= One-sided interval averaging operator
{title2=$A_r^+,A_r^-$}
For $r>0$, the forward average is $r^{-1}\int_x^{x+r}f(t)\,dt$ and the backward average is $r^{-1}\int_{x-r}^xf(t)\,dt$, 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 $x$.
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