Data di Pubblicazione:
2006
Citazione:
On the constants for multiplication in Sobolev spaces / C. Morosi, L. Pizzocchero. - In: ADVANCES IN APPLIED MATHEMATICS. - ISSN 0196-8858. - 36:4(2006), pp. 319-363. [10.1016/j.aam.2005.09.002]
Abstract:
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.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Sobolev spaces; inequalities; pointwise multiplication
Elenco autori:
C. Morosi, L. Pizzocchero
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