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

TL;DR

This paper extends the Born-Jordan pseudodifferential calculus by applying Shubin's global symbol classes, clarifying its relationship with Weyl quantization and establishing asymptotic expansions and regularity results.

Contribution

It introduces a new framework using Shubin classes for Born-Jordan operators, providing clearer connections to Weyl quantization and deriving asymptotic and hypoellipticity properties.

Findings

Derived asymptotic expansions for symbols transitioning between Born-Jordan and Weyl quantizations

Established regularity and global hypoellipticity results for the operators

Clarified the conceptual relationship between Born-Jordan and Weyl quantizations

Abstract

We apply Shubin's theory of global symbol classes $\Gamma_{\rho}^{m}$ 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.