Global $${\mathcal {M}}-$$ M - Hypoellipticity, Global $${\mathcal {M}}-$$ M - Solvability and Perturbations by Lowe
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Global M−Hypoellipticity, Global M−Solvability and Perturbations by Lower Order Ultradifferential Pseudodifferential Operators Igor Ambo Ferra1 · Gerson Petronilho2 · Bruno de Lessa Victor3 Received: 15 March 2020 / Accepted: 25 September 2020 / Published online: 18 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We introduce a new class of ultradifferentiable pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity, in ultradifferentiable classes, with a finite loss of derivatives of a system of pseudodifferential operators, is stable under perturbations by lower order pseudodifferential operators whose order depends on the loss of derivatives. The key point in our study is our definition of loss of derivatives. We also give an easy proof of the fact that if a system of pseudodifferential operators is globally M-hypoelliptic then its transpose is globally solvable in T N . Finally we present an application of our results. DM Keywords Ultradifferentiable classes · Global hypoellipticity with loss of derivatives · Solvability · System of M-ultradifferentiable pseudodifferential operators · N -dimensional torus · Lower order perturbations Mathematics Subject Classification 35H10 · 35H20 · 35S05
Communicated by Fabio Nicola.
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Gerson Petronilho [email protected] Igor Ambo Ferra [email protected] Bruno de Lessa Victor [email protected]
1
Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Alameda da Universidade, s/n, Bairro Anchieta, São Bernardo do Campo, SP 09606-045, Brazil
2
Departamento de Matemática, Universidade Federal de São Carlos (UFSCar), São Carlos, SP 13569-905, Brazil
3
Departamento de Matemática, Universidade Federal do Paraná, Curitiba, PR 82590-300, Brazil
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Journal of Fourier Analysis and Applications (2020) 26:85
1 Introduction If A is an elliptic properly supported pseudodifferential operator of order m in a loc implies u ∈ H loc C ∞ manifold then Au ∈ H(s) (s+m) (cf. Hörmader, [8], Theorem loc implies 18.1.29). There are cases where A is not elliptic but the condition Au ∈ H(s) loc loc loc u ∈ H(s+m−r ) for some r > 0. When Au ∈ H(s) implies that u ∈ H(s+m−r ) for some r ≥ 0 we say that the operator A is hypoelliptic with loss of r derivatives. There are many results in the literature on this problem. For instance, we refer the reader to Parmegiani [12], Ferra and Petronilho [6], Chinni and Cordaro [5] and to the references in these papers. Our main purpose in this paper is to study the above problem in the periodic ultradifferentiable frame, which includes the periodic Gevrey case, as P. D. Cordaro suggested to us. We start by recalling that in the paper “On Global Analytic and Gevrey Hypoellipticity on the Torus and the Métivier Inequality”, see [5], the authors G. Chinni and P. D. Cordaro introduced a new theory about analytic pseudodifferential operators on the N -dimensional torus T N . One question analyzed by them is the following: assum
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