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

TL;DR

This paper demonstrates that under weighted conditions with Muckenhoupt weights, the weak-type (1,1) boundedness of averaging operators can be extended from individual operators to their sums, addressing longstanding open problems.

Contribution

It establishes that weighted weak-type (1,1) bounds for sequences of operators imply similar bounds for their sum under Muckenhoupt weight assumptions, a significant advancement in harmonic analysis.

Findings

Weighted bounds hold for sums of operators under A_1 weights.

Extension of weak-type (1,1) results to weighted settings.

Addresses open problems in harmonic analysis related to operator sums.

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 $$ T_j:L^1\longrightarrow L^{1, \infty}, $$ 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.