Rearrangement invariant spaces

Rearrangement invariant spaces are studied in much of the literature, see for example Lindenstrauss and Tzafriri (1977). However, we will work with a definition that is a little less restrictive. A rearrangement invariant space on the random variables is a quasi-normed Banach space ${\cal L}$ of random variables such that $1 \in {\cal L}$, and if $X^* \le Y^*$ and $Y \in {\cal L}$, then $X \in {\cal L}$ and ${\left\Vert X\right\Vert}_{\cal L}\le {\left\Vert Y\right\Vert}_{\cal L}$. Obviously the spaces $L_p$ for $0<p\le \infty$ are rearrangement invariant spaces.

Given a rearrangement invariant space ${\cal L}$, we define the quasi-constant of ${\cal L}$ to be the least constant $K>0$ such that ${\left\Vert X+Y\right\Vert}_{\cal L}\le K({\left\Vert X\right\Vert}_{\cal L}+ {\left\Vert Y\right\Vert}_{\cal L})$ for all $X,Y \in {\cal L}$. Notice that if $X^*(2t) \le Y^*(t)$, and $Y \in {\cal L}$, then $X$ may be written as the sum of two disjoint random variables $Y_1$ and $Y_2$ with $Y_1^*(t),Y_2^*(t) \le Y^*(t)$, and hence ${\left\Vert X\right\Vert}_{\cal L}\le 2K {\left\Vert Y\right\Vert}_{\cal L}$.

Given two rearrangement invariant spaces ${\cal L}$ and ${\cal M}$, we will say that ${\cal L}$ embeds into ${\cal M}$ if there is a positive constant $c$ such that if $X \in {\cal L}$, then $X \in {\cal M}$ and ${\left\Vert X\right\Vert}_{\cal M}\le c {\left\Vert X\right\Vert}_{\cal L}$. We will call the least such $c$ the embedding constant of ${\cal L}$ into ${\cal M}$.

Theorem 7.1   Let $p_0>0$, and let ${\cal L}$ be a rearrangement invariant space such that ${\cal L}$ embeds into $L_p$, and $L_q$ embeds into ${\cal L}$, where $q \ge p \ge p_0$. Then there is a positive constant $c$, depending only upon the quasi-constant of ${\cal L}$, the embedding constants, $p_0$ and $q/p$, such that for any sequence of Banach valued independent random variables $(X_n)$

$$c^{-1}({\left\Vert U\right\Vert}_p + {\left\Vert\ell\right\Vert}_... ... L}\le c({\left\Vert U\right\Vert}_p + {\left\Vert\ell\right\Vert}_{\cal L}) .$$

Proof: Let us first obtain the left hand side inequality. It follows by hypothesis that ${\left\Vert U\right\Vert}_{\cal L}\ge c_1^{-1} {\left\Vert U\right\Vert}_p$, where $c_1$ is the embedding constant of ${\cal L}$ into $L_p$. Furthermore, $U \ge {\frac{1}{ 2}} M$, and by Proposition 2.1, $\ell(t) \le M^*(2t)$. Hence ${\left\Vert U\right\Vert}_{\cal L}\ge (4K)^{-1} {\left\Vert\ell\right\Vert}_{\cal L}$, where $K$ is the quasi-constant of ${\cal L}$.

Now let us obtain the right hand inequality. By Corollary 3.2, we have that there is a universal positive $c_2$ for $0 \le t \le 1$

$$U^*(t) I_{0\le t \le 2^{-2q/p}} \le c_2 {\frac{2q }{ p}} (U^*(t^{p/2q}) + M^*(t/2)).$$

Now $U^*(t) I_{0\le t \le 2^{-2q/p}} \ge U^*(2^{2q/p} t)$, and hence

$${\left\Vert U\right\Vert}_{{\cal L}} \le (2K)^{\lceil 2q/p \rceil... ...o U^*(t^{p/2q})\mathclose\Vert}_{\cal L}+ {\left\Vert M\right\Vert}_{\cal L}) .$$

To finish the proof, suppose that ${\left\Vert U\right\Vert}_p = \lambda$. Then it is easily seen that $U^*(t) \le \lambda t^{-1/p}$. Thus, if $c_3$ is the embedding constant of $L_q$ into ${\cal L}$, then

$${\mathopen\Vert t\mapsto U^*(t^{p/2q})\mathclose\Vert}_{\cal L} \le c_3 {\mathopen\Vert t\mapsto U^*(t^{p/2q})\mathclose\Vert}_q$$

$$= c_3 \left(\int_0^1 (U^*(t^{p/2q}))^q \, dt \right)^{1/q}$$

$$\le c_3 \lambda \left(\int_0^1 t^{-1/2} \, dt\right)^{1/q}$$

$$= 2^{1/q} c_3 \lambda .$$