How to get the under-eye area more compact and slightly lifted?

No problem and no need for the expense, downtime and risks of surgery at this time.
The upper cheek region is a major support for the under-eyes. So, when there is a deficit of volume in that region, it leads to under-eye hollows and a poor transition between the upper cheeks and the infraorbital regions. That being the case, the skilled use an an appropriate robust volumzing filler for the cheeks can secondarily improve the under-eye issues. And, voila, the results are typically immediate and the treatment usually engenders little to no significant downtime. You would be wise to consult with an experienced, board certified cosmetic dermatologist filler expert. Hope this helps and best of luck.

It is frequently seen in younger patients to have hollowness underneath the eyelids due to small cheek bones. Malar hypoplasia, how this condition is called, can be corrected with small cheek or periorbital implants or adding the volume by means of fat grafts. In person consultation with board certified plastic surgeon with experience in this technique is recommended. Good luck.