Two Weights Weak Type For Fractional Integral Operators And Commutators

In this paper, we mainly study the conditions sufficient for two weights weak type inequalities for fractional integral operators and commutators.In the first chapter, we mainly introduce the conceptions of fractional integral and commutators, then summarize the celebrated results and relative conclusions.In the second chapter, we mainly give sufficient conditions for two weights weak (p,q) type inequalities for fractional integral operators, that is Ap-type condition relate to Orlicz function.Let 0<α<n, 1<p<n/α,1/q=1/p—α/n. For Orlicz functionΦ, satisfyΔ2条件, and complementary functionΨ∈Bp.A pair of weights (u,v), if exists r> 1, for any cube Q, Then fractional integral operator Iαsatisfies the weak (p, q) type inequality.Attention, this type condition are true for any function with satisfying the Bp condition andΔ2 condition which we define in the paper.At last, for a fixed Orlicz function m∈N+, we give the sufficient condition for two weights weak.(p,p) type inequalities for the higher order commutators of fractional integral operators.Let 0<a<n, 1<p<∞, b∈BMO(Rn), a pair of weights (u,v), and m∈N+. If for some r> 1 and any cub e Q, where Then the higher order commutators of fractional integral operators Iαb,m satisfies the weak (p, p) type inequality.