Improved bounds for the bilinear spherical maximal operators

Yaryong Heo, Sunggeum Hong, Chan Woo Yang

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8 Citations (Scopus)


In this paper we study the bilinear multiplier operator of the form Ht(f, g)(x) = ZRdZRd m(tξ, tη) e2πit|(ξ,η)| fb(ξ) gb(η) e2πix(ξ+η) dξdη, 1 ≤ t ≤ 2 where m satisfies the Marcinkiewicz-Mikhlin-Hörmander's derivative conditions. And by obtaining some estimates for Ht, we establish the Lp1(Rd) × Lp2(Rd) → Lp(Rd) estimates for the bi(sub)linear spherical maximal operators M(f, g)(x) = sup t>0ZS2d−1 f(x − ty) g(x − tz) dσ2d(y, z) which was considered by Barrionevo et al in [1], here σ2d denotes the surface measure on the unit sphere S2d−1. In order to investigate M we use the asymptotic expansion of the Fourier transform of the surface measure σ2d and study the related bilinear multiplier operator Ht(f, g). To treat the bad behavior of the term e2πit|(ξ,η)| in Ht, we rewrite e2πit|(ξ,η)| as the summation of e2πit√N2+|2aN(tξ, tη)'s where N's are positive integers, aN(ξ, η) satisfies the Marcinkiewicz-Mikhlin-Hörmander condition in η, and supp(aN(·, η)) ⊂ {ξ: N ≤ |ξ| < N + 1}. By using these decompositions, we significantly improve the results of Barrionevo et al in [1].

Original languageEnglish
Pages (from-to)397-434
Number of pages38
JournalMathematical Research Letters
Issue number2
Publication statusPublished - 2020

Bibliographical note

Funding Information:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology NRF-2018R1D1A1B07042871, NRF-2017R1A2B4002316, and NRF-2016R1D1A1B01014575.

Publisher Copyright:
© 2020 International Press of Boston, Inc.. All rights reserved.

ASJC Scopus subject areas

  • General Mathematics


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