The present paper is devoted to obtain numerical estimations for the equivalences between the Hardy–Littlewood norms of Zygmund’s spaces, \(L_{\exp }\) and \(L\log L,\) and the Luxemburg norms associated to concrete Young functions that define these spaces. Moreover, for a (finite) measure we compute the equivalence constants between the Hardy–Littlewood norms of Zygmund’s spaces and the norms as associate (Köthe-dual) spaces. It is also proved that, for each \(0<r<1,\) the quasinorm of the r-convexification \(L^r_{\exp },\) of \(L_{\exp },\) is equivalent to a norm. In the opposite, the quasinorm of the r-convexification \(L^r\log L,\) of \(L\log L,\) is not equivalent to a norm. In the atomic case, the r-convexification \(L^r\log L\) has a separating dual. We analyse the weak compactness of the multiplication operators from \(L^{\infty }\) to \(L_{\exp }\) and from \(L\log L\) to \(L^1.\) From the weak compactness of the embeddings follows the reflexivity of certain Lions–Peetre interpolated spaces.
