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

• Autorzy: María Carro , Leonardo Colzani , Gord Sinnamon
• Języki publikacji: 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.

Kategorie tematyczne

• 47A30: Norms (inequalities, more than one norm, etc.)
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 182
• Numer: 1
• Strony: 1-27

Daty

wydano

2007

Twórcy

autor

María Carro

• Departament de Matemàtica, Aplicada i Anàlisi, Universitat de Barcelona, E-08071 Barcelona, Spain

autor

Leonardo Colzani

• Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, 20126 Milano, Italy

autor

Gord Sinnamon

• Department of Mathematics, University of Western Ontario, N6A 5B7, London, Canada

Identyfikatory

DOI

10.4064/sm182-1-1