The trace fractional Laplacian and the mid-range fractional Laplacian

In this paper we introduce two new fractional versions of the Laplacian. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the classical fractional Laplacian as the mean value (in the sphere) of the 1-dimensional fractional Laplacians in lines with directions in the sphere. To obtain this second new fractional operator we just replace the mean value by the mid-range of 1-dimensional fractional Laplacians with directions in the sphere. For these two new fractional operators we prove a comparison principle for viscosity sub and supersolutions and then we obtain existence and uniqueness for the Dirichlet problem, that turns out to be nonlinear. Strong maximum and comparison principles also hold. Finally, we prove that for the first operator we recover the classical Laplacian in the limit as $s\nearrow 1$, while for the second operator we obtain the sum of the smallest and the largest classical Hessian eigenvalues.