New results on multiplication in Sobolev spaces

We consider the Sobolev (Bessel potential spaces H^l(R^d, C) and their standard norms || ||_l (with l integer or noninteger). We are interested in the unknown sharp constant K_{l m n d} in the inequality || f g ||_l <= K_{l m n d} || f ||_m || g||_n (f in H^m(R^d,C), g in H^n(R^d,C); 0 <= l <= m <= n, m + n - l > d/2); we derive upper and lower bounds K{+}_{l m n d}, K^{-}_{l m n d} for this constant. As examples, we give a table of these bounds for d=1, d=3 and many values of l, m, n; here the ratio K^{-}_{l m n d}/K^{+}_{l 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_{l, b l, c l, d} when 1 <= b <= c and l -> +Infinity. As an example, from this analysis we obtain the l -> +Infinity limiting behavior of the sharp constant K_{l, 2 l, 2 l, d}; a second example concerns the l -> + Infinity limit for K_{l, 2 l, 3 l, d}. The present work generalizes our previous paper Morosi and Pizzocchero (2006) [16], entirely devoted to the constant K_{l m n d} in the special case l = m = n; many results given therein can be recovered here for this special case.