Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions (Q2331718)

The authors develop a theory of pseudodifferential operators (\(\Psi\)do) of infinite order for global classes of ultradifferentiable functions. They consider the class \(S_\omega({\mathbb R}^d)\) introduced by \textit{G. Björck} [Ark. Mat. 6, 351--407 (1966; Zbl 0166.36501)], which is defined in terms of a weight function \(\omega\) and consists of all \(u\in L^1({\mathbb R}^d)\) such that \(u\) and its Fourier transform \(\widehat{u}\) are smooth functions and, for each \(\lambda>0\) and \(\alpha\in{\mathbb N}^d_0,\) \[ \sup_{x\in{\mathbb R}^d}e^{\lambda\omega(x)}|D^\alpha u(x)|<+\infty,\ \ \sup_{\xi\in{\mathbb R}^d}e^{\lambda\omega(\xi)}|D^\alpha\widehat{u}(\xi)|<+\infty.\] Section~2 explains the basic properties of this class of functions and Section~3 introduces the appropriate classes of symbols and amplitudes. For a given amplitude \(a(x,y,\xi)\), Theorem 3.7 proves that the iterated integral \[ A(f)(x) := \int_{\mathbb{R}^d} \Big(\int_{\mathbb{R}^d} e^{i(x-y)\xi} a(x,y,\xi) f(y)\, dy \Big) d\xi\] gives a continuous linear map \(A:S_\omega({\mathbb R}^{d}) \to S_\omega({\mathbb R}^{d}).\) Then it is proved that the previous operator can be extended to the corresponding class of tempered ultradistributions. Section~4 develops the symbolic calculus, which permits to address the composition of operators in Section~5. To achieve this goal, it is important to know the behaviour of the kernel of a \(\Psi\)do outside the diagonal. This is the content of Theorem 5.2. The estimates obtained in that theorem are valid in the complement of a band around the diagonal, which is an improvement over previously known results even in the Gevrey setting, where only estimates in the complement of a conical neighbourhood of the diagonal were known.