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### 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. document.getElementById('cloak74e5e37d91a936a9c36c6b86d03e79a2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy74e5e37d91a936a9c36c6b86d03e79a2 = 'sh&#101;ldy&#111;mbr&#111;s&#105;' + '&#64;'; addy74e5e37d91a936a9c36c6b86d03e79a2 = addy74e5e37d91a936a9c36c6b86d03e79a2 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text74e5e37d91a936a9c36c6b86d03e79a2 = 'sh&#101;ldy&#111;mbr&#111;s&#105;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak74e5e37d91a936a9c36c6b86d03e79a2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy74e5e37d91a936a9c36c6b86d03e79a2 + '\'>'+addy_text74e5e37d91a936a9c36c6b86d03e79a2+'<\/a>';

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).