On the constants for some Sobolev imbeddings

We consider the imbedding inequality ‖ ‖ Lr(R d ) ≤ Sr,n,d ‖ ‖Hn(Rd); H n(R) is the Sobolev space (or Bessel potential space) of L2 type and (integer or fractional) order n. We write down upper bounds for the constants Sr,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 = ∞, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on Sr,n,d for n > d/2, 2 < r < ∞; 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.