Regularity Property And Harnack’s Inequality For Fractional Operators

In the last decades the theory of diffusion semigroups have been used successfully in the development of a theory of Harmonic Analysis associated to several Laplacians. This theory has meanly dealt with Lp and Hp boundedness of operators like Riesz po-tentials, Riesz transforms, Litlewood-Paley functions, etc. After a very short account of the use of semigroup theory in Harmonic Analysis, we arrive to the use of the theory in PDEs. The fractional Laplacian has become one of the most famous operators in the last five years, after the celebrated work of L. Caffarelli and L. Silvestre. And now, the theory related with fractional operators becomes a hot topic in Harmonic Analysis and PDEs.The main aim of this thesis is to develop the regularity theory and Haranck’s inequalities for fractional operators by using semigroup theory. We use harmonic extension, Carleson measure and Poisson semigroups to characterize the Holder space associated with Schrodinger operator. With this characterization, we give a simple proof of the regularity theorem of the fractional Schrodinger operator. Because of the Campanato type description of the Holder space associated with Schrodinger operator, we develop a T1-theorem for a class operators. With this T1-theorem. we get some regular estimates of some operators related with Schrodinger operator. And Harnack’s inequality is an important tool to get the regularity property of the solution of an equation in PDEs. We use the idea of L. Caffarelli and L. Silvestre’s extension problem, semigroup theory and a transference method to prove some Harnack’s inequalities for some fractional operators. By introducing fractional derivatives, we get some results related with Littlewood-Paley-Stein theory on semigroups.The thesis contains five chapters. The first chapter recalled the development of the fractional operators, Littlewood-Paley-Stein theory and the noncommutative martingale theory. In Chapter2, we characterized the Holder space associated with Schrodinger operators by using harmonic extension and Carleson measure, and with that characterization, we proved the regularity of fractional Schrodinger operators. We proved some regular estimates by using a T1-theorem in Chapter3. In Chapter4, by using the extension problem and a transference method that we developed, we proved some Harnack’s inequalities for some fractional operators. The fifth chapter introduced the fractional derivatives into the Littlewood-Paley functions, and we characterized some geometric properties by using the boundedness of the functions.