The area under my eyes is very hollow.  Is it possible to have a fat transfer in this area?

Yes, it is possible to have fat transfer to hollow areas under the eyes.  Other alternatives include injection of hyaluronic acid filler (Restylane, for example) and lower eyelid surgery (blepharoplasty) with fat transposition.  The fat transposition procedure involves re-positioning your own fat from the lower eyelids into the areas of hollowness, thus providing a more youthful contour between the lower eyelids and cheeks.