Weighted multilinear Poincare inequalities for vector fields of Hormander type

TL;DR

This paper introduces weighted multilinear Poincaré inequalities for vector fields of Hormander type in subelliptic settings, providing new tools for analysis when classical inequalities fail for certain p-values.

Contribution

It develops a framework for weighted multilinear Poincaré inequalities in subelliptic contexts, extending classical results to cases where p<1.

Findings

Established multilinear representation formulas

Proved weighted estimates for multilinear potential operators

Extended inequalities to subelliptic spaces of homogeneous type

Abstract

As the classical $(p,q)$-Poincar\'e inequality is known to fail for $0 < p < 1$, we introduce the notion of weighted multilinear Poincar\'e inequality as a natural alternative when $m$-fold products and $1/m < p$ are considered. We prove such weighted multilinear Poincar\'e inequalities in the subelliptic context associated to vector fields of H\"ormader type. We do so by establishing multilinear representation formulas and weighted estimates for multilinear potential operators in spaces of homogeneous type.