New Hardy inequalities and behaviour of linear elliptic equations

In this study, we want to emphasize the role of some Hardy inequalities in the blow-up phenomena of the very weak solution of a linear equation in the sense of Brezis. Thus we present here some new Hardy inequalities related to some extended Sobolev spaces such that Sobolev–Hardy spaces, Sobolev–Zygmund spaces, or other non-standard weighted spaces. Firstly we apply those results then provide two applications of these inequalities. Secondly we improve recent results by showing that the blow-up phenomena of the gradient can also occur in Hardy spaces. The Hardy inequalities for Sobolev–Zygmund spaces are obtained via an integral formula estimating the oscillation in a ball of radius r of a general function u in the usual Sobolev space. This formula involves the notion of relative rearrangement. We shall give a pointwise estimate for the solution u of linear equation −Δu=−div(F) for a bounded function F, using the distance function δ.