On the Laplacian and fractional Laplacian in exterior domains, and applications to the dissipative quasi-geostrophic equation

In this work, we develop an extension of the generalized Fourier transform for exterior domains due to T. Ikebe and A. Ramm for all dimensions n ≥ 2 to study the Laplacian, Delta, and fractional Laplacian operators in such a domain O.;Using the harmonic extension approach due to L. Caffarelli and L. Silvestre, we can obtain a localized version of the operator Lambda = (--Delta) 1/2, so that it is precisely the square root of the Laplacian as a self-adjoint operator in L2(O) with Dirichlet boundary conditions.;In turn, this allowed us to obtain a maximum principle for solutions of the dissipative two-dimensional quasi-geostrophic equation in the exterior domain, which we apply to prove decay results using an adaptation of the Fourier Splitting method of M.E. Schonbek.