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

Tom 182 / 2007

María Carro, Leonardo Colzani, Gord Sinnamon Studia Mathematica 182(2007), 1-27 MSC: 47A30, 46E30. DOI: 10.4064/sm182-1-1


Let $X$ be a quasi-Banach rearrangement invariant space and let $T$ be an $(\varepsilon, \delta)$-atomic operator for which a restricted type estimate of the form $\Vert T\chi_E\Vert_X\le D(|E|)$ 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 $f\in L^1$ such that $\|f\|_\infty\le 1$, in the sense that $\Vert Tf\Vert_X\le D(\|f\|_1)$. 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 $X={\mit\Lambda}^q(w)$ and their weak version.


  • María CarroDepartament de Matemàtica
    Aplicada i Anàlisi
    Universitat de Barcelona
    E-08071 Barcelona, Spain
  • Leonardo ColzaniDipartimento di Matematica e Applicazioni
    Università di Milano – Bicocca
    20126 Milano, Italy
  • Gord SinnamonDepartment of Mathematics
    University of Western Ontario
    N6A 5B7, London, Canada

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