The H\"older-Poincar\'e Duality for $L_{q,p}$-cohomology

TL;DR

This paper establishes a duality relationship between reduced $L_{q,p}$-cohomology and interior $L_{p',q'}$-cohomology on Riemannian manifolds, extending classical Poincaré duality to a broader functional setting.

Contribution

It proves a new version of Poincaré duality specifically for reduced $L_{q,p}$-cohomology on Riemannian manifolds, linking it with interior $L_{p',q'}$-cohomology.

Findings

Establishes duality between $L_{q,p}$-cohomology and interior $L_{p',q'}$-cohomology.

Extends classical Poincaré duality to $L_{q,p}$-cohomology.

Provides a framework for understanding cohomological dualities in non-compact settings.

Abstract

We prove the following version of Poincare duality for reduced $L_{q,p}$-cohomology: For any $1<q,p<\infty$, the $L_{q,p}$-cohomology of a Riemannian manifold is in duality with the interior $L_{p',q'}-cohomology for $1/p+1/p'=1$, $1/q+1/q'=1$.