Composition of haar paraproducts: The random case

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

doi:10.1007/s10476-009-0101-9

Composition of haar paraproducts: The random case

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

School of Mathematics

Research output: Contribution to journal › Article › peer-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 h_I = h_I⁰ be the L²-normalized Haar function adapted to I, the superscript 0 denoting that it has integral zero. Set h_I¹ = 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 L²(R), permitting in particular both paraproducts to be unbounded.

Original language	English (US)
Pages (from-to)	1-13
Number of pages	13
Journal	Analysis Mathematica
Volume	35
Issue number	1
DOIs	https://doi.org/10.1007/s10476-009-0101-9
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).

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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.",

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AU - Lacey, Michael T.

AU - Li, Xiaochun

AU - Wick, Brett D.

N1 - 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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N2 - 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.

AB - 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.

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Composition of haar paraproducts: The random case

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