Data di Pubblicazione:
2004
Citazione:
Quantitative functional calculus in Sobolev spaces / C. Morosi, L. Pizzocchero. - In: JOURNAL OF FUNCTION SPACES AND APPLICATIONS. - ISSN 0972-6802. - 2:3(2004), pp. 279-231. [10.1155/2004/832750]
Abstract:
In the
framework of Sobolev (Bessel potential) spaces
$H^n(\reali^d, \reali~ \mbox{or}~\complessi)$,
we consider the nonlinear Nemytskij operator
sending a function
$x \in \reali^d \mapsto f(x)$
into a
composite function $x \in \reali^d \mapsto G(f(x), x)$.
Assuming sufficient smoothness for $G$, we give a
"tame" bound on the $H^n$ norm of this composite function in terms
of a linear function of the $H^n$ norm of $f$, with a coefficient
depending on $G$ and on the $H^a$ norm of $f$, for all integers
$n, a, d$ with $a > d/2$. In comparison
with previous results on this subject, our bound is fully explicit,
allowing to estimate quantitatively the $H^n$ norm of
the function $x \mapsto G(f(x),x)$. When applied to the case $G(f(x), x) =
f^2(x)$, this bound agrees with a previous result of ours on the pointwise
product of functions in Sobolev spaces.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Sobolev spaces, inequalities, Nemytskij operators.
Elenco autori:
C. Morosi, L. Pizzocchero
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