curvature method whose L 2-boundedness still implies rectifiability
AU - Chunaev, Petr
AU - Mateu, Joan
AU - Tolsa, Xavier
PY - 2019/10/1
Y1 - 2019/10/1
N2 - © 2019, The Hebrew University of Jerusalem. The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the L2(μ)-boundedness of certain singular integral operators to the geometric properties of the support of measure μ, e.g., rectifiability. It can be applied, however, only if Menger curvature-like permutations directly associated with the kernel of the operator are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the L2(μ)-boundedness of the operator still implies that the support of μ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors–David regularity conditions.
AB - © 2019, The Hebrew University of Jerusalem. The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the L2(μ)-boundedness of certain singular integral operators to the geometric properties of the support of measure μ, e.g., rectifiability. It can be applied, however, only if Menger curvature-like permutations directly associated with the kernel of the operator are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the L2(μ)-boundedness of the operator still implies that the support of μ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors–David regularity conditions.
U2 - 10.1007/s11854-019-0043-5
DO - 10.1007/s11854-019-0043-5
M3 - Article
VL - 138
SP - 741
EP - 764
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
SN - 0021-7670
ER -