Around Fatou Theorem and Its Converse on Certain Lie Groups

Abstract

A classical result due to Fatou relates the radial and nontangential behaviour of the Poisson integral of suitable measures on the real line with certain differentiability properties of the measure. Loomis proved the converse of Fatou's theorem for positive measures on the real line. Rudin and Ramey-Ullrich later extended the results of Loomis in higher dimensions. In the first part of the thesis, we have proved generalizations of the result of Rudin, involving a large class of approximate identities generalizing the Poisson kernel. We have then used it to show that the analogue of Rudin's result holds for certain positive eigenfunctions of the Laplace-Beltrami operator on real hyperbolic spaces. In the second part of the thesis, we have proved the analogues of the result of Ramey-Ullrich, regarding nontangential convergence of Poisson integrals, for certain positive eigenfunctions of the Laplace-Beltrami operator of Harmonic NA groups. We have also proved similar results for positive solutions of the heat equation on stratified Lie groups.