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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On some nonuniform cases of the weighted Sobolev and Poincaré inequalities


Authors: F. I. Mamedov and R. A. Amanov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 447-463
MSC (2000): Primary 46E35
Published electronically: April 7, 2009
MathSciNet review: 2454455
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Abstract: Weighted inequalities $ \Vert f\Vert _{q,\nu,B_0}\le C\sum^{n}_{j=1}\Vert f_{xj}\Vert _{p,\omega_j,B_0}$ of Sobolev type $ (\supp f\subset B_0)$ and of Poincaré type $ (\bar f_{\nu,B_0}=0)$ are studied, with different weight functions for each partial derivative $ f_{x_j}$, for parallelepipeds $ B_0\subset E_n, n\ge 1$. Also, weighted inequalities $ \Vert f\Vert _{q,\nu}\le C\Vert Xf\Vert _{p,\omega}$ of the same type are considered for vector fields $ X=\{X_j\}$, $ j=1, \ldots, m$, with infinitely differentiable coefficients satisfying the Hörmander condition.


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Additional Information

F. I. Mamedov
Affiliation: Institute of Mathematics and Mechanics, National Academy of Sciences, Azerbaidzhan, and Dichle University, Diyarbakir, Turkey
Email: farman-m@mail.ru

R. A. Amanov
Affiliation: Institute of Mathematics and Mechanics, National Academy of Sciences, Azerbaidzhan
Email: rabilamanov@hotmail.com

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01055-3
PII: S 1061-0022(09)01055-3
Keywords: Sobolev and Poincar\'e inequalities, Carnot-Carath\'eodory metric, Besicovitch property
Received by editor(s): June 14, 2006
Published electronically: April 7, 2009
Additional Notes: The work of the first author was supported in part by INTAS (grant no. 8792)
Article copyright: © Copyright 2009 American Mathematical Society