On a pointwise estimate for a weighted rough singular integral operator in stratified Lie groups and applications

In this article, we present a new pointwise estimate for a weighted rough singular integral operator in the setting of stratified Lie groups. This operator, TΩ,, is based on a kernel $Ω$ and a weight , where the kernel satisfies a natural size condition and a cancellation property with respect to the weight . Moreover, we do not assume any kind of regularity on these objects. This weighted rough singular integral operator, applied to a function f, is estimated through a combination of information involving a weighted maximal function of the gradient of f and a weighted Morrey space. We also deduce from this pointwise estimate some new weighted functional inequalities and, as an application, we obtain a uniqueness result for a rough version of the stationary Navier-Stokes equation over the Heisenberg group.