### Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Thursday, July 15, 19:40 ~ 20:10 UTC-3

## Fefferman-Stein inequalities for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree

### Sheldy Javier Ombrosi

#### Universidad Nacional del Sur, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we present weighted endpoint estimates for the Hardy-Littlewood maximal function on the infinite rooted $k$-ary tree. Namely, the following Fefferman-Stein estimate \[ w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}dx\qquad s>1 \] is settled and moreover it is shown it is sharp, in the sense that it does not hold in general if $s=1$. This result is a generalization of the unweighted case ($w\equiv 1$) independently obtained by Naor-Tao and Cowling-Meda-Setti. Some examples of non trivial weights such that the weighted weak type $(1,1)$ estimate holds are provided. A strong Fefferman-Stein type estimate and as a consequence some vector valued extensions are obtained.

Joint work with Israel Rivera-Ríos (Universidad Nacional del Sur) and Martín Dario Safe (Universidad Nacional del Sur).