Dual of $L \log L(\mathbb{R})$ Consider the space$$L\log L(\mathbb{R})=\left \{f\in L^1(\mathbb{R}):\int \limits _\mathbb{R}|f(x)|\log ^+|f(x)|\,dx<\infty \right \}.$$Is it known what its dual and predual spaces are? Also any reference on its properties would be great!
I learnt of this space and its relationship with the Hardy-Littlewood maximal operator in Stein's paper in Studia Math, and I would like to discover further properties.
 A: Copied from Chapter 3 in:
Edgar, G. A.; Sucheston, Louis, Stopping times and directed processes, Encyclopedia of Mathematics and Its Applications. 47. Cambridge: Cambridge University Press. xii, 428 p. (1992). ZBL0779.60032.
Terminology matches that.  In particular, we use the Luxembourg norm for Orlicz spaces.  For more, you can read that chapter.

First, $L\log L$ is the Orlicz space $L_\Phi$ for Orlicz function $\Phi$ defined by
$$
\Phi(x) = \begin{cases}
0, & x \le 1,\\
x\log x, & x > 1 .
\end{cases}
\tag1$$
This Orlicz function is "finite" in the sense that it does not take the value $\infty$.  It satisfies the $\Delta_2$-condition at $\infty$ but not at $0$.
Consider also the "heart" $H_\Phi$ of the Orlicz space $L_\Phi$.    In case of finite Orlicz functions (as this is), $H_\Phi$ is the closure in $L_\Phi$ of the integrable simple functions.   For $\Phi$ defined by $(1)$, $H_\Phi$ is the space $R_1$ studied by Fava.  A simple description: for a measurable function $f : X \to \mathbb R$,
$$
f \in R_1 \quad\Longleftrightarrow\quad f \in L\log L\text{ and }
\mu\{\omega : |f(\omega)| \ge a\} < \infty \text{ for all } a > 0.
$$
Now $(1)$ satisfies $\Delta_2$ at $\infty$, so in case $\mu$ is a finite measure, $R_1(\mu) = L\log L(\mu)$;  Theorem 2.1.17(2).
The conjugate of Orlicz function $(1)$ is the Orlicz function $\Psi$ given by
$$
\Psi(x) = \begin{cases}
x, & x\le 1
\\
e^{x-1}, & x > 1
\end{cases}
\tag2$$
The Orlicz function $\Psi$ is finite.  It satisfies $\Delta_2$ at $0$
but not at $\infty$.
[Warning: $\{f : \int \Psi(f)\;d\mu < \infty\}$ is not a linear space, so in particular it is not the Orlicz space $L_\Psi$.]
For dual of $L\log L$:  $(2)$ is finite, so $H_\Phi^* = L_\Psi$; Theorem 2.2.11; So in case $\mu(X) < \infty$, $L_\Phi^* = L_\Psi$.
For pre-dual of $L\log L$:  $(1)$ is finite, so $H_\Psi^* = L_\Phi$;  Theorem 2.2.11.
