On maximal functions generated by Hörmander-type spectral multipliers

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-Honzík-Seeger \cite{GHS2006} to obtain an optimal $\sqrt{\log (1+N)}$ bound in $L^p$ for the maximal function ${\sup }_{1\le i\le N}\mathrm{\mid }{m}_i(L)f\mathrm{\mid }$ for multipliers $m_i,1\le i\le N,$ with uniform estimates. Based on this, we establish sufficient conditions on the bounded Borel function $m$ such that the maximal function $M_{m,L}f(x)={\sup }_{t>0}\mathrm{\mid }m(tL)f(x)\mathrm{\mid }$ is bounded on $L^p(X)$. The applications include Schrödinger operators with inverse square potential, Scattering operators, Bessel operators and Laplace-Beltrami operators.