Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles

This survey paper deals with strengthened forms of classical Sobolev inequalities, involving remainder terms depending on the distance from the family of extremals, and with analogues for Hardy inequalities, where extremals do not exist, but can be replaced by "virtual" extremals. An account of the stability of isoperimetric and symmetrization inequalities, on which these Sobolev and Hardy inequalities rely, is provided as well.