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Quantitative functional calculus in Sobolev spaces | | Carlo Morosi
; Livio Pizzocchero
; | | Date: |
23 May 2003 | | Journal: | Journal of Function Spaces and Applications (JFSA) 2 (2004), 279-321 | | Subject: | Functional Analysis; Mathematical Physics MSC-class: 46E35, 26D10, 47A60 | math.FA math-ph math.MP | | Affiliation: | Politecnico di Milano), Livio Pizzocchero (Univ. di Milano | | Abstract: | In the framework of Sobolev (Bessel potential) spaces $H^n(
eali^d,
eali {or} complessi)$, we consider the nonlinear Nemytskij operator sending a function $x in
eali^d mapsto f(x)$ into a composite function $x in
eali^d mapsto G(f(x), x)$. 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. | | Source: | arXiv, math.FA/0305331 | | Services: | Forum | Review | PDF | Favorites | |
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