A $T1$ criterion for Schrödinger-Calderón-Zygmund operators with exponential decay

We establish the boundedness of exponential Schr\"odinger-Calder\'on-Zygmund operators on weighted $BMO_\rho^\alpha(w)$ spaces via a $T1$ criterion, where the weights belong to classes that capture the exponential decay of the operators, and $\rho$ is a critical radius function. Specifically, we prove that the boundedness of such an operator $T$ on $BMO_\rho^\alpha(w)$ is equivalent to a natural oscillation condition on $T1$ over sub-critical balls. The weight classes considered, introduced in connection with $\rho$, include and extend the classical $A_p^\rho$ weights, and are well-adapted to the exponential decay of the kernels. As applications, we derive weighted endpoint estimates for several operators associated to the generalized Schr\"odinger operator $\mathcal{L}_\mu=-\Delta+\mu$, including Riesz transforms, Laplace transform-type multipliers, maximal operators for the heat and Poisson semigroups, Littlewood-Paley functions and fractional integral operators. When $d\mu(x)=V(x)dx$, the results above extend the known endpoint estimates to larger classes of weights.