Mixed weak type estimates: Examples and counterexamples related to a problem of E. Sawyer

We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.Comment: Colloquium Mathematicum, to appea

Optimalidad en la conjetura débil de Muckenhoupt-Wheeden

En el año 2009 conjuntamente con A. Lerner y C. P\'erez probamos que la dependencia en relación a la constante [w]_A_1 de un peso $w$ en el tipo débil (1,1) de la cualquier operador de Calderón-Zygmund se puede controlar por [w]_A_1 x log([w]_A_1+e). Que la dependencia fuese lineal se conocía como conjetura débil de Muckenhoupt y Wheeden. Posteriormente, F. Nazarov, A. Reznikov, V. Vasyunin y A. Volberg probaron que no es posible dependencia lineal en general, de hecho probaron que la dependencia debía ser al menos [w]_A_1 x log^{1/3}([w]_A_1+e) para la Transformada Martingala y conjeturaron que nuestra estimación debería ser óptima. Finalmente en un trabajo reciente conjuntamente con A. Lerner y F. Nazarov probamos la optimalidad de la estimación por [w]_A_1 x log([w]_A_1+e) para la Transformada de Hilbert. En esta charla daremos una idea general de como obtener este resultado.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A boundedness criterion for general maximal operators

We consider maximal operators Mβ with respect to a basis β. In the case when Mβ satisfies a reversed weak type inequality, we obtain a boundedness criterion for Mβ on an arbitrary quasiBanach function space X. Being applied to specific β X this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M+ Lp/w

On pointwise and weighted estimates for commutators of Calder\'on-Zygmund operators

In recent years, it has been well understood that a Calder\'on-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a locally integrable function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.Comment: V3: Lemma 5.1 is corrected. We would like to thank Irina Holmes for pointing out an error in the previous versio