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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) ofL^2 type and (integer or fractional) order n. We write down upper bounds forthe constants S_{r, n, d}, using an argument previously applied in theliterature in particular cases. We prove that the upper bounds computed in thisway are in fact the sharp constants if (r=2 or) n > d/2, r=infinity, andexhibit the maximising functions. Furthermore, using convenient trialfunctions, 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 anumber of examples, thus characterizing the sharp constants with littleuncertainty.

On the constants for some Sobolev imbeddings