二进双参数仿积的加权有界性

Weighted Boundedness of Dyadic Bi-parameter Paraproduct

为了证明双参数双线性的Coifman-Meyer乘子算子定理,一种二进双参数仿积∏(f,g)(x,y)=R∈RΣ1|R|1/2<f,ΦR1>,<g,ΦR2>ΦR3(x,y)被引入,其Lr有界性被证明,即‖∏(f,g)‖Lr茱‖f‖Lp‖f‖Lq,其中1/r=1/p+1/q,q<∞.但目前仍没有相应的加权有界性结果.利用对偶原理研究了∏(f,g)的加权有界性,即成立‖∏(f,g)‖Lr(ω)茱‖f‖Lp(ω)‖g‖Lp(ω),其中1/r=1/p+1/q,1<p,q<∞,ω∈Ar(R×R).

To prove bi-linear and bi-parameter Coifman-Meyer multiplier theorem, Mathematicians have introduced the following dyadic bi-parameter paraproduct1123∏（f, g）（x, y） =Σ1/2f, ΦR, g, ΦRΦR（x, y）R∈R R r and have proved its L boundedness, that is ‖∏（f, g）‖rL‖f‖pL‖f‖q, L where 1/r = 1/p ＋ 1/q, q ∞. But there is no any result of weighted estimates of the above paraproduct. In this paper, the weighted boundedness of ∏（ f, g）via duality was investigated. that is, ‖∏（ f, g）‖r L（ω）‖f‖p L（ω）‖g‖p L（ω）, where 1/r = 1/p ＋ 1/q, 1 p, q ∞,ω∈Ar（R × R）.