The doctoral dissertation in the field of Mathematics will be examined at the Faculty of Science, Forestry and Technology, Joensuu Campus.
What is the topic of your doctoral research? Why is it important to study the topic?
This thesis contains several new results on classical linear operators acting on spaces of analytic functions in the unit disc. The classical linear operators involved in the thesis are the difference of composition operators, integral operators, and small Hankel operators. We are mainly interested in Hardy spaces, weighted Bergman and Dirichlet spaces induced by (two-sides) doubling weights. The problem of depicting the boundedness and compactness of the difference of composition operators on those weighted Bergman spaces is unsolved. It turned out that characterizing the bounded and compact integral operators from those spaces to the space of bounded analytic functions is very hard. There is no related literature to give a complete characterization of bounded small Hankel operators on weighted Bergman spaces with doubling weights. For these reasons, the projects of the thesis are meaningful and important.
What are the key findings or observations of your doctoral research?
Bounded and compact differences of two composition operators acting on weighted Bergman space are characterized in terms of a new characterization of Carleson measure for this weighted Bergman space. The boundedness and compactness of integral operators acting from Hardy, weighted Bergman, and Dirichlet spaces into the space of bounded analytic functions are also investigated. We also characterize the bounded small Hankel operator on weight Bergman spaces, which is used to get a certain weak factorization of these weighted Bergman spaces.
What are the key research methods and materials used in your doctoral research?
To characterize bounded and compact differences of two composition operators acting on weighted Bergman space, the key methods are the novel characterization of Carleson measure for these weighted Bergman spaces and some useful inequality such as Khinchine’s inequality. To study integral operators from Hardy spaces, weighted Bergman and Dirichlet spaces to the space of bounded analytic functions, we need many duality relations, a decomposition of weighted Bergman and Dirichlet spaces, and some results on universal Césaro basis of polynomials. To study the small Hankel operators on weighted Bergman spaces, we need many tools such as atomic decomposition of weighted Bergman spaces, some estimate of Bergman kernel, and complicated applications of the dualities of different weighted Bergman spaces.
The doctoral dissertation of Fanglei Wu, PhD, entitled Classical linear operators on weighted Bergman and Dirichlet spaces will be examined at the Faculty of Science, Forestry and Technology, Joensuu Campus. The opponent will be Professor Hyungwoon Koo, Korea University, and the custos will be Docent Janne Heittokangas, University of Eastern Finland. Language of the public defence is English.
For more information, please contact:
Fanglei Wu, firstname.lastname@example.org, tel. 046 639 4127