Note on Poincaré $L^p$ Type Inequality for Differential Forms on Semialgebraic Sets

C. R. Math. Rep. Acad. Sci. Canada Vol. 34 (1) 2012, pp. 23–32
(Received: 2011-06-02 )

Leonid Shartser, Department of Mathematics, University of Toronto, 40 St. George St., Toronto, ON M5S 2E4; e-mail: shartl@math.toronto.edu

Abstract/Résumé:

We study local and global Poincaré type \(L^p\) inequalities on a compact semialgebraic subset of \(\mathbb{R}^n\) for \(p\gg 1\). As a consequence, we obtain an isomorphism between \(L^p\) cohomology and singular cohomology of a normal compact semialgebraic set. The global inequality is derived from the local one, while the local inequality is proved by means of a semialgebraic Lipschitz deformation retraction with estimates on its derivatives.

On étudie les inégalités locales et globales de type \(L^p\) de Poincaré sur un sous-ensemble compact semialgébrique de \(\mathbb{R}^n\) pour \(p\gg 1\). Par conséquent, nous obtenons un isomorphisme entre la cohomologie \(L^p\) et la cohomologie singulière d’un ensemble normal compact semialgébrique. L’inégalité globale est dérivée de la locale, tandis que l’inégalité locale est prouvée au moyen d’une rétraction de déformation semialgébrique Lipschitz avec des estimations sur ses dérivés.

Keywords:

AMS Subject Classification: Semialgebraic sets and related spaces 14P10

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