Mathematics – Functional Analysis
Scientific paper
2009-02-04
Advances in Applied Mathematics 44 (2010) 393-432
Mathematics
Functional Analysis
LaTex, 47 pages
Scientific paper
10.1016/j.aam.2009.11.006
We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \leqs K_{ell m n d} || f ||_{m} || g ||_n (f in H^m(R^d, C), g in H^n(R^d, 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 K_{ell, 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 K_{ell, 2 ell, 2 ell, d}; a second example concerns the ell -> + Infinity limit for K_{ell, 2 ell, 3 ell, d}. The present work generalizes our previous paper [16], entirely devoted to the constant K_{ell m n d} in the special case ell = m = n; many results given therein can be recovered here for this special case.
Morosi Carlo
Pizzocchero Livio
No associations
LandOfFree
New results on multiplication in Sobolev spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with New results on multiplication in Sobolev spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and New results on multiplication in Sobolev spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-553522