Balanced metrics on Cartan and Cartan-Hartogs domains

This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] (Kaehler-Einstein submanifolds of the infinite dimensional projective space, to appear in Mathematische Annalen) we also provide the first example of complete, Kaehler-Einstein and projectively induced metric g such that $\alpha g$ is not balanced for all $\alpha >0$.Comment: 11 page

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. 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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. 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Holomorphic linearization of commuting germs of holomorphic maps

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of \C^n fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.Comment: 24 pages; final version with erratum (My original paper failed to cite the work of L. Stolovitch [ArXiv:math/0506052v2]); J. Geom. Anal. 201

Berezin quantization of homogeneous bounded domains

We prove that a homogeneous bounded domain admits a Berezin quantization

Spaces of anisotropic ultradifferentiable functions and local solvability for semilinear partial differential equations

We propose constructions of multiparameter depending scales of Banach spaces of anisotropic ultra-differentiable functions and investigate their invariance under changes of the variables. We prove new local solvability results in such spaces for some classes of linear and semilinear partial differential equations with multiple characteristics

Evolution Pde With Elliptic Dissipative Terms: Critical Index For Singular Initial Data, Self-similar Solutions And Analytic Regularity [edp D'évolution Avec Dissipation Elliptique : L'indice Critique Pour Des Données Initiales Singulières, Solutions Auto-similaires Et Régularité Analytique]

We investigate the influence of the elliptic dissipative terms of evolution equations in ℝn and double-struck T signn on the critical Lp, 1 ≤ p ≤ ∞, index of the singularity of the initial data, the analytic regularity for positive time and the existence of self-similar solutions. © Académie des Sciences/Elsevier, Paris.32714146Bekhiranov, D., The initial value problem for the generalized Burgers' equation (1996) Differ. Integ. Eq., 9, pp. 1253-1265Biagioni, H.A., Cadeddu, L., Gramchev, T., Parabolic equations with conservative nonlinear term and singular initial data, Proc. 2nd World Congress of Nonlinear Analysts (1997) Nonlin. Anal. TMA, 30, pp. 2489-2496. , Athens, Greece, July 10-17, 1996Brézis, H., Friedman, H., Nonlinear parabolic equations involving measures as initial conditions (1983) J. Math. Pures Appl., 62, pp. 73-97Brézis, H., Cazenave, T., Nonlinear heat equation with singular initial data (1996) J. Anal. Math., 68, pp. 276-304Cannone, M., Planchon, F., Self-similar solutions for Navier-Stokes equations in ℝ3, Commun (1996) Partial Differ, Eq., 21, pp. 179-193Chemin, J.-Y., Fluides parfaits incompressibles (1995) Astérisque, 230, pp. 1-177Dix, D., Nonuniqueness and uniqueness in thé initial value problem for Burgers' equation (1996) SIAM J. Math. Anal., 27, pp. 709-724Ferrari, A., Titi, E., Gevrey regularity for nonlinear analytic parabolic equations (1998) Commun. Partial Differ. Eq., 23, pp. 1-16Foias, C., Temam, R., Gevrey class regularity for the solutions of the Navier-Stokes equations (1989) J. Funct. Anal., 87, pp. 359-369Kozono, H., Yamazaki, M., Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data (1994) Commun. Partial Differ. Eq., 19, pp. 959-1014Levermore, D., Oliver, M., Distribution-valued initial data for the complex Ginzburg-Landau equation (1997) Commun. Partial Differ. Eq., 22, pp. 39-49Planchon, F., Convérgence des solutions des équations de Navier-Stokes vers des solutions auto-similaires (1996) Séminaire X-EDP, 95-96Ribaud, F., (1996) Analyse de Littlewood-Paley Pour la Résolution d'Équations Paraboliques Semi-linéaires, , Thèse de Docteur en Sciences, OrsayBollerman, P., Doelman, A., Van Harten, A., Titi, E., Analyticity for essentially bounded solutions to strongly parabolic semilinear systems (1996) SIAM J. Math. Anal., 27, pp. 424-44