  We study the recovery conditions of weighted mixed $\ell_2/\ell_p\,(0<p\leq
1)$ minimization for block sparse signal reconstruction from compressed
measurements when partial block support information is available. We show that
the block $p$-restricted isometry property (RIP) can ensure the robust
recovery. Moreover, we present the sufficient and necessary condition for the
recovery by using weighted block $p$-null space property. The relationship
between the block $p$-RIP and the weighted block $p$-null space property has
been established. Finally, we illustrate our results with a series of numerical
experiments.
