Addition to Theorems 4.6.5: Relation (4.25) can be supplemented by a similar inequality for the L^p-norms. This inequality can be easily obtained by following the proof of Theorem 4.7.1 (that is, iteration of Brezis-Lieb lemma) in the case q=p without assumption of cocompact embedding.
(thanks to a comment by Mizuho Okumura)