A simple proof of Kotake-Narasimhan theorem in some classes of ultradifferentiable functions

[EN] We give a simple proof of a general theorem of Kotake-Narasimhan for elliptic operators in the setting of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We follow the ideas of Komatsu. Based on an example of Metivier, we also show that the ellipticity is a necessary condition for the theorem to be true.C. Boiti and D. Jornet were partially supported by the INdAM-GNAMPA Projects 2014 and 2015. D. Jornet was partially supported by MINECO, Project MTM2013-43540-PBoiti, C.; Jornet Casanova, D. (2017). A simple proof of Kotake-Narasimhan theorem in some classes of ultradifferentiable functions. Journal of Pseudo-Differential Operators and Applications. 8(2):297-317. https://doi.org/10.1007/s11868-016-0163-yS29731782Boiti, C., Jornet, D.: The problem of iterates in some classes of ultradifferentiable functions. Oper. Theory Adv. Appl. Birkhauser Basel 245, 21–33 (2015)Boiti, C., Jornet, D.: A characterization of the wave front set defined by the iterates of an operator with constant coefficients. arXiv:1412.4954Boiti, C., Jornet, D., Juan-Huguet, J.: Wave front set with respect to the iterates of an operator with constant coefficients. Abstr. Appl. Anal. 2014, 1–17 Article ID 438716 (2014). doi: 10.1155/2014/438716Bolley, P., Camus, J., Mattera, C.: Analyticité microlocale et itérés d’operateurs hypoelliptiques. Séminaire Goulaouic-Schwartz, 1978–1979, Exp No. 13, École Polytech, PalaiseauBonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways of define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990)Fernández, C., Galbis, A.: Superposition in classes of ultradifferentiable functions. Publ. Res. I Math. Sci. 42(2), 399–419 (2006)Jornet Casanova, D.: Operadores Pseudodiferenciales en Clases no Casianalíticas de Tipo Beurling. Universitat Politècnica de València (2004). doi: 10.4995/Thesis/10251/54953Juan-Huguet, J.: Iterates and hypoellipticity of partial differential operators on non-quasianalytic classes. Integr. Equ. Oper. Theory 68, 263–286 (2010)Juan-Huguet, J.: A Paley–Wiener type theorem for generalized non-quasianalytic classes. Stud. Math. 208(1), 31–46 (2012)Komatsu, H.: A characterization of real analytic functions. Proc. Jpn Acad. 36, 90–93 (1960)Komatsu, H.: On interior regularities of the solutions of principally elliptic systems of linear partial differential equations. J. Fac. Sci. Univ. Tokyo Sect. 1, 9, 141–164 (1961)Komatsu, H.: A proof of Kotaké and Narasimhan’s theorem. Proc. Jpn Acad. 38(9), 615–618 (1962)Kotake, T., Narasimhan, M.S.: Regularity theorems for fractional powers of a linear elliptic operator. Bull. Soc. Math. Fr. 90, 449–471 (1962)Kumano-Go, H.: Pseudo-Differential Operators. The MIT Press, Cambridge, London (1982)Langenbruch, M.: P-Funktionale und Randwerte zu hypoelliptischen Differentialoperatoren. Math. Ann. 239(1), 55–74 (1979)Langenbruch, M.: Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partielle Differentialgleichungen. J. Reine Angew. Math. 311/312, 57–79 (1979)Langenbruch, M.: On the functional dimension of solution spaces of hypoelliptic partial differential operators. Math. Ann. 272, 217–229 (1985)Langenbruch, M.: Bases in solution sheaves of systems of partial differential equations. J. Reine Angew. Math. 373, 1–36 (1987)Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 3. Dunod, Paris (1970)Métivier, G.: Propriété des itérés et ellipticité. Commun. Part. Differ. Eq. 3(9), 827–876 (1978)Nelson, E.: Analytic vectors. Ann. Math. 70, 572–615 (1959)Newberger, E., Zielezny, Z.: The growth of hypoelliptic polynomials and Gevrey classes. Proc. Am. Math. Soc. 39(3), 547–552 (1973)Oldrich, J.: Sulla regolarità delle soluzioni delle equazioni lineari ellittiche nelle classi di Beurling. (Italian) Boll. Un. Mat. Ital. (4) 2, 183–195 (1969)Petzsche, H.-J., Vogt, D.: Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions. Math. Ann. 267(1), 17–35 (1984

Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

[EN] We use techniques from time-frequency analysis to show that the space S(omega )of rapidly decreasing omega-ultradifferentiable functions is nuclear for every weight function omega(t) = o(t) as t tends to infinity. Moreover, we prove that, for a sequence (M-p)(p) satisfying the classical condition (M1) of Komatsu, the space of Beurling type S-(M)p when defined with L-2 norms is nuclear exactly when condition (M2)' of Komatsu holds.We thank the reviewer very much for the careful reading of our manuscript and the comments to improve the paper. The first three authors were partially supported by the Project FFABR 2017 (MIUR), and by the Projects FIR 2018 and FAR 2018 (University of Ferrara). The first and third authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The research of the second author was partially supported by the project MTM2016-76647-P and the grant BEST/2019/172 from Generalitat Valenciana. The fourth author is supported by FWF-project J 3948-N35.Boiti, C.; Jornet Casanova, D.; Oliaro, A.; Schindl, G. (2021). Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis. Collectanea mathematica. 72(2):423-442. https://doi.org/10.1007/s13348-020-00296-0S423442722Asensio, V., Jornet, D.: Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4), 3477–3512 (2019)Aubry, J.-M.: Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices. J. London Math. Soc. 2(78), 392–406 (2008)Björck, G.: Linear partial differential operators and generalized distributions. Ark. Mat. 6(21), 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, 920–944 (2017)Boiti, C., Jornet, D., Oliaro, A.: The Gabor wave front set in spaces of ultradifferentiable functions. Monatsh. Math. 188(2), 199–246 (2019)Boiti, C., Jornet, D., Oliaro, A.: About the nuclearity of $$\cal{S}_{(M_{p})}$$ and $$\cal{S}_{\omega }$$. In: Boggiatto, P., et al. (eds.) Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis, pp. 121–129. Birkhäuser, Cham (2020)Boiti, C., Jornet, D., Oliaro, A.: Real Paley-Wiener theorems in spaces of ultradifferentiable functions. J. Funct. Anal. 278(4), 108348 (2020)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. Result. Math. 17, 206–237 (1990)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators on non-quasianalytic classes of Beurling type. Studia Math. 167(2), 99–131 (2005)Fernández, C., Galbis, A., Jornet, D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Anal. Appl. 340(2), 1153–1170 (2008)Franken, U.: Weight functions for classes of ultradifferentiable functions. Results Math. 25, 50–53 (1994)Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)Gröchenig, K., Leinert, M.: Wiener’s Lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17(1), 1–18 (2004)Gröchenig, K., Zimmermann, G.: Spaces of Test Functions via the STFT. J. Funct. Spaces Appl. 2(1), 25–53 (2004)Heinrich, T., Meise, R.: A support theorem for quasianalytic functionals. Math. Nachr. 280(4), 364–387 (2007)Hörmander, L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser, Boston (1994)Janssen, A.J.E.M.: Duality and Biorthogonality for Weyl-Heisenberg Frames. J. Fourier Anal. Appl. 1(4), 403–436 (1995)Komatsu, H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect IA Math. 20, 25–105 (1973)Langenbruch, M.: Hermite functions and weighted spaces of generalized functions. Manuscripta Math. 119(3), 269–285 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Petzsche, H.J.: Die nuklearität der ultradistributionsräume und der satz vom kern I. Manuscripta Math. 24, 133–171 (1978)Pietsch, A.: Nuclear Locally Convex Spaces. Springer, Berlin (1972)Pilipović, S., Prangoski, B., Vindas, J.: On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution. J. Math. Pures Appl. 116, 174–210 (2018)Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific Publishing Co. Inc, River Edge, NJ (1993)Rodino, L., Wahlberg, P.: The Gabor wave front set. Monatsh. Math. 173, 625–655 (2014)Schmets, J., Valdivia, M.: Analytic extension of ultradifferentiable Whitney jets. Collect. Math. 50(1), 73–94 (1999

