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

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...

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Bibliographic Details
Published in:Science China. Mathematics 2017-08, Vol.60 (8), p.1477-1502
Main Authors: Si, ZengYan, Xue, QingYing, Yabuta, Kôzô
Format: Article
Language:English
Subjects:
Applications of Mathematics
Commutators
Fourier analysis
Kernels
Mathematics
Mathematics and Statistics
Operators (mathematics)
Regularity
乘子算子
傅立叶
傅里叶
双线性
多线性
平方函数
运算符
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Summary: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.
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-016-0324-3