Dyadic harmonic analysis beyond doubling measures

Nondoubling Calderón–Zygmund theory: a dyadic approach

“…It resembles the classical one although the irregularity of Σ forces to subtract the average over the ancestors of the maximal cubes instead of the actual maximal cubes. This is very similar to the Calderón-Zygmund-Gundy decomposition in [22] but goes beyond it, since we can deduce the weak type (1, 1) bound for Calderón-Zygmund operators from it. In summary, this new Calderón-Zygmund type decomposition -theorem 3.2 in the text-unifies the decompositions in [22,31] and admits potential applications in the noncommutative setting.…”

“…The proof of theorem 3.2 is essentially the same than that of theorem A in [22]. However, the additional properties satisfied by the filtration Σ yield…”

“…On the other hand, the problem with the study of Haar shifts associated to a dyadic latticeD -when the complexity is nonzero; otherwise, the problem was understood long ago via Gundy's martingale decomposition-is related to the interaction between cubes an their dyadic ancestors. The 'right' decomposition in this case, found in [22], reads as follows: given f ∈ L 1 (µ) and a height λ > 0, if {Q j } j is the family of maximal dyadic cubes with respect to the property |f | Q > λ, then…”

“…Lerner's results go far beyond the fact that they yield a very simple approach to the A 2 conjecture. Unfortunately, despite some isolated exceptions like the remarkable use of random dyadic lattices by Nazarov, Treil and Volberg in [26] or the recent analysis of Haar shifts in [22], dyadic and more elaborated probabilistic techniques have not been explored systematically in the context of nondoubling measures.…”

“…However, as we just indicated above, dyadic covering lemmas are not available in our setting. Also, the negative results in [22] suggest that even if they were available, boundedness of high complexity dyadic shifts by sparse operators is not to be expected. Therefore, we are forced to introduce a modification in our formula which includes a nicely behaved maximal operator…”