Infinite pseudo-differential operators on \(W_M(R^n)\) space (Q2872889)

The authors study the boundedness of pseudo-differential operators on the spaces \(W_M(\mathbb{R}^n)\) of \textit{I. M. Gel'fand} and \textit{G. E. Shilov} [Generalized functions. Vol.\,3: Theory of differential equations. New York-San Francisco-London: Academic Press (1967; Zbl 0355.46017)]. Denoting by \(\Omega\) the Young dual of the weight function \(M\), the symbols \(\sigma\) are assumed to satisfy NEWLINE\[NEWLINE|D^\alpha_x D^\beta_\eta \sigma(x,\eta)|\leq C_{\alpha\beta}(1+ |u|)^{m-|\beta|}\exp\Omega(t),NEWLINE\]NEWLINE where \(x\in\mathbb{R}^n\), \(\eta\in\mathbb{C}^n\), \(\eta= u+ it\). In particular, in the real domain the symbols belong to the standard class \(S^m_{1,0}\). The authors prove that NEWLINE\[NEWLINE\sigma(x,D): W_M(\mathbb{R}^n)\to W_M(\mathbb{R}^n)NEWLINE\]NEWLINE continuously. Related boundedness results are obtained in the frame of Gel'fand-Shilov ultradistributions and \(L^p\)-Gevrey spaces.