Composition of haar paraproducts: The random case

Dmitriy Bilyk, Michael T. Lacey, Xiaochun Li, Brett D. Wick

Research output: Contribution to journalArticlepeer-review

Abstract

When is the composition of paraproducts bounded? This is an important, and difficult question. We consider randomized variants of this question, finding nonclassical characterizations. For dyadic interval I, let hI = hI0 be the L2-normalized Haar function adapted to I, the superscript 0 denoting that it has integral zero. Set hI1 = hI, the superscript 1 denoting a nonzero integral. A (classical dyadic) paraproduct with symbol b is one of the operators. Here, ε, δ ∈ {0, 1}, with one of the two being zero and the other one. We characterize when certain randomized compositions B(b, B(β, ·)) are bounded operators on L2(R), permitting in particular both paraproducts to be unbounded.

Original languageEnglish (US)
Pages (from-to)1-13
Number of pages13
JournalAnalysis Mathematica
Volume35
Issue number1
DOIs
StatePublished - Mar 2009

Bibliographical note

Funding Information:
Research supported in part by a National Science Foundation DMS Grant #0801036 (1); by a National Science Foundation Grant (2); by a National Science Foundation DMS Grants #0456976 and # 0801154 (3); by a National Science Foundation DMS Grant # 0752703 and the Fields Institute (4).

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