Hi,
Leg length discrepency (LLD ) is a fairly common occurence amongst the population. Management depends on a few factors, including:
1. Symptoms. Does it actually cause symptoms in the foot, leg, or back somewhere else.
2. The actual LLD.
A 1cm LLD is actually managed by orthotics. A visit to the podiatrist will help the problem. They can custom make an insole to correct the discrepency in height (1cm), and wearing it should help the problem. There is NO indication for any invasive option at a LLD of 1cm. Orthotics are actually suitable for LLD till 5cm.
Symptoms wise, like you mentioned, should usually get better once the LLD is resolved. As the body weight distribution and mechanics become more normal, the pressure on the back and other leg should decrease, and hence alleviate the symptoms. If you still are having lower back pain despite the correction of the LLD, it would then be important to find out why the pain is persistent. There would be other causes of low back pain to exclude as well.
Hope this helps
Dr Sean Ng