Abstract. We prove admissible convergence to the boundary of functions that are harmonic on a subset of a homogeneous tree by means of a discrete Green formula and an analogue of the Lusin area function.

In analogy with the setting of homogeneous, locally finite trees we introduce the Radon transforms on affine, locally finite buildings with diagram of type Ãn. We study the problem of the inversion of these transforms and in some cases we show how inversion algorithms and formulas are available.

We define the Radon transform for functions on the set of chambers of affine, locally finite, rank three buildings. We investigate the problem of the inversion of this transform. Explicit inversion formulas are exhibited for functions which fulfill required summability conditions.

No abstract

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.