A sharp form of an embedding into exponential and double exponential spaces

Let Ω be a bounded domain in Rn, n⩾2. In the well-known paper (Indiana Univ. Math. J. 20 (1971) 1077) Moser found the smallest value of K such thatsup ∫Ω exp(( f(x)/K)n/(n−1)) : f∈W1,n0(Ω),||∇f||Ln⩽1<∞.We extend this result to the situation in which the underlying space Ln is replaced by the Zygmund space Ln logα L (α