On weak-type (1, 1) for averaging type operators

It is known that, due to the fact that L1,∞ not a Banach space, if (Tj)j sequence of bounded operators so thatTj:L1⟶L1,∞, with norm less than or equal ||Tj|| and ∑j||Tj||<∞, nothing can be said about operator T=∑jTj. This origin many difficult open problems. However, we assume thatTj:L1(u)⟶L1,∞(u),∀u∈A1, φ(||u||A1)||Tj||, where φ nondecreasing function A1 Muckenhoupt class weights, then prove essentially,T:L1(u)⟶L1,∞(u),∀u∈A1. We shall see this case interesting problems in Harmonic Analysis.