Riesz transforms, spectral multipliers and Hardy spaces on graphs

In this thesis we consider a connected locally finite graph G that possesses the Cheeger isoperimetric property. We define a decreasing one parameter family of Hardy-type spaces associated with the standard nearest neighbour Laplacian on G. We show that the space with parameter ½ is the space of all integrable functions whose Riesz transform is integrable. We show that if G has bounded geometry and the parameter is an integer, the corresponding Hardy-type space admits an atomic decomposition. We also show that if G is a homogeneous tree and the parameter is not an integer, the corresponding Hardy-type space does not admit an atomic decomposition. Furthermore, we consider the Hardy-type spaces defined in terms of the heat and Poisson maximal operators, and we analyse their relationships with the family of spaces defined previously. We also show that the space associated with the heat maximal operator is properly contained in the one associated with the heat maximal operator, a phenomenon which has no counterpart in the euclidean setting. Applications to the purely imaginary powers of the Laplacian are also given. Finally, we characterise, for every p, the class of spherical multipliers on the p-integrable functions on homogeneous trees in terms of Fourier multipliers on the torus. Furthermore we give a sharp sufficient condition on spherical multipliers on the product of homogeneous trees.