Global poincaré inequalities for greenʹs operator applied to the solutions of the nonhomogeneous A-harmonic equation

Both local and global Ar-weighted Poincaré inequalities for Greenʹs operator applied to the solutions of the nonhomogeneous A-harmonic equation in John domains are established. As applications of the Poincaré inequalities for Greenʹs operator, the Ar-weighted Sobolev imbedding inequalities for Greenʹs operator in John domains are also proved.