Bergman projections on weighted Fock spaces in several complex variables

Area of Science:

• Complex Analysis
• Functional Analysis
• Several Complex Variables

Background:

• The study of plurisubharmonic functions and their associated Fock spaces is crucial in complex analysis.
• Bergman projections are fundamental operators in function space theory.
• Understanding the boundedness of operators on various function spaces is a key area of research.

Purpose of the Study:

• To establish the boundedness of the Bergman projection on pth Lebesgue and Fock spaces under specific conditions related to a plurisubharmonic function.
• To investigate the properties of Bergman projections in the context of generalized Fock spaces.
• To derive estimates for the distance induced by the function and the norm of the Bergman kernel.

Main Methods:

• Utilizing properties of real-valued plurisubharmonic functions with uniformly comparable Hessian eigenvalues.
• Applying techniques from functional analysis to analyze the boundedness of the Bergman projection.
• Deriving estimates for norms and distances within the defined function spaces.

Main Results:

• The Bergman projection is proven to be bounded from the pth Lebesgue space Lp to itself for 1 < p < ∞.
• Bergman projections are shown to be well-defined and bounded on Fock spaces associated with the function ϕ.
• Estimates for the distance induced by ϕ and the L2-norm of the Bergman kernel are obtained.

Conclusions:

• The research establishes key boundedness properties of the Bergman projection on Lebesgue and Fock spaces.
• The findings contribute to the understanding of operator theory in the context of complex analysis.
• The derived estimates provide further insights into the structure of these function spaces and their associated kernels.