An explicit expression for singular integral operators with non-necessarily doubling measures

Abstract

We study singular integral operators with Hilbert-valued kernels in the setting of Rn with non-necessarily doubling measures. We obtain an explicit formula for these operators following a similar approach as in Macias et al. (Adv Math 93:25–60, 1992). By using this formula and a result due to Krein we get a T1-theorem in this context. Finally, we develop a theory for antisymmetric kernels and we apply the results to the oscillation operators related to the Riesz transform.