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.
