On some nonuniform cases of weighted Sobolev and Poincaré inequalities

Weighted inequalities $\|f\|_{q,\nu,B_0}\le C\sum^{n}_{j=1}\|f_{xj}\|_{p,\omega_j,B_0}$ of Sobolev type $(\operatorname{supp}f\subset B_0)$ 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 $X=\{X_j\}$, $j=1,\dots,m$, with infinitely differentiable coefficients satisfying the Hörmander condition.