Spectral multipliers for functions of fixed $K$-type on $L^p(SL(2,\mathbb{R}))$

TL;DR

This paper establishes an $L^p$ spectral multiplier theorem for $K$-invariant functions of a sublaplacian on $SL(2,R)$, providing insights into the spectral analysis of fixed $K$-type functions on the group.

Contribution

It proves a new spectral multiplier theorem for fixed $K$-type functions on $SL(2,R)$, extending harmonic analysis tools to this setting.

Findings

Derived the joint $L^p$ spectrum of $L$ and the $K$-derivative on $SL(2,R)$

Established $L^p$ boundedness of spectral multipliers for the $K$-invariant sublaplacian

Extended spectral analysis techniques to functions of fixed $K$-type

Abstract

We prove an $L^p$ spectral multiplier theorem for functions of the $K$-invariant sublaplacian $L$ acting on the space of functions of fixed $K$-type on the group $SL(2,\mathbb{R}).$ As an application we compute the joint $L^p(SL(2,\mathbb{R}))$ spectrum of $L$ and the derivative along $K$.