34 research outputs found
Tameness of holomorphic closure dimension in a semialgebraic set
Given a semianalytic set S in a complex space and a point p in S, there is a
unique smallest complex-analytic germ at p which contains the germ of S, called
the holomorphic closure of S at p. We show that if S is semialgebraic then its
holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic
filtration by the holomorphic closure dimension. As a consequence, every
semialgebraic subset of a complex vector space admits a semialgebraic
stratification into CR manifolds satisfying a strong version of the condition
of the frontier.Comment: Published versio
Obstructions to embeddability into hyperquadrics and explicit examples
We give series of explicit examples of Levi-nondegenerate real-analytic
hypersurfaces in complex spaces that are not transversally holomorphically
embeddable into hyperquadrics of any dimension. For this, we construct
invariants attached to a given hypersurface that serve as obstructions to
embeddability. We further study the embeddability problem for real-analytic
submanifolds of higher codimension and answer a question by Forstneri\v{c}.Comment: Revised version, appendix and references adde
Theorems on existence and global dynamics for the Einstein equations
This article is a guide to theorems on existence and global dynamics of
solutions of the Einstein equations. It draws attention to open questions in
the field. The local-in-time Cauchy problem, which is relatively well
understood, is surveyed. Global results for solutions with various types of
symmetry are discussed. A selection of results from Newtonian theory and
special relativity that offer useful comparisons is presented. Treatments of
global results in the case of small data and results on constructing spacetimes
with prescribed singularity structure or late-time asymptotics are given. A
conjectural picture of the asymptotic behaviour of general cosmological
solutions of the Einstein equations is built up. Some miscellaneous topics
connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living
Rev. Rel. 5 (2002)
Global regularity in ultradifferentiable classes
Se estudia la w-regularidad de soluciones de ciertos operadores que son globalmente hipoelĂpticos 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 analĂticas 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, 153â181 (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, 411â425 (2012)Albanese A.A., Zanghirati L.: Global hypoellipticity and global solvability in Gevrey classes on the nâdimensional torus. J. Differ. Equ. 199, 256â268 (2004)Albanese A.A., Popivanov P.: Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions. Ann. Mat. Pura e Appl. 185, 395â409 (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, 65â81 (2006)Baouendi M.S., Goulaouic C.: Nonanalyticâhypoellipticity for some degenerate elliptic operators. Bull. Am. Math. Soc. 78, 483â486 (1972)Bergamasco A.P.: Remarks about global analytic hypoellipticity. Trans. Am. Math. Soc. 351, 4113â4126 (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, 425â444 (2007)Braun R.W., Meise R., Taylor B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17, 206â237 (1990)Chen W., Chi M.Y.: Hypoelliptic vector fields and almost periodic motions on the torus . Commun. Part. Differ. Equ. 25, 337â354 (2000)Christ M.: Certain sums of squares of vector fields fail to be analytic hypoelliptic. Commun. Part. Differ. Equ. 16, 1695â1707 (1991)Christ M.: A class of hypoelliptic PDE admitting non-analytic solutions. Contemp. Math. Symp. Complex Anal. 137, 155â168 (1992)Christ M.: Intermediate optimal Gevrey exponents occur. Commun. Part. Differ. Equ. 22, 359â379 (1997)Cordaro P.D., Himonas A.A.: Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus. Math. Res. Lett. 1, 501â510 (1994)Cordaro P.D., Himonas A.A.: Global analytic regularity for sums of squares of vector fields. Trans. Am. Math. Soc. 350, 4993â5001 (1998)Dickinson D., Gramchev T., Yoshino M.: Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena. Proc. Edinb. Math. Soc. 45, 731â759 (2002)FerĆandez C., Galbis A., Jornet D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Appl. Math. 340, 1153â1170 (2008)Gramchev T., Popivanov P., Yoshino M.: Some note on Gevrey hypoellipticity and solvability on torus. J. Math. Soc. Jpn. 43, 501â514 (1991)Gramchev T., Popivanov P., Yoshino M.: Some examples of global Gevrey hypoellipticity and solvability. Proc. Jpn. Acad. 69, 395â398 (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, 145â172 (1993)Greenfield S., Wallach N.: Global hypoellipticity and Liouville numbers. Proc. Am. Math. Soc. 31, 112â114 (1972)Greenfield S.: Hypoelliptic vector fields and continued fractions. Proc. Am. Math. Soc. 31, 115â118 (1972)Hanges N., Himonas A.A.: Singular solutions for sums of squares of vector fields. Commun. Part. Differ. Equ. 16, 1503â1511 (1991)Hanges N., Himonas A.A.: Analytic hypoellipticity for generalized BaouendiâGoulaouic operators. J. Funct. Anal. 125, 309â325 (1994)Helfer B.: Conditions nĂ©cessaires dâhypoanalyticitĂ© puor des opĂ©rateurs invariants Ă gauche homogĂšnes sur un groupe nilpotent graduĂ©. J. Differ. Equ. 44, 460â481 (1982)Himonas A.A.: On degenerate elliptic operators of infinite type. Math. Z. 220, 449â460 (1996)Himonas A.A.: Global analytic and Gevrey hypoellipticity of sublaplacians under diophantine conditions. Proc. Am. Math. Soc. 129, 2001â2007 (2000)Himonas A.A., Petronilho G.: Global hypoellipticity and simultaneous approximability. J. Funct. Anal. 170, 356â365 (2000)Himonas A.A., Petronilho G.: Propagation of regularity and global hypoellipticity. Mich. Math. J. 50, 471â481 (2002)Himonas A.A., Petronilho G.: On Gevrey regularity of globally C â hypoelliptic operators. J. Differ. Equ. 207, 267â284 (2004)Himonas A.A., Petronilho G.: On C â and Gevrey regularity of sublaplacians. Trans. Am. Math. Soc. 358, 4809â4820 (2006)Himonas A.A., Petronilho G., dos Santos L.A.C.: Regularity of a class of subLaplacians on the 3âdimensional torus. J. Funct. Anal. 240, 568â591 (2006)Hörmander L.: Hypoelliptic second order differential equations. Acta Mat. 119, 147â171 (1967)Langenbruch M.: Ultradifferentiable functions on compact intervals. Math. Nachr. 140, 109â126 (1989)Meise R.: Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals. J. Reine Angew. Math. 363, 59â95 (1985)The Lai P., Robert D.: Sur un problĂ©me aus valeurs propres non linĂ©aire. Israel J. Math. 36, 169â186 (1980)Petronilho G.: On Gevrey solvability and regularity. Math. Nachr. 282, 470â481 (2009)Petzsche H.-J.: Die NuklearitĂ€t der UltradistributionsrĂ€ume und der Satz von Kern I. Manuscripta Math. 24, 133â171 (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, 2577â2583 (1996
Theorems on existence and global dynamics for the Einstein equations
This article is a guide to theorems on existence and global dynamics of
solutions of the Einstein equations. It draws attention to open questions in
the field. The local in time Cauchy problem, which is relatively well
understood, is surveyed. Global results for solutions with various types of
symmetry are discussed. A selection of results from Newtonian theory and
special relativity which offer useful comparisons is presented. Treatments of
global results in the case of small data and results on constructing spacetimes
with prescribed singularity structure are given. A conjectural picture of the
asymptotic behaviour of general cosmological solutions of the Einstein
equations is built up. Some miscellaneous topics connected with the main theme
are collected in a separate section.Comment: 54 pages, submitted to Living Reviews in Relativit