$L^p$ spectral multipliers on the free group $N_{3,2}$

TL;DR

This paper establishes Mihlin-H"ormander type spectral multiplier theorems for the homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent group, requiring a certain differentiability order for the multipliers.

Contribution

It extends spectral multiplier results to the specific setting of the free group $N_{3,2}$ with new differentiability conditions.

Findings

Proves Mihlin-H"ormander type theorem for $L$ on $N_{3,2}$

Determines the differentiability order $s > 3$ needed for boundedness

Provides a framework for spectral multipliers on free 2-step nilpotent groups

Abstract

Let $L$ be the homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent group $N_{3,2}$ on 3 generators. We prove a theorem of Mihlin-H\"ormander type for the functional calculus of $L$, where the order of differentiability $s > 6/2$ is required on the multiplier.