Some relations between fractional Laplace operators and Hessian operators

After recalling the many representations of the fractional Laplace operator and some of its important properties, some recent results (proved in a joint work with Bruno Franchi and Igor Verbitsky) about the relations between the k-Hessian energy of the k-Hessian operator of a k convex function vanishing at infinity and the fractional energy of a particular fractional operator will be introduced. Moreover we shall recall an integration by parts formula for the fractional Laplace operator giving a new simpler proof.