On the constants for some Sobolev imbeddings

We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the constants S_{r, n, d}, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if (r=2 or) n > d/2, r=infinity, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on S_{r,n,d} for n > d/2, 2 < r < infinity; in many cases these are close to the previous upper bounds, as illustrated by a number of examples, thus characterizing the sharp constants with little uncertainty.