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

Joachim Toft¹

¹ Department of Mathematics and Systems Engineering Växjö University S-351 95 Växjö, Sweden

Studia Mathematica, Tome 179 (2007) no. 1, pp. 63-80

Résumé

Let ${\rm WF}_*$ be the wave front set with respect to $C^\infty$, quasi analyticity or analyticity, and let $K$ be the kernel of a positive operator from $C_0^\infty$ to $\mathscr D'$. We prove that if $\xi\ne0$ and $(x,x,\xi ,-\xi )\notin {\rm WF}_*(K)$, then $(x,y,\xi ,-\eta )\notin {\rm WF}_*(K)$ and $(y,x,\eta ,-\xi )\notin {\rm WF}_*(K)$ for any $y,\eta$. We apply this property to positive elements with respect to the weighted convolution $$ u*_B\varphi (x)=\int u(x-y)\varphi (y)B(x,y)\, dy, $$ where $B\in C^\infty$ is appropriate, and prove that if $(u*_B\varphi ,\varphi )\ge 0$ for every $\varphi \in C_0^\infty$ and $(0,\xi )\notin {\rm WF}_*(u)$, then $(x,\xi )\notin {\rm WF}_*(u)$ for any $x$.

Détail

DOI : 10.4064/sm179-1-6

Keywords: * wave front set respect infty quasi analyticity analyticity kernel positive operator infty mathscr prove notin * eta notin * eta notin * eta apply property positive elements respect weighted convolution u* varphi int x y varphi y where infty appropriate prove u* varphi varphi every varphi infty notin * notin * nbsp