# Some Calculations of Orlicz Cohomology and Poincar\'e--Sobolev--Orlicz Inequalities

## TL;DR

This paper computes Orlicz cohomology for fundamental Riemannian manifolds and explores its connection to Poincaré--Sobolev--Orlicz inequalities, advancing understanding of geometric analysis in these contexts.

## Contribution

It provides explicit calculations of Orlicz cohomology on key manifolds and discusses their relationship with Poincaré--Sobolev--Orlicz inequalities, a novel link in geometric analysis.

## Key findings

- Calculated Orlicz cohomology for the real line, hyperbolic plane, and ball.
- Established relationships between Orlicz cohomology and Poincaré--Sobolev--Orlicz inequalities.
- Enhanced understanding of geometric inequalities in Riemannian manifolds.

## Abstract

We carry out calculations of Orlicz cohomology for some basic Riemannian manifolds (the real line, the hyperbolic plane, the ball). Relationship between Orlicz cohomology and Poincar\'e--Sobolev--Orlicz-type inequalities is discussed.