Geodesic
L^2-bounded singular integrals on a purely unrectifiable set in R^d
Joan Mateu1 ; Laura Prat1
1 Universitat Autònoma de Barcelona, Departament de Matemàtiques and Centre de Reserca Matemàtica
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 187-200

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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 $\mathbf{R}^d$.
Keywords: Purely unrectifiable set, singular integral operator, Cantor type set, T(1)-theorem

Joan Mateu 1 ; Laura Prat 1

1 Universitat Autònoma de Barcelona, Departament de Matemàtiques and Centre de Reserca Matemàtica 
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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/