Topics in Harmonic Analysis; commutators and directional singular integrals
This dissertation focuses on two main topics: commutators and maximal directional operators. Our first topic will also distinguish between two cases: commutators of singular integral operators and BMO functions and commutators of fractional integral operators and a BMO class that comes from changing the underlying measure. Commutators are not only interesting for its own sake, but they have been broadly studied because of their connection to PDEs. Our first result gives us a new way of characterizing the class BMO. Assuming that the commutator of the Hilbert transform in dimension 1 (or a Riesz transform in dimensions 2 and higher) and the symbol b satisfy an Llog L-type of modular inequality on the endpoint with constant B, we can bound the BMO norm of the symbol by a fixed multiple of B; thus providing an endpoint version of the classical result of Coifman, Rochberg and Weiss for commutators of Calderón-Zygmund operators and BMO. We also studied commutators of fractional integrals and BMO. In this case, we were interested in finding quantitave two-weights estimates for the iterated version of these operators. We extended the known sharp inequalities for the commutator of first order to the iterated case and also provided a new proof of the previous results. Lastly, we studied maximal directional operators. Specifically, we considered a singular integral operator that commutes with translations and studied the maximal directional operator that arises from it. We proved that for any subset of cardinality N of a lacunary set of directions we can bound the Lp(Rn)-norm of the operator by the sharp bound √log N, thus completing some previous results on the Hilbert transform on low dimensions.