Littlewood-Paley sets and sums of permuted lacunary sequences

by Sidney. Trudeau

Institution: McGill University
Department: Department of Mathematics and Statistics.
Degree: PhD
Year: 2009
Keywords: Littlewood-Paley theory.; Fourier series.
Record ID: 1854465
Full text PDF: http://digitool.library.mcgill.ca/thesisfile115883.pdf


Let {Ij} be an interval partition of the integers, f(x) a function on the circle group T and S(f) = (sum |f j|2)1/2 where fˆ j = fˆ cIj . In their 1995 paper, Hare and Klemes showed that, for fixed p ∈ (1, infinity), there exist lambdap > 1 and Ap, Bp > 0 such that if l(Ij+1)/ l(Ij) ≥ lambdap, where l(Ij) is the length of the interval Ij, then Ap∥ f∥p ≤ ∥S( f)∥p ≤ Bp∥ f∥p. That is, {Ij} is a Littlewood-Paley (p) partition. Since the intervals need not be adjacent, these partitions may be viewed as permutations of lacunary intervals. Partitions like these can be induced by subsets of sums of permuted lacunary sequences. In this thesis, we present two main results. First, complementary to the aforementioned work of Hare and Klemes who proved that sums of permuted lacunary sequences were Littlewood-Paley (p) partitions (for large enough ratio), we prove the surprising result that there are sums of permuted lacunary sequences of fixed ratio that cannot be obtained by iterating sums of permuted lacunary sequences of larger ratio finitely many times. The proof of this statement is based on the ideas developed in the 1989 paper of Hare and Klemes, especially with respect to the definition of a tree and to the theorem on the equivalency of a finitely generated partition and the absence of certain trees. These special sums may then be viewed as the critical test case for further progress on the conjecture of Hare and Klemes that sums of permuted lacunary sequences are Littlewood-Paley (p) partitions for any p. Secondly, we use the non-branching case of the method of Hare and Klemes developed in their 1992 and 1995 papers, and further developed by Hare in a general setting in 1997, to prove a result of Marcinkiewicz on iterated lacunary sequences in the case p = 4. This shows that the method introduced by Hare and Klemes can potentially be adapted to partitions other than those they were originally applied to. As well, in considering the proof given by Hare and Klemes (and by Hare in a general setting) that lacunary sequences are Littlewood-Paley (4) partitions, we present a slight variation on one of the computations which may be useful in regard to sharp versions of some of these computations, but otherwise follows the same pattern as that of the above papers. Finally, we prove an elementary property of the finite union of lacunary sequences.