Born-Jordan Pseudo-Differential Operators with Symbols in the Shubin Classes

We apply Shubin’s theory of global symbol classes ${\mathrm{\Gamma }}_{\rho }^m$ to the Born-Jordan pseudodifferential calculus we have previously developed. This approach has many conceptual advantages, and makes the relationship between the conflicting Born-Jordan and Weyl quantization methods much more limpid. We give, in particular, precise asymptotic expansions of symbols allowing to pass from Born-Jordan quantization to Weyl quantization, and vice-versa. In addition we state and prove some regularity and global hypoellipticity results.