Riesz transforms for the Weinstein operator

In this paper we study the Riesz transforms Rw related to the Weinstein operators Δw=∑i=1dxd−2α−1(∂/∂xi)(xd2α+1(∂/∂xi)). We develop for Rw a theory that runs parallel to the one for the Euclidean Riesz Transform. It is proved that the Riesz–Weinstein transform in coordinates i=1,…,d, Rwi is actually a Calderon–Zygmund singular integral operator in the sense of the associated space of homogeneous type. Moreover, our Riesz–Weinstein transform can be written as a principal value.