Scale invariance structures of the critical and the subcritical Hardy inequalities and their improvements

• DOI: 10.1007/s00526-017-1166-0
• Publication Date: 2017-04-26T11:31:19Z
• Authors: Megumi Sano; Futoshi Takahashi
• Keywords: Ball (mathematics)

In this paper, we study classical Hardy inequalities, both in the subcritical case on the whole space and the critical case on a ball. Two Hardy inequalities are quite different from each other in view of their forms, scaling structures and optimal constants. Nevertheless we show that, when the exponents are chosen appropriately, both inequalities are equivalent at least in the radial setting. A transformation which connects two inequalities is a key in our argument. As an application, we improve the critical Hardy inequality on a ball by using the improved subcritical inequality on the whole space.