The green formula and H p spaces on trees

Abstract

A theory o f H e spaces on (not necessarily homogeneous) trees was developed in [KPT]. These spaces are defined by means of certain maximal or square function operators associated with a nearest neighbour transition operator P which is very regular: i.e., for some 6o with 0 < 60 < 89 we assume the i _ 60 for every pair of neighbours x, y. The regularity bounds 60 < p ( x , y ) < condition yields uniform estimates for transience o f P, and therefore gives a good control over several operators defined on the boundary f2 of T, by means of which one defines the H p spaces. As observed long ago in the homogeneous case [Ta], a nearest neighbour transition operator on an arbitrary tree gives rise to a natural martingale structure on its boundary (see [KPT] for details). This martingale inherits the strong regularity property, which is heavily used in [KPT] to prove that the different definitions of H p are equivalent. The approach of [KPT] is in the same spirit o f the construction o f H p spaces by means of martingales, developed in [BGS] in a very general setting. On the other hand, another independent approach to H p theory in smooth domains (say, the disc or the upper half plane) was established in [FS]. Some of its operators, as the Lusin area integral, are defined in terms of differential operators, and the equivalence between different constructions o f H p is based upon good lambda inequalities. An important tool is the Green formula, which is used to transform the area integral of a harmonic function over smooth compact domains to a line integral over their boundary curves. The method of [FS] has been extended to semisimple symmetric spaces of rank one [Ci].