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Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that

$$\int_{\mathbb{T}^2} J^s (\partial^3_x w) (J^s w)dxdy + \int_{\mathbb{T}^2} J^s (\partial_x \partial_y^2 w)dxdy=0.$$

I could not reach this result. I used the integration by parts (assuming $J^s$ not intact) by differentiation and integration and obtained

$$\int_{\mathbb{T}^2} J^s (\partial^3_x w) (J^s w)dxdy + \int_{\mathbb{T}^2} J^s (\partial_x \partial_y^2 w)dxdy=\int_{\mathbb{T}^2} J^s (\Delta w) J^s(\partial_xw)dxdy. $$

Any help is appraised.

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    $\begingroup$ you might want to disclose which paper? $\endgroup$
    – Carlo Beenakker
    Commented Apr 12, 2022 at 19:02
  • $\begingroup$ arxiv.org/pdf/1809.02027.pdf $\endgroup$
    – Mr. Proof
    Commented Apr 13, 2022 at 18:12
  • $\begingroup$ Please do not link to pdfs on arxiv - link to the abstract instead (I don't follow pdf links when on mobile, for example). Also why not give the name of the paper as link text? $\endgroup$
    – Dirk
    Commented Apr 14, 2022 at 16:24
  • $\begingroup$ The last integration in your solution equals zero because both integrations on left are already zero. $\endgroup$
    – Mr. Proof
    Commented Apr 30, 2022 at 11:18

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