Geodesic
Differential operators of infinite order with symbols of Gevrey class
Izvestiya. Mathematics , Tome 39 (1992) no. 2, pp. 1085-1096.

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Differential operators of infinite order with symbols that are infinitely differentiable functions (of Gevrey class) in some domain in $\mathbf R^n$ are considered. With the help of such operators a generalized Fourier transform of infinitely differentiable functions is constructed. For these operators a criterion for the of solvability of the Cauchy problem in some subclasses of exponential functions is proved. The results are similar to those of Dubinskii [1] for differential operators of infinite order with symbols analytic in some Runge domain in $\mathbf C^n$. 
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     title = {Differential operators of infinite order with symbols of {Gevrey} class},
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O. V. Odinokov. Differential operators of infinite order with symbols of Gevrey class. Izvestiya. Mathematics , Tome 39 (1992) no. 2, pp. 1085-1096. http://geodesic.mathdoc.fr/item/IM2_1992_39_2_a6/

[1] Dubinskii Yu. A., “Algebra psevdodifferentsialnykh operatorov s kompleksnymi argumentami i ee prilozheniya”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 29, VINITI AN SSSR, M., 1986, 109–150 | MR

[2] Komatsu H., “Ultradistributions. I. Structure theorems and a characterization”, Journal Fac. Sci. Univ. Tokyo Sect. IA Math., 20:1 (1973), 25–105 | MR | Zbl

[3] Odinokov O. V., Differentsialnye operatory beskonechnogo poryadka i zadacha Koshi v kompleksnoi oblasti, Dis. $\dots$ kand. fiz.-matem. nauk, M., 1988