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I have a question concerning rest-frame vs observed frame. In the observer frame, I can define an apparent magnitude and an absolute magnitude (as the apparent magnitude if the object were at d=10pc). From the apparent magnitude I can calculate the flux of the galaxy in the observed filter.

Now let's say I have a set of galaxies from which I know the rest-frame absolute magnitudes. Is there such a thing as a rest-frame apparent magnitude or a rest-frame flux?

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It would be a bit pointless.

The apparent magnitude is the magnitude that object appears to be. It can measured purely from Earth based observations.

The rest-based absolute magnitude is calculated using an estimate of the object's distance and radial velocity. It is a physical property of the object and so it is intrinsically interesting.

The "rest based apparent magnitude" would neither be something that can be measured, nor an intrinsic property of the object, hence it is not an interesting variable to calculate.

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