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Orbits and global unique continuation for systems of vector fields. (English) Zbl 0917.58004
Let $${\mathcal M}$$ be a smooth, connected paracompact manifold and let $${\mathcal V}$$ be a locally integrable vector subbundle of the complexified tangent bundle $$\mathbb{C} T{\mathcal M}$$ of $${\mathcal M}$$. The main purpose of the present paper is to explore the global unique continuation property of distribution solutions of $${\mathcal V}$$, i.e., the distributions $$u$$ on $${\mathcal M}$$ such that $$Lu=0$$ whenever $$L$$ is a section of $${\mathcal V}$$. More precisely, the authors are interested to determine conditions on $${\mathcal M}$$ and/or the structure $${\mathcal V}$$ such that if $$u$$ is a solution on $${\mathcal M}$$ that vanishes on an open subset of $${\mathcal M}$$, then $$u$$ vanishes on $${\mathcal M}$$. The closely related problem of the structure of the Sussmann orbits of $${\mathbb{R}}\mathcal V$$ is also studied in this paper.

##### MSC:
 58A30 Vector distributions (subbundles of the tangent bundles) 32V99 CR manifolds 35N10 Overdetermined systems of PDEs with variable coefficients 58J99 Partial differential equations on manifolds; differential operators
##### Keywords:
unique continuation; involutive structures; CR manifolds; orbits
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##### References:
 [1] M. S. Baouendi and L. P. Rothschild, Cauchy-Riemann functions on manifolds of higher codimension in complex space,Invent. Math. 101 (1990), 45–56. · Zbl 0712.32009 · doi:10.1007/BF01231495 [2] M.S. Baouendi, L. P. Rothschild, and J.-M. Trepreau, On the geometry of analytic discs attached to real manifolds.J. Differential Geom. 39 (1994), no. 2, 379–405. · Zbl 0821.32014 [3] M. S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of vector fields,Ann. Math. 113 (1981), 387–421. · Zbl 0491.35036 · doi:10.2307/2006990 [4] J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégenerés,Ann. Inst. Fourier Grenoble 10 (1969), 277–304. · Zbl 0176.09703 · doi:10.5802/aif.319 [5] F. Cardoso and J. Hounie, First-order linear PDE’s and uniqueness in the Cauchy problem,J. Diff. Eq. 33 (1979), 239–248. · Zbl 0404.35020 · doi:10.1016/0022-0396(79)90090-1 [6] K. Diederich and J. E. Fornaess, Pseudoconvex domains with real analytic boundary,Ann. of Math. 107 (1978), 371–384. · Zbl 0378.32014 · doi:10.2307/1971120 [7] J.J. Duistermaat and L. Hörmander, Fourier integral operators, II,Acta Math. 128 (1972), 183–269. · Zbl 0232.47055 · doi:10.1007/BF02392165 [8] V. Guillemin and S. Sternberg,Geometric Asymptotics, Mathematical Surveys, vol. 14, American Mathematical Society, 1976. [9] N. Hanges and J. Sjöstrand, Propagation of analyticity for a class of nonmicrocharacteristic operators,Ann. of Math. 116(1982), 559–577. · Zbl 0537.35007 · doi:10.2307/2007023 [10] N. Hanges and F. Treves, Propagation of holomorphic extendability of CR functions,Math. Ann. 263 (1983), 157–177. · Zbl 0507.32009 · doi:10.1007/BF01456878 [11] L. Hörmander,Linear Partial Differential Operators, Springer-Verlag, Berlin, 1969. [12] J. Hounie, Globally hypoelliptic vector fields on compact surfaces,Comm. PDE 7 (1982), 343–370. · Zbl 0588.35064 · doi:10.1080/03605308208820226 [13] L. R. Hunt, The complex Frobenius Theorem and uniqueness of solutions to the tangential Cauchy-Riemann equations,J. Diff. Eq. 27 (1978), 214–233. · Zbl 0359.35025 · doi:10.1016/0022-0396(78)90031-1 [14] L. R. Hunt, J. C. Polking, and M. J. Strauss, Unique continuation for solutions to the induced Cauchy-Riemann equations,J. Diff. Eq. 23 (1977), 436–447. · Zbl 0337.35001 · doi:10.1016/0022-0396(77)90121-8 [15] S. Lojasiewicz, Ensembles semianalytiques,Notes, Inst. Hautes Études, Bres-sur-Yvette, 1965. [16] S. Lojasiewicz, Triangulation of semianalytic sets,Ann. Scuola Norm. Sup. Pisa 18 (1964), 449–474. [17] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras,J. Math. Soc. Japan 18 (1966), 398–404. · Zbl 0147.23502 · doi:10.2969/jmsj/01840398 [18] L. Nirenberg and F. Treves, Solvability of a first-order linear partial differential equation,Comm. Pure Appl. Math. XVI (1963), 355–377. [19] Schwartz, L.,Théorie des Distributions, Hermann, 1966. [20] M. J. Strauss and F. Treves, First-order linear PDEs and uniqueness in the Cauchy problem,J. Diff. Eq. 15 (1974), 195–209. · Zbl 0266.35009 · doi:10.1016/0022-0396(74)90094-1 [21] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions,Trans. Amer. Math. Soc. 180 (1973), 171–188. · Zbl 0274.58002 · doi:10.1090/S0002-9947-1973-0321133-2 [22] H. J. Sussmann, Real-analytic desingularization and subanalytic sets: An elementary approach,Trans. Amer. Math. Soc. X (1990), 417–461. · Zbl 0696.32005 [23] J.-M. Trepreau, Sur la propagation des singularités dans les variétés CR,Bull. Soc. Math. Fr. 118 (1990), 403–450. · Zbl 0742.58053 [24] F. Treves, Approximation and representation of functions and distributions annihilated by a system of complex vector fields,Centre de Mathematiques Ecole Polytechnique, Paliseau, France, 1981. [25] F. Treves,Hypo-Analytic Structures, Princeton University Press, Princeton, NJ, 1992. · Zbl 0787.35003 [26] F. Treves, Integral representation of solutions of first order linear partial differential equations I,Ann. Scuola Norm. Sup. Pisa 3 (1976), 1–35. [27] E. C. Zachmanoglou, Propagation of the zeroes and uniqueness in the Cauchy problem for first order partial differential equations,Arch Ratl. Mech. Anal 38 (1970), 178–188. · Zbl 0199.15903 · doi:10.1007/BF00251658
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