LisMath Seminar 

13/07/2023, 09:30 — 10:00 — Online
Frederico Toulson, Instituto Superior Técnico, Universidade de Lisboa

Peaks and Variations

One can define the class of non tangential Hardy-Littlewood maximal operators as follows:

$$ M f(x) = \sup_{|x-y|< \alpha r} \frac{1}{2r}\int_{y-r}^{y+r} |f(t)|dt,
$$ where the supremum is taken in $r>0$ and $y$ for a given parameter $\alpha\in [0,1]$. This gives us a maximal averaging operator in $\mathbb R$. In this talk, we will explore how this operator behaves when applied to functions of finite total variation. One would expect that, by taking averages, the function becomes smoother in some sense, and thus, total variation should decrease.

We will present what is known for each $\alpha$, and give a sketch for an alternative proof of the result in [1]. We finish by showing how one could adapt this result for a discrete version of the same operator.

References:

[1] J. Ramos, Sharp total variation results for maximal functions, Ann. Acad. Sci. Fenn. Math. 44 (2019) 41-64

[2] E. Carneiro and D. Moreira, On the regularity of maximal operators, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4395–4404.

[3] J. Kinnunen, The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math. 100 (1997), 117–124.

[4] O. Kurka, On the variation of the Hardy-Littlewood maximal function, Ann. Acad. Sci. Fenn. Math. 40 (2015), 109–133.

[5] J. Bober, E. Carneiro, K. Hughes, and L. B. Pierce, On a discrete version of Tanaka’s theorem for maximal functions, Proc. Amer. Math. Soc. 140 (2012), 1669-1680.