Global regularity in ultradifferentiable classes

Se estudia la w-regularidad de soluciones de ciertos operadores que son globalmente hipoelípticos en el toro N-dimensional. Se aplican estos resultados para probar la w-regularidad global de ciertas clases de sublaplacianos. En este sentido, se extiende trabajo previo en el contexto de la clases analíticas y de Gevrey. Se dan varios ejemplos de w-hipoelipticidad local y global.The research of the authors was partially supported by MEC and FEDER Project MTM2010-15200.Albanese, AA.; Jornet Casanova, D. (2014). Global regularity in ultradifferentiable classes. Annali di Matematica Pura ed Applicata. 193(2):369-387. https://doi.org/10.1007/s10231-012-0279-5S3693871932Albanese A.A., Jornet D., Oliaro A.: Quasianalytic wave front sets for solutions of linear partial differential operators. Integr. Equ. Oper. Theory 66, 153–181 (2010)Albanese A.A., Jornet D., Oliaro A.: Wave front sets for ultradistribution solutions of linear pertial differential operators with coefficients in non-quasianalytic classes. Math. Nachr. 285, 411–425 (2012)Albanese A.A., Zanghirati L.: Global hypoellipticity and global solvability in Gevrey classes on the n–dimensional torus. J. Differ. Equ. 199, 256–268 (2004)Albanese A.A., Popivanov P.: Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions. Ann. Mat. Pura e Appl. 185, 395–409 (2006)Albanese A.A., Popivanov P.: Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators. Ann. Univ. Ferrara 52, 65–81 (2006)Baouendi M.S., Goulaouic C.: Nonanalytic–hypoellipticity for some degenerate elliptic operators. Bull. Am. Math. Soc. 78, 483–486 (1972)Bergamasco A.P.: Remarks about global analytic hypoellipticity. Trans. Am. Math. Soc. 351, 4113–4126 (1999)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, 425–444 (2007)Braun R.W., Meise R., Taylor B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17, 206–237 (1990)Chen W., Chi M.Y.: Hypoelliptic vector fields and almost periodic motions on the torus $${\mathbb{T}^n}$$ . Commun. Part. Differ. Equ. 25, 337–354 (2000)Christ M.: Certain sums of squares of vector fields fail to be analytic hypoelliptic. Commun. Part. Differ. Equ. 16, 1695–1707 (1991)Christ M.: A class of hypoelliptic PDE admitting non-analytic solutions. Contemp. Math. Symp. Complex Anal. 137, 155–168 (1992)Christ M.: Intermediate optimal Gevrey exponents occur. Commun. Part. Differ. Equ. 22, 359–379 (1997)Cordaro P.D., Himonas A.A.: Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus. Math. Res. Lett. 1, 501–510 (1994)Cordaro P.D., Himonas A.A.: Global analytic regularity for sums of squares of vector fields. Trans. Am. Math. Soc. 350, 4993–5001 (1998)Dickinson D., Gramchev T., Yoshino M.: Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena. Proc. Edinb. Math. Soc. 45, 731–759 (2002)Ferńandez C., Galbis A., Jornet D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Appl. Math. 340, 1153–1170 (2008)Gramchev T., Popivanov P., Yoshino M.: Some note on Gevrey hypoellipticity and solvability on torus. J. Math. Soc. Jpn. 43, 501–514 (1991)Gramchev T., Popivanov P., Yoshino M.: Some examples of global Gevrey hypoellipticity and solvability. Proc. Jpn. Acad. 69, 395–398 (1993)Gramchev T., Popivanov P., Yoshino M.: Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients. Rend. Sem. Univ. Pol. Torino 51, 145–172 (1993)Greenfield S., Wallach N.: Global hypoellipticity and Liouville numbers. Proc. Am. Math. Soc. 31, 112–114 (1972)Greenfield S.: Hypoelliptic vector fields and continued fractions. Proc. Am. Math. Soc. 31, 115–118 (1972)Hanges N., Himonas A.A.: Singular solutions for sums of squares of vector fields. Commun. Part. Differ. Equ. 16, 1503–1511 (1991)Hanges N., Himonas A.A.: Analytic hypoellipticity for generalized Baouendi–Goulaouic operators. J. Funct. Anal. 125, 309–325 (1994)Helfer B.: Conditions nécessaires d’hypoanalyticité puor des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué. J. Differ. Equ. 44, 460–481 (1982)Himonas A.A.: On degenerate elliptic operators of infinite type. Math. Z. 220, 449–460 (1996)Himonas A.A.: Global analytic and Gevrey hypoellipticity of sublaplacians under diophantine conditions. Proc. Am. Math. Soc. 129, 2001–2007 (2000)Himonas A.A., Petronilho G.: Global hypoellipticity and simultaneous approximability. J. Funct. Anal. 170, 356–365 (2000)Himonas A.A., Petronilho G.: Propagation of regularity and global hypoellipticity. Mich. Math. J. 50, 471–481 (2002)Himonas A.A., Petronilho G.: On Gevrey regularity of globally C ∞ hypoelliptic operators. J. Differ. Equ. 207, 267–284 (2004)Himonas A.A., Petronilho G.: On C ∞ and Gevrey regularity of sublaplacians. Trans. Am. Math. Soc. 358, 4809–4820 (2006)Himonas A.A., Petronilho G., dos Santos L.A.C.: Regularity of a class of subLaplacians on the 3–dimensional torus. J. Funct. Anal. 240, 568–591 (2006)Hörmander L.: Hypoelliptic second order differential equations. Acta Mat. 119, 147–171 (1967)Langenbruch M.: Ultradifferentiable functions on compact intervals. Math. Nachr. 140, 109–126 (1989)Meise R.: Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals. J. Reine Angew. Math. 363, 59–95 (1985)The Lai P., Robert D.: Sur un probléme aus valeurs propres non linéaire. Israel J. Math. 36, 169–186 (1980)Petronilho G.: On Gevrey solvability and regularity. Math. Nachr. 282, 470–481 (2009)Petzsche H.-J.: Die Nuklearität der Ultradistributionsräume und der Satz von Kern I. Manuscripta Math. 24, 133–171 (1978)Tartakoff D.: Global (and local) analyticity for second order operators constructed from rigid vector fields on product of tori. Trans. Am. Math. Soc. 348, 2577–2583 (1996