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 $(T_j{)}_j$ is a sequence of bounded operators so that $$T_j:L^1⟶L^{1,\infty },$$ 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 $$T_j:L^1(u)⟶L^{1,\infty }(u),\forall u\in A_1,$$ with norm less than or equal to $\phi (||u|{|}_{A_1})||T_j||$, where $\phi$ is a nondecreasing function and $A_1$ the Muckenhoupt class of weights, then we prove that, essentially, $$T:L^1(u)⟶L^{1,\infty }(u),\forall u\in A_1.$$ We shall see that this is the case of many interesting problems in Harmonic Analysis.