Regularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods

Mele, Claudio; Oliaro, Alessandro

doi:10.1016/j.jde.2021.03.034

Cited by 5 publications

(3 citation statements)

References 13 publications

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“…See [4] for a study of ω-regularity of linear partial differential operators with polynomial coefficients using quadratic transformations (cf. [21] for the non-isotropic case).…”

Section: Parametrices and ω-Regularitymentioning

confidence: 99%

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

Asensio

Jornet

2019

RACSAM

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We develop a theory of pseudodifferential operators of infinite order for the global classes Sω of ultradifferentiable functions in the sense of Björck, 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. IntroductionThe local theory of pseudodifferential operators grew out of the study of singular integral operators, and developed after 1965 with the systematic studies of , Hörmander [13], and others. Since then, several authors have studied pseudodifferential operators of finite or infinite order in Gevrey classes in the local sense; we mention, for instance, [12,21]. We refer to Rodino [19] for an excellent introduction to this topic, and the references therein.Gevrey classes are spaces of (non-quasianalytic) ultradifferentiable functions in between real analytic and C ∞ functions. The study of several problems in general classes of ultradifferentiable functions has received much attention in the last 60 years. Here, we will work with ultradifferentiable functions as defined by Braun, Meise and Taylor [5], which define the classes in terms of the growth of the derivatives of the functions, or in terms of the growth of their Fourier transforms (see, for example, Komatsu [15] and Björck [2], or [5], for two different points of view to define spaces of ultradifferentiable functions and ultradistributions; and [4] for a comparison between the classes defined in [5] and [15]).In [10], a full theory of pseudodifferential operators in the local sense is developed for ultradifferentiable classes of Beurling type as in [5], and it is proved that the corresponding operators are ω-pseudo-local, and the product of two operators is given in terms of a suitable symbolic calculus. In [9,11] the same authors construct a parametrix for such operators and study the action of the wave front set on them (see also [1] for a different point of view). On the other hand, very recently, Prangoski [18] studies pseudodifferential operators of global type and infinite order for ultradifferentiable classes of Beurling and Roumieu type in the sense of Komatsu, and later,in [8], a parametrix is constructed for such operators. See [18,17] and the references therein for more examples of pseudodifferential operators in global classes (e.g., in Gelfand-Shilov classes).Our aim is to study pseudodifferential operators of global type and infinite order in classes of ultradifferentiable functions of Beurling type as introduced in [5]. Hence, the right setting is the class S ω as introduced by Björck [2]. We follow the lines of Prangoski [18] and Shubin [20], but from the point of view of [10], in such a way that our proofs simplify the ones of [18]. Moreover, we clarify the role of some kind of entire functions [6,16] that become crucial throughout the text.The paper is organized as follows. First, in Section 2, we introduce our setting, we give some useful results about the class S ...

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“…See [4] for a study of ω-regularity of linear partial differential operators with polynomial coefficients using quadratic transformations (cf. [21] for the non-isotropic case).…”

Section: Parametrices and ω-Regularitymentioning

confidence: 99%

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

Asensio

Jornet

2019

RACSAM

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“…See [4] for a study of ω-regularity of linear partial differential operators with polynomial coefficients using quadratic transformations (cf. [21] for the non-isotropic case). We use the well-known method of the construction of a parametrix for the symbol of the operator, using symbolic calculus.…”

Section: Parametrices and ω-Regularitymentioning

confidence: 99%

Quantizations and Global Hypoellipticity for Pseudodifferential Operators of Infinite Order in Classes of Ultradifferentiable Functions

Asensio

2022

Mediterr. J. Math.

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We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global ω-hypoellipticity and global ω-regularity of these classes of pseudodifferential operators.

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“…See [13] for a study of ω-regularity of linear partial differential operators with polynomial coefficients using quadratic transformations (cf. [53] for the non-isotropic case).…”

Section: Global Regularitymentioning

confidence: 99%

Global pseudodifferential operators in spaces of ultradifferentiable functions

López¹

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Voldria començar aquesta secció donant les gràcies, per descomptat, al director de tesi David Jornet per tot l'esforç i ajuda, no només durant aquest període, sinó també des que vaig estudiar el màster. Et voldria donar les gràcies especialment per haver sentit que només tu em podies ajudar a resoldre els problemes que anaven sorgint. Crec que mai t'ho he dit, i em sembla un bon lloc per dir-t'ho.Vull agrair especialment a José Bonet el seu recolçament constant i les seues bones paraules durant aquest temps. I també m'agradaria donar les gràcies a Carmen Fernández i Antonio Galbis per l'interés en els resultats de la tesi i els seus ànims. A tots us done les gràcies pels comentaris que han fet millorar la qualitat d'aquesta tesi.Aquesta tesi ha estat desenvolupada a l'Institut Universitari de Matemàtica Pura i Aplicada. Voldria donar les gràcies als directors José Calabuig i Alfred Peris i als seus equips directius per la seua amabilitat i comprensió amb mi. Quisiera también agradecer a Minerva por su inestimable ayuda, a Nuria por sus consejos y comentarios, a Víctor por los buenos momentos, a Nazaret por los viajes, y a los compañeros y becarios que me han ofrecido su colaboración, ayuda, y, en definitiva, que han amenizado este proceso en la sala del café.I am very thankful to Armin Rainer for accepting me to visit the Fakultät für Mathematik, and I am grateful to Gerhard Schindl for his predisposition to collaborate with me. Despite the time we faced, I enjoyed it a lot. As you once told me, I may tell my 'unusual visit in Vienna 2020' to my greatv grandchildren within many years. I would like to thank all my colleagues, who created a great atmosphere in the faculty during that enjoyable time.Vorrei ringraziare Chiara Boiti, non soltanto per la disponibilità e collaborazione ma sopratutto per le belle passeggiate della domenica. Mi piacerebbe inoltre esprimere la mia gratitudine ad Alessandro Oliaro per l'incoraggiamento e l'interese dimostratimi. Grazie davvero a tutti e due del prezioso aiuto.Durante estos años he tenido la fortuna de impartir docencia en l'Escola Tècnica Superior d'Enginyeria Informàtica y en l'Escola Tècnica Superior d'Enginyeria Industrial. A todos los profesores de las escuelas les doy las gracias por su ayuda, y por echarme unos cuantos años menos de los que tengo. En especial, a Antonio Hervás y María Carmen Pedraza les agradezco su inestimable ayuda por ponerme las cosas tan fáciles. A Elena Alemany y a Carles Bivià su simpatía y su tenacidad durante este difícil último curso. A todos los compañeros de las asignaturas os doy, de corazón, las gracias por haberme hecho sentir uno más.En darrer lloc, i no per això menys important, voldria tindre un record per als familiars i amics que m'han ajudat i animat durant aquest camí. No ha sigut fàcil. Moltes gràcies. vi de ondas con el frente de ondas de Gabor definido mediante la STFT y damos aplicaciones a la regularidad de las cuantizaciones de Weyl.x ResumEn aquesta tesi estudiem operadors pseudodiferencials, que són operadors integrals de...

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Regularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods

Cited by 5 publications

References 13 publications

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

Quantizations and Global Hypoellipticity for Pseudodifferential Operators of Infinite Order in Classes of Ultradifferentiable Functions

Global pseudodifferential operators in spaces of ultradifferentiable functions

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