# An identity about Bessel potential operators

I'm reading this paper where I encounter below equality, i.e.,

\begin{aligned} & \left|\int_{\mathbb{R}^d}\left\{\ell_s^2\left(a_s^{\gamma^2}-a_s^{\gamma^1}\right)_{i j} \partial_i \partial_j p_{s, t}^{\gamma^1}(\cdot, y)\right\}(x) \, \mathrm dx\right| \\ =& \left|\int_{\mathbb{R}^d}\left[(1-\Delta)^{\frac{\delta}{2}}\left\{\ell_s^2\left(a_s^{\gamma^2}-a_s^{\gamma^1}\right)_{i j}\right\}(x)\right] \cdot\left[\partial_i \partial_j(1-\Delta)^{-\frac{\delta}{2}} p_{s, t}^{\gamma^1}(\cdot, y)(x)\right] \mathrm{d} x\right|. \end{aligned} \label{1}\tag{1}

Above, $$\delta \in (0, 1)$$ and $$\ell_s^2, a_s^{\gamma^2}, a_s^{\gamma^1}, p_{s, t}^{\gamma^1}(\cdot, y)$$ are real-valued functions on $$\mathbb{R}^d$$. Also, $$\partial_i$$ is the partial derivative w.r.t. the $$i$$-th coordinate. I got that $$(1-\Delta)^{\frac{\delta}{2}}$$ and $$(1-\Delta)^{-\frac{\delta}{2}}$$ are Bessel potential operators. It seems to me \eqref{1} follows from the ''property'' $$\int_{\mathbb R^d} f(x) g(x) \, \mathrm dx = \int_{\mathbb R^d} [(1-\Delta)^{\frac{\delta}{2}} f (x)] [(1-\Delta)^{-\frac{\delta}{2}} g (x) ] \, \mathrm dx. \label{2}\tag{2}$$

Could you confirm if my '''guess'' is fine and where I can find a reference for \eqref{2}?

Thank you so much for your help!