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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 196, Issue 5, pp 1927–1960 | Cite as

A.e. convergence and 2-weight inequalities for Poisson-Laguerre semigroups

  • G. Garrigós
  • S. Hartzstein
  • T. Signes
  • B. Viviani
Article
  • 79 Downloads

Abstract

We find optimal decay estimates for the Poisson kernels associated with various Laguerre-type operators L. From these, we solve two problems about the Poisson semigroup \(e^{-t\sqrt{L}}\). First, we find the largest space of initial data f so that \(e^{-t\sqrt{L}}f(x)\rightarrow f(x)\) at \({\,a.e.\,}x\). Secondly, we characterize the largest class of weights w which admit 2-weight inequalities of the form \(\Vert \sup _{0<t\le t_0}|e^{-t\sqrt{L}}f|\,\Vert _{L^p(v)}\lesssim \Vert f\Vert _{L^p(w)}\), for some other weight v.

Keywords

Laguerre expansions Poisson integral Heat semigroup 2-weight problem Fractional laplacian 

Mathematics Subject Classification

33C45 35C15 40A10 42C10 47D06 

Notes

Acknowledgements

We wish to thank José Luis Torrea for many conversations around these topics at the earlier stages of this work. We thank the anonymous referees for their careful reading and their useful suggestions to improve the presentation of this paper. First and third authors were partially supported by Grants MTM2013-40945-P, MTM2013-42220-P and MTM2014-57838-C2-1-P from MINECO (Spain), and grants 19368/PI/14 and 19378/PI/14 from Fundación Séneca, (Región de Murcia, Spain). Second and fourth authors were partially supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Universidad Nacional del Litoral (Argentina).

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • G. Garrigós
    • 1
  • S. Hartzstein
    • 2
  • T. Signes
    • 1
  • B. Viviani
    • 2
  1. 1.Departamento de MatemáticasUniversidad de MurciaMurciaSpain
  2. 2.IMAL (UNL-CONICET) y FIQ (Universidad Nacional del Litoral) CCT CONICET Santa FeSanta FeArgentina

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