New results on multiplication in Sobolev spaces

AbstractWe consider the Sobolev (Bessel potential) spaces Hℓ(Rd,C), and their standard norms ‖ ‖ℓ (with ℓ integer or non-integer). We are interested in the unknown sharp constant Kℓmnd in the inequality ‖fg‖ℓ⩽Kℓmnd‖f‖m‖g‖n (f∈Hm(Rd,C), g∈Hn(Rd,C); 0⩽ℓ⩽m⩽n, m+n−ℓ>d/2); we derive upper and lower bounds Kℓmnd± 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 Kℓmnd−/Kℓmnd+ 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 Kℓ,bℓ,cℓ,d when 1⩽b⩽c and ℓ→+∞. As an example, from this analysis we obtain the ℓ→+∞ limiting behavior of the sharp constant Kℓ,2ℓ,2ℓ,d; a second example concerns the ℓ→+∞ limit for Kℓ,2ℓ,3ℓ,d. The present work generalizes our previous paper Morosi and Pizzocchero (2006) [16], entirely devoted to the constant Kℓmnd in the special case ℓ=m=n; many results given therein can be recovered here for this special case