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• # Artykuł - szczegóły

## Studia Mathematica

2007 | 182 | 1 | 1-27

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

EN

### Abstrakty

EN
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.

1-27

wydano
2007

### Twórcy

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