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!


1 Answer 1


Since Bessel potential operators are multiplicative in Fourier space, you can use Plancherel theorem first, then it is quite clear.


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