The thesis is composed of four articles. In the first one, 'Hardy spaces for the Laplacian with lower order perturbations ", we consider the Hardy spaces of harmonic functions for the Laplacian with gardient or Schrödinger perturbations, under appropriate Kato conductions. We show the representation theorem for the Hardy spaces on bounded smooth domains in euclidiean spaces for the Laplacian and the fractional Laplacian by means of the Hardy-Stein type identities. In the third article, " Boundary behavior of alpha-harmonic functions on the complement of the sphere and hyperplan ", we study the properties of harmonic functions of the representation theorems for appropriate Hardy spaces and the Fatou theorems. We also obtain explicit formulas for the Martin kernel of the complement of a sphere, and for the Green function, Martin kernel and harmonic measure for the complement of a hyperplane. The fourth article, " Martin representation, Relative Fatou Theorem and Hardy spaces for fractional Laplacian with a gradient perturbation ", concerns the potential theory for the fractional Laplacian perturbated by gradient on bounded smooth domains. Here we show the existence of the Martin kernel and the Martin representation for appropriate harmonic functions. The relativ Fatou theorem and the representation theorem for Hardy spaces are also proved.
