Abstract. Our main goal in this work is to further improve the mixed norm estimates due to Fournier [13], and also Algervik and Kolyada [1], to more general rearrangement invariant (r.i.) spaces. In particular we find the optimal domains and the optimal ranges for these embeddings between mixed norm spaces and r.i. spaces.

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We improve the Sobolev-type embeddings due to Gagliardo [20] and Nirenberg [28] 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 W 1 L p by Poornima [31], O'Neil [29] and Peetre [30] (1 ≤ p < n), and by Hansson [21], Brezis and Wainger [12] and Maz'ya [26] (p = n) can be further strengthened by considering mixed norms on the target spaces.(see Definition 3.2) and then, using an iterated form of Hölder's inequality, completed the proof; i.e.,where n ′ denotes the conjugate exponent of n, i.e., 1/n + 1/n ′ = 1.Later, a new approach based on properties of mixed norm spaces was introduced by Fournier [19] and was subsequently developed, via different methods, by various authors, including Blei and Fournier [8], Milman [27], Algervik and Kolyada [2] and Kolyada [24,25]. To be more precise, the central part of Fournier's work was to study embeddings between mixed norm spaces and Lorentz spaces L p,q (see Sections 2 for further details on Lorentz spaces). Specifically, he proved that R(L 1 , L ∞ ) ֒→ L n ′ ,1 (I n ), 2010 Mathematics Subject Classification. 28A35, 46E30, 46E35.

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