Abstract
We investigate global solvability, in the framework of smooth functions and Schwartz distributions, of certain sums of squares of vector fields defined on a product of compact Riemannian manifolds T × G, where G is further assumed to be a Lie group. As in a recent article due to the authors, our analysis is carried out in terms of a system of left-invariant vector fields on G naturally associated with the operator under study, a simpler object which nevertheless conveys enough information about the original operator so as to fully encode its solvability. As a welcome side effect of the tools developed for our main purpose, we easily prove a general result on propagation of regularity for such operators.
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References
G. Araújo, Global regularity and solvability of left-invariant differential systems on compact Lie groups, Ann. Global Anal. Geom. 56 (2019), 631–665.
G. Araújo, I. A. Ferra and L. F. Ragognette, Global hypoellipticity of sums of squares on compact manifolds, arXiv:2005.04484 [math.AP]
I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Orlando, FL, 1984.
S. J. Greenfield and N. R. Wallach, Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc. 31 (1972), 112–114.
S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields, Topology 12 (1973), 247–254.
S. J. Greenfield and N. R. Wallach, Remarks on global hypoellipticity, Trans. Amer. Math. Soc. 183 (1973), 153–164.
A. A. Himonas and G. Petronilho, Global hypoellipticity and simultaneous approximability, J. Fúnct. Anal. 170 (2000), 356–365.
A. A. Himonas and G. Petronilho, Propagation of regularity and global hypoellipticity, Michigan Math. J. 50 (2002), 471–481.
L. Hörmander, Linear Partial Differential Operators, Springer, New York, 1969.
J. Hounie, Globally hypoelliptic and globally solvable first-order evolution equations, Trans. Amer. Math. Soc. 252 (1979), 233–248.
Y. Kannai, An unsolvable hypoelliptic differential operator, Israel J. Math. 9 (1971), 306–315.
A. W. Knapp, Lie Groups Beyond an Introduction, Birkhaäser, Boston, MA, 1996.
H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366–383.
G. Köthe, Topological Vector Spaces. II, Springer, New York-Berlin, 1979.
G. Petronilho, Global solvability and simultaneously approximable vectors, J. Differential Equations 184 (2002), 48–61.
G. Petronilho, Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application, Trans. Amer. Math. Soc. 363 (2011), 6337–6349.
M. Ruzhansky and V. Turunen, Pseudo-Differential Operators and Symmetries, Birkhäuser, Basel, 2010.
F. Treves, Topological Vector Spaces, Distributions and Kernels, Dover, Mineola, NY, 2006.
F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, New York-Berlin, 1983.
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This work was supported by the São Paulo Research Foundation (FAPESP, grants 2016/13620-5 and 2018/12273-5).
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Araújo, G., Ferra, I.A. & Ragognette, L.F. Global solvability and propagation of regularity of sums of squares on compact manifolds. JAMA 148, 85–118 (2022). https://doi.org/10.1007/s11854-022-0223-6
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DOI: https://doi.org/10.1007/s11854-022-0223-6