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 u^q_k/(1 + |j|)α(1 + |k- j|)λ(1 + |k|)β,(0.1)vj =∑ k ∈Zn u^p_k/(1 + |j|)β(1 + |k- j|)λ(1 + |k|),where u, v > 0, 1 < p, q < ∞, 0 < λ < n, 0 ≤α + β≤ n- λ,1p+1<λ+αnand1p+1+1q+1≤λ+α+βn:=λˉn. We first show that positive solutions of(0.1) have the optimal summation interval under assumptions that u ∈ lp+1(Zn) and v ∈ lq+1(Zn). Then we show that problem(0.1) has no positive solution if 0 <λˉ pq ≤ 1 or pq > 1 and max{(n-)(q+1)pq-1,(n-λˉ)(p+1)pq-1} ≥λˉ.
This paper is concerned with the pointwise estimates for the sharp function of two kinds of maximal commutators of multilinear singular integral operators T_(Σb)~*and T_(Πb)~*,which are generalized by a weighted BMO function 6 and a multilinear singular integral operator T,respectively.As applications,some commutator theorems are established.
In this paper, some endpoint estimates for the generalized multilinear fractional integrals I_(α,m) on the non-homogeneous metric spaces are established.
In this note, the author prove that maximal Bocher-Riesz commutator Bbδ , generated by operator Bδ , and function b ∈ BMO(ω) is a bounded operator from Lp(μ) into Lp (ν), where ω∈ (μν-1 )1 p , μ , ν∈ Ap for 1 < p < ∞. The proof relies heavily on the pointwise estimates for the sharp maximal function of the commutator Bbδ, .
Let H_2 =(-△)~2 +V^2 be the Schrodinger type operator,where V satisfies reverse Holder inequality.In this paper,we establish the L^P boundedness for V^2H_2^(-1),H_2^(-1)V^2,VH_2^(-1/2)and H_2^(-1/2)V,and that of their commutators.We also prove that H_2^(-1)V^2,H_2^(-1/2)V are bounded from BMO_L to BMO_L.
