Global wave-front sets of Banach, Fr{é}chet and Modulation space types, and pseudo-differential operators

Abstract

We introduce global wave-front sets ${\mathrm{WF}}_{\mathcal{B}}(f)$, $f\in {\mathcal{S}}^{\prime }({\text{R}}^d)$, with respect to suitable Banach or Fréchet spaces $\mathcal{B}$. An important special case is given by the modulation spaces $\mathcal{B}=M(\omega ,\mathcal{B})$, where $\omega$ is an appropriate weight function and $\mathcal{B}$ is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to ${\mathrm{WF}}_{\mathcal{B}}(f)$. In particular, we prove that micro locality and microellipticity hold for a class of globally defined pseudo-differential operators ${\mathrm{Op}}_t(a)$, acting continuously on the involved spaces.

51 pages, mistakes and typos correction, reorganized materials

References in corpus (2)

• Wave-front sets of Banach function types
• Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces

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