Gårding inequality in large dimension
The main result of the paper is an analog of the sharp Gårding inequality for some classes of semiclassical pseudodifferential operators. The authors prove the following result. Given nonnegative real numbers \(\rho_j\) and \(\delta_j\), \(0\leq j\leq n\), let \(F\) be a bounded nonnegative real-valued continuous function defined on \(\mathbb R^{2n}\) satisfying for a constant \(M>0\), for all \((\alpha ,\beta )\in \mathbb N^{2n}\), \(0\leq \alpha_j\leq m\), \(0\leq \beta_j\leq m\), \(0\leq j\leq n\), for all \((x,\xi )\in \mathbb R^{2n}\), \(| \partial_x^\alpha\partial_\xi^\beta F(x,\xi )| \leq M\rho^\alpha\delta^\beta\). Then, for all \(f\in L^2(\mathbb R^n)\), \[ \langle Op_h^w(F)f,f\rangle \geq -225\pi Kh\times \rho \cdot \delta \times \prod\limits_{j=1}^n(1+225\pi Kh\rho_j\delta_j)\times\| f\|^2, \] where\ \(K=\sup_{j\leq n}\max \{ 1,(\rho_j\delta_j)^3\} \) and for \(h>0\), \(Op_h^w(F)\) denotes the operator formally defined by \[ Op_h^w(F)(\varphi )(x)=(2\pi h)^{-n}\int\limits_{\mathbb R^{2n}}e^{ \frac{i}{h}(x-y)\cdot \xi}F(\frac{x+y}{2},\xi )\varphi (y)dyd\xi,\quad\varphi \in S. \]
scientific article; zbMATH DE number 6308415
scientific article