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Prescribing analytic singularities for solutions of a class of vector fields on the torus. (English) Zbl 1077.35004
This paper deals with the analytic singularities of the operator \[ L= \partial_t+ (a(t)+ ib(t))\partial_x \] acting on distributions on the torus \(\mathbb{T}^2_{t,x}\). The functions \(a\), \(b\) are real valued, real analytic on \(\mathbb{T}^1_t\). We shall formulate some of the main results of the paper. Assume that \(b\) changes sign and \(\Sigma\) is any subset of the set of the local extrema of the local primitives of \(b\). Then there exists a solution \(u\in D'(\mathbb{T}^2)\) of \(Lu= f\in C^\omega(\mathbb{T}^2)\) such that the \(t\)-projection of its analytic singular support is \(\Sigma\). Moreover, for any \(\tau\in\Sigma\) and any closed \(F\subset\mathbb{T}^1_x\) one can find \(u\in D'(\mathbb{T}^2)\) such that \(Lu\in C^\omega(\mathbb{T}^2)\) and \(\text{sing\,supp}_A(u)= \{\tau\}\times F\). The results here proposed are sharp, i.e. if \(t\) is neither a local minimum nor a local maximum, then every \(u\in D'(\mathbb{T}^2)\), for which \(Lu\in C^\omega(\mathbb{T}^2)\), is real analytic in \((t,x)\).

MSC:
35A20 Analyticity in context of PDEs
35H10 Hypoelliptic equations
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[1] M. S. Baouendi and F. Trèves, A microlocal version of Bochner’s tube theorem, Indiana Univ. Math. J. 31 (1982), no. 6, 885 – 895. · Zbl 0505.32013 · doi:10.1512/iumj.1982.31.31060 · doi.org
[2] Adalberto P. Bergamasco, Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4113 – 4126. · Zbl 0932.35046
[3] Adalberto P. Bergamasco, Paulo D. Cordaro, and Pedro A. Malagutti, Globally hypoelliptic systems of vector fields, J. Funct. Anal. 114 (1993), no. 2, 267 – 285. · Zbl 0777.58041 · doi:10.1006/jfan.1993.1068 · doi.org
[4] Adalberto P. Bergamasco, Wagner V. L. Nunes, and Sérgio Luís Zani, Global analytic hypoellipticity and pseudoperiodic functions, Mat. Contemp. 18 (2000), 43 – 57 (English, with English and Portuguese summaries). VI Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 1999). · Zbl 0979.35036
[5] Adalberto P. Bergamasco, Wagner V. L. Nunes, and Sérgio Luís Zani, Global properties of a class of overdetermined systems, J. Funct. Anal. 200 (2003), no. 1, 31 – 64. · Zbl 1034.32024 · doi:10.1016/S0022-1236(02)00055-1 · doi.org
[6] Detta Dickinson, Todor Gramchev, and Masafumi Yoshino, Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena, Proc. Edinb. Math. Soc. (2) 45 (2002), no. 3, 731 – 759. · Zbl 1032.37010 · doi:10.1017/S001309150000064X · doi.org
[7] Stephen J. Greenfield, Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc. 31 (1972), 115 – 118. · Zbl 0229.35024
[8] Stephen J. Greenfield and Nolan R. Wallach, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc. 31 (1972), 112 – 114. · Zbl 0229.35023
[9] A. Alexandrou Himonas, Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2061 – 2067. · Zbl 0984.35050
[10] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients.
[11] J. Hounie, Globally hypoelliptic vector fields on compact surfaces, Comm. Partial Differential Equations 7 (1982), no. 4, 343 – 370. · Zbl 0588.35064 · doi:10.1080/03605308208820226 · doi.org
[12] Abdelhamid Meziani, Hypoellipticity of nonsingular closed 1-forms on compact manifolds, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1255 – 1269. · Zbl 1017.58014 · doi:10.1081/PDE-120005837 · doi.org
[13] Johannes Sjöstrand, Singularités analytiques microlocales, Astérisque, 95, Astérisque, vol. 95, Soc. Math. France, Paris, 1982, pp. 1 – 166 (French).
[14] François Trèves, Analytic-hypoelliptic partial differential equations of principal type, Comm. Pure Appl. Math. 24 (1971), 537 – 570. · Zbl 0222.35014 · doi:10.1002/cpa.3160240407 · doi.org
[15] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. · Zbl 0565.35079
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