PhD Oral Exam - Shahaboddin Shaabani, Mathematics and Statistic

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

The results presented in this thesis concern several aspects of the theory of real Hardy spaces and BMO. The first problem we address is the extension problem for Hardy spaces, for which we provide a complete resolution. Specifically, we investigate: for which open subsets Ω can every element of H^(p ) (Ω), be extended to an element of H^p(R^n ), with comparable norms? We give a complete geometric characterization of such domains.

We then turn our attention to the behavior of maximal operators on the space BMO. In this context, we study four questions, including the discontinuity of the Hardy--Littlewood maximal operator, its boundedness on VMO, and the unboundedness of both the strong and directional maximal operators on BMO. This part concludes with a counterexample demonstrating the failure of the Fubini property for this class of functions.

The final part of the thesis focuses on paraproducts and their operator norms on Hardy spaces. We establish sharp lower bounds for the norms of these operators acting on various types of Hardy spaces, both in the one-parameter and multi-parameter settings. These results yield an alternative characterization of Hardy spaces as admissible symbol classes for such operators.