Georgia Institute of Technology College of Sciences

The first part, consists on a result in the area of commutators. The classic result by Coifman, Rochber and Weiss, stablishes a relation between a BMO function, and the commutator of such a function with the Hilbert transform. The result obtained for this thesis, is in the two parameters setting (with obvious generalizations to more than two parameters) in the case where the BMO function is matrix valued. The second part of the thesis corresponds to domination of operators by using a special class called sparse operators. These operators are positive and highly localized, and therefore, allows for a very efficient way of proving weighted and unweighted estimates. Three main results in this area will be presented: The first one, is a sparse version of the celebrated $T1$ theorem of David and Journé: under some conditions on the action of a Calderón-Zygmund operator $T$ over the indicator function of a cube, we have sparse control.. The second result, is an application of the sparse techniques to dominate a discrete oscillatory version of the Hilbert transform with a quadratic phase, for which the notion of sparse operator has to be extended to functions on the integers. The last resuilt, proves that the Bochner-Riesz multipliers satisfy a range of sparse bounds, we work with the ’single scale’ version of the Bochner-Riesz Conjecture directly, and use the ‘optimal’ unweighted estimates to derive the sparse bounds.