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}


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

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