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 -