This talk consists of two main parts. Firstly we will give a brief overview of the recent progress concerning endpoint spaces (Hardy, BMO) associated to operators. We mainly focus on a series of characterizations of endpoint spaces and their applications.

In the second part, we will study the endpoint space of functions with bounded square functions associated to a self-adjoint operator. We establish some exponential-square integrability and apply it to obtain several estimates such as two-weighted estimates for square functions, eigenvalue estimates for Schrodinger operators on Lipschitz domains. This part is based on a joint work with P. Chen, X.T. Duong and L.X. Yan.