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

Abstract

It is known that, due to the fact that $L^{1, \infty}$ is not a Banach space, if $\left(T_j\right)_j$ is a sequence of bounded operators so that $$ T_j: L^1 \longrightarrow L^{1, \infty} $$ with norm less than or equal to $\left\|T_j\right\|$ and $\sum_j\left\|T_j\right\|<\infty$, 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 $$ T_j: L^1(u) \longrightarrow L^{1, \infty}(u), \quad \forall u \in A_1 $$ with norm less than or equal to $\varphi\left(\|u\|_{A_1}\right)\left\|T_j\right\|$, where $\varphi$ is a nondecreasing function and $A_1$ the Muckenhoupt class of weights, then we prove that, essentially, $$ T: L^1(u) \longrightarrow L^{1, \infty}(u), \quad \forall u \in A_1 $$ We shall see that this is the case of many interesting problems in Harmonic Analysis.