Littlewood-Paley theorem, Nikolskii inequality, Besov spaces, Fourier and spectral multipliers on graded Lie groups

TL;DR

This paper extends classical harmonic analysis results like Littlewood-Paley and Nikolskii inequalities to Besov spaces on graded Lie groups, establishing embeddings and multiplier theorems.

Contribution

It introduces new inequalities and embedding results for Besov spaces on graded Lie groups, along with conditions for multiplier boundedness.

Findings

Proved a Nikolskii type inequality on graded Lie groups

Established a Littlewood-Paley theorem for these groups

Derived multiplier theorems for spectral and Fourier multipliers

Abstract

In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse H\"older inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.