The Green formula and \(H^ p\) spaces on trees
It is known that, under suitable regularity conditions, the Hardy spaces of harmonic functions on a (not necessarily homogeneous) tree, associated with the area function over cones, are equivalent to those associated to the nontangential maximal function, and independent of the width of the cone. This was proved in [\textit{A. Korányi}, \textit{M. A. Picardello} and \textit{M. H. Taibleson}, Symp. Math. 29, 205-254 (1987; Zbl 0637.31004)] by a probabilistic approach, which makes use of martingales on the boundary of the tree. Here we give a new, independent proof, inspired by the differential approach which is typical of the continuous setting. The main tools are the Green formula and good lambda inequalities. This paper is aimed at showing that -- surprisingly! -- differential and geometric methods can be used successfully in this discrete context.
scientific article; zbMATH DE number 737186
scientific article