On the constants for multiplication in Sobolev spaces

Abstract

For n > d / 2 , the Sobolev (Bessel potential) space H n ( R d , C ) is known to be a Banach algebra with its standard norm ‖ ‖ n and the pointwise product; so, there is a best constant K n d such that ‖ f g ‖ n ⩽ K n d ‖ f ‖ n ‖ g ‖ n for all f , g in this space. In this paper we derive upper and lower bounds for these constants, for any dimension d and any (possibly noninteger) n ∈ ( d / 2 , + ∞ ) . Our analysis also includes the limit cases n → ( d / 2 ) + and n → + ∞ , for which asymptotic formulas are presented. Both in these limit cases and for intermediate values of n, the lower bounds are fairly close to the upper bounds. Numerical tables are given for d = 1 , 2 , 3 , 4 , where the lower bounds are always between 75 and 88% of the upper bounds.