The Boundedness Of Multilinear Operators On Weighted Morrey Spaces

The Morrey spaces were introduced by Morrey to investigate the local of solutions to the second order elliptic partial differential equations. These spaces can see as the expansion of the Lebesgue spaces and play an important role in the research of the local behavior of solutions to the partial differential equations. To study the boundedness of all kind of operators on this space is natural and meaningful.The multilinear operators are very important operators in harmonic analysis and have a lot of applications in the partial differential equations and complex analysis. In this paper, we will discuss the boundedness on of the weighted Morrey spaces for the multilinear Hardy-Littlewood maximal operator, multilinear Calderon-Zygmund operator, the commutators of multilinear Calderon-Zygmund operator with BMO functions, the multilinear fractional integral operators and so on.In section1, we introduce the definitions of weighted Morrey spaces Lp,k(w,u) and the multilinear Hardy-Littlewood maximal operators M., multilinear Calderon-Zygmund operators T, the commutators Tbâ†'of the multilinear Calderon-Zygmund operators with BMO functions, the multilinear fractional integral operators (?)α and so on. We also give some lemmas and notation which we need in this paper.In section2, we obtain the following results on the multilinear Hardy-Littlewood maximal operators:Let1<pj<∞,j=1,…,m,w∈AminjPj,0≤k≤1, then M.:Lp1,k(w)×…×Lpm,k(w)â†'Lp,k(w), namelyLet1≤pj<∞, for some j, pj=1, w∈A1, then for all t>0and any cube Q, We also get the similar results for the multilinear Calderon-Zygmund operators in section2and the commutators of the multilinear Calderon-Zygmund operators with BMO functions in section3.In section4, we obtain the following results on the multilinear fractional integral operators:Let0<α<mn and (?)α is the multilinear fractional integral operators. (1)Let1<p1,…,pm<∞,1/m<p<n/α,1/q=1/p-α/n,w∈Aminjpj,q,0<k<p/q.Then there exists a constant C>0,such that(2)If for some j,pj=1,1/q=1/p-α/n,w∈A1,q,0<k<p/q,then for all t>0and any cube Q, We get the similar results for the multilinear maximal operators and the commutators of the multilinear fractional integral operators with BMO functions.