Weighted norm inequalities for spectral multipliers without Gaussian estimates

TL;DR

This paper establishes sharp weighted $L^p$ bounds for spectral multipliers and their commutators associated with non-negative self-adjoint operators, without relying on Gaussian heat kernel estimates.

Contribution

It provides new weighted inequalities for spectral multipliers and commutators without assuming Gaussian bounds on heat kernels.

Findings

Sharp weighted $L^p$ estimates for spectral multipliers

Boundedness of commutators with BMO functions

No Gaussian heat kernel bounds required

Abstract

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$. By spectral theory, we can define the operator $F(L)$, which is bounded on $L^2(X)$, for any bounded Borel function $F$. In this paper, we study the sharp weighted $L^p$ estimates for spectral multipliers $F(L)$ and their commutators $[b, F(L)]$ with BMO functions $b$. We would like to emphasize that the Gaussian upper bound condition on the heat kernels associated to the semigroups $e^{-tL}$ is not assumed in this paper.