A Hilbert transform that takes several functions

Question

While playing with some PDE I came across a singular integral that looks something like $$T(f_1,f_2,\ldots,f_n)(x)=p.v.\int_{-\infty}^\infty\frac{(f_1(x)-f_1(y))(f_2(x)-f_2(y))\cdots(f_n(x)-f_n(y))}{(x-y)^n}dy$$ where the functions are really nice, say Schwartz. Clearly if $n=1$ then $T$ is just a multiple of the Hilbert transform. Therefore I'm hoping one could write this as some kind of combination of $n$ Hilbert transforms, but I don't see how. The closest thing I've found is the Calderon commutators which differ in that there are $n+1$ factors of $(x-y)$ in the denominator. In that case, Coifman, McIntosh, Meyer were able to write it in terms of $H$ (except they did it in French).