Using explicit representations of the Brownian motions on hyperbolic spaces, we show that their almost sure convergence and the central limit theorems for the radial components as time tends to infinity can be easily obtained. We also give a straightforward strategy to obtain explicit expressions for the limit distributions or Poisson kernels.
This is a report on some joint work with Aobing Li on Liouville type theorems for some conformally invariant fully nonlinear equations.
In this paper we present some Liouville type theorems for solutions of differential inequalities involving the φ-Laplacian. Our results, in particular, improve and generalize known results for the Laplacian and the p-Laplacian, and are new even in these cases. Phragmen-Lindeloff type results, and a weak form of the Omori-Yau maximum principle are also discussed.
For certain non compact Riemannian manifolds with ends which may or may not satisfy the doubling condition on the volume of geodesic balls, we obtain Littlewood-Paley type estimates on (weighted) spaces, using the usual square function defined by a dyadic partition.
We study the weak type (1,1) and the -boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions and ℋ respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold M. Without any assumption on M, we observe that and ℋ are bounded in , 1 < p ≤ 2. We also consider modified Littlewood-Paley-Stein functions that take into account the positivity of the bottom of the spectrum. Assuming that...
Let be a long range metric perturbation of the Euclidean Laplacian on , . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group where has a suitable development at zero (resp. infinity).