Baena Miret, SergiCarro Rossell, María Jesús2025-01-202025-01-202023-05-150022-1236https://hdl.handle.net/2445/217655It 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.24 p.application/pdfengcc by (c) Sergi Baena Miret et al., 2023http://creativecommons.org/licenses/by/3.0/es/Anàlisi harmònicaAnàlisi funcionalTeoria d'operadorsTransformacions de FourierHarmonic analysisFunctional analysisOperator theoryFourier transformationsinfo:eu-repo/semantics/article2025-01-20info:eu-repo/semantics/openAccess