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Failure of analytic hypoellipticity in a class of differential operators

Ovidiu Costin, Rodica D. Costin (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

For the hypoelliptic differential operators P = ∂ x 2 + x k ∂ y - x l ∂ t 2 introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P also fail to be analytic hypoelliptic (except for ( k , l ) = ( 0 , 1 ) ), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

Fourier-Wigner transforms and Liouville's theorems for the sub-Laplacian on the Heisenberg group

Aparajita Dasgupta, M. W. Wong (2010)

Banach Center Publications

The sub-Laplacian on the Heisenberg group is first decomposed into twisted Laplacians parametrized by Planck's constant. Using Fourier-Wigner transforms so parametrized, we prove that the twisted Laplacians are globally hypoelliptic in the setting of tempered distributions. This result on global hypoellipticity is then used to obtain Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group.

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