An Adams Inequality for Vector Valued Maps

In this paper, we generalize the Adams inequality of the case of scalar functions to the case of vector functions.When n=2k, we prove that the Adams inequalities under two equivalent Sobolev norms are both true.Moreover, the two supremums of the inequalities are equal to each other.This kind of vector valued inequality can be used in some systems of partial differential equations arising out of geometric and physical problems to which are not suitable to apply the inequality for scalar functions.