The two options for under eye hollows are surgery to better frame the eye or fillers. Surgery is a longer lasting solution, but would require downtime. Fillers may be problematic if not performed by an experienced and expert injector. Consult with 2 - 3 board certified plastic surgeons to understand your options.
Lower lid hollowness can be treated in 2 main ways:
1. surgery to re drape the deeper tissues so that the eye socket is better framed. This is a long-lasting solution but requires surgery and has a 1-2 week downtime.
2. Fillers can also be used to camouflage this area. These effects are immediate but usually only last 1 year or so. My favorite filler for this area is Juvederm.