So I found this operator while messing around with the equations of 2D incompressible fluid mechanics (it relates to the pressure Hessian). Specifically, if we call it $K$, then $$ K\Delta p = (p_{xx} - p_{yy}) + 2ip_{xy}.$$ Thus it's related to the Riesz transforms, $K = (R_1+iR_2)^2$.

It's a unitary operator and looks kind of familiar, but I haven't been able to find anything in a search. Does it have a name? Are its properties well-known? I'm especially interested in anything like a product rule to simplify $K(fg) - fK(g) - gK(f)$.