Born-Jordan pseudodifferential operators with symbols in the Shubin classes

Abstract

We apply Shubin’s theory of global symbol classes Γ ρ m \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 us to pass from Born-Jordan quantization to Weyl quantization and vice versa. In addition we state and prove some regularity and global hypoellipticity results.