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Germ hypoellipticity and loss of derivatives


Author: Gregorio Chinni
Journal: Proc. Amer. Math. Soc. 140 (2012), 2417-2427
MSC (2010): Primary 35H10, 35A27
DOI: https://doi.org/10.1090/S0002-9939-2011-11252-8
Published electronically: November 15, 2011
MathSciNet review: 2898704
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Abstract: We prove hypoellipticity in the sense of germs for the operator

$\displaystyle \mathcal {P}= L_{q}\overline {L}_{q} + \overline {L}_{q}t^{2k}L_{q} +Q^{2}, $

where

$\displaystyle L_{q}=D_{t}+it^{q-1}\sqrt {-\Delta _{x}}$$\displaystyle \quad \text {and}\quad Q = x_{1}D_{2}-x_{2}D_{1}, $

even though it fails to be hypoelliptic in the strong sense. The primary tool is an a priori estimate.

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Additional Information

Gregorio Chinni
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italia
Email: chinni@dm.unibo.it

DOI: https://doi.org/10.1090/S0002-9939-2011-11252-8
Received by editor(s): February 23, 2011
Published electronically: November 15, 2011
Communicated by: Franc Forstneric
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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