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I am studying the use of the commutator for finding the estimate of energy. During my looking through many papers I found that this paper contains a possible typo. Here is the archive version which has the same prospective typo!

Another question about the commutator. When I espand the follawing commutator $[J^s,u]\partial_x u$ the last two terms are cancel eachohter! Are my calculations right?

\begin{align} [J^s,u]\partial_x u &= J^s(u\, \partial_x u)- u J^s(\partial_x u)\\ &=J^s(u) \partial_x u + u J^s(\partial_x u) - u J^s(\partial_x u)\\ &=J^s(u) \partial_x u \end{align}

Is the above right?

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    $\begingroup$ If $s=2k$ is an even integer, then $J^s=(1-\Delta)^k$. Check your computation in this case. Is it right? $\endgroup$
    – Piero D'Ancona
    Commented Jun 6, 2022 at 10:54

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The computation is not right in general. $J^s$ does not satisfy the Leibniz rule. The idea is that at least one of the derivatives hits the function $w$ in the paper, So the authors used the commutator to get rid of it.

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