What procedure can lift lateral cheek and correct hollow lower eyelid?

 A properly done mini lift that elevates the SMAS tissue high up on the cheek would do it.  It will need to be combined with a lower lid procedure to remove excess skin and sometimes fat graft the tear trough area.  

The changes that occur with aging in the cheek and around the eyes are multifactorial in nature, and thus a definitive solution may require a series of different procedures. The descent of soft tissues in the cheek can be treated with an endoscopic mid face lift with endotine fixation.  The access incisions for this procedure are in the temple behind the hairline, and in the mouth between the cheek and the gums. Through these incisions, the soft tissue of the cheek can be completely freed from its attachments to the bone and completely elevated. Once the soft tissue has been placed in its proper position, it can be fixated with an endotine device  to lock it in place.  The endotine dissolves completely after several months, once scar tissue at the operative site has formed to maintain the new elevated position of the soft tissue of the cheeks.Once the soft tissue volume and position has been reestablished in the cheeks, attention can be turned to the lower eyelid, where aging changes include redundancy of the lower eyelid skin; relaxation of the orbital septum and increases in preseptal fat, which can cause bulging of the eyelids or eye "bags"; and loss of lower eyelid support. The tear trough deformity at the medial aspect of the lid/cheek junction can be exacerbated by loss of subcutaneous fat and thinning of the skin with age, as well as tightness at the insertion of the orbiculares oculi muscle at the malar bone. These changes can be addressed by a conservative resection of preseptal fat in the eyelid; conservative excision of lower eyelid skin; surgical release of the arcus marginalis (the insertion of the orbiculares oculi muscle onto bone); transposition of the resected preseptal fat to camouflage the tear trough defect; and a lateral canthopexy to increase the tendinous support of the lower eyelid, which will also partially correct the lengthening of the eyelid that occurs with age.Lastly, meticulous and precise fat transfer in multiple planes, from preperiosteal to subcutaneous, can provide additional camouflage of the tear trough defect, the final step in an integrated surgical plan to treat loss of cheek volume and a visible tear trough deformity in a patient desiring a natural rejuvenation of the mid face and periorbital region.

I find fillers work best in this area, and/or fat with PRP.  I suggest a formal consultation. There are skin tightening procedures as well that can be used , in place of surgery if you are a candidate for a non invasive approach.  Best, Dr. Emer.