Sharp weighted $L^p$ estimates for spectral multipliers

TL;DR

This paper establishes sharp weighted $L^p$ bounds for spectral multipliers and their commutators associated with a self-adjoint operator generating a Gaussian-bounded semigroup, improving previous results.

Contribution

It provides new sharp weighted inequalities for spectral multipliers and their commutators, extending and refining earlier work in the field.

Findings

Sharp weighted $L^p$ inequalities for spectral multipliers.

Improved bounds for commutators with BMO functions.

Enhanced understanding of spectral multiplier behavior under Gaussian bounds.

Abstract

Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that operator $L$ generates the analytic semigroup $\{e^{-tL}\}_{t>0}$ whose kernels satisfy the standard Gaussian upper bounds. This paper investigates the sharp weighted inequalities for spectral multipliers $F(L)$ and their commutators with BMO functions. The obtained results in this paper can be considered to be improvements of those in \cite{DSY}[Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers, {\it J. Funct. Anal.}, {\bf 260} (2011), 1106-1131].