Dual of $L \log L(\mathbb{R})$

Question

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.

Answer 1

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.

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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.