The author considers Lorentz type spaces \({\mathcal L}_{\sigma}^{p,r}\) defined in terms of iterated rearrangements of functions of several variables (\(\sigma\) is a permutation of \(\{1,\dots,n\}\)). Further, he studies Fourier-Gagliardo mixed norm spaces \({\mathcal V}({\mathbb R}^n)\) closely related to Sobolev spaces \(W_1^1({\mathbb R}^n)\) and proves an estimate of \(\|f\|_{{\mathcal L}_{\sigma}^{n',1}}\) via \(\|f\|_{\mathcal V}\) with the sharp constant where \(n'=\frac{n}{n-1}\). In particular, this gives a refinement of the known Sobolev type inequalities for the space \(W_1^1({\mathbb R}^n)\).