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

It is known that, due to the fact that $L^{1, \infty}$ is not a Banach space, if $(T_j)_j$ is a sequence of bounded operators so that $$T_j:L^1\longrightarrow L^{1, \infty},$$ with norm less than or equal to $||T_j||$ and $\sum_j ||T_j||<\infty$, nothing can be said about the operator $T=\sum_j T_j$. This is the origin of many difficult and open problems. However, if we assume that $$T_j:L^1(u)\longrightarrow L^{1, \infty}(u), \qquad \forall u\in A_1,$$ with norm less than or equal to $\varphi(||u||_{A_1})||T_j||$, where $\varphi$ is a nondecreasing function and $A_1$ the Muckenhoupt class of weights, then we prove that, essentially, $$T:L^1(u) \longrightarrow L^{1, \infty}(u), \qquad \forall u\in A_1.$$ We shall see that this is the case of many interesting problems in Harmonic Analysis.