We construct an example of a purely unrectifiable measure $\mu$ in $\mathbf{R}^d$ for which the singular integrals associated to the kernels $K(x)=P_{2k+1}(x)/|x|^{2k+d}$, with $k\geq 1$ and $P_{2k+1}$ a homogeneous harmonic polynomial of degree $2k+1$, are bounded in $L^2(\mu)$. This contrasts starkly with the results concerning the Riesz kernel $x/|x|^d$ in $\mathbf{R}^d$.
@article{AFM_2021_46_1_a10,
author = {Joan Mateu and Laura Prat},
title = {L^2-bounded singular integrals on a purely unrectifiable set in {R^d}},
journal = {Annales Fennici Mathematici},
pages = {187--200},
year = {2021},
volume = {46},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a10/}
}
TY - JOUR
AU - Joan Mateu
AU - Laura Prat
TI - L^2-bounded singular integrals on a purely unrectifiable set in R^d
JO - Annales Fennici Mathematici
PY - 2021
SP - 187
EP - 200
VL - 46
IS - 1
UR - http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a10/
LA - en
ID - AFM_2021_46_1_a10
ER -
%0 Journal Article
%A Joan Mateu
%A Laura Prat
%T L^2-bounded singular integrals on a purely unrectifiable set in R^d
%J Annales Fennici Mathematici
%D 2021
%P 187-200
%V 46
%N 1
%U http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a10/
%G en
%F AFM_2021_46_1_a10
Joan Mateu; Laura Prat. L^2-bounded singular integrals on a purely unrectifiable set in R^d. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 187-200. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a10/
