T1 criterions for generalised Calder\'on--Zygmund type operators on Hardy and BMO spaces associated to Schr\"odinger operators and applications

Suppose $L=-\Delta+V$ is a Schr\"odinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse H\"older class $RH_\sigma$ with $\sigma\geq n/2$. The main aim of this paper is to provide necessary and sufficient conditions in terms of $T1$ criteria for a generalised Calder\'on--Zygmund type operator with respect to $L$ to be bounded on Hardy spaces $H^p_L(\mathbb{R}^n)$ and on BMO type spaces BMO$_L^\alpha(\mathbb{R}^n)$ associated with $L$. As applications, we prove the boundedness for several singular integral operators associated to $L$. Our approach is flexible enough to prove the boundedness of the Riesz transforms related to $L$ with $n/2 \leq \sigma <n$ which were investigated in \cite{MSTZ} under the stronger condition $\sigma\geq n$. Thus our results not only recover existing results in \cite{MSTZ} but also contains new results in literature.