- Harmonic Analysis
- Complex Analysis
- Operator Theory.
Maria’s research interests lie in harmonic analysis, and especially in the theory of weights for singular integrals. Her initial work was concerned with weighted sharp inequalities for singular integral operators in terms of the weight’s constant. This question is of significant importance in the study of the regularity of solutions to the Beltrami equation and it has received a lot of attention in the recent years.
A very important question closely related to the previous one is the boundedness properties of singular integrals with respect to those of the Hardy-Littlewood maximal function in a weighted context. Maria and collaborators have provided counterexamples that show that, outside the classical realm of weights, one does not have the classical implications. Namely, that boundedness of the maximal function does not necessarily imply boundedness of the singular integral. There are a number of interesting questions in this particular area that still remain open.
Maria’s more recent interests also include operator theory, and more precisely the application of classical weighted theory to understand compositions of operators in spaces of analytic functions like the Bergman spaces.