Optimal Gevrey classes for the existence of solution operators for linear partial differential operators in three variables (Q1883377)

The existence of a continuous linear solution operator is investigated for a constant coefficient linear partial differential operator acting on all infinitely differentiable functions or \(\omega\)-ultradifferentiable functions of Beurling type on \({\mathbb R}^n\). Herein the space of all \(\omega\)-ultradifferentiable functions of Beurling type on \({\mathbb R}^n\) is defined as \[ \begin{multlined} {\mathcal E}_\omega({\mathbb R}^n):= \left\{f\in C^\infty({\mathbb R}^n): \text{ for each }K\subset{\mathbb R}^n\text{ compact and each }\right.\\ \left. m\in \mathbb{N},\;\sup_{\alpha\in\mathbb{N}^n_0}\sup_{x\in K}|f^{(\alpha)}(x)|\exp\left(-m\varphi^* \left(\frac{|\alpha|}{m}\right)\right)<\infty\right\}\end{multlined} \] where \(\varphi^*(y) := \sup_{x>0}(xy - \omega(\exp(x)),\;y\geq 0\). Here, \(\omega\) is a weight function and an example of one is given which satisfies \(\int^\infty_0\frac{\omega(t)}{1+t^2}\,dt <\infty\). The paper then proves that there is an optimal weight function in the sense that a solution operator exists for a weight \(\sigma\) iff \(\omega = O(\sigma)\), provided that such an operator exists for at least one weight. The big \(O\) stands for its Landau definition. Furthermore, the optimal class is either a Gevrey class of rational exponent or the class of all infinitely differentiable functions.