TY - JOUR
T1 - Failure of L 2 boundedness of gradients of single layer potentials for measures with zero low density
AU - Conde-Alonso, José M.
AU - Mourgoglou, Mihalis
AU - Tolsa, Xavier
PY - 2019/2/8
Y1 - 2019/2/8
N2 - © 2018, Springer-Verlag GmbH Germany, part of Springer Nature. Consider a totally irregular measure μ in R n+1 , that is, the upper density lim supr→0μ(B(x,r))(2r)n is positive μ-a.e. in R n+1 , and the lower density lim infr→0μ(B(x,r))(2r)n vanishes μ-a.e. in R n+1 . We show that if Tμf(x)=∫K(x,y)f(y)dμ(y) is an operator whose kernel K(· , ·) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Hölder continuous coefficients, then T μ is not bounded in L 2 (μ). This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform.
AB - © 2018, Springer-Verlag GmbH Germany, part of Springer Nature. Consider a totally irregular measure μ in R n+1 , that is, the upper density lim supr→0μ(B(x,r))(2r)n is positive μ-a.e. in R n+1 , and the lower density lim infr→0μ(B(x,r))(2r)n vanishes μ-a.e. in R n+1 . We show that if Tμf(x)=∫K(x,y)f(y)dμ(y) is an operator whose kernel K(· , ·) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Hölder continuous coefficients, then T μ is not bounded in L 2 (μ). This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform.
U2 - 10.1007/s00208-018-1729-1
DO - 10.1007/s00208-018-1729-1
M3 - Article
VL - 373
SP - 253
EP - 285
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
ER -