On the characterization of harmonic functions on weighted spaces and the Heat equation

We study the solutions of convolution equations f ∗ μ = f for radial probability measures μ on Rⁿ and on symmetric spaces on rank 1. By using spectral analysis and synthesis we deal with the characterization of harmonic functions by the mean-value property in the bounded case. For weighted L¹-spaces we use results of Beurling and Rudin on the primary ideals of the disk algebra to prove that f ∗ e^-|x| = f, ∈L¹(e^-|x|) implies that/is harmonic if and only if n ≤ 8. Similar ideas are applied to the Hyperbolic spaces. Our results are applied to the investigation of periodic solutions of the Heat equation.