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

María Carro; Leonardo Colzani; Gord Sinnamon

María Carro ; Leonardo Colzani ; Gord Sinnamon

Studia Mathematica, Tome 178 (2007), p. 1-27 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X be a quasi-Banach rearrangement invariant space and let T be an (ε,δ)-atomic operator for which a restricted type estimate of the form $∥ T χ_{E} ∥_{X} \leq D (| E |)$ 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 ∈ L¹ such that ${| | f | |}_{\infty} \leq 1$ , in the sense that $∥ T f ∥_{X} \leq D (| | f | | ₁)$ . 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 $X = Λ^{q} (w)$ and their weak version.

Publié le : 2007-01-01

Zbl 1182.47009

EUDML-ID : urn:eudml:doc:284907

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     title = {From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces},
     journal = {Studia Mathematica},
     volume = {178},
     year = {2007},
     pages = {1-27},
     zbl = {1182.47009},
     language = {en},
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María Carro; Leonardo Colzani; Gord Sinnamon. From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces. Studia Mathematica, Tome 178 (2007) pp. 1-27. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-1-1/