Characterization of matrix valued BMO by commutators and sparse domination of operators

In the first part of this thesis, we characterize the space of matrix-valued, two-parameters BMO functions by using commutators with the Hilbert transform. The second part deals with domination of certain operators, by using a class of positive forms which have better boundedness properties and are highly localized. We present a sparse version of the T1 Theorem of David and Journé, the sparse control of a discrete version of the Hilbert transform with an oscillatory term with a quadratic phase, and the sparse domination of the Bochner-Riesz multipliers.Ph.D