On constructions of isometric embeddings of nonseparable $L^p$ spaces, $0p\le 2$

Jolanta Grala-Michalak, and Artur Michalak. "On constructions of isometric embeddings of nonseparable $L^p$ spaces, $0 p \le 2$." Commentationes Mathematicae 48.2 (2008): null. <http://eudml.org/doc/291949>.

@article{JolantaGrala2008,
abstract = {Let $J$ be an infinite set. Let $X$ be a real or complex $\sigma $-order continuous rearrangement invariant quasi-Banach function space over $(\lbrace 0, 1\rbrace ^J,\ \mathcal \{B\}^J,\ \lambda _J)$, the product of $J$ copies of the measure space $(\lbrace 0, 1\rbrace ,\ 2^\{0,1\},\ \frac\{1\}\{2\} \delta _0 + \frac\{1\}\{2\}\delta _1)$. We show that if $0 p 2$ and $X$ contains a function $f$ with the decreasing rearrangement $f^∗$ such that $f^∗(t) t^\{-\frac\{1\}\{p\}\}$ for every $t\in (0, 1)$, then it contains an isometric copy of the Lebesgue space $L^p (\lambda _J)$. Moreover, if $X$ contains a function $f$ such that $f^∗(t) \sqrt\{|\text\{ln\}(t)|\}$ for every $t\in (0, 1)$, then it contains an isometric copy of the Lebesgue space $L^2(\lambda _J)$.},
author = {Jolanta Grala-Michalak, Artur Michalak},
journal = {Commentationes Mathematicae},
keywords = {$L^p$-spaces},
language = {eng},
number = {2},
pages = {null},
title = {On constructions of isometric embeddings of nonseparable $L^p$ spaces, $0 p \le 2$},
url = {http://eudml.org/doc/291949},
volume = {48},
year = {2008},
}

TY - JOUR
AU - Jolanta Grala-Michalak
AU - Artur Michalak
TI - On constructions of isometric embeddings of nonseparable $L^p$ spaces, $0 p \le 2$
JO - Commentationes Mathematicae
PY - 2008
VL - 48
IS - 2
SP - null
AB - Let $J$ be an infinite set. Let $X$ be a real or complex $\sigma $-order continuous rearrangement invariant quasi-Banach function space over $(\lbrace 0, 1\rbrace ^J,\ \mathcal {B}^J,\ \lambda _J)$, the product of $J$ copies of the measure space $(\lbrace 0, 1\rbrace ,\ 2^{0,1},\ \frac{1}{2} \delta _0 + \frac{1}{2}\delta _1)$. We show that if $0 p 2$ and $X$ contains a function $f$ with the decreasing rearrangement $f^∗$ such that $f^∗(t) t^{-\frac{1}{p}}$ for every $t\in (0, 1)$, then it contains an isometric copy of the Lebesgue space $L^p (\lambda _J)$. Moreover, if $X$ contains a function $f$ such that $f^∗(t) \sqrt{|\text{ln}(t)|}$ for every $t\in (0, 1)$, then it contains an isometric copy of the Lebesgue space $L^2(\lambda _J)$.
LA - eng
KW - $L^p$-spaces
UR - http://eudml.org/doc/291949
ER -