New results on multiplication in Sobolev spaces

Abstract

We consider the Sobolev (Bessel potential) spaces H ℓ ( R d , C ) , and their standard norms ‖ ‖ ℓ (with ℓ integer or non-integer). 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 ∈ H m ( R d , C ) , g ∈ 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 ± 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 → + ∞ . As an example, from this analysis we obtain the ℓ → + ∞ limiting behavior of the sharp constant K ℓ , 2 ℓ , 2 ℓ , d ; a second example concerns the ℓ → + ∞ 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.