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We study Sobolev embeddings in the Sobolev space W 1,p(·) (Ω) with variable exponent satisfying 1 p(x) n. Since the exponent is allowed to reach the values 1 and n, we need to introduce new techniques, combining weak-and strong-type estimates, and a new variable exponent target space scale which features a space of exponential type integrability instead of L ∞ at the upper end. 2000 Mathematics Subject Classification. 46E35.doi:10.5565/publmat_52208_05 fatcat:6dcb7hgkbzgvtoofmfjr4q6gei