I am reading POincare inequality from Evans ( can be found here : http://anhngq.wordpress.com/2010/02/22/the-poincare-inequality/ ) and i am having some trouble understanding it . I would be glad if someone could explain the renormalizing step ? Specially i didn't understand how average of $v_k=0$ and few steps there .
I am also interested to know under what conditions does Poincare inequality hold ? Suppose if i take a constant function then it doesn't hold .