Consider a non-doubling space like a non-doubling manifold with ends which is formed by connecting (smoothly) a copy of  with , . Grigoryan and Saloff-Coste studied the heat kernel of the Laplace Beltrami operator in this setting. Duong-Li-Bui-Wick recently proved that some holomorphic functional calculus is weak type (1,1) and bounded on  in this setting.

Question: how do we define appropriate weights in this setting to study singular integrals in this non-doubling setting? The usual Muckenhoupt weights need the underlying space to be doubling and we do not know any work about weighted estimates in non-doubling setting.