On Hardy-Sobolev-Maz'ya inequalities

This survey talk focuses on a class of Hardy–Sobolev-type inequalities with cylindrical weights, as introduced by V. Maz'ya in his seminal book on Sobolev spaces.
We will mainly concentrate on the existence (or nonexistence) of extremals and their symmetry properties.
Although the topic was intensively studied and a abundant literature emerged during the first decade of this century,
several intriguing and fundamental questions remain open to this day.