TY - JOUR

T1 - Non existence of principal values of signed riesz transforms of non integer dimension

AU - Ruiz De Villa, Aleix

AU - Tolsa, Xavier

PY - 2010/7/20

Y1 - 2010/7/20

N2 - In this paper we prove that, given s ≥ 0, and a Borel non zero measure μ in Rm, if for μ-almost every x ∈ Rm the limit mathematical equation presented exists and 0 < lim sup r→0μ(B(x; r)) /rs < ∞, then s in an integer. In particular, if E ⊂ Rm is a set with positive and bounded s-dimensional Hausdorff measure Hs and for H s-almost every x ∈ E the limit mathematical equation presented exists, then s is an integer. ©, Vol. 59, No. 1 (2010).

AB - In this paper we prove that, given s ≥ 0, and a Borel non zero measure μ in Rm, if for μ-almost every x ∈ Rm the limit mathematical equation presented exists and 0 < lim sup r→0μ(B(x; r)) /rs < ∞, then s in an integer. In particular, if E ⊂ Rm is a set with positive and bounded s-dimensional Hausdorff measure Hs and for H s-almost every x ∈ E the limit mathematical equation presented exists, then s is an integer. ©, Vol. 59, No. 1 (2010).

KW - Hausdorff measures

KW - Riesz transforms

U2 - https://doi.org/10.1512/iumj.2010.59.3884

DO - https://doi.org/10.1512/iumj.2010.59.3884

M3 - Article

VL - 59

SP - 115

EP - 130

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

ER -