I am seeing loss of volume under the mouth on both sides. Is this something that can be corrected? (Photo)

This volume loss can be easily restored with conservative placement of fillers.  I would recommend seeing a facial plastic surgeon in consultation to discuss your options.  

Based on this photo it seem to be a minor deformity which can be easily corrected with injection of filler. In person evaluation is recommended. Good luck.

Hello and thank you for your question and the picture.
You have a small amount of volume loss around the mouth on each side, which can be corrected with filler. 
An in-person examination and further discussion would help better assess your area of concern and help decide which type of filler to use.
Good luck!
Olivier A. Deigni, MD, MPH
Houston Plastic Surgeon

You are wise to consider local treatments in the context of your whole face.Filling under the mouth has to be subtle or your lips and chin will appear different and you have it almost perfect now.Will placed Sculptra may be an option but an in person consult would be best to see your face in 3D.

you have mind or creases below the corners of your mouth.These appear to be fairly mild to me. They may benefit well from judicious use of fillers. Please understand that there is a in Normas variability among the talent and expertise of providers offering treatments with fillers. You could be ecstatic with the results from one provider while Fila the results from another would be a  these appear to be fairly mild to me. They may benefit well from judicious use of fillers. Please understand that there is a in Normas variability among the talent and expertise of providers offering treatments with fillers. You could be ecstatic with the results from one provider while Fiola the results from another would be a mistake.  I suggest several thorough and patient person a person consultations with senior experienced board-certified plastic surgeon who do their own injections with filler. Best,Mats Hagstrom,MD