A+ CATEGORY SCIENTIFIC UNIT

Weighted norm estimates for the maximal operator of the Laguerre functions heat diffusion semigroup

Volume 172 / 2006

R. Macías, C. Segovia, J. L. Torrea Studia Mathematica 172 (2006), 149-167 MSC: 42A45, 42B15, 42B20, 42B25, 42C10. DOI: 10.4064/sm172-2-3

Abstract

We obtain weighted $L^{p}$ boundedness, with weights of the type $y^{\delta }$, $\delta >-1$, for the maximal operator of the heat semigroup associated to the Laguerre functions, $\{ {\cal L}_k^\alpha\}_k$, when the parameter $\alpha $ is greater than $-1$. It is proved that when $-1< \alpha < 0$, the maximal operator is of strong type $( p,p) $ if $ p>1 $ and $2( 1+\delta ) /( 2+\alpha ) < p< 2( 1+\delta)/( -\alpha )$, and if $\alpha \geq 0$ it is of strong type for $ 1< p\leq \infty $ and $2( 1+\delta ) /( 2+\alpha ) < p$.

The behavior at the end points of the intervals where there is strong type is studied in detail and sharp results about the existence or not of strong, weak or restricted types are given.

Authors

  • R. MacíasIMAL-FIQ
    CONICET–Universidad Nacional del Litoral
    Güemes 3450
    3000 Santa Fe, Argentina
    e-mail
  • C. SegoviaInstituto Argentino de Matemática
    (IAM)–CONICET
    Saavedra 15
    1083 Buenos Aires, Argentina
    e-mail
  • J. L. TorreaDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad Autónoma de Madrid
    28049 Madrid, Spain
    e-mail

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