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

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2017,（8）
• 页码范围: 1477-1502
• 页数: 26页
• 语种: 英文
• DOI: 10.1007/s11425-016-0324-3

摘要

Let n ≥1 and ζm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by ζm(f1,f2)(x) =(∫∞0|∫(Rn)2 e2πix·(ξ1+ξ2)m(tξ1,tξ2)(f)1 (ξ1)(f)2(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 υ→ω =Π2i=1 ωp/pii and each ωi is a nonnegative function on Rn.In this paper,we show that under some condition on m,ζm is bounded from Lp1 (ω1) × Lp2(ω2) to Lp(v→ω) if p0 ＜ p1,p2 ＜ ∞ with 1/p =1/p1 + 1/p2.Moreover,if p0 ＞ 2n/s and p1 =p0 or p2 =p0,then ζm is bounded from Lp1 (ω1) × Lp2 (ω2) to Lp,∞ (υ→ω).The weighted end-point L log L type estimate and strong estimate for the commutators of ζm 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.