On the analogue of the Fefferman–Stein theorem on graphs with the Cheeger property

In this paper, we consider connected locally finite graphs g that possess the Cheeger isoperimetric property. We investigate the Hardy spaces H-R(1)(g), H-H(1)(g) and H-P(1)(g), defined in terms of the Riesz transform, the heat and the Poisson maximal operator on g, respectively. Quite surprisingly, we prove that contrary to what happens in the Euclidean case, these three spaces are distinct. In addition, we prove that if g is an homogeneous tree, then H-R(1)(g) does not admit an atomic decomposition. Applications to the boundedness of the purely imaginary powers of the nearest neighbour Laplacian and of the associated Riesz transform are given.