A generalized Poincaré inequality for a class of constant coefficient differential operators
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Abstract:
We study first order differential operators $\mathcal {P} = \mathcal {P}(D)$ with constant coefficients. The main question is under what conditions the following full gradient $L^p$ estimate holds: \[ \|D(f-f_0)\|_{L^p} \leq C \|\mathcal {P} f\|_{L^p},\ \textrm {for some } f_0 \in \ker \mathcal {P}.\] We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.References
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Additional Information
- Derek Gustafson
- Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13210
- Email: degustaf@syr.edu
- Received by editor(s): October 11, 2009
- Received by editor(s) in revised form: February 15, 2010, and May 13, 2010
- Published electronically: March 23, 2011
- Communicated by: Matthew J. Gursky
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2721-2728
- MSC (2010): Primary 35A99; Secondary 35B45, 58J10
- DOI: https://doi.org/10.1090/S0002-9939-2011-10607-5
- MathSciNet review: 2801612