Iterated rearrangements and Gagliardo–Sobolev type inequalities

In this paper we consider Lorentz type spaces L σ p , r defined in terms of iterated rearrangements of functions of several variables (σ is a permutation of { 1 , … , n } ). Further, we study Fournier–Gagliardo mixed norm spaces V ( R n ) closely related to Sobolev spaces W 1 1 ( R n ) . We prove estimate of ‖ f ‖ L σ n ′ , 1 via ‖ f ‖ V with the sharp constant. In particular, this gives a refinement of the known Sobolev type inequalities for the space W 1 1 ( R n ) .