The operator $(\partial_x+i\partial_y)(\partial_x-i\partial_y)^{-1}$

Question

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 $K(fg) - fK(g) - gK(f)$.

Answer 1

This is (the adjoint of) the Beurling transform (also known as the Beurling-Ahlfors transform). It's also a Calderon-Zygmund operator and a pseudodifferential operator of order 0, so all the usual theory of those operators (as found e.g. in Stein's "Harmonic analysis", or his earlier text "Singular integrals") would apply. In particular, the expression $K(fg)-fK(g)-gK(f)$ is basically a paraproduct of f and g and can be handled by the Coifman-Meyer theory of such paraproducts (as covered for instance in Taylor's "Tools for PDE", or the more recent "Harmonic Analysis" by Muscalu and Schlag).