The space of smooth vector fields on $\mathbb{R}^2$ and open unit disc $\mathbb{D}$ are denoted by $\chi^{\infty} (\mathbb{R}^2)$ and $\chi^{\infty}(\mathbb{D})$, respectively. A vector field on $\mathbb{R}^2$ or $\mathbb{D}$ is regarded as a complex function $f(z),\;z\in \mathbb{C}$. By a holomorphic vector field on these complex sets, we mean a vector field whose corresponding complex function is a holomorphic function. The space of these holomorphic vector fields is denoted by "Hol"accordingly.
Inspired by the Cauchi integral formula, we consider the following linear map $$T:\chi^{\infty}(\mathbb{R}^2)\to \chi^{\infty}(\mathbb{D})\\ T(f)(w)=1/2\pi i \int_{|z|=1} \frac{f(z)dz}{z-w}$$
Obviously the restriction of $T$ to $Hol(\mathbb{R}^2)$ preserves the usual Lie bracet of vector fields since it acts as identity, or better to say, as restriction operator. Simillarly the operator $T$ restricts to a Lie algebra morphism on the sub Lie algebra of $\chi^{\infty}(\mathbb{R}^2)$ described as follows: $$\tilde{H}=\{f\in \chi^{\infty}(\mathbb{R}^2) \mid \partial f/\partial \bar z =0,\text{on}\; \mathbb{D}\}$$ Is there a subvector space or Lie sub algebra $Z$ of $\chi^{\infty}(\mathbb{R}^2)$ which properly contains $\tilde{H}$ such that the restriction of $T$ to $Z$ is a Lie algebra morphism? What is a description of the kernel of $T$?Is it closed under complex multiplication?
Is there any relation between the dynamics of a vector field $f$ restricted to$\mathbb{D}$ and the dynamics of $T(f)$? Under which conditions these later vector field satisfy $[f, T(f)]=0$?
