Dominación sparse para conmutadores y estimaciones cuantitativas

Dada una funci ́on localmente integrable y un operador lineal T definimos el commutator [b, T ] como [b, T ]f (x) = b(x)T f (x) − T (bf )(x). En esta charla presentaremos resultados de dominación sparse para conmutadores y sus iteraciones para los casos en que T es un una integral singular rough, un operador A-Hörmander o un operador de Calderón-Zygmund. Suponiendo adicionalmente que b ∈ BMO o alguna clase análoga mostraremos la versatilidad de las técnicas de dominación sparse para obtener estimaciones cuantitativas. Esta charla está basada en trabajos conjuntos con A. Lerner, S. Ombrosi, C. P ́erez, L. Roncal y G. Ibañez-Firnkorn.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A quantitative approach to weighted Carleson Condition

Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea in the 80's for the operator $$\mathcal{M}f(x,t)=\sup_{x\in Q,\,l(Q)\geq t}\frac{1}{|Q|}\int_{Q}|f(x)|dx \qquad x\in\mathbb{R}^{n}, \, t \geq0$$ are obtained. As a consequence, some sufficient conditions for the boundedness of $\mathcal{M}$ in the two weight setting in the spirit of the results obtained by C. P\'erez and E. Rela and very recently by M.T. Lacey and S. Spencer for the Hardy-Littlewood maximal operator are derived. As a byproduct some new quantitative estimates for the Poisson integral are obtained

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