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<sub>XE</sub>||x ≤ -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 ∈ L<sup>1</sup> such that ||f||∞ ≤ 1, in the sense that ||Tf||x ≤ D(||f||<sub>1</sub>)- 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<sup>q</sup> (w) and their weak version. © Instytut Matematyczny PAN, 2007.
Carro, M., Colzani, L., Sinnamon, G. (2007). From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces. STUDIA MATHEMATICA, 182(1), 1-27 [10.4064/sm182-1-1].
From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces
COLZANI, LEONARDO;
2007
Abstract
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 ||TXE||x ≤ -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||∞ ≤ 1, in the sense that ||Tf||x ≤ 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 = Λq (w) and their weak version. © Instytut Matematyczny PAN, 2007.
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