Heavy tailed approximate identities and $\sigma$-stable Markov kernels

AIMAR, HUGO; GOMEZ, IVANA; MORANA, FEDERICO

POTENTIAL ANALYSIS

SPRINGER

Lugar: Berlin; Año: 2018 vol. 48 p. 473 - 493

0926-2601

Theaim of this paper is to present some results relating the properties ofstability, concentration and approximation to the identity of convolutionthrough not necessarily mollification type families of heavy tailed Markovkernels. A particular case is provided by the kernels K t obtained as the tmollification of L σ(t) selected from the family L={Lσ:Lσ?(ξ)=e−|ξ|σ,0<σ<2},by a given function σ with values in the interval (0,2). We show thata basic Harnack type inequality, introduced by C. Calderón in the convolutioncase, becomes at once natural to the setting and useful to connect the conceptsof stability, concentration and approximation of the identity. Some of thegeneral results are extended to spaces of homogeneous type since most of theconcepts involved in the theory are given in terms of metric and measure.