  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.
