Some weighted norm inequalities for a one-sided version of $g_{\lambda }^*$
Volume 176 / 2006
Studia Mathematica 176 (2006), 21-36
MSC: Primary 42B25, 26A33.
DOI: 10.4064/sm176-1-2
Abstract
We study the boundedness of the one-sided operator $g_{\lambda ,\varphi }^+$ between the weighted spaces $L^p(M^{-}w)$ and $L^p(w)$ for every weight $w.$ If $\lambda = 2/p$ whenever $1< p< 2,$ and in the case $p=1$ for $\lambda >2,$ we prove the weak type of $g_{\lambda ,\varphi }^+.$ For every $\lambda >1$ and $p=2,$ or $\lambda > 2/p$ and $1< p< 2,$ the boundedness of this operator is obtained. For $p>2$ and $\lambda >1,$ we obtain the boundedness of $g_{\lambda ,\varphi }^+$ from $L^p((M^{-})^{[p/2]+1} w)$ to $L^p(w),$ where $(M^{-})^{k}$ denotes the operator $M^{-}$ iterated $k$ times.
