Sharp estimates for homogeneous semigroups in homogeneous spaces. Applications to PDEs and fractional diffusion in RN

In this paper, we analyze evolution problems associated to homogenous operators. We show that they have an homogenous associated semigroup of solutions that must satisfy some sharp estimates when acting on homogenous spaces and on the associated fractional power spaces. These sharp estimates are determined by the homogeneity alone. We also consider fractional diffusion problems and Schrödinger type problems as well. We apply these general results to broad classes of PDE problems including heat or higher order parabolic problems and the associated fractional and Schrödinger problems or Stokes equations. These equations are considered in Lebesgue or Morrey spaces