Exact Constants for Embeddings of Sobolev Spaces in $$L_1$$

In this paper, we study the problem of embedding constants of Sobolev spaces $$\mathring{W}_p^n[0,1]$$ in the space $$L_1[0,1]$$ . For arbitrary $$n \in \mathbb{N}$$ and $$1 \le p \le \infty$$ , we prove that the embedding constants are equal to $$({1}/{n!})\|T_{n, p'}\|_{L_{p'}[0,1]}$$ , where $$T_{n, p'}$$ is a polynomial of degree $$n$$ with leading coefficient $$1$$ , deviating least from zero in the space $$L_{p'}[0,1]$$ , $$1/p+1/p'=1$$ . Using this equality, explicit formulas for the embedding constants for $$p=1$$ and $$p=\infty$$ , as well as an explicit asymptotic formula for $$p\in (1, \infty )$$ and $$n\to \infty$$ , 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 $$p=2$$ .