On the Exponential Integrability of Conjugate Functions

We relate the exponential integrability of the conjugate function $${\tilde{f}}$$ f ~ to the size of the gap in the essential range of f. Our main result complements a related theorem of Zygmund.

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.