Let ω, ω' be appropriate weight functions and B be an invariant BF-space. We introduce the wave-front set WF_{FB(ω)}(f) with respect to weighted Fourier Banach space FB(ω). We prove that usual mapping properties for pseudo-differential operators Op(a) with symbols a in S_{ρ,0}(ω') hold for such wave-front sets. In particular we prove WF_{FB(ω/ω')}(Op(a)f) ⊆ WF_{FB(ω)}(f) and WF_{FB(ω)}(f) ⊆ WFFB(ω/ω') (Op(a)f) union Char(a). Here Char(a) is the set of characteristic points of a.
Local wave-front sets of Banach and Fréchet types,and pseudo-differential operators
CORIASCO, Sandro;
2013-01-01
Abstract
Let ω, ω' be appropriate weight functions and B be an invariant BF-space. We introduce the wave-front set WF_{FB(ω)}(f) with respect to weighted Fourier Banach space FB(ω). We prove that usual mapping properties for pseudo-differential operators Op(a) with symbols a in S_{ρ,0}(ω') hold for such wave-front sets. In particular we prove WF_{FB(ω/ω')}(Op(a)f) ⊆ WF_{FB(ω)}(f) and WF_{FB(ω)}(f) ⊆ WFFB(ω/ω') (Op(a)f) union Char(a). Here Char(a) is the set of characteristic points of a.
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