On some nonuniform cases of the weighted Sobolev and Poincaré inequalities

Mamedov, F.; Amanov, R.

doi:10.1090/S1061-0022-09-01055-3

On some nonuniform cases of the weighted Sobolev and Poincaré inequalities
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by F. I. Mamedov and R. A. Amanov
Translated by: A. Plotkin

St. Petersburg Math. J. 20 (2009), 447-463

DOI: https://doi.org/10.1090/S1061-0022-09-01055-3

Published electronically: April 7, 2009

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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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