Mixed-class, mixed-carrier itineraries are one of the biggest gray areas in the DOT regulation for baggage limits. There's no good way to know in advance.

The originating carrier might do any of the following:

(a) use United's business class rule for luggage
(b) use United's economy rule for luggage
(c) use their own economy rule for luggage

You might ask for experiences in the forum of whichever airline you'd be starting the trip with.