Regularity of General Maximal and Minimal Functions

Abstract

In this paper, our object of investigation is the endpoint regularity of the following general maximal operator,

$$\begin{aligned}{} & {} \widetilde{\mathcal {M}}_\Phi f(x)=\sup \limits _{r,s\ge 0\atop r+s>0}\Phi (r+s)\int _{x-r}^{x+s}|f(y)|\textrm{d}y, \end{aligned}$$

and minimal operator,

$$\begin{aligned}{} & {} \widetilde{m}_\Phi f(x)=\inf \limits _{r,s\ge 0\atop r+s>0}\Phi (r+s)\int _{x-r}^{x+s}|f(y)|\textrm{d}y, \end{aligned}$$

where $\Phi (t):(0,\infty )\rightarrow (0,\infty )$ is a non-increasing continuous function and satisfies $B_q:=\sup _{t>0}t\Phi (t)^q<\infty$ for some $q\ge 1$. We prove that if $f:\mathbb {R}\rightarrow \mathbb {R}$ is a function of bounded variation, then

$$\begin{aligned}{} & {} \max \{\textrm{Var}_q(\widetilde{\mathcal {M}}_\Phi f),\textrm{Var}_q(\widetilde{m}_\Phi f)\}\le (8B_q)^{1/q}\textrm{Var}(f). \end{aligned}$$

Here, $\textrm{Var}_q(f)$ denotes the q-variation of f and $\textrm{Var}_q(f)=\textrm{Var}(f)$ when $q=1$. Similar results are proved for the discrete versions of the above operators.