A generalized pseudo-differential operator on Gel'fand--Shilov space and Sobolev space
Pseudodifferential operators are linear operators \(A\) defined by expressions of the type \[ (Af)(x)= A(x, D)f(x)= (2\pi)^{-{n\over 2}} \int_{\mathbb{R}^n} e^{i(x,\xi)} a(x,\xi)\widehat f(\xi)\,d\xi, \] where \(f(x)\) has support in \(\Omega\subset \mathbb{R}^n\), while \(\widehat f(\xi)\) is its Fourier transform. The function \(a(x,\xi)\), satisfying suitable estimates in \(\Omega_x\times \mathbb{R}^n_\xi\), is called the symbol of operator \(A\). The class \(S^m_\lambda(\Omega)\), with \(\Omega\) an open set in \(\mathbb{R}^n\), consists of \(a(x,\xi)\in C^\infty(\Omega\times \mathbb{R}^n)\) satisfying conditions \[ |D^\alpha_x D^\beta_\xi a(x,\xi)|\leq c^{|\alpha|+|\beta|+ 1}\alpha!\beta![\lambda(\xi)]^{m-|\beta|} \] for every \(\alpha\), \(\beta\), \(x\in K\Subset\Omega\), \(\xi\in\mathbb{R}^n\) with \(\lambda(\xi)\geq B|\beta|\). For \(a(x,\xi)\in S^m_\lambda(\Omega)\), the pseudodifferential operator \(A\) is a continuous linear mapping from \(C^\infty_0(\Omega)\) to \(C^\infty(\Omega)\). The authors discuss extensions to more general classes of symbols \(S(m,\lambda,\mathbb{R}^{2n})\).
generalised pseudodifferential operators
scientific article; zbMATH DE number 5151970
scientific article