Weighted positivity of second order elliptic systems

Integral inequalities that concern the weighted positivity of a differential operator have important applications in qualitative theory of elliptic boundary value problems. Despite the power of these inequalities, however, it is far from clear which operators have this property. In this paper, we study weighted integral inequalities for general second order elliptic systems in R-n (n >= N >= 3) and prove that, with a weight, smooth and positive homogeneous of order 2-n, the system is weighted positive only if the weight is the fundamental matrix of the system, possibly multiplied by a semi-positive definite constant matrix.