Off-diagonal estimates for the first order commutators in higher dimensions

Yaryong Heo, Sunggeum Hong, Chan Woo Yang

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1 Citation (Scopus)


In this paper we study natural generalizations of the first order Calderón commutator in higher dimensions d≥2. We study the bilinear operator Tm which is given by Tm(f,g)(x):=∬R2d[∫01m(ξ+tη)dt]fˆ(ξ)gˆ(η)e2πix⋅(ξ+η)dξdη. Our results are obtained under two different conditions of the multiplier m. The first result is that when K∈S∩Lloc 1(Rd∖{0}) is a regular Calderón-Zygmund convolution kernel of regularity 0<δ≤1, T maps Lp(Rd)×Lq(Rd) into Lr(Rd) for all 1<p,q≤∞, [Formula presented] as long as [Formula presented]. The second result is that when the multiplier m∈Cd+1(Rd∖{0}) satisfies the Hörmander derivative conditions |∂ξ αm(ξ)|≤Dα|ξ|−|α| for all ξ≠0, and for all multi-indices α with |α|≤d+1, Tm maps Lp(Rd)×Lq(Rd) into Lr(Rd) for all 1<p,q≤∞, [Formula presented] as long as [Formula presented]. These two results are sharp except for the endpoint case [Formula presented]. In case d=1 and K(x)=1/x, it is well-known that T maps Lp(R)×Lq(R) into Lr(R) for 1<p,q≤∞, [Formula presented] as long as r>1/2. In higher dimensional case d≥2, in 2016, when Kˆ(ξ)=ξj/|ξ|d+1 is the Riesz multiplier on Rd, P. W. Fong, in his Ph.D. Thesis [9], obtained ‖T(f,g)‖r≤C‖f‖p‖g‖q for 1<p,q≤∞ as long as r>d/(d+1). As far as we know, except for this special case, there has been no general results for the off-diagonal case r<1 in higher dimensions d≥2. To establish our results we develop ideas of C. Muscalu and W. Schlag [18,19] with new methods.

Original languageEnglish
Article number108652
JournalJournal of Functional Analysis
Issue number7
Publication statusPublished - 2020 Oct 15

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-2016R1D1A1B0101457 .

Publisher Copyright:
© 2020 Elsevier Inc.


  • Commutators
  • Multilinear operators
  • Multilinear singular integral operators

ASJC Scopus subject areas

  • Analysis


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