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St. Petersburg Mathematical Journal

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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
DOI: https://doi.org/10.1090/S1061-0022-09-01055-3
Published electronically: April 7, 2009
MathSciNet review: 2454455
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Abstract: Weighted inequalities $\|f\|_{q,\nu ,B_0}\le C\sum ^{n}_{j=1}\|f_{xj}\|_{p,\omega _j,B_0}$ of Sobolev type $(\mathrm {supp}f\subset B_0)$ 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 $\|f\|_{q,\nu }\le C\| Xf\|_{p,\omega }$ 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

Keywords: Sobolev and Poincaré inequalities, Carnot-Carathéodory 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