Counterexamples for Boundedness of Pseudodifferential Operators

This is the classical version of pseudodifferential operators that is used in the investigation of partial differential operators, cf. [21]. In the language of physics, the Kohn–Nirenberg correspondence and its relatives such as the Weyl correspondence are methods of quantization. In the language of engineering, they are time-varying filters. The Kohn–Nirenberg correspondence is usually analyzed using methods from hard analysis. The problems arising from the theory of partial differential equations suggest using the classical Hörmander symbol classes S ρ,δ(R ), which are defined in terms of differentiability conditions [21], [31]. On the other hand, if we introduce the time-frequency shifts MωTxf(t) = e f(t− x) , (1) then we can write σ(X,D) as a formal superposition of time-frequency shifts: