The refinement and generalization of Hardy’s inequality in Sobolev space

摘要

In this paper, we refine the proof of Hardy’s inequality in (Evans in Partial Differential Equations, , Hardy in Inequalities, ) and extend Hardy’s inequality from two aspects. That is to say, we extend the integral estimation function from u|x| to u|x|σ with suitable σ>0 and extend the space dimension from n≥3 to n≥2. Hardy’s inequality in (Evans in Partial Differential Equations, , Hardy in Inequalities, ) is the special case of our results.