Boundedness of Weyl quantization (Q1284275)

The paper concerns different problems in the general theory of the Weyl quantization of a pseudodifferential symbol \(a(x,\eta)\): \[ a^w(x,D)u(x)= (2\pi)^{-n}\iint e^{i(x- y)\eta} a((x+ y)/2,\eta) u(y)dy d\eta. \] First, boundedness in function spaces is considered, when \(a(x,\eta)\in S^0_{\rho,\delta}(\mathbb{R}^n)\), \(0\leq \rho\), \(\delta\leq 1\), \(\delta<1\). We report in particular the following result, concerning the Triebel-Lizorkin spaces \(F^s_{pq}\): The operator \(a^w(x,D)\) is bounded from \(F^s_{pq}\) to \(L^p\) for \(s> n+2\lambda\delta\), where \(\lambda= n/(1-\delta)2+1\). The authors also recapture a result of E. M. Stein, concerning boundedness from \(H^s\) to \(H^s\) for \(s>0\), in the limit case when the symbol \(a(x,\eta)\) is in \(S^0_{1,1}(\mathbb{R}^n)\). In the second part of the paper, the authors define almost diagonalizable operators as maps of the form \(a^w(x,D)= A(x,D)+ R(x,D)\) where \(R(x,0)\) is negligible and \(A\) preserves the \(L^p\) spectrum. It is proved that if \(a(x,\eta)\in S^0_{\rho,\delta}(\mathbb{R}^n)\), \(0\leq \rho\), \(\delta\leq 1\), \(\delta<1\), then \(a^w(x,D)\) is almost diagonalizable.