# Interlude: Atomic decomposition of L(log L)^r

This is going to be a shorter post about a technical fact that will be used in concluding the proof of the Tao-Wright lemma.
What we are going to see today is an atomic decomposition of the Orlicz spaces of $L (\log L)^r$ type. Surprisingly, I could find no classical references that explicitely state this useful little fact – some attribute it to Titchmarsh, Zygmund and Yano; indeed, something resembling the decomposition can be found for example in Zygmund’s book (Volume II, page 120). However, I could only find a proper statement together with a proof in a paper of Tao titled “A Converse Extrapolation Theorem for Translation-Invariant Operators“, where he claims it is a well-known fact and proves it in an appendix (the paper is about reversing the implication in an old extrapolation theorem of Yano [1951], a theorem that tells you that if the operator norms $\|T\|_{L^p \to L^p}$ blow up only to finite order as $p \to 1^{+}$, then you can “extrapolate” this into an endpoint inequality of the type $\|Tf\|_{L^1} \lesssim \|f\|_{L(\log L)^r}$).

Briefly stated, the result is as follows. We will consider only $L(\log L)^r ([0,1])$, that is the Orlicz space of functions on $[0,1]$ with Orlicz/Luxemburg norm

$\displaystyle \|f\|_{L(\log L)^r ([0,1])} = \inf \Big\{\mu > 0 \text{ s.t. } \int_{0}^{1} \frac{|f(x)|}{\mu} \Big(\log \Big(2 + \frac{|f(x)|}{\mu}\Big)\Big)^{r} \,dx \leq 1 \Big\}.$

Our atoms will be quite simply normalised characteristic functions: that is, for any measurable set $E \subset [0,1]$ we let $a_E$ denote the atom associated to $E$, given by

$\displaystyle a_E := \frac{\mathbf{1}_E}{\|\mathbf{1}_E\|_{L(\log L)^r}};$

obviously $\|a_E\|_{L(\log L)^r} = 1$.
The statement is then the following.

Atomic decomposition of $L(\log L)^r$:
Let $f \in L(\log L)^{r}([0,1])$. Then there exist measurable sets $(E_j)_j$ and coefficients $(\alpha_j)_j$ such that

$\displaystyle f = \sum_{j} \alpha_j a_{E_j}$

and

$\displaystyle \sum_{j} |\alpha_j| \lesssim \|f\|_{L(\log L)^r}.$

# Lorentz spaces basics & interpolation

(Updated with endpoint ${q = \infty}$)

I’ve written down an almost self contained exposition of basic properties of Lorentz spaces. I’ve found the sources on the subject to leave something to be desired, and I grew a bit confused at the beginning. Therefore this relatively short note (I might be ruining someone’s assignments out there, but I think the pros of writing down everything in one place balance the cons).

1. Lorentz spaces

In the following take ${1< p,q < \infty}$ otherwise specified, and ${(X, |\cdot|)}$ a ${\sigma}$-finite measure space with no atoms.

The usual definition of Lorentz space is as follows:

Definition 1 The space ${L^{p,q}(X)}$ is the space of measurable functions ${f}$ such that

$\displaystyle \|f\|_{L^{p,q}(X)}:= \left(\int_{0}^{\infty}{t^{q/p}{ f^\ast (t)}^q}\,\frac{dt}{t}\right)^{1/q} < \infty,$

where ${f^\ast}$ is the decreasing rearrangement [I] of ${f}$. If ${q=\infty}$ then define instead

$\displaystyle \|f\|_{L^{p,\infty}(X)} := \sup_{t}{t^{1/p} f^\ast (t)} < \infty.$