### Abstract

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.

Original language | English (US) |
---|---|

Pages (from-to) | 1739-1785 |

Number of pages | 47 |

Journal | Advances in Mathematics |

Volume | 225 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2010 |

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### Keywords

- Bilinear multipliers
- Oscillatory integrals

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*225*(4), 1739-1785. https://doi.org/10.1016/j.aim.2010.03.032

**Bilinear oscillatory integrals and boundedness for new bilinear multipliers.** / Bernicot, Frédéric; Germain, Pierre.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 225, no. 4, pp. 1739-1785. https://doi.org/10.1016/j.aim.2010.03.032

}

TY - JOUR

T1 - Bilinear oscillatory integrals and boundedness for new bilinear multipliers

AU - Bernicot, Frédéric

AU - Germain, Pierre

PY - 2010/11

Y1 - 2010/11

N2 - We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.

AB - We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.

KW - Bilinear multipliers

KW - Oscillatory integrals

UR - http://www.scopus.com/inward/record.url?scp=77956232653&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956232653&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.03.032

DO - 10.1016/j.aim.2010.03.032

M3 - Article

VL - 225

SP - 1739

EP - 1785

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -