Brezis–Marcus Problem and Its Generalizations

Certain Hardy inequalities in domains of Euclidean space contain sharp but unreachable constants. V. G. Maz'ya and many other authors used this fact to improve the corresponding inequalities by adding new integral terms. In this paper, a survey of results in this direction initiated by H. Brezis and M. Marcus is presented. Also, we give some generalizations of Brezis–Marcus-type inequalities to the case of Rellich-type inequalities with weights that are powers of the distance from a point to the boundary of the domain. Generalizations to the case of conformally invariant integral inequalities in simply connected and doubly connected planar domains of hyperbolic type are discussed.