Boundedness of fractional elliptic and parabolic operators on Lebesgue and Hölder spaces. A semigroup approach

Resumen

The connecting thread of this thesis is the semigroup language, a unifying and general technique to formulate and analyze fundamental properties of fractional operators. We have used this approach to deal with di erent problems. The rst chapter is devoted to the discrete fractional derivatives. We have de ned them via semigroups and we have proved that they approximate the continuous fractional derivatives. We have also obtained comparison and maximum principles and regularity results for the fractional powers. These results also allow us to prove the pointwise coincidence of the Marchaud and Gr unwald-Letnikov derivatives. On the second chapter we consider Schr odinger operators on Rn with n 3, that is, L = �� + V , where V is a nonnegative potential satisfying a reverse H older inequality. We have found the appropriated pointwise de nition of Lipschitz (or H older) classes in the Schr odinger setting for 0 < < 2. Secondly, we have de ned, for every > 0, new Lipschitz spaces adapted to L by means of the heat and Poisson semigroups. We prove that in fact these spaces do coincide with the ones de ned pointwise. Moreover, we use these new de nitions of Lipschitz spaces via semigroups to get regularity results of fractional powers of Schr odinger operators. On the third chapter we deal with the Hermite operator, a Schr odinger operator for which are known a lot of interesting properties. These properties have allowed us to get better results in this case than for general Schr odinger operators. We have got a complete characterization of Lipschitz spaces adapted to the Hermite operator (also in the parabolic case) and we have obtained regularity results for the Hermite fractional operators in those spaces. Finally, the last chapter is devoted to the study of the classical solvability of the parabolic Bessel di erential equation and the boundedness on (mixed) weighted Lp spaces of the associated Riesz transforms.