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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Degenerate Sobolev spaces and regularity of subelliptic equations


Authors: Eric T. Sawyer and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 362 (2010), 1869-1906
MSC (2000): Primary 35B65, 35D10, 35H20, 46E35
Published electronically: October 30, 2009
MathSciNet review: 2574880
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Abstract: We develop a notion of degenerate Sobolev spaces naturally associated with nonnegative quadratic forms that arise from a large class of linear subelliptic equations with rough coefficients. These Sobolev spaces allow us to make the widest possible definition of a weak solution that leads to local Hölder continuity of solutions, extending our results in an earlier work, where we studied regularity of classical weak solutions. In cases when the quadratic forms arise from collections of rough vector fields, we study containment relations between the degenerate Sobolev spaces and the corresponding spaces defined in terms of weak derivatives relative to the vector fields.


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Additional Information

Eric T. Sawyer
Affiliation: Department of Mathematics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Richard L. Wheeden
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04756-4
PII: S 0002-9947(09)04756-4
Received by editor(s): September 6, 2007
Published electronically: October 30, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.