I'm having trouble with answering the following question:
Use the Fourier transform to compute the commutator $$[x,|D_x|^\alpha], \alpha > 0.$$ Here, $|D_x|^\alpha$ is rigorously viewed as a Fourier multiplier (I believe), and $\alpha$ is a multi-index.
Please no full solutions! Just hints. I have already done some examples. Mainly, I am wondering:
- Why is the Fourier transform even needed to solve this problem?
- Is there a way in which this simplifies nicely?
- Is there any way to write this other than just a sum of convolutions?