The Green formula and \(H^ p\) spaces on trees
type: Item
instance of: scholarly article
title: The Green formula and \(H^ p\) spaces on trees
zbMATH Open document ID: 0819.31004
DOI: 10.1007/BF02571903
author: Fausto Di Biase
author: Massimo A. Picardello
published in: Mathematische Zeitschrift
publication date: 1995-08-24T00:00:00Z
full work available at URL: https://eudml.org/doc/174728
review text: 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.
Mathematics Subject Classification ID: 31C20
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 43A99
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 737186
zbMATH Keywords: trees
zbMATH Keywords: boundary of finite subtree
zbMATH Keywords: spaces of homogeneous type
zbMATH Keywords: Hardy spaces of harmonic functions
zbMATH Keywords: area function
zbMATH Keywords: nontangential maximal function
zbMATH Keywords: Green formula
zbMATH Keywords: good lambda inequalities
MaRDI profile type: Publication
cites work: Extrapolation and interpolation of quasi-linear operators on martingales
cites work: A Maximal Function Characterization of the Class H p
cites work: Q4769263
cites work: Parabolic maximal functions associated with a distribution
cites work: Q3709253
cites work: Q2548398
cites work: Q3822887
cites work: Q3342799
cites work: \(H^p\) spaces of several variables
cites work: Sur les groupes hyperboliques d'après Mikhael Gromov. (On the hyperbolic groups à la M. Gromov)
cites work: Q3758200
cites work: Q3142876
cites work: Harmonic analysis on \(n\)-dimensional vector spaces over local fields. III: Multipliers
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description: scientific article; zbMATH DE number 737186