Local integration by parts and Pohozaev identities for higher order fractional Laplacians

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-\mathrm{\Delta }{)}^s$ with $s>1$. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in (0,1)$. As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-\mathrm{\Delta }{)}^s\varphi =\lambda \varphi$ in $\mathrm{\Omega }$, $\varphi \equiv 0$ in ${\mathbb{R}}^n\setminus \mathrm{\Omega }$.