## Abstract

In this paper we study the bilinear multiplier operator of the form H^{t}(f, g)(x) = ^{Z}_{Rd}^{Z}_{Rd} m(tξ, tη) e^{2}πit|(ξ,η)^{|} f^{b}(ξ) gb(η) e^{2}πix(ξ+η^{)} dξdη, 1 ≤ t ≤ 2 where m satisfies the Marcinkiewicz-Mikhlin-Hörmander's derivative conditions. And by obtaining some estimates for H^{t}, we establish the Lp^{1}(R^{d}) × Lp^{2}(R^{d}) → L^{p}(R^{d}) estimates for the bi(sub)linear spherical maximal operators M(f, g)(x) = sup _{t>0}^{Z}_{S}2d−_{1} f(x − ty) g(x − tz) dσ_{2d}(y, z) which was considered by Barrionevo et al in [1], here σ_{2d} denotes the surface measure on the unit sphere S^{2d−1}. In order to investigate M we use the asymptotic expansion of the Fourier transform of the surface measure σ_{2d} and study the related bilinear multiplier operator H^{t}(f, g). To treat the bad behavior of the term e^{2}πit|(ξ,η)^{|} in H^{t}, we rewrite e^{2}πit|(ξ,η)^{|} as the summation of e^{2}πit√^{N2+}|η^{|2}a_{N}(tξ, tη)'s where N's are positive integers, a_{N}(ξ, η) satisfies the Marcinkiewicz-Mikhlin-Hörmander condition in η, and supp(a_{N}(·, η)) ⊂ {ξ: N ≤ |ξ| < N + 1}. By using these decompositions, we significantly improve the results of Barrionevo et al in [1].

Original language | English |
---|---|

Pages (from-to) | 397-434 |

Number of pages | 38 |

Journal | Mathematical Research Letters |

Volume | 27 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2020 |

## ASJC Scopus subject areas

- Mathematics(all)