From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces
Volume 182 / 2007
Studia Mathematica 182 (2007), 1-27
MSC: 47A30, 46E30.
DOI: 10.4064/sm182-1-1
Abstract
Let $X$ be a quasi-Banach rearrangement invariant space and let $T$ be an $(\varepsilon, \delta)$-atomic operator for which a restricted type estimate of the form $\Vert T\chi_E\Vert_X\le D(|E|)$ for some positive function $D$ and every measurable set $E$ is known. We show that this estimate can be extended to the set of all positive functions $f\in L^1$ such that $\|f\|_\infty\le 1$, in the sense that $\Vert Tf\Vert_X\le D(\|f\|_1)$. This inequality allows us to obtain strong type estimates for $T$ on several classes of spaces as soon as some information about the galb of the space $X$ is known. In this paper we consider the case of weighted Lorentz spaces $X={\mit\Lambda}^q(w)$ and their weak version.
