People with partial hearing loss typically experience the loss in some frequency range -- so if someone's hearing, like in your example, had low response in higher frequencies, the hearing aid would be finely tuned to amplify the exact range of high frequencies necessar.
Some of the more expensive hearing aids contain digital signal processors, so they could plausibly be programmed to do what you suggest. However, the result wouldn't sound the way you probably expect.
Simply translating peaks in the upper frequencies down to the lower frequencies would result in a muddy, unintelligible sound -- kind of the way outside noises sound when you're underwater, except louder.
DSPs can also transpose *pitches*, which is more like what I think you mean -- so, for instance, a woman's high voice becomes deeper. Still, it doesn't sound really like a man's voice, and might also be difficult to understand, unless a component of the signal (called the "formant") is also changed.
All this takes a lot more processing power than simply filtering and amplifying frequencies, though, so it may not be an option for the DSP chips that are built into hearing aids.
Also, since this technique would necessarily mask the *real* sounds in the lower range, I'm not sure how helpful it would be for most partially hearing folks.