Characterizations of Lipschitz functions via the commutators of maximal function in Orlicz spaces on stratified Lie groups

TL;DR

This paper characterizes when certain maximal commutators are bounded in Orlicz spaces on stratified Lie groups, linking these bounds to Lipschitz space membership, and provides new characterizations of Lipschitz subclasses.

Contribution

It establishes necessary and sufficient conditions for boundedness of maximal commutators in Orlicz spaces on stratified Lie groups, connecting these to Lipschitz space properties and offering new characterizations.

Findings

Boundedness conditions for maximal commutators in Orlicz spaces.

Characterizations of Lipschitz space subclasses on stratified Lie groups.

New criteria linking Lipschitz regularity to operator boundedness.

Abstract

We give necessary and sufficient conditions for the boundedness of the maximal commutators $M_{b}$, the commutators of the maximal operator $[b, M]$ and the commutators of the sharp maximal operator $[b, M^{\sharp}]$ in Orlicz spaces $L^{\Phi}(\mathbb{G})$ on any stratified Lie group $\mathbb{G}$ when $b$ belongs to Lipschitz spaces $\dot{\Lambda}_{\beta}(\mathbb{G})$. We obtain some new characterizations for certain subclasses of Lipschitz spaces $\dot{\Lambda}_{\beta}(\mathbb{G})$.