A weighted weak-type multilinear gradient inequality

Abstract

We characterize the weights w,v1,v2,…,vm for which the weak-type multilinear gradient inequality

‖∏i=1mfi‖p,∞;w≤C∏i=1m‖x⋅∇fi(x)‖pi,vi holds for all f1,f2,…,fm∈Cc∞(Rn) in the case 1p=1p1+1p2+⋯+1pm.