New results on multiplication in Sobolev spaces
Abstract
We consider the Sobolev (Bessel potential) spaces Hell(Rd, C), and their standard norms || ||ell (with ell integer or noninteger). We are interested in the unknown sharp constant Kell m n d in the inequality || f g ||ell ≤s Kell m n d || f ||m || g ||n (f in Hm(Rd, C), g in Hn(Rd, C); 0 <= ell <= m <= n, m + n - ell > d/2); we derive upper and lower bounds K+ell m n d, K-ell m n d for this constant. As examples, we give a table of these bounds for d=1, d=3 and many values of (ell, m, n); here the ratio K-ell m n d/K+ell m n d ranges between 0.75 and 1 (being often near 0.90, or larger), a fact indicating that the bounds are close to the sharp constant. Finally, we discuss the asymptotic behavior of the upper and lower bounds for Kell, b ell, c ell, d when 1 <= b <= c and ell -> + Infinity. As an example, from this analysis we obtain the ell -> + Infinity limiting behavior of the sharp constant Kell, 2 ell, 2 ell, d; a second example concerns the ell -> + Infinity limit for Kell, 2 ell, 3 ell, d. The present work generalizes our previous paper [16], entirely devoted to the constant Kell m n d in the special case ell = m = n; many results given therein can be recovered here for this special case.