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

Dog filariosis in the Lazio region (Central Italy): first report on the presence of Dirofilaria repens

BACKGROUND: Epidemiological investigations were carried out in the Lazio Region to assess the status of canine filariosis and to evaluate the actual risk for veterinary and medical public health. METHODS: Since August 2001 to June 2003, a total of 972 canine blood samples, collected in public kennels and from private owners animals of the 5 Provinces of the Region, were tested. The presence of filarial parasites was evaluated by microscopy and bio-molecular techniques; the species identification was performed by means of the same diagnostic tools. RESULTS: A total of 17/972 (1.75%; 95%CI 1.06%–2.85%) blood samples were parasitized by D. repens,13 out them drawn by dogs resident in the Province of Roma, and 4 in the other provinces. Multivariate analysis was performed in order to evaluate the association between filariosis and risk factors. The origin from coastal territories seems to be a significant risk factor to acquire the infection. CONCLUSION: This is the first report of canine filariosis in the Lazio Region, where D. repens was before reported only in foxes. The risk of human zoonotic infection is stressed, and the absence of other filarial species is discusse

Evaluation of Oxfendazole, Praziquantel and Albendazole against Cystic Echinococcosis: A Randomized Clinical Trial in Naturally Infected Sheep

Cystic Echinococcosis (CE) is a near-cosmopolitan parasitic zoonosis that causes economic losses in many regions of the world. This parasitic infection can be regarded as an emerging or re-emerging disease causing considerable losses in livestock production. CE is produced by the larval cystic stage (hydatid) of the dog parasite Echinococcus granulosus. After infective eggs are ingested, cysts develop mainly in lungs and liver of humans and animals (sheep, cattle, pigs, horses, etc). Infected people may require surgery and/or Albendazole-based chemotherapy. In this study, we evaluated the effects of Oxfendazole alone (an antiparasitic drug used in animals), Oxfendazole plus Praziquantel, and Albendazole plus Praziquantel against hydatid cysts in sheep over 4 to 6 weeks of treatment. All of the treatments in this study were efficacious in killing the larval stages and, therefore, in minimizing the risk of a dog acquiring new infections (taenias). These treatment schemes can be added to control measures in animals and eventually could be used for the treatment of human infection. Further investigations on different schedules of monotherapy or combined chemotherapy are needed, as well as studies to evaluate the safety and efficacy of Oxfendazole in humans