On the constants for multiplication in Sobolev spaces

For $n > d/2$, the Sobolev (Bessel potential) space $H^n(\reali^d, \complessi)$ 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 $\| f g \|_{n} \leqs K_{n d} \| f \|_{n} \| g \|_{n}$ 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) $n \in (d/2, + \infty)$. Our analysis also includes the limit cases $n \vain (d/2)^{+}$ and $n \vain + \infty$, 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.