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.
In the first part of the thesis, we discuss the boundedness of inhomogeneous singular integral operators suitable for local Hardy spaces as well as their commutators. First, we consider the equivalence of different localizations of a given convolution operator by giving minimal conditions on the localizing functions; in the case of the Riesz transforms this results in equivalent characterizations of $h^1$. Then, we provide weaker integral conditions on the kernel of the operator and sufficient and necessary cancellation conditions to ensure the boundedness on local Hardy spaces for all values of $p$. Finally, we introduce a new class of atoms and use them to establish the boundedness of the commutators of inhomogeneous singular integral operators with $\bmo$ function.
In the second part of the thesis, we investigate periodic solutions of a class of stochastic partial differential equations driven by degenerate noises with regime-switching. First, we consider the existence and uniqueness of solutions to the equations. Then, we discuss the existence and uniqueness of periodic measures for the equations. In particular, we establish the uniqueness of periodic measures by proving the strong Feller property and irreducibility of semigroups associated with the equations. Finally, we use the stochastic fractional porous medium equation as an example to illustrate the main results.