2
$\begingroup$

In the Handbook of the Geometry of Banach Spaces, vol 1, page 855, Johnson and Schechtman say that if $M$ is an Orlicz function, the following are equivalent:

  • The unit vector basis of the Orlicz space $\ell_M$ is $p$-convex and $2$-concave;
  • $M(\left|t\right|^\frac{1}{p})$ is equivalent to a convex function and $M(t^{\frac{1}{2}})$ is equivalent to a concave function on $[0, \infty)$

And they give as reference Application de l'étude de certaines formes linéaires aléatoires au plongement d'espaces de Banach dans des espaces $L^p$, by J. Bretagnolle and D. Dacunha-Castelle, which I cannot understand since I don't speak French.

So my question is: how does one prove this result?

Thank you!

Improve this question
$\endgroup$
3
  • $\begingroup$ Would you mind including, what an Orlicz-function is? (I could think of different meanings…) $\endgroup$ Commented Nov 10, 2017 at 11:22
  • $\begingroup$ An Orlicz function $M : \mathbb{R} \rightarrow \mathbb{R}$ is an even convex function such that $M(0) = 0$ and $\lim_{t \rightarrow \infty} M(t) = \infty$. $\endgroup$ Commented Nov 10, 2017 at 12:11
  • $\begingroup$ Sorry, I forgot to tell that $M(t) \geq 0$ for every $t$. $\endgroup$ Commented Nov 10, 2017 at 23:57

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.