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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/230526
On the Bergman Kernel and the Suita Conjecture
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The primary object of study in this thesis is the Bergman kernel of a domain in the complex plane. This object can be examined from multiple perspectives, naturally fostering a deep interplay between complex analysis, functional analysis, and potential theory. The work begins with a foundational overview of reproducing kernel Hilbert spaces (RKHS), before specializing to unweighted Bergman spaces. Here, we detail the extremal properties of the Bergman kernel, the Bergman projection, and their critical role in establishing the boundary regularity of conformal mappings. The thesis then develops essential concepts from potential theory, exploring the Bergman kernel’s strong connections with objects such as the Green’s function and the logarithmic capacity. These potential-theoretic tools are subsequently used to characterize domains with a trivial Bergman space. This mathematical synergy culminates in the final chapter, which is dedicated to the Suita conjecture. We present Zbigniew B locki’s purely onedimensional proof of this geometric inequality which establishes a sharp bound of the logarithmic capacity of a domain by its Bergman kernel.
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Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2026. Director: Matteo Levi
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FERNÁNDEZ-CATUXO ORTIZ, Tomás. On the Bergman Kernel and the Suita Conjecture. [consulted: 28 of October of 2026]. Available at: https://hdl.handle.net/2445/230526