Le module de continuité des valeurs au bord des fonctions propres et des formes automorphes
Abstract
Let Γ be a Fuchsian group acting on the hyperbolic plane ℍ. Any eigenfunction of the Laplacian, any automorphic form on an hyperbolic surface ℍ/Γ induce a distribution on the boundary ∂ℍ. This distribution is the derivative of a certain order of a fonction F on ∂ℍ : the derivative of order 1 in the case of bounded eigenfunctions, the derivative of order 
in the case of cuspidal automorphic forms of weight k. For cuspidal eigenfunctions (resp. for cuspidal automorphic forms) the optimal Hölder exponent of F at a point ξ ∈ ∂ℍ can be computed exactly when ℍ/Γ has finite volume : this exponent depends only on the eigenvalue of the function (resp. on the weight of the automorphic form) and on the fact that the ray oξ be recurrent or not ; in particular F is not differentiable at any point, except possibly at the cusps, depending on the eigenvalue (resp. on the weight). For regular but non-cuspidal automorphic forms, there is a continuous family of possibilities for the modulus of continuity. We study in details the case of automorphic forms of weight 
(like the Jacobi theta function, whose associate function F on ∂ℍ has imaginary part the Riemann function 
: using properties of the geodesic flow such as the Khintchine-Sullivan theorem, we show that at almost all ξ a modulus of continuity is 
, for any є > 0.
© de Gruyter 2010
Articles in the same Issue
- Inhomogeneous Strichartz estimates
- Some characterizations of finite groups in which semipermutability is a transitive relation
- Fine gradings on the Lie algebra
- Harmonic analysis on a finite homogeneous space II: The Gelfand–Tsetlin decomposition
- Regularity results for the gradient of solutions of linear elliptic systems with VMO-coefficients and L1,λ data
- Amplitude inequalities for Differential Graded modules
- Homogeneous Lagrangian submanifolds of positive Euler characteristic
- Kostant convexity for affine buildings
- Algebraic monodromy and obstructions to formality
- Capacity and potentials on curves
- New characterizations of pseudo-coherent rings
- Le module de continuité des valeurs au bord des fonctions propres et des formes automorphes
Articles in the same Issue
- Inhomogeneous Strichartz estimates
- Some characterizations of finite groups in which semipermutability is a transitive relation
- Fine gradings on the Lie algebra
- Harmonic analysis on a finite homogeneous space II: The Gelfand–Tsetlin decomposition
- Regularity results for the gradient of solutions of linear elliptic systems with VMO-coefficients and L1,λ data
- Amplitude inequalities for Differential Graded modules
- Homogeneous Lagrangian submanifolds of positive Euler characteristic
- Kostant convexity for affine buildings
- Algebraic monodromy and obstructions to formality
- Capacity and potentials on curves
- New characterizations of pseudo-coherent rings
- Le module de continuité des valeurs au bord des fonctions propres et des formes automorphes
