On $L^p$ inequality for differential forms and $L^p$ cohomology of a semialgebraic set for $p>>1$

TL;DR

This paper establishes an $L^p$ Poincaré inequality for differential forms on compact semialgebraic sets for large $p$, linking $L^p$ cohomology with classical singular cohomology.

Contribution

It introduces a method to derive global $L^p$ inequalities using Lipschitz deformation retractions and double complex techniques, connecting $L^p$ cohomology to singular cohomology.

Findings

Proves local $L^p$ inequalities via Lipschitz deformation retractions.

Extends local inequalities to global ones using double complex methods.

Shows isomorphism between $L^p$ cohomology and singular cohomology for semialgebraic pseudomanifolds.

Abstract

We study Poincar\'e type $L^p$ inequality on a compact semialgebraic subset of $\R^n$ for $p>>1$. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives. Then, we extend the local inequality to a global inequality by employing double complex technique. As a consequence we obtain an isomorphism between $L^p$ cohomology and singular cohomology of a normal compact semialgebraic pseudomanifold.