Optimal Rearrangement Invariant Sobolev Embeddings in Mixed Norm Spaces

We improve the Sobolev-type embeddings due to Gagliardo (Ric Mat 7:102-137, 1958) and Nirenberg (Ann Sc Norm Sup Pisa 13:115-162, 1959) in the setting of rearrangement invariant (r.i.) spaces. In particular, we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space (WLp)-L-1 by Poornima (Bull Sci Math 107(3):253-259, 1983), O'Neil (Duke Math J 30:129-142, 1963) and Peetre (Ann Inst Fourier 16(1):279-317, 1966) (1 <= p < n), and by Hansson (Math Scand 45(1):77-102, 1979, Brezis and Wainger (Commun Partial Differ Equ 5(7):773-789, 1980) and Maz'ya (Sobolev spaces, 1985) (p = n) can be further strengthened by considering mixed norms on the target spaces.