New results on multiplication in Sobolev spaces

We consider the Sobolev (Bessel potential) spaces H^s, and their standard norms | . |_s (with s integer or non-integer). For any dimension d, we are interested in the unknown sharp constant K_smn in the inequality |f g|_s <=K_smn |f |_m |g|_n ( f in H^m, g in H^n; 0<=s<=n, m + n − s > d/2); we derive upper and lower bounds K± for this constant. As examples, we give a table of these bounds for d = 1, d = 3 and many values of (s,m,n); here the ratio K−/K+ 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 Ks,bs,cs,d when 1<= b<= c and s → +∞. As an example, from this analysis we obtain the s → +∞ limiting behavior of the sharp constant Ks,2s,2s; a second example concerns the s → +∞ limit for Ks,2s,3s. The present work generalizes a previous paper of ours (2006), entirely devoted to the constant Ksmn in the special case s = m = n; many results given therein can be recovered here for this special case.