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


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.


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

Mathematics Subject Classification

33C45 35C15 40A10 42C10 47D06 



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).


  1. 1.
    Abu-Falahah, I., Macías, R., Segovia, C., Torrea, J.L.: Transferring strong boundedness among Laguerre orthogonal systems. Proc. Indian Acad. Sci. 119(2), 203–220 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Carleson, L., Jones, P.: Weighted Norm Inequalities and A Theorem of Koosis. Mittag-Leffler Inst., report n. 2 (1981)Google Scholar
  3. 3.
    Chicco-Ruiz, A., Harboure, E.: Weighted norm inequalities for heat-diffusion Laguerres semigroups. Math. Z. 257, 329–354 (2007)Google Scholar
  4. 4.
    García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Publishing Co., Amsterdam (1985)zbMATHGoogle Scholar
  5. 5.
    Garrigós, G.: A weak 2-weight problem for the Poisson-Hermite semigroup. In: Martín-Reyes et al. (eds.) Advanced Courses in Mathematical Analysis, vol. VI, pp. 153–171. World Scientific 2016. doi: 10.1142/9789813147645_0006
  6. 6.
    Garrigós, G., Hartzstein, S., Signes, T., Torrea, J.L., Viviani, B.: Pointwise convergence to initial data of heat and Laplace equations. Trans. Am. Math. Soc. 368(9), 6575–6600 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hartzstein, S.I., Torrea, J.L., Viviani, B.E.: A note on the convergence to initial data of Heat and Poisson equation. Proc. Am. Math. Soc. 141, 1323–1333 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lebedev, N.: Special Functions and Their Applications. Prentice-Hall Inc., Englewood CIiffs, N.J. (1965)Google Scholar
  9. 9.
    Liu, L., Sjögren, P.: On the global Gaussian Lipschitz space. In: Proceedings of the Edinburgh Mathematical Society, pp. 1–14 (2017). doi: 10.1017/S0013091516000390
  10. 10.
    Muckenhoupt, B.: Poisson integrals for Hermite and Laguerre expansions. Trans. Am. Math. Soc. 139, 231–242 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nowak, A.: Heat-diffusion and Poisson integrals for Laguerre and special Hermite expansions on weighted \(L^p\) spaces. Studia Math. 158(3), 239–268 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nowak, A., Stempak, K.: Weighted estimates for the Hankel transform transplantation operator. Tohoku Math. J. 58(2), 277–301 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rubio de Francia, J.L.: Weighted norm inequalities and vector valued inequalities. In: Ricci F., Weiss G. (eds.) Harmonic Analysis. Lecture Notes in Mathematics, vol. 908, pp. 86–101. Springer, Berlin, Heidelberg (1982)Google Scholar
  14. 14.
    Stein, E., Shakarchi, R.: Real Analysis. Princeton University Press, Princeton, New Jersey (2005)Google Scholar
  15. 15.
    Stempak, K.: Heat diffusion and Poisson integrals for Laguerre expansions. Tohoku Math. J. 46, 83–104 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Stinga, P., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differ. Equ. 35(11), 2092–2122 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Szegö, G.: Orthogonal Polynomials. American Mathematical Society Colloquium Publications, vol. XXIII (1939)Google Scholar
  18. 18.
    Thangavelu, S.: Lectures on Hermite and Laguerre expansions, Math Notes 42, Princeton University Press (1993)Google Scholar

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

Personalised recommendations