On the bilinear square Fourier multiplier operators and related multilinear square functions  

作　　者：SI ZengYan XUE QingYing YABUTA Kz 

机构地区：[1]School of Mathematics and Information Science, Henan Polytechnic University [2]School of Mathematical Sciences, Beijing Normal University,Laboratory of Mathematics and Complex Systems, Ministry of Education [3]Research Center for Mathematical Sciences, Kwansei Gakuin University

出　　处：《Science China Mathematics》2017年第8期1477-1502,共26页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 11401175, 11501169 and 11471041);the Fundamental Research Funds for the Central Universities (Grant No. 2014KJJCA10);Program for New Century Excellent Talents in University (Grant No. NCET-13-0065);Grantin-Aid for Scientific Research (C) (Grant No. 15K04942);Japan Society for the Promotion of Science

摘　　要：Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm（f1, f2）（x） =（∫0^∞︱∫（Rn）^2）e^2πix·（ξ1＋ξ2））m（tξ1, tξ2）f1（ξ1）f2（ξ2）dξ1dξ2︱^2dt/t）^1/2.Let s be an integer with s ∈ [n ＋ 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏i^2=1ω^i^p/p） and each ωi is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p（νω） if p0 〈 p1, p2 〈 ∞ with 1/p = 1/p1 ＋ 1/p2. Moreover,if p0 〉 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p,∞（νω）. The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e^(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)/t) ^(1/2).Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏_i^2=1ω_i^(p/pi) and each ω_i is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^(p1)(ω_1) × L^(p2)(ω_2) to L^p(ν_ω) if p0 < p1, p2 < ∞ with 1/p = 1/p1 + 1/p2. Moreover,if p0 > 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^(p1)(ω_1) × L^(p2)(ω_2) to L^(p,∞)(ν_ω). The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.

关 键 词：multilinear square functions Fourier multiplier operator multiple weights COMMUTATORS 

分 类 号：O175[理学—数学]