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

Suppose $L=-\mathrm{\Delta }+V$ is a Schr"odinger operator on ${\mathbb{R}}^n$ with a potential $V$ belonging to certain reverse H"older class $R{H}_{\sigma }$ with $\sigma \ge n\mathrm{/}2$. 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 $H_L^p({\mathbb{R}}^n)$ 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 $n\mathrm{/}2\le \sigma <n$ which were investigated in \cite{MSTZ} under the stronger condition $\sigma \ge n$. Thus our results not only recover existing results in \cite{MSTZ} but also contains new results in literature.