One-sided interval averaging operator ($A_r^{+},A_r^{-}$)

For $r>0$, the forward average is $r^{-1}{\int }_x^{x+r}f(t)dt$ and the backward average is $r^{-1}{\int }_{x-r}^{x}f(t)dt$, both with positive interval length in the denominator. They are linear operators. Their absolute values are bounded by the uncentered maximal function: enlarge the interval to put $x$ in its interior and pass to the limit. The same boundary-enlargement argument works regardless of the value of the chosen representative at $x$.