Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft. "Wave front set for positive operators and for positive elements in non-commutative convolution algebras." Studia Mathematica 179.1 (2007): 63-80. <http://eudml.org/doc/286203>.

@article{JoachimToft2007,
abstract = {Let WF⁎ be the wave front set with respect to $C^\{∞\}$, quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C₀^\{∞\}$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u∗_\{B\}φ(x) = ∫ u(x-y)φ(y)B(x,y)dy$, where $B ∈ C^\{∞\}$ is appropriate, and prove that if $(u∗_\{B\}φ,φ) ≥ 0$ for every $φ ∈ C₀^\{∞\}$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.},
author = {Joachim Toft},
journal = {Studia Mathematica},
keywords = {distribution kernel; positive operator},
language = {eng},
number = {1},
pages = {63-80},
title = {Wave front set for positive operators and for positive elements in non-commutative convolution algebras},
url = {http://eudml.org/doc/286203},
volume = {179},
year = {2007},
}

TY - JOUR
AU - Joachim Toft
TI - Wave front set for positive operators and for positive elements in non-commutative convolution algebras
JO - Studia Mathematica
PY - 2007
VL - 179
IS - 1
SP - 63
EP - 80
AB - Let WF⁎ be the wave front set with respect to $C^{∞}$, quasi analyticity or analyticity, and let K be the kernel of a positive operator from $C₀^{∞}$ to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution $u∗_{B}φ(x) = ∫ u(x-y)φ(y)B(x,y)dy$, where $B ∈ C^{∞}$ is appropriate, and prove that if $(u∗_{B}φ,φ) ≥ 0$ for every $φ ∈ C₀^{∞}$ and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.
LA - eng
KW - distribution kernel; positive operator
UR - http://eudml.org/doc/286203
ER -