This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar's inequality and the sharp maximal flmction estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.
In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/(1 + |j|)^α(1 + |k- j|)^λ(1 + |k|)^β,(0.1)vj =∑ k ∈Zn uk^p/(1 + |j|)^β(1 + |k- j|)^λ(1 + |k|)^α,where u, v 〉 0, 1 〈 p, q 〈 ∞, 0 〈 λ 〈 n, 0 ≤α + β≤ n- λ,1/p+1〈λ+α/n and 1/p+1+1/q+1≤λ+α+β/n:=λ^-/n. We first show that positive solutions of(0.1) have the optimal summation interval under assumptions that u ∈ l^p+1(Z^n) and v ∈ l^q+1(Z^n). Then we show that problem(0.1) has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{(n-λ^-)(q+1)/pq-1,(n-λ^-)(p+1)/pq-1} ≥λ^-.
This paper is concerned with the pointwise estimates for the sharp function of two kinds of maximal commutators of multilinear singular integral operators T∑b^* and TПb^* which are generalized by a weighted BMO function b and a multilinear singular integral operator T, respectively. As applications, some commutator theorems are established.
In this note, the author prove that maximal Bocher-Riesz commutator Bδ,*^b generated by operator Bδ,* and function b∈ BMO(ω) is a bounded operator from L^p(μ) into L^p(ν), where w∈ (μν^- 1)^1/p,μ,v ∈ Ap for 1 〈 p 〈 ∞. The proof relies heavily on the pointwise estimates for the sharp maximal function of the commutator Bδ,*^b.
. Let H2 = (-△)2 + V2 be the Schr6dinger type operator, where V satisfies reverse HSlder inequality. In this paper, we establish the Lp boundedness for V2 H2- 1, H21 V2, VH2- 1/2 and Hfl/2V, and that of their commutators. We also prove that H^IV2, Hfl/2V are bounded from BMOL to BMOL.