Commutator estimates and a sharp form of Gårding's inequality (Q1567126)

The authors consider the class of pseudodifferential operators \[ p(x,\partial)u(x)= {1\over (2\pi)^n} \iint e^{i(x- x').\xi}p(x,\xi, x') u(x')dx' d\xi\tag{1} \] with amplitude \(p(x,\xi, x')\) in the class \(S^m_{\rho,\delta,\lambda}\). This means that the derivatives of the function \(p(x,\xi, x')\) satisfy the estimate \[ |\partial^\alpha_\xi \partial^\beta_x \partial^{\beta'}_{x'} p(x, \xi,x')|\leq C_{\alpha, \beta,\beta'} \lambda(\xi)^{m- \rho|\alpha|+ \delta|\beta+ \beta'|}, \] where \(\lambda(\xi)\) is a weight function modeled after the basic weight \((1+ |\xi|^2)^{1/2}\). This class \(S^m_{\rho, \delta,\lambda}\) was considered by \textit{N. Jacob} [Osaka J. Math. 26, No. 4, 857-879 (1989; Zbl 0725.35112)]. When the function \(p\) in (1) does not depend on \(x'\) the authors obtain the following Gårding inequality \[ \text{Re}(p(x, \partial)u,u)\geq -C\|u\|_{{m-1\over 2},\lambda} \] for any \(u\) in the Schwartz class \(S\). They assume that \(\text{Re }p(x,\xi)\geq 0\), and they also assume that \(\partial^\beta_x p(x,\xi)\) belongs to \(S^m_{1,\delta, \lambda}\) with \(\delta< 1\), and \(0\leq|\beta|\leq 2\). With \(\|u\|_{{m-1\over 2},\lambda}\) the authors indicate the norm in a Sobolev-type space defined in terms of the weight \(\lambda(\xi)\).