The maximal operator function $P^*$ is defined in this way:
DEF: If $f\in L^{p}(\mathbb{T})$, $P^*f(x)=sup_{0<r<1}|P_{r}*f(x)|$.Where $P_{r}(t)=\sum_{I=-\infty}^{\infty}r^{|i|}\phi_{i}(t)$ is the kernel of Poisson.
But now, with this maximal operator, is possible to demonstrate that, for $1\le p<\infty$, there is a constant $c_{p}>0$ such that, for all $f\in L^p(\mathbb{T})$:
$||P^*f||_{p}\le c_{p}||f||_{p}$?
Thanks.