Improved bounds for the bilinear spherical maximal operators

Yaryong Heo, Sunggeum Hong, Chan Woo Yang

Research output: Contribution to journalArticlepeer-review

Abstract

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
Volume27
Issue number2
DOIs
Publication statusPublished - 2020

ASJC Scopus subject areas

  • Mathematics(all)

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