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

R. Macías¹ ; C. Segovia² ; J. L. Torrea³

¹ IMAL-FIQ CONICET–Universidad Nacional del Litoral Güemes 3450 3000 Santa Fe, Argentina
² Instituto Argentino de Matemática (IAM)–CONICET Saavedra 15 1083 Buenos Aires, Argentina
³ Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid 28049 Madrid, Spain

Studia Mathematica, Tome 172 (2006) no. 2, pp. 149-167

Résumé

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.

Détail

DOI : 10.4064/sm172-2-3

Keywords: obtain weighted boundedness weights type delta delta maximal operator heat semigroup associated laguerre functions cal alpha parameter alpha greater proved alpha maximal operator strong type delta alpha delta alpha alpha geq strong type leq infty delta alpha behavior end points intervals where there strong type studied detail sharp results about existence strong weak restricted types given