Weighted Inequalities For Multilinear Potential Type Operators

LetΦbe a non-negative locally integrable function on Rⁿ. The potential type operators associated toΦis defined byLet A be a function whose derivatives of order one belong to the space BMO(Rⁿ). Define the multilinear potential type integral operator T_Φ^A asIn this paper, we discuss the two-weights inequalities for T_Φ^A whenΦsatisfies some weak growth conditions. The main results are the following.(i) If the pair of wights (u,Ï…) is such that for some r>1 and any cube Q,where B₁(t) = t^p'(1+ log⁺t)^p',1< p≤q<∞.Then the multilinear potential type operators T_Φ^A satisfies the following two-weight weak type (p, q) inequality,(ii) Let B(t) =t(1+ log⁺t). Then there exists constant C > 0 such that for any weightω, we have(iii) Letδ>0, for any weightω, we have...