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Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces

Philippe Jaming (1999)

Colloquium Mathematicae

We study harmonic functions for the Laplace-eltrami operator on the real hyperbolic space n . We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball n . We then study the Hardy spaces H p ( n ) , 0

Inequalities involving heat potentials and Green functions

Neil A. Watson (2015)

Mathematica Bohemica

We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set E whose supports are compact polar subsets of E . We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set E , we prove...

Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Patie Pierre (2009)

Annales de l'I.H.P. Probabilités et statistiques

We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x>0, γ≥0 and fa smooth function on ℜ + , 𝐋 ( γ ) f ( x ) = x - α ( σ 2 x 2 f ' ' ( x ) + ( σ γ + b ) x f ' ( x ) + ∫ 0 ∞ f e - r x - f ( x ) e - r γ + x f ' ( x ) r 𝕀 { r ≤ 1 } ν ( d r ) ) , ( 0 . 1 ) where the coefficients b ∈ ℜ ,σ≥0 and the measure ν, which satisfies the integrability condition ∫0∞(1∧r2)ν(dr)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L(γ) is known to be the infinitesimal generator of a positive...

Jump processes, ℒ-harmonic functions, continuity estimates and the Feller property

Ryad Husseini, Moritz Kassmann (2009)

Annales de l'I.H.P. Probabilités et statistiques

Given a family of Lévy measures ν={ν(x, ⋅)}x∈ℝd, the present work deals with the regularity of harmonic functions and the Feller property of corresponding jump processes. The main aim is to establish continuity estimates for harmonic functions under weak assumptions on the family ν. Different from previous contributions the method covers cases where lower bounds on the probability of hitting small sets degenerate.

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