Pseudodifferential operators with homogeneous symbols
The authors define homogeneous symbols as symbols \(a(x,\xi)\) in \(C^\infty(\mathbb{R}^n\times (\mathbb{R}^n/0))\) satisfying \[ |D^\alpha_x D^\beta_\xi a(x,\xi)|\leq C_{\alpha\beta}|\xi|^{m- |\beta|+|\alpha|} \] for all \(\alpha\), \(\beta\), with constants \(C_{\alpha\beta}\) independent of \(x\in\mathbb{R}^n\), \(\xi\in \mathbb{R}^n/0\). The corresponding class is denoted \(\dot S^m_{1,1}\). Examples of symbols in \(\dot S^m_{1,1}\) independent of \(x\) are the reciprocals of elliptic homogeneous polynomials. The authors prove that the corresponding pseudodifferential operators are bounded in the frame of the homogeneous Triebel-Lizorkin spaces \(\dot F^{\alpha,q}_p(\mathbb{R}^n)\) and Besov-Lipschitz spaces \(\dot B^{\alpha, q}_p(\mathbb{R}^n)\).
scientific article; zbMATH DE number 1548318