Iterated rearrangements and Gagliardo–Sobolev type inequalities

In this paper we consider Lorentz type spaces 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(Rn) closely related to Sobolev spaces . We prove estimate of via ‖f‖V with the sharp constant. In particular, this gives a refinement of the known Sobolev type inequalities for the space .