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Spaces defined by the level function and their duals

100%

Sinnamon G.

Studia Mathematica

1994

tom 111

nr 1

19-52

The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of $L^p$. These domains are shown to be Banach spaces which, although closely tied to $L^p$ spaces, are not reflexive. A related construction is given which characterizes their dual spaces.

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

51%

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

2 Studia Mathematica

2 Sinnamon G.

1 Carro M.

1 Colzani L.

1 2007

1 1994

Wyniki wyszukiwania

Spaces defined by the level function and their duals

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