Using a recent result of Chernyavskaya and Shuster we show that the maximal operator determined by
M
[
y
]
=
−
y
+
q
y
M[y]=-y+qy
on
[
a
,
∞
)
[a,\infty )
,
a
>
−
∞
a>-\infty
, where
q
≥
0
q\ge 0
and the mean value of
q
q
computed over all subintervals of
R
\mathbb {R}
of a fixed length is bounded away from zero, shares several standard “limit-point at
∞
\infty
" properties of the
L
2
L^2
case. We also show that there is a unique solution of
M
[
y
]
=
0
M[y]=0
that is in all
L
p
[
a
,
∞
)
L^p[a, \infty )
,
p
=
[
1
,
∞
]
p=[1,\infty ]
.