Weighted Estimates For Multilinear Fractional Strong Maximal Operators

In this paper, a strong version of fractional maximal operator MαR is introduced, and we get the strong (Lp(Rn), Lq(Rn)) estimate, distribution estimate and weighted estimate for it, then we generalize it to multilinear case and get various weighted estimates.The multilinear fractional strong maximal operator is defined as where the supremum is taken over all rectangles with sides parallel to the coordinate axes. When m=1, it is the strong fractional maximal operator MαR.We get the results for the fractional strong maximal operator as follows:(1) When 1<p<∞, 1/q=1/p-α/n, then(2) There exists constant C> 0, such that for any λ> 0,(3) If (v, ω) satisfies for some r> 1, and v satisfies condition (A), then(4) Set 1< p, q< ∞,Φ is a Young function such that Mα,ΦR is bounded from Lp(Rn) to Lq(Rn), (v,ω) is weight function, where v satisfies condition (A), and then MαR is bounded from Lp(ω) to Lq(v).For the multilinear fractional strong maximal operator, we get the the following results:(1) When (?)j ∈A((?),q)*, then(2) When weight function (v,(?)) satisfies for some r> 1, and v satisfies condition (A), then(3) If v satisfies condition (A), and then MαR is bounded from LP1(ω1)×…× Lpm(ωm) to Lq(v).