Characterization of pseudodifferential operators and applications

Time-Frequency Analysis of Sjöstrand’s Class

“…This is the Wiener property of M ∞, 1 . For the classical symbol classes results of this type go back to Beals [3].…”

Spectral Invariance of Non-Smooth Pseudo-Differential Operators

“…A fundamental result in the theory of pseudodifferential operators allows one to directly conclude that the inverse of a pseudodifferential operator with a symbol in the Hörmander class S 0 ρ,δ (R n × R n ), 0 ≤ δ ≤ ρ ≤ 1, δ < 1, which is invertible as an operator on L 2 (R n ), is again a pseudodifferential operator in the same symbol-class. This important statement was derived by Beals [5] and Ueberberg [20]. Their proof even showed that the same holds for all Bessel potential spaces H s 2 (R n ), s ∈ R, see Definition 1 for the definition of these spaces, and that the spectrum is independent of the choice of the space.…”

“…In analogy to the proof in the smooth case of Beals and Ueberberg, c.f. [20], [5], we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator.…”

On a class of hyperbolic equations in Weyl–Hörmander calculus

“…This is the kind of fact that has led to the introduction of more general classes as the ϕ − Φ Beals-Fefferman classes (cf. [BF74], [Bea77]) and the S(m, g)-classes of Hörmander. If L is subelliptic with order of subellipticity 1 ≤ τ ≤ 2, we associate to L the Hörmander metric g defined by…”

“…[Bea77]). The metric g with τ = 1 has also been applied in [CCX93] for the study of parametrices for sum of squares satisfying the Hörmander condition of order 2.…”