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

María Carro, Leonardo Colzani, and Gord Sinnamon. "From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces." Studia Mathematica 182.1 (2007): 1-27. <http://eudml.org/doc/284907>.

@article{MaríaCarro2007,
abstract = {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.},
author = {María Carro, Leonardo Colzani, Gord Sinnamon},
journal = {Studia Mathematica},
keywords = {rearrangement inequality; extrapolation theory; restricted weak type estimates; -atomic operator; galb},
language = {eng},
number = {1},
pages = {1-27},
title = {From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces},
url = {http://eudml.org/doc/284907},
volume = {182},
year = {2007},
}

TY - JOUR
AU - María Carro
AU - Leonardo Colzani
AU - Gord Sinnamon
TI - From restricted type to strong type estimates on quasi-Banach rearrangement invariant spaces
JO - Studia Mathematica
PY - 2007
VL - 182
IS - 1
SP - 1
EP - 27
AB - 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.
LA - eng
KW - rearrangement inequality; extrapolation theory; restricted weak type estimates; -atomic operator; galb
UR - http://eudml.org/doc/284907
ER -