Principal values for the signed Riesz kernels of non-integer dimension
This paper consists of a study of the \(\alpha\)-Riesz transform \(R^\alpha\) of a positive Radon measure \(\mu\) on \(\mathbb R^n\) for \(0<\alpha<1\). If \(\mu\) has compact support, the author first proves that the \(L^2(\mu)\)-boundedness of \(R^\alpha\) implies the \(\mu\)-almost everywhere existence of the principal values of the \(\alpha\)-Riesz transform of \(\mu\). The proof follows some ideas given by \textit{X. Tolsa} in [J. Reine Angew. Math. 502, 199--235 (1998; Zbl 0912.42009)]. Then, the author shows that the \(\mu\)-measure of the set where the principal value of \(R^\alpha\) does not exist is positive when \(\mu\) has a positive and finite upper \(\alpha\)-density \(\mu\)-almost everywhere. This is done by adapting a powerful result of \textit{F. Nazarov, S. Treil} and \textit{A. Volberg} [Acta Math. 190, No. 2, 151--239 (2003; Zbl 1065.42014)] on the \(L^2(\mu)\)-boundedness of singular integrals on non-homogeneous spaces.
scientific article; zbMATH DE number 5934874
scientific article