Here are a few problems range vent hoods have and some answers on how to fix them:
Most range vent hoods are designed to be wired directly into the household power supply, so if you have no power, check your fuse box or circuit breaker panel.
If the light is coming on, but the fan isn’t, your problem is with the fan switch or the fan motor. You can access these from under the hood, without removing it. Check the fan switch (in the on position) for continuity. If there is continuity, then replace the fan motor.
The first thing to check are the filters on your range vent hood. If these are full of grease, they could stop the flow of air and the fan may sound like it is only running at one speed. Once your filters are clean, you can check the fan switch and the fan motor to see if they are the problem. You can get to these from under the hood, without removing it.
Many vent hoods circulate air, but not all push the air outside of the home. You can tell if yours is a circulate-only, a vent-type hood, or a convertible hood by close examination of it.
Models with vent openings near the top front that the air blows through could be a circulate-only type of hood or a convertible type hood.
Models that have duct work coming out the top or back are probably a vent-type hood. Close examination will reveal what type you have.
If your hood has the vents in the front and the metal ductwork, then it is convertible. Careful inspection of the control panel may show a switch or lever to change where the range vent hood directs the air.