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 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 , a theorem that tells you that if the operator norms blow up only to finite order as , then you can “extrapolate” this into an endpoint inequality of the type ).
Briefly stated, the result is as follows. We will consider only , that is the Orlicz space of functions on with Orlicz/Luxemburg norm
Our atoms will be quite simply normalised characteristic functions: that is, for any measurable set we let denote the atom associated to , given by
The statement is then the following.
Atomic decomposition of :
Let . Then there exist measurable sets and coefficients such that