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 -> +
Infinity, 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.