While flying at 3000 ft, I get a clearance that says "maintain 3000
until established on the localizer, cleared ILS...." If I am already
at 3000, what exactly is the purpose of "maintain 3000"?

The purpose of the "maintain 3000" is to caution the pilot that the
approach clearance does not imply that he can start descending right
away.

In this context, I take the "maintain 3000 until established" to mean
"maintain at least 3000 until establshed".

If it were an instruction to descend, it would have been issued as
"descend and maintain".

"Maintain" means "don't go higher or lower. There may be terrain lower.
There may be traffic higher.

Jose
--
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