Microlocal analysis in the Denjoy-Carleman class (Q2731053)

Let \(\Omega\) be an open set in \(\mathbb{R}^n\) and \(M_p\) \((p=0,1,2,\dots)\) be a sequence of positive numbers satisfyingNEWLINENEWLINENEWLINE(i) \(p!\leq Ch^p M_p\),NEWLINENEWLINENEWLINE(ii) \(M^2_p\leq M_{p-1} M_{p+2}\),NEWLINENEWLINENEWLINE(iii) \(M_{p+1}\leq CH^p M_p\), for some positive constants \(C\), \(h\), \(H\).NEWLINENEWLINENEWLINEDenote by \(C^M(\Omega)\) the Denjoy-Carleman class of all \(\phi\in C^\infty(\Omega)\) such that on each compact \(K\subset \Omega\), NEWLINE\[NEWLINE\sup_{x\in K} |\partial^\alpha\phi(x)|\leq Ch^{|\alpha|} M_{|\alpha|}\qquad (\alpha\in N^n_0),NEWLINE\]NEWLINE where \(\partial^\alpha= \partial^{\alpha_1}_1 \partial^{\alpha_2}_2\cdots \partial^{\alpha_n}_n\), \(\partial_j= {\partial\over\partial x_j}\) for \(\alpha\in N^n_0\). Let \(F(x)\) denote all \(\phi\in C^\infty(\Omega)\) (for some neighbourhood \(\Omega\) of \(K\)) with NEWLINE\[NEWLINE|\phi|_{k, h}= \sup{|\partial^\alpha \phi(x)|\over h^{|\alpha|}\alpha!} \exp k|x|<\inftyNEWLINE\]NEWLINE for some \(k\), \(h\) and let \(F'(K)\) be its strong dual. For \(u\in F'(K)\) denote by \(WF_M(u)\) the complement of the set \((x_0,\xi_0)\), \(\xi_0\neq 0\) such that there exists a neighborhood \(\Gamma\) of \(\xi_0\) with NEWLINE\[NEWLINE|u_y(\exp[-|\xi|(x- y)^2/2- i\langle y,\xi\rangle])|\leq C\exp[-M\gamma|\xi|]NEWLINE\]NEWLINE for all \(x\in U\) and \(\xi\in\Gamma\) with \(|\xi|\geq N\).NEWLINENEWLINENEWLINELet \(A'(\mathbb{R}^n)\) denote the space of all analytic functionals in \(\mathbb{R}^n\). In this paper the authors characterize all \((x_0,\xi_0)\not\in WF_M(u)\) for a given \(u\in F'(K)\) and also show that for \(u\in A'(\mathbb{R}^n)\) with \(\text{supp }u\subset \{x_n\geq 0\}\) if \((0,\xi)\not\in WF_M(u)\) \((\xi_0\neq 0)\) then \((0(\xi_0',\mu))\not\in WF_M(u)\) for all \(\mu\) with \(|\mu|\leq |\xi_{0n}|\) where \(\xi_0= (\xi_0', \xi_{0n})\).NEWLINENEWLINENEWLINEFinally, the authors state and prove a quasi-analytic version of Holmgren's uniqueness theorem.