First-order Cauchy-Riemann operator (${\partial }_{\bar{z}}$)

The first-order Cauchy-Riemann operator on two real variables is ${\partial }_{\bar{z}}=\frac{1}{2}({\partial }_{x_1}+i{\partial }_{x_2})$. It is a scalar complex elliptic differential operator, since its principal symbol has nonzero modulus at every nonzero real frequency.