Exact Constants for Embeddings of Sobolev Spaces in L₁

In this paper, we study the problem of embedding constants of Sobolev spaces Wₚⁿ[0,1] in the space L₁[0,1] . For arbitrary n ∈ N and 1 ≤ p ≤ ∞ , we prove that the embedding constants are equal to (1/n!)\|Tn, p'\|_Lp'[0,1] , where Tn, p' is a polynomial of degree n with leading coefficient $$1$$ , deviating least from zero in the space Lp'[0,1] , $$1/p+1/p'=1$$ . Using this equality, explicit formulas for the embedding constants for $$p=1$$ and p=∞ , as well as an explicit asymptotic formula for p∈ (1, ∞ ) and n→ ∞ , are obtained. It is shown that the corresponding extremal functions are symmetric with respect to the midpoint of the interval. These results generalize the previously known case $$p=2$$ .