The best I can do is provide some general advice, since I'm missing detail and I'm not a physical therapist. If you have pain working your abdominals, it is typically unusual to have pain in your lower back.
You'll want to make sure of the following things:
- Correct technique: a common problem is hyper-extending the lower back, which increases stress in the lumbar spine. Make sure your hips are in a neutral position, or even tuck your pelvis forward while performing abdominal work. If you can't tell that your abdominals are even engaging in the exercise, then you most likely are not doing it right.
- Pinched nerves: your sciatic nerve extends from your lumbar spine into your leg. If your hip bones are pinching the nerve, that can cause pain. If you regularly keep anything in your back pocket, try moving it to the front pocket so that it doesn't shift your weight to one side. Beyond that, see a doctor to get it diagnosed. If possible use physical therapy to resolve broken movement patterns.
- Nocebo effect: a nocebo is basically your body manufacturing pain because you expect it. It's similar to a placebo except the impact is negative instead of positive. Fixing these issues require retraining your body to not expect pain. Typically low intensity work with higher reps.
If I were a betting man, I'd assume you are performing exercise incorrectly. One problem I had with laying leg raises is precisely what I pointed out (hyper-extending the lumbar when my legs were in the bottom position). To fix that, I started performing the laying leg lifts on a bench with my legs hanging in free space. I also slowed down the exercise so I could control the lower and upper position better. It also had the benefit of keeping the abs under constant tension. With planks, I had to do them incline (elbows on a bench) for a while, focusing on tucking my pelvis forward by engaging the abs directly.
- Slow down!
- Get someone who knows the exercise to coach you, or at the very least take video of you performing it.
- Find your pain free range of motion.
- Determine if that range of motion is less than expected or more than necessary.
Abs work best by bracing the body for work, so if you are moving too fast to keep them fully engaged then you are losing out on much of the benefit of those exercises.
If your pain free range of motion is not complete for the exercise, work on increasing it slowly over time.
Use the feedback from your friend/coach/video to help you improve your technique.