Remarks on the Hardy type inequalities with remainder terms in the   framework of equalities
  We study the Hardy type inequalities in the framework of equalities. We present equalities which immediately imply Hardy type inequalities by dropping the remainder term. Simultaneously we give a characterization of the class of functions which makes the remainder term vanish. A point of our observation is to apply an orthogonality properties in general Hilbert space, and which gives a simple and direct understanding of the Hardy type inequalities as well as the nonexistence of nontrivial extremizers. 
Shuji Machihara, Tohru Ozawa, Hidemitsu Wadade
math.AP