Abstract: In this paper, the authors first introduce a new
weight: $A^{\lambda_{3}}_{r}(\lambda_{1},\lambda_{2},\Omega)$-weight,and prove the local weighted integral inequalities for conjugate
${\cal A}$ -harmonic tensors. Then, as an application of the local
result, the authors prove a global weighted integral inequality for
conjugate ${\cal A}$-harmonic tensors in a bounded domain $\Omega$,
which can be regarded as generalizations of the classical results.
Finally, the authors give some applications of the above results to
quasiregular mappings.

Key words: Conjugate ${\cal A}$-harmonic tensor, $A^{\lambda_{3}}_{r}(\lambda_{1},\lambda_{2},\Omega)$-weight, Weighted integral inequality, Quasiregular mapping