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

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Carro M., Colzani L., Sinnamon G.

Studia Mathematica

2007

tom 182

nr 1

1-27

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 $∥Tχ_{E}∥_{X} ≤ D(|E|)$ 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 $X = Λ^{q}(w)$ and their weak version.

Ograniczanie wyników

1 Studia Mathematica

1 Carro M.

1 Colzani L.

1 Sinnamon G.

1 2007

Wyniki wyszukiwania

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