Local Fatou Theorem and Area Theorem for Symmetric Spaces of Rank One

1976 ◽  
Vol 224 (1) ◽  
pp. 157 ◽  
Author(s):  
A. Koranyi ◽  
R. B. Putz
Keyword(s):  
Symmetric Spaces ◽  
Area Theorem ◽  
Fatou Theorem ◽  
Rank One

scholarly journals An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

2021 ◽  
Author(s):  
Laura Atanasi ◽  
Massimo A. Picardello
Keyword(s):  
Harmonic Functions ◽  
Symmetric Spaces ◽  
Cartesian Product ◽  
Measurable Subset ◽  
Hyperbolic Spaces ◽  
Area Theorem ◽  
Area Function ◽  
Homogeneous Trees ◽  
Rank One ◽  
Higher Rank

AbstractFor 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.

scholarly journals Polar actions on rank-one symmetric spaces

1999 ◽  
Vol 53 (1) ◽  
pp. 131-175 ◽  
Author(s):  
Fabio Podestà ◽  
Gudlaugur Thorbergsson
Keyword(s):  
Symmetric Spaces ◽  
Polar Actions ◽  
Rank One
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