Course higher education credits 5
Course is normally given Period 4. Course first taught 2013
Graduate school Mathematics
Department Mathematical Sciences
Course start 2013-04-08
Course end 2013-05-17
Starting with the classical Hilbert and Riesz transforms and their links to the Laplacian, we introduce principal value integrals. Then regularity conditions for more general singular integral kernels are discussed. Both convolution kernels and kernels depending on two variables are considered. An interpolation theorem is given. Via the Calderón-Zygmund decomposition, the weak type 1,1 and strong type L^p estimates for singular integral operators are deduced. The space BMO is introduced, and it is proved that the operators map bounded functions into BMO. Then the T1 theorem for two-variable kernels is stated, and proved to the extent that time allows. This will require tools like Cotlar's lemma and Carleson measures.