Dichotomies for Lorentz spaces

@article{SzymonGłąb2013,
abstract = {Assume that L p,q, $L^\{p_1 ,q_1 \} ,...,L^\{p_n ,q_n \} $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \lbrace (f_1 ,...,f_n ) \in L^\{p_\{1,\} q_1 \} \times \cdots \times L^\{p_n ,q_n \} :f_1 \cdots f_n \in L^\{p,q\} \rbrace $. We prove the following dichotomy: either $E = L^\{p_1 ,q_1 \} \times \cdots \times L^\{p_n ,q_n \} $ or E is σ-porous in $L^\{p_1 ,q_1 \} \times \cdots \times L^\{p_n ,q_n \} $, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E = L^\{p_1 ,q_1 \} \times \cdots \times L^\{p_n ,q_n \} $ or E is meager. This is a generalization of the results for classical L p spaces.},
author = {Szymon Głąb, Filip Strobin, Chan Yang},
journal = {Open Mathematics},
keywords = {Lorentz spaces; Integration; Baire category; Porosity; integration; porosity},
language = {eng},
number = {7},
pages = {1228-1242},
title = {Dichotomies for Lorentz spaces},
url = {http://eudml.org/doc/269740},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Szymon Głąb
AU - Filip Strobin
AU - Chan Yang
TI - Dichotomies for Lorentz spaces
JO - Open Mathematics
PY - 2013
VL - 11
IS - 7
SP - 1228
EP - 1242
AB - Assume that L p,q, $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \lbrace (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \rbrace $. We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L p spaces.
LA - eng
KW - Lorentz spaces; Integration; Baire category; Porosity; integration; porosity
UR - http://eudml.org/doc/269740
ER -