Higher-order multilinear Poincar\'e and Sobolev inequalities in Carnot groups

TL;DR

This paper introduces higher-order multilinear Poincaré and Sobolev inequalities in Carnot groups and applies them to establish weighted Leibniz rules in Campanato-Morrey spaces.

Contribution

It develops new higher-order multilinear inequalities in the setting of Carnot groups, extending classical results to a non-commutative geometric context.

Findings

Established weighted Leibniz rules in Campanato-Morrey spaces

Extended Poincaré and Sobolev inequalities to higher-order multilinear forms in Carnot groups

Provided foundational tools for analysis in sub-Riemannian geometry

Abstract

The notions of higher-order weighted multilinear Poincar\'e and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibnitz-type rules in Campanato-Morrey spaces are established.