Abstract
We present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions to sums of squares of vector fields satisfying the Hörmander condition.
Similar content being viewed by others
References
Albano, P., Bove, A.: Wave front set of solutions to sums of squares of vector fields. Mem. Am. Math. Soc 221, 1039 (2013)
Albano, P., Bove, A., Chinni, G.: Minimal microlocal gevrey regularity for “sums of squares”. Int. Math. Res. Not. 12, 2275–2302 (2009)
Albano, P., Bove, A., Mughetti, M.: Analytic hypoellipticity for sums of squares and the treves conjecture. J. Funct. Anal. 274(10), 2725–2753 (2018)
Baouendi, M.S., Goulaouic, C.: Nonanalytic-hypoellipticity for some degenerate operators. Bull. Am. Math. Soc. 78, 483–486 (1972)
Bender, C.M., Wang, Q.: A class of exactly solvable eigenvalue problems. J. Phys. A 34, 9835–9847 (2001)
Berezin, F.A., Shubin, M.A.: The Schrödinger Equation, Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991)
Bolley, P., Camus, J., Nourrigat, J.: La condition de Hörmander-Kohn pour les opérateurs pseudo-différentiels. Commun. Partial Differ. Equ. 7(2), 197–221 (1982)
Bove, A., Mughetti, M.: Analytic and Gevrey hypoellipticity for a class of pseudodifferential operators in one variable. J. Differ. Equ. 255(4), 728–758 (2013)
Bove, A., Mughetti, M.: On a new method of proving Gevrey hypoellipticity for certain sums of squares. Adv. Math. 293, 146–220 (2016)
Bove, A., Mughetti, M.: Analytic hypoellipticity for sums of squares and the treves conjecture II. Anal. PDE 10(7), 1613–1635 (2017)
Bove, A., Mughetti, M.: Analytic hypoellipticity for sums of squares in the presence of symplectic non treves strata. J. Inst. Math. Jussieu. https://doi.org/10.1017/S1474748018000580
Bove, A., Mughetti, M.: Gevrey Regularity for a Class of Sums of Squares of Monomial Vector Fields, preprint, (2019)
Bove, A., Tartakoff, D.S.: Optimal non-isotropic Gevrey exponents for sums of squares of vector fields. Commun. Partial Differ. Equ. 22(7–8), 1263–1282 (1997)
Bove, A., Tartakoff, D.S.: Gevrey hypoellipticity for sums of squares of vector fields in \(\mathbb{R}^{2}\) with quasi-homogeneous polynomial vanishing. Indiana Univ. Math. J. 64, 613–633 (2015)
Bove, A., Treves, F.: On the Gevrey hypo-ellipticity of sums of squares of vector fields. Ann. Inst. Fourier (Grenoble) 54, 1443–1475 (2004)
Chanillo, S., Helffer, B., Laptev, A.: Nonlinear eigenvalues and analytic hypoellipticity. J. Funct. Anal. 209(2), 425–443 (2004)
Chinni, G.: (Semi)-global analytic hypoellipticity for a class of “sums of squares” which fail to be locally analytic hypoelliptic, to appear in Proc. Amer. Math. Soc., https://doi.org/10.1090/proc/14464
Chinni, G.: On the sharp Gevrey regularity for a generalization of the Métivier operator, preprint (2019)
Christ, M.: Certain sums of squares of vector fields fail to be analytic hypoelliptic. Commun. Partial Differ. Equ. 16, 1695–1707 (1991)
Christ, M.: Intermediate optimal Gevrey exponents occur. Commun. Partial Differ. Equ. 22(3–4), 359–379 (1997)
Cordaro, P.D., Hanges, N.: A new proof of Okaji’s theorem for a class of sum of squares operators. Ann. Inst. Fourier (Grenoble) 59(2), 595–619 (2009)
Derridj, M.: Un problème aux limites pour une classe d’opérateurs du second ordre hypoelliptiques. Ann. Inst. Fourier (Grenoble) 21(4), 99–148 (1971)
Ehrenpreis, L.: Solutions of some problems of division IV. Am. J. Math 82, 522–588 (1960)
Fefferman, C., Phong, D.H.: The uncertainty principle and sharp Gårding inequalities. Commun. Pure Appl. Math. 34, 285–331 (1981)
Grigis, A., Sjöstrand, J.: Front d’onde analytique et sommes de carrés de champs de vecteurs. Duke Math. J. 52(1), 35–51 (1985)
Grušin, V.V.: On a class of Hypoelliptic operators. Math. USSR Sb. 12, 458–476 (1970)
Grušin, V.V.: On a class of elliptic pseudodifferential operators degenerating at a submanifold. Mat. Sbornik 84(2), 163–195 (1971)
Gundersen, G.G.: A class of anharmonic oscillators whose eigenfunctions have no recurrence relations. Proc. Am. Math. Soc. 58, 109–113 (1976)
Hanges, N., Himonas, A.A.: Singular solutions for sums of squares of vector fields. Commun. Partial Differ. Equ. 16, 1503–1511 (1991)
Hanges, N., Himonas, A.A.: Singular solutions for a class of Grusin type operators. Proc. Am. Math. Soc. 124(5), 1549–1557 (1996)
