## 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 h_{I} = h_{I}^{0} be the L^{2}-normalized Haar function adapted to I, the superscript 0 denoting that it has integral zero. Set h_{I}^{1} = h_{I}, 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 L^{2}(R), permitting in particular both paraproducts to be unbounded.

Original language | English (US) |
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Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Analysis Mathematica |

Volume | 35 |

Issue number | 1 |

DOIs | |

State | Published - 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).