Weighted norm estimates for the maximal operator of the Laguerre functions heat diffusion semigroup
Volume 172 / 2006
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.