Hanges, N., Himonas, A.A.: Non-analytic hypoellipticity in the presence of symplecticity. Proc. Am. Math. Soc. 126(2), 405–409 (1998)
Helffer, B.: Sur l’hypoellipticité des opérateurs pseudodifférentiels à caractéristiques multiples (perte de 3/2 dérivées). Mémoires de la S. M. F. 51–52, 13–61 (1977)
Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques. Astérisque 112, 9–197 (1984)
Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
Hörmander, L.: Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients. Commun. Pure Appl. Math. 24, 671–704 (1971)
Hörmander, L.: The Analysis of Partial Differential Operators, I. Springer, New York (1985)
Hörmander, L.: The Analysis of Partial Differential Operators. Springer Verlag III, New York (1985)
Hoshiro, T.: Failure of analytic hypoellipticity for some operators of \(X^{2}+Y^{2}\) type. J. Math. Kyoto Univ. 35(4), 569–581 (1995)
Kohn, J.J.: Pseudo-differential operators and non-elliptic problems, 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, pp. 158–165. Edizioni Cremonese, Roma (1968)
Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18, 443–492 (1965)
Łojasiewicz, S.: Ensembles semi-analytiques, cours à Orsay, saisie en LaTeX par M. Coste, Juillet (1965)
Métivier, G.: Une Classe d’Opérateurs Non Hypoelliptiques Analytiques. Indiana Univ. Math. J. 29, 823–860 (1980)
Métivier, G.: Hypoellipticité analitique sur des groupes nilpotents de rang 2. Duke Math. J. 47, 195–221 (1980)
Métivier, G.: Non-hypoellipticité Analytique pour \(D_{x}^{2} + (x^{2} + y^{2}) D_{y}^{2}\), C. R. Acad. Sci. Paris Sér. I Math. 292(7), 401–404 (1981)
Métivier, G.: Analytic hypoellipticity for operators with multiple characteristics. Comm. Partial Differ. Equ. 6(1), 1–90 (1981)
Mughetti, M.: Regularity properties of a double characteristics differential operator with complex lower order terms. J. Pseudo-Differ. Oper. Appl. 5(3), 343–358 (2014)
Mughetti, M.: On the spectrum of an anharmonic oscillator. Trans. Am. Math. Soc. 367, 835–865 (2015)
Ōkaji, T.: Analytic hypoellipticity for operators with symplectic characteristics. J. Math. Kyoto Univ. 25(3), 489–514 (1985)
Oleĭnik, O. A.: On the analyticity of solutions of partial differential equations and systems, Colloque International CNRS sur les Équations aux Dérivées Partielles Linéaires (Univ. Paris- Sud, Orsay, 1972), pp. 272–285. Astérisque, 2 et 3. Societé Mathématique de France, Paris (1973)
Oleĭnik, O.A., Radkevič, E.V.: The analyticity of the solutions of linear partial differential equations , (Russian) Mat. Sb. (N.S.), 90(132), 592–606 (1973)
Rothschild, L., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3–4), 247–320 (1976)
Sjöstrand, J.: Analytic wavefront set and operators with multiple characteristics. Hokkaido Math. J. 12, 392–433 (1983)
Sjöstrand, J.: Singularités analytiques microlocales. Astérisque 95 (1982)
Tartakoff, D.S.: On the local real analyticity of solutions to \(\Box _{b}\) and the \(\bar{\partial }\)-Neumann problem. Acta Math. 145, 117–204 (1980)
Tartakoff, D.S.: Elementary proofs of analytic hypoellipticity for \(\Box _{b}\) and the \(\overline{\partial }\)-Neumann problem, Analytic solutions of partial differential equations (Trento, 1981), 85–116; Astérisque, 89–90. Soc. Math. France, Paris (1981)
Trèves, F.: Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the \(\bar{\partial }\)-Neumann problem. Commun. Partial Differ. Equ. 3(6–7), 475–642 (1978)
Treves, F.: Introduction to Pseudodifferential and Fourier Integral Operators, vol. 1. Plenum Press, New York (1980)
Treves, F., Symplectic geometry and analytic hypo-ellipticity, in Differential equations, La Pietra,: (Florence). In: Proc. Sympos. Pure Math. 65, Amer. Math. Soc. Providence, vol. 1999, pp. 201–219 (1996)
Treves, F.: On the analyticity of solutions of sums of squares of vector fields, Phase space analysis of partial differential equations, Bove, Colombini, Del Santo ed.’s, pp. 315–329, Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA (2006)
Treves, F.: Aspects of Analytic PDE, book in preparation
Zworski, M.: Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of Nick Hanges.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bove, A., Mughetti, M. Analytic regularity for solutions to sums of squares: an assessment. Complex Anal Synerg 6, 18 (2020). https://doi.org/10.1007/s40627-020-00055-8
Published:
DOI: https://doi.org/10.1007/s40627-020-00055-8