Pseudodifferential operators of Beurling type and the wave front set

Abstract

We investigate the action of pseudodifferential operators of Beurling type on the wave front sets. More precisely, we show that these operators are microlocal, that is, preserve or reduce wave front sets. Some consequences on micro-hypoellipticity are derived.

Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators

“…Therefore, if ω is a non-quasianalytic weight function, an equivalent definition of wave front set is given by the following proposition. Combining Proposition 3.7 with Proposition 3.3 and Lemmas 3.1 and 3.2, we recover the definition of wave front set in the Gevrey setting, [20, p. 36], and in the nonquasianalytic Beurling setting, [11].…”

“…We point out that it is necessary only to state and prove Lemma 4.4 appropriately. But, we also present another proof of the Roumieu version based on an application of Theorem 4.1 and of the following proposition, which is an extension to the quasianalytic case of [11,Proposition 2] and could be of independent interest.…”

“…Proceeding in a similar way as in Example 1 in [11], we can constructṽ ∈ E * (R n−1 ) satisfying W F * (ṽ) = {(x , ξ ) ∈ R n−1 × R n−1 \ {0} : x = 0, ξ = (0, . .…”

“…The Roumieu version is obtained as a consequence of the Beurling case and the description of Roumieu wave front set of a classical distribution as the closure of the union of all the Beurling wave front sets contained in it. Such a description is proved for arbitrary weight functions, quasianalytic or not, in Proposition 4.5 (see [11] for a version for non-quasianalytic weight functions). To show our inclusion results, Theorem 4.1 and Theorem 4.8, we need to assume that the weight functions are equivalent to subadditive weight functions (see [18,19]).…”

Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes

“…It is known that partial differential operators with ultradifferentiable coefficients, or even pseudodifferential operators of ultradifferentiable type, reduce in most cases the singular support and the wave front set of ultradistributions, 1, 3–5, 14, 16–18, 22, 23. The main aim of this paper is to prove a suitable converse result, more precisely, we prove that if P = P ( x , D ) is a linear partial differential operator with ultradifferentiable coefficients on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega \subset \mathbb {R}^N$\end{document} then the following inclusion holds for all ultradistributions with compact support \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$u\in \mathcal {E}^\prime _{\omega }(\Omega )$\end{document}, where Σ is the characteristic manifold of P , i.e., the set where the principal symbol of P vanishes.…”