Wave front set for positive operators and for positive elements in non-commutative convolution algebras
Volume 179 / 2007
Studia Mathematica 179 (2007), 63-80
MSC: Primary 35A18, 47B65, 35A21; Secondary 43A35, 35S05.
DOI: 10.4064/sm179-1-6
Abstract
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 $x$.
