On maximal functions generated by H\"ormander-type spectral multipliers

TL;DR

This paper establishes optimal bounds for maximal functions generated by spectral multipliers of certain operators on metric spaces, extending to various important operators like Schrödinger and Bessel operators.

Contribution

It introduces a new approach using the Doob transform to obtain sharp $L^p$ bounds for maximal functions associated with spectral multipliers.

Findings

Achieved an optimal $ oot{ ext{log}}{ ext{(1+N)}}$ bound in $L^p$ for the maximal function.

Provided sufficient conditions for boundedness of maximal functions $M_{m,L}$ on $L^p$.

Applied results to Schrödinger, scattering, Bessel, and Laplace-Beltrami operators.

Abstract

Let $(X,d,\mu)$ be a metric space with doubling measure and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. We assume that there exists an $L$-harmonic function $h$ such that the semigroup $\exp(-tL)$, after applying the Doob transform related to $h$, satisfies the upper and lower Gaussian estimates. In this paper we apply the Doob transform and some techniques as in Grafakos-Honz\'ik-Seeger \cite{GHS2006} to obtain an optimal $\sqrt{\log(1+N)}$ bound in $L^p$ for the maximal function $\sup_{1\leq i\leq N}|m_i(L)f|$ for multipliers $m_i,1\leq i\leq N,$ with uniform estimates. Based on this, we establish sufficient conditions on the bounded Borel function $m$ such that the maximal function $M_{m,L}f(x) = \sup_{t>0} |m(tL)f(x)|$ is bounded on $L^p(X)$. The applications include Schr\"odinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators.