Time-frequency Analysis of Born-Jordan Pseudodifferential Operators

TL;DR

This paper conducts a time-frequency analysis of Born-Jordan pseudodifferential operators, exploring their boundedness on modulation spaces using the Cohen kernel and advanced function space techniques.

Contribution

It introduces a detailed analysis of Born-Jordan operators via modulation and Wiener amalgam spaces, establishing boundedness criteria on modulation spaces.

Findings

Derived necessary and sufficient conditions for boundedness.

Analyzed the Cohen kernel within modulation and Wiener amalgam spaces.

Provided new insights into symbol classes for Born-Jordan operators.

Abstract

Born-Jordan operators are a class of pseudodifferential operators arising as a generalization of the quantization rule for polynomials on the phase space introduced by Born and Jordan in 1925. The weak definition of such operators involves the Born-Jordan distribution, first introduced by Cohen in 1966 as a member of the Cohen class. We perform a time-frequency analysis of the Cohen kernel of the Born -Jordan distribution, using modulation and Wiener amalgam spaces. We then provide sufficient and necessary conditions for Born-Jordan operators to be bounded on modulation spaces. We use modulation spaces as appropriate symbols classes.