I have been studying [this preprinted paper][1] and the references there in. I believe that there are some typos; However, since I am not an expert yet, so I would like to make sure I am correct.

 1. I think that there must be missing $\| g\|_{L^2}^2$ in the fourth inequality in the proof of theorem 2.5.
 2. I do not understand how the author made the restriction in the paragraph after equation (2.5)? in particular $ l \le 2j+2$. 
 3. Why he fixed $\Lambda = 2^{2j+5}$

Thanks in Advance.

  [1]: https://arxiv.org/pdf/2107.01482.pdf