The [Orlicz space](http://en.wikipedia.org/wiki/Birnbaum%25E2%2580%2593Orlicz_space) associated to the convex function $e^{x^2}-1$ arises frequently in probabilistic problems (being in this space implies that a function has sub-Gaussian tails). Does this space have a more concise name?



Does the Orlicz space associated to $e^{x^2}-1$ have a name?
One often denotes $\psi\_\alpha (x) = e^{x^\alpha} - 1$ for $\alpha \ge 1$, and the corresponding Orlicz space is $L\_{\psi\_\alpha}$. An estimate on the $L\_{\psi\_\alpha}$ norm is also often referred to simply as a $\psi\_\alpha$ estimate. Of course, the case $\alpha = 2$ which you asked about is the one most often of interest.


One might also call this space $\exp(L^\alpha)$, in analogy with $L^p$ and $L\log L$, etc., but I've never seen this done.


Added: Here are a few references where the $\psi\_\alpha$ and $L\_{\psi\_\alpha}$ notations are used:


1. Ledoux and Talagrand, *Probability in Banach Spaces*, p. 10 and chapter 11
2. Klartag, "[Uniform almost sub-gaussian estimates for linear
functionals on convex sets](http://www.math.tau.ac.il/~klartagb/papers/psitwo.pdf)"
3. Giannopoulos, Pauoris, and Valettas, "[$\Psi\_\alpha$-estimates for marginals of log-concave probability measures](http://www.math.tamu.edu/~grigoris/log-concave-marginals.pdf)"


