From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces

María Carro¹ ; Leonardo Colzani² ; Gord Sinnamon³

¹Departament de Matemàtica Aplicada i Anàlisi Universitat de Barcelona E-08071 Barcelona, Spain
²Dipartimento di Matematica e Applicazioni Università di Milano – Bicocca 20126 Milano, Italy
³Department of Mathematics University of Western Ontario N6A 5B7, London, Canada

Studia Mathematica, Tome 182 (2007) no. 1, pp. 1-27

Résumé

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.

Détail

DOI : 10.4064/sm182-1-1

Keywords: quasi banach rearrangement invariant space varepsilon delta atomic operator which restricted type estimate form vert chi vert positive function every measurable set known estimate extended set positive functions infty sense vert vert inequality allows obtain strong type estimates several classes spaces soon information about galb space known paper consider weighted lorentz spaces mit lambda their weak version