Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

[EN] We develop a theory of pseudodifferential operators of infinite order for the global classes S. of ultradifferentiable functions in the sense of Bjorck, following the previous ideas given by Prangoski for ultradifferentiable classes in the sense of Komatsu. We study the composition and the transpose of such operators with symbolic calculus and provide several examples. The first author was partially supported by the project GV Prometeo 2017/102, and the second author by the project MTM2016-76647-P. This article is part of the PhD. Thesis of V. Asensio. The authors are very grateful to the two referees for the careful reading and their suggestions and comments, which improved the paper. Asensio, V.; Jornet Casanova, D. (2019). Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3477-3512. https://doi.org/10.1007/s13398-019-00710-8 Albanese, A.A., Jornet, D., Oliaro, A.: Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non-quasianalytic classes. Math. Nachr. 285(4), 411–425 (2012) Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6, 351–407 (1966) Boiti, C., Jornet, D., Oliaro, A.: Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms. J. Math. Anal. Appl. 446(1), 920–944 (2017) Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14(3), 425–444 (2007) Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17(3–4), 206–237 (1990) Braun, R.W.: An extension of Komatsu’s second structure theorem for ultradistributions. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40(2), 411–417 (1993) Cappiello, M.: Fourier integral operators of infinite order and applications to SG-hyperbolic equations. Tsukuba J. Math. 28(2), 311–361 (2004) Cappiello, M., Pilipović, S., Prangoski, B.: Parametrices and hypoellipticity for pseudodifferential operators on spaces of tempered ultradistributions. J. Pseudo-Differ. Oper. Appl. 5(4), 491–506 (2014) Fernández, C., Galbis, A., Jornet, D.: $$\omega $$-hypoelliptic differential operators of constant strength. J. Math. Anal. Appl. 297(2), 561–576 (2004). Special issue dedicated to John Horváth Fer