On the constants for some Sobolev imbeddings

2000/11/20 by C. Morosi, L. Pizzocchero · 1 citation
Mathematics · Physics and Astronomy · #math.FA #math-ph #math.MP

paper · pdf

published as Journal of Inequalities and Applications 6, 665-679 (2001) · 12 pages, to appear in the Journal of Inequalities and Applications

arxiv created 2000/11/20 · arxiv updated 2009/11/30

Abstract

We consider the imbedding inequality || f ||_L^r(R^d) <= S_r,n,d || f ||_Hⁿ(R^d); Hⁿ(R^d) 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 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.

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