When is this formula valid $\int_{\mathbb R^N} (-\Delta)^sf g =\int_{\mathbb R^N} (-\Delta)^s g f $ with $s\in (0, 1)?$ I am aware of the integration by parts formula holds for $f, g$ in $L^2(\mathbb R^N)$. Is it valid for a larger class of functions which are not in $L^2$ but $\int_{\mathbb R^N} (-\Delta)^sf g$ and $\int_{\mathbb R^N} (-\Delta)^sg f$ are finite.
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2$\begingroup$ The formula is valid for $C_c^\infty$ functions, and then it can be extended continuously to $(f, g) \in \mathcal{D}_X \times \mathcal{D}_Y$ whenever $C_c^\infty$ is the core of $(-\Delta)^s$ on $X$ (with domain $\mathcal{D}_X$) and on $Y$ (with domain $\mathcal{D}_Y$), and $Y$ is continuously embedded into $X^*$ (perhaps with a weak* topology in $X^*$). Therefore, $X$ can be any of the $L^p$ spaces ($p \in (1, \infty)$), with $Y = X^*$, or a $C_0$ space, with, say, $Y = L^1$. Is that what you need? If yes, would you like a more detailed answer? $\endgroup$– Mateusz KwaśnickiCommented Jul 3, 2018 at 8:28
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$\begingroup$ Thank for the help. Thats exactly what I was looking for. $\endgroup$– sadiazCommented Jul 4, 2018 at 9:09
